Trends and Breaks in Cointegrated VAR Models

Transcription

1 Trends and Breaks in Cointegrated VAR Models Håvard Hungnes Thesis for the Dr. Polit. degree Department of Economics, University of Oslo Defended March 17, 2006 Research Fellow in the Research Department at Statistics Norway, Unit for macroeconomics, POB 8131 Dep, N-0033 Oslo, Norway. Homepage:

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3 3 Preface In July 2000 I attended a PhD Course in econometrics at the University of Oslo. The course was given by David F. Hendry. Half the course was dedicated forecasting in cointegrated vector autoregressive models. In a simple cointegrated vector autoregressive model he showed how to rewrite the system in deviations about means. In this formulation one parameter vector has an interpretation as growth rate parameters. This result was not new. It was also known from Søren Johansen s version of Granger s representation theorem that the drift in the series is a function of the parameters in the cointegrated vector autoregressive model. However, it was the course by David F. Hendry that inspired this thesis. Having finished the course I asked myself two questions. First, since the growth rates are functions of the parameters in the system, why don t we report them? These parameters describes the steady state growth in the variables in the system, and therefore they describe long term properties of the system. Second, why don t we impose restrictions on these growth rates? In a system we could expect some variables to grow and some not to grow. Why not test such restrictions? In March 2001 I presented a very preliminary version of my first paper at Norges Bank (the central bank of Norway. At this time I was unsure if my idea was interesting. Gunnar Bårdsen was appointed as a discussant on my presentation. He gave me positive feedback on my idea, and some good advice on how to improve my paper. Without this feedback I probably never would have started with this thesis. After the presentation at Norges Bank, I have presented my papers several times. My second presentation was in June 2001 at the European University Institute in Florence, Italy. This was a workshop organized by Søren Johansen and Katarina Juselius. Søren Johansen then gave me some valuable comments both on my manuscript as well as on my presentation. (Søren Johansen has also given me valuable comments on the two other papers in my thesis. In January 2002 I was admitted at the Doctoral Program at the University of Oslo. At that time my first paper was almost finished, and I had attended several courses. As supervisors I got Ragnar Nymoen (first supervisor at the Department of Economics and Anders Rygh Swensen (second supervisor at the Department of Mathematics, both at the University of Oslo. They have, with their different backgrounds, given me a broad feedback on the papers in this thesis. In addition to those mentioned above, I have also received valuable comments on my papers from Eilev S. Jansen and Bent Nielsen. Terje Skjerpen at Statistics Norway

4 4 Preface deserves special thanks. He has carefully read various versions of the papers in this thesis and checked the mathematics as well as the grammar. His suggestions have been very helpful. I would also like to thank Statistics Norway for allowing me to take the PhD as a part of my job. In particular since the topic of my thesis is a bit outside what is defined as the core of research in the Research Department at Statistics Norway. Special thanks to Ådne Cappelen, Director of Research at Statistics Norway, for this opportunity. Håvard Hungnes August 2005

5 Contents 0 Introduction Background Cointegration and Error Correction Models without deterministic variables Models with deterministic variables Long-run properties in cointegrated VAR models The papers Growth Rates Structural Breaks Factor Demand GRaM Conclusions and further work References Growth Rates Introduction Growth rates and cointegration mean levels The linear switching algorithm Application: Danish money demand Conclusions and suggestions for further work References A Appendix Structural Breaks Introduction Model formulation Conventional formulation of cointegrated VAR Alternative formulation of cointegrated VAR Structural breaks

6 6 Contents 2.3 Estimation Restrictions The estimation problem Alternative formulations of the estimation problem Distribution of the likelihood tests Empirical illustrations Uncovered interest parity Money demand in unified Germany Conclusions References Factor Demand Introduction Theoretical model Factor demand relationships Growth and growth rates Empirical analysis Distribution of cointegrating rank test Factor demand with Cobb-Douglas Technology Factor demand with CES Technology Conclusions References A Appendix A B Appendix B C Appendix C A GRaM 83 A.1 Introduction A.1.1 System requirements A.1.2 Installation A.1.3 User license and citation A.2 Theoretical introduction A.2.1 Introduction A.2.2 Model formulation and Granger s representation theorem A.3 GRaM tutorial A.3.1 Introduction A.3.2 Formulation A.3.3 Imposing restrictions on gamma

7 Contents 7 A.3.4 Imposing other (coefficient restrictions A.3.5 Graphical output A.3.6 Using GRaM together with PcGive and PcGets A.3.7 The Batch editor A.4 Technical documentation A.4.1 The estimation problem without restrictions A.4.2 The estimation problem with restrictions A.4.3 Conditional Reduced Rank Regression A.4.4 Starting values A.4.5 Simulation A.4.6 Switching A.4.7 Standard errors A.4.8 Distribution for the likelihood tests References

8 8 Contents

9 Chapter 0 Introduction 9

10 10 Chapter 0. Introduction 0.1 Background In analysing a dynamic econometric model we are often interested in identifying and testing long-run properties. The cointegrating vectors are examples of long run relationships between different variables. However, also the underlying growth rates (i.e. steady state growth rates can be identified in cointegrated vector autoregressive (VAR models. The growth rates tell us how much to expect (unconditionally the variables in the system to grow from one period to the next. When the system is used for forecasting, the vector of growth rates is very important in providing good forecasts. In fact, as the forecasting horizon approaches infinity, the forecast will be determined by this vector only. In this thesis I show how the deterministic terms in a cointegrated VAR model can be decomposed into interpretable components. The corresponding coefficients describe the long run (steady state growth rates for the variables, and possibly shifts in level and growth rates (the latter depending on the type of deterministic variables that are included in the system. Combined with the coefficients for the cointegrating vectors, they also describe level and trends (and possibly shifts in these in the cointegrating vectors. This Introduction is organized as follows: Section 0.2 describes how deterministic variables are included in the conventional cointegrating vector autoregressive model and shows that it might be difficult to interpret the effect of the different deterministic variables in that model. In Section 0.3 I show how the deterministic variables can be introduced in such a way that the interpretation of the corresponding coefficients then becomes clear. In Section 0.4 I describe the different papers in the thesis, with special focus on how the empirical formulation in the papers relates to the formulation in Section 0.3. Section 0.5 sums up. 0.2 Cointegration and Error Correction Strict stationarity describes a stochastic process whose joint distribution of observations is not a function of time, i.e. the joint distribution of (X t1, X t2,..., X tk is the same as the distribution of ( X t1 +h, X t2 +h,..., X tk +h. Weak stationarity (or covariance stationarity describes a process where the first two moments are not functions of time. A stochastic process is called integrated of order d, I(d, if it is weakly stationary after differencing d times, but not weakly stationary after differencing d 1 times. We call Y t cointegrated with cointegrating vector β = 0 if β Y t is I(0. The cointegrat-

11 0.2. Cointegration and Error Correction 11 ing rank is the number of linearly independent cointegrating vectors, and the space spanned by the cointegrating vectors is the cointegrating space. Granger (1981 introduced the term cointegration Models without deterministic variables Let Y t be an n-dimensional vector of variables that are integrated of order one at most. α and β are matrices of dimension n r (where r is the number of cointegrating vectors and β Y t is an r 1 vector where all elements are I(0. Furthermore, Γ i (i = 1, 2,..., p 1 are n n matrices of coefficients, where p is the number of lags. is the difference operator. The errors ε t are assumed to be independent and identically distributed Gaussian white noise (ε t iidn (0, Ω. Y t = αβ Y t 1 + p 1 Γ i Y t i + ε t (1 i=1 Condition Assume that n r of the roots of the characteristic polynomial A (z = (1 z I n αβ z p 1 Γ i (1 z z i i=1 are equal to 1 and the remaining roots are outside the complex unit circle. Theorem (Granger representation theorem without deterministic variables Under Condition 0.2.1, Y t in (1 has the representation Y t = C t i=1 ε i + ι + B t, (2 where C = β ( α Γβ 1 α with Γ = I n p 1 i=1 Γ i. The process B t is stationary with zero expectation. The level coefficients ι depends on initial values in such a way that β ι = 0. (3 The Granger representation theorem can also be interpreted as a multivariate Beveridge-Nelson decomposition, where C t i=1 ε i describes the permanent effects and B t represents the temporary effects (see Beveridge and Nelson (1981 and Hansen (2005.

12 12 Chapter 0. Introduction Models with deterministic variables The system in (1 implies that there is no drift in any of the time series in Y (since there is no trend in (2. Furthermore, it implies that the cointegrating space cancels out the levels in the time series (see (3. This is seldom realistic for macro time series, which often have drift, seasonality and possibly shift in level or drift. We can allow for drift, seasonality etc. by including deterministic variables in (1. Let Dt be a vector of deterministic variables and Φ the corresponding coefficient matrix. Y t = αβ p 1 Y t 1 + Γ i Y t i + ΦDt + ε t (4 i=1 With this general vector of deterministic variables, results from Johansen (1991, 1996 and Hansen (2005 show that the system in (4 has the Granger representation with τ = CΦ Y t = C t i=1 t i=1 ε i + ι + τ + B t, (5 D i + C(LΦ (D t D 0, (6 where C(L is a lag polynomial (see Johansen (1996 or Hansen (2005 for indirect definition of this lag polynomial and β ι can be expressed as a function of the parameters in (4. As we see from (6, the effect on the endogenous variable of the deterministic variables is quite complicated. First, it is difficult to see how the different deterministic variables affect the process of the endogenous variables. Second, it is near impossible to impose restrictions on the parameters in (4 in order to test various relationships between the deterministic and endogenous variables (described in the expression for τ. The effect of the deterministic variables on the endogenous variables is quite complicated even if Dt includes only an intercept and a trend. If the trend is restricted to enter the cointegarting space only we can write ΦDt = δ 0 + αρ 1 t. Under these choices of deterministic variables, (4 has the Granger representation (5 where τ = γt with γ = Cδ 0 + (CΓ I β ( β β 1 ρ1. (7 As we see, the coefficients for trends in variables (γ are complicated, non-linear functions of the parameters in (4. If we are interested in imposing restrictions on these trends, it could be very difficult in the system formulated as in (4.

13 0.3. Long-run properties in cointegrated VAR models Long-run properties in cointegrated VAR models In the previous section we saw that by including deterministic variables in the form of ΦD t in (1, it becomes difficult to interpret the effect of the deterministic variables on the different endogenous variables. This problem is due to the complicated relationship between the parameters in (4 and the parameters describing the effect of the different deterministic variables on the endogenous variables (see (6. The problem of interpretation remains even if we include only a few deterministic variables (see e.g. (7. My recommendation is therefore to include the deterministic variables in a different manner. Let D t be a vector of q deterministic variables, such as trend and seasonally dummies. I then write the system as Y t γ D t = α ( β (Y t 1 γd t 1 µ + where γ is now an n q matrix. p 1 Γ i [ Y t i γ D t i ] + ε t, (8 i=1 Note that D t is different from Dt in the parametrization in Section Hence, in the example where ΦDt = δ 0 + αρ 1 t, we have D t = t. Theorem (Granger representation theorem Under Condition 0.2.1, Y t in (8 has the representation Y t = C t i=1 ε i + ι + γd t + B t, (9 where C = β ( α Γβ 1 α with Γ = I n p 1 i=1 Γ i. The process B t is stationary with zero expectation. The level coefficients ι depends on initial values in such a way that µ = β ι. (10 Proof. See Hungnes (2005b. Note that the coefficient matrix γ is included in both (8 and (9. From (9 it is obvious that γ describes the effect of the deterministic variables on the endogenous variables, and that each element in γ has an interpretation. 0.4 The papers The thesis consists of three papers plus a documentation of GRaM. GRaM is an Ox program for estimating the systems in the thesis. It might be noted that some of the

14 14 Chapter 0. Introduction systems in Hungnes (2002 cannot be estimated with the present version of GRaM, since the present version of GRaM does not allow for restrictions on the matrix of adjustment parameters (α. To reproduce the estimation results in Hungnes (2002 another (less user friendly Ox program must be used Growth Rates Restricting Growth Rates in Cointegrated VAR Models, Hungnes (2002, Chapter 1. The formulation of the system is somewhat different in this paper compared to the other papers in the thesis. To make it comparable, let D t = t in (8. In addition we impose the coefficient restriction β γ = 0. In Hungnes (2002 also (centered seasonal dummies are allowed for. However, to ease the exposition the seasonally dummies are ignored here. Since only one deterministic variable is included in D t, γ is a vector. And since the deterministic variable included in D t is the trend, γ represents the trend for the endogenous variables in the system. Furthermore, since the variables often are logtransformed, γ represents the growth rates of the untransformed series of the endogenous variables. The main contribution in the paper is to allow for restriction on the vector of growth rate parameters, γ. Therefore we can restrict some variables not to grow and allow other variables to grow. As an example, an empirical analysis of money demand in Denmark is conducted. In this analysis four variables are used; (logs of real money, (logs of real income, the bond rate, and the deposit rate. We would expect real money and real income to grow. However, there might not be a trend in the two interest rates. In the paper I test these restrictions on the vector of growth rates combined with restrictions on the cointegrating space (β and the matrix of adjustment parameters (α. To estimate the systems a switching algorithm is used. This is an extension of the linear switching algorithm in Boswijk (1995. The advantage of such a switching algorithm is that the likelihood value is increased in each step. However, the disadvantage is that there is no guarantee that the algorithm will reach the global maximum. Therefore, careful selection of starting values is important. The problem of obtaining good starting values is more difficult if there are more cointegrating vectors Structural Breaks Identifying Structural Breaks in Cointegrated VAR Models, Hungnes (2005b, Chapter 2.

15 0.4. The papers 15 In this paper I generalize the system to allow for the whole set of deterministic variables. The formulation in (8 is suitable for analyses of structural breaks, both in the variables and in the cointegrating space. In the paper I investigate different kinds of structural breaks in two different data sets. In one of the data sets I look at money demand in Germany in a period covering the German re-unification. Two deterministic variables of special importance are included in the analysis: one step dummy and one broken trend. The shift dummy picks up level shifts in the time series and in the cointegrating space. The broken trend picks up trend shifts in the time series and in the cointegrating space. The empirical results indicate that there is a significant level shift in the data series, but there is no trend shift in the data series. If there is no trend shift in the time series, there can not be a trend shift in the cointegrating space. However, an interesting question is if there is a significant shift in the level of the cointegrating space. If there is no significant level shift in the cointegrating space, the cointegrating space also represents a cobreaking space, see Hendry and Mizon (1998. The result in the paper is not clear; the cointegrating space may also represent a cobreaking space. If so, there is neither a level break nor a trend break in the cointegrating vectors Factor Demand A Factor Demand System when there are Technological Changes in Production, Hungnes (2005a, Chapter 3. In this paper the analytical framework is extended by inclusion of an exogenous variable in D t. By including exogenous variables in D t the effect from those variables on the endogenous variables are identified. The exogenous variables are not included exactly in the same way as in conventional conditional systems, as investigated in Harbo et al. (1998. This turns out to have an important effect on the critical values of the cointegrating rank test. In the conventional conditional system, the critical values depend on the number of exogenous variables. However, when including the exogenous variables in D t (as I do, I argue that the critical values are independent of the number of exogenous variables. A simulation experiment supports this claim. In the paper I look at demand of input factors. One approach to secure that the optimal cost shares for the different input factors are not trending, is to impose many restricted cointegrating relationships. However, due to technological changes there might not be that many cointegrating relationships. An alternative approach to secure that the cost shares are not following a drift and that the scale-elasticity is treated consistently, is to impose the restrictions on γ.

16 16 Chapter 0. Introduction The empirical results show that there are clearly fewer cointegrating relationships than there are input factors in my model of one Norwegian industry sector. The reduced cointegrating rank is consistent with technological changes. Conventional estimation approaches might undermine the extent of technological changes. This paper differs from the other two above by having the main focus on a practical econometric problem GRaM Identifying the Deterministic Components in Cointegrated VAR Models using GRaM for Ox Professional - User Manual and Documentation, Hungnes (2005c, Appendix A. This is the documentation of the Ox program used for estimating the systems. The program utilizes the OxPack interface in Ox, and is therefore menu driven and easy to use. GRaM estimates systems of the form in (8. GRaM also allows for different types of hypothesis testing. Both linear tests on the cointegrating space (β and on the growth rates (vec (γ are allowed for. The program does no allow for restrictions on the individual cointegrating vectors (but only on the space spanned by the cointegrating vectors. Neither does the program allow for restrictions on the loading matrix, α. These limitations are helpful since they make the maximization of the likelihood function easier. The program must simulate in order to estimate the system. Three different simulations algorithm are implemented in the program. Furthermore, five different methods for getting starting values are included. The user can choose between the different algorithms and different methods for providing starting values. Hungnes (2005c is both a user manual and a documentation of the different algorithms in GRaM. A tutorial is included in Hungnes (2005c, showing step by step how to use the program. GRaM also generates batch code witch make it easy to document and reproduce estimation results. The batch code is also documented in Hungnes (2005c. The documentation also shows how GRaM can be combined with other programs in the OxMetrix family (such as PcGive and PcGets. 0.5 Conclusions and further work The specification of the system as in (8 has some important advantages. First, it identifies interpretable coefficients for the deterministic variables. Second, it allows testing

17 References 17 for all the different hypothesis regarding the deterministic variables. On the other hand, one problem with the approach suggested here, is that it involves a more complicated maximizing problem. However, the provided software makes it feasible to use the model specification also for non-experts. References Beveridge, S. and C. R. Nelson (1981, A new approach to decompositions of time series into permanent and transitory components with particular attention to measurement of the business cycle, Journal of Monetary Economics 7, Boswijk, H. P. (1995, Identifibility of cointegrated systems, Discussion Paper ti , Tinberger Institute, University of Amsterdam. Granger, C. W. J. (1981, Some properties of time series data and their use in econometric model specification, Journal of Econometrics pp Hansen, P. R. (2005, Granger s representation theorem: A closed-form expression for I(1 processes, Econometrics Journal 8, Harbo, I., S. Johansen, B. Nielsen and A. Rahbek (1998, Asymptotic inference on cointegrating rank in partial systems, Journal of Business and Economic Statistics 16, Hendry, D. F. and G. E. Mizon (1998, Exogeneity, causality, and co-breaking in economic policy analysis of a small econometric model of money in the UK, Empirical Economics 23, Hungnes, H. (2002, Restricting growth rates in cointegrated VAR models, Revised version of Discussion Papers 309, Statistics Norway. (Downloadable at Hungnes, H. (2005a, A demand system for input factors when there are technological changes in production. Hungnes, H. (2005b, Identifying structural breaks in cointegrated VAR models, Discussion Papers 422, Statistics Norway. (Downloadable at Hungnes, H. (2005c, Identifying the deterministic components in GRaM for Ox Professional - user manual and documentation.

18 18 Chapter 0. Introduction Johansen, S. (1991, Estimating and hypothesis testing of cointegration vectors in gaussian vector autoregressive models, Econometrica 59, Johansen, S. (1996, Likelihood-based Inference in Cointegrated Vector Autoregressive Models, 2nd printing. Oxford: Oxford University Press.

19 Chapter 1 Restricting Growth Rates in Cointegrated VAR Models 19

20 20 Chapter 1. Growth Rates Abstract In a cointegrated vector autoregressive model the intercept parameters can be decomposed into growth rate parameters and cointegration mean parameters. The growth rate parameters have important economic interpretations and may be equally important to identify and conduct hypothesis testing on as the cointegration vectors and the matrix of adjustment parameters. Here we develop a linear switching algorithm for estimating (possibly restricted growth rates as a part of the cointegration analysis. An example with Danish money demand illustrates the method. Keywords: Johansen procedure, cointegrated VAR, growth rates, cointegration mean levels, linear switching algorithm, money demand. JEL classification: C32, C51, C52, E41. Acknowledgments: Thanks to Gunnar Bårdsen, Eilev S. Jansen, Søren Johansen, Bjørn Naug, Terje Skjerpen and Anders Rygh Swensen for valuable comments on various versions of the paper. An earlier version of the paper was presented at the Econometric Society European Meeting (ESEM conference in Venice (August 25-28, 2002.

21 1.1. Introduction Introduction Cointegrated vector autoregressive (VAR model is a powerful tool in analysing time series. Granger s representation theorem (see Engle and Granger, 1987 shows that cointegrated time series can be represented in an equilibrium correction vector autoregressive model. Furthermore, Johansen (1988 shows that canonical correlation technique combined with reduced rank regression technique can be used to estimate such models. These techniques are implemented in standard time series packages such as PcGive (see Doornik and Hendry, 2001 and Cats in Rats (see Hansen and Juselius, The cointegration vectors can be identified and over-identified restrictions can be tested against economic theory. A lot of work has been accomplished in estimating long-run cointegration relationships in economics. However, other parameters in a cointegrated VAR model also have long-run economic interpretations. By rewriting the equilibrium correction form of the VAR model (VEqCM, we can identify the underlying growth of the variables as well as the longrun mean levels of the cointegration relationships. Within the VEqCM the intercepts can either be restricted to lie in the cointegration space, or not. If the intercepts are not restricted to lie in the cointegration space (commonly referred to that the intercept is included unrestricted, they allow the system to have both growth and cointegration mean levels. If, however, the intercepts are restricted to lie in the cointegration space ( restricted, for short, there is no growth in the system, (see Johansen and Juselius, The growth rates tell us how much to expect (unconditionally the variables in the system to grow from one period to the next. When the system is used for forecasting, the vector of growth rates is very important in providing good forecasts. In fact, as the forecasting horizon approaches infinity, the forecast will be determined by this vector only, see Clements and Hendry (1999, pp There are also variables we do not believe will grow over time. If the interest rate or the inflation rate is assumed to be I(1, we may not want to allow them to grow. Especially not if we want to use the system for forecasting. However, restricting the intercepts to lie in the cointegration space may be too restrictive, as the system may include variables we believe do grow over time. We then want to restrict some of the variables to have no growth and let other variables in the system grow. We develop an estimation procedure in which we allow restrictions in the system on some or all of the growth rates. The cointegration mean levels may also have economic interpretations. In a system with (the logs of consumption and income, the intercept in the cointegration vector

22 22 Chapter 1. Growth Rates can be interpreted as the equilibrium savings ratio if the income elasticity is unity. A system with nominal interest rate and inflation (both assumed to be I(1, where the cointegration mean can be interpreted as the equilibrium real interest rate, is another example. Sometimes we may want to restrict the cointegration mean level. Assume we are testing the law of one price, and are analysing a system with an unrestricted intercept to allow the prices to grow over time. 1 We may find that p e p is the cointegration relationship, (where p and p are the domestic and foreign price respectively, and e the exchange rate, all variables measured in logs, and want to test the strict version of the law of one price. This implies testing if the cointegration mean level is equal to zero. To achieve this, we have to decompose the intercepts in the system in growth rates and a cointegration mean level, and test if the mean level is equal to zero. This can also be achieved by the estimation procedure presented here. The paper is organized as follows: In section 1.2 we show how the growth rates and cointegration mean levels can be estimated. In section 1.3 a linear switching algorithm is presented. The switching algorithm we derive here is an extension of the linear switching algorithm in Boswijk (1995. In section 1.4 the method is applied on money demand in Denmark. Section 1.5 concludes. Throughout the paper we define the orthogonal complement of the full column rank matrix A as A such that A A = 0 and (A, A has full rank. (The orthogonal complement of a nonsingular matrix is 0, and the orthogonal complement of a zero matrix is an identity matrix of a suitable dimension. 1.2 Growth rates and cointegration mean levels In this section we look at some properties of the cointegrated VAR model. In particular, we focus on how the growth rates and cointegration mean levels can be estimated. In (1.1 X t is an n-dimensional vector of non-stationary I(1 variables, η is a vector of intercepts, α and β are matrixes of dimension n r (where r is the number of cointegration vectors and β X t is I(0. Furthermore, Γ i is an n n matrix of coefficients and is the difference operator. D t is a vector of centered seasonal dummies and impulse 1 The law of one price states that one product shall have the same price in two different regions. Let P be the price of the product in one of the regions and P the price in the other region. Furthermore, let E be the exchange rate (if the two regions lie in two different countries. Then the law of one price states that P = A E P, where A is a constant capturing the difference in the price levels due to transportation costs etc. The strict version of the law of one price states that A = 1, i.e. there are no differences in the prices of the product in the two regions.

23 1.2. Growth rates and cointegration mean levels 23 dummies. The residual ε is assumed to be white noise Gaussian (ε t N (0, Ω. X t = η + αβ X t 1 + p 1 Γ i X t i + sd t + ε t, t = 1, 2,..., T (1.1 i=1 The system grows at the unconditional rates E [ X] = γ with long run (cointegration mean levels E [ β X ] = µ apart from terms involving seasonal dummies (and other impulse dummies. Then we can rewrite the relationship as X t γ = α ( β X t 1 µ + p 1 Γ i ( X t i γ + sd t + ε t. (1.2 i=1 β. Notice that β γ = 0. To prove this, we pre-multiply the left hand side of (1.2 with β ( X t γ = ( β X t β X t 1 β γ (1.3 We see that since E [ X] = γ the unconditional expectation of the left hand side of (1.3 equals zero. And since E [ β X ] = µ the unconditional expectation of the right hand side of (1.3 will equal zero only if β γ = 0. By comparing equation (1.1 and (1.2 we see that where Γ = η = Γγ αµ, (1.4 ( I p 1 i=1 Γ i. For given estimates of the coefficients in (1.1 we find 2 γ = Cη (1.5 and µ = ( α α 1 α (ΓC I n η, (1.6 where C = β ( α Γβ 1 α. These properties are known from Granger s representation theorem, see Engle and Granger (1987 and Johansen ( To simplify the notation we apply the following definition in this footnote: For a matrix A with dimension n m (m n with full column rank, we define A = A (A A 1. To find (1.5, pre-multiply (1.4 with α and use I = ββ + β β ; α η = α Γβ β γ + α Γββ γ = ( α Γβ β γ since β γ = 0. The n r matrix ( α Γβ must have full rank, or some of the variables in the system are I(2. Therefore; β γ = ( α Γβ 1 α η or, since γ = ββ γ + β β γ = β β γ, pre-multiplying with β gives (1.5. To find (1.6, rearrange (1.4 to αµ = Γγ η, pre-multiply with α and apply (1.5.

24 24 Chapter 1. Growth Rates 1.3 The linear switching algorithm When no restrictions are imposed on the growth rates γ and the cointegration mean levels µ, the vector of intercepts δ in (1.1 have n variation free elements, and these could be estimated with ordinary least square. However, when we restrict γ and/or on µ, another estimation procedure must be used. Since we are imposing restrictions on the growth parameters and cointegration mean levels as a part of the cointegration analysis, our estimation procedure must also allow for restrictions on α and β. We extend the linear switching algorithm in Boswijk (1995 to also involve restrictions on the growth rates. We define β = ( β, µ and X t = (X t, 1, so restrictions on the cointegration mean levels can be imposed on β. The algorithm in Boswijk (1995 allows for linear restrictions on α and β. The restrictions on the cointegration vectors can be written as R β vecβ = c β or vecβ = H β φ + h β, (1.7 where H β = ( 1 R β and h β = H β (H β H β cβ. Since we are stacking the cointegration vectors into one vector, we can allow for restrictions between the cointegration vectors as well as within them. Similarly, restrictions on the adjustment parameters can be written R αvecα = 0 or vecα = H α ϕ, (1.8 where H α = (R α. Here the intercepts are excluded, since we normally only test exclusion restrictions on α. 3 The restrictions on γ are a bit more complex, since - in addition to the restrictions we want to place on γ - the cointegration vector also imposes restrictions on γ. The restrictions we want to impose on γ can be written as R γγ = c γ, whereas the restrictions imposed by the cointegration vectors can be expressed as β γ = 0. In a compact (, notation, these restrictions involve (β, R γ γ = 0, c γ which equivalently can be written as γ = H γ ψ + h γ, (1.9 1 (. where H γ = (β, R γ and h γ = H γ (H γh γ 0, c γ Before we present the log likelihood function, we must define some variables. We first define Z t = vec ( X t, X t 1,..., X t p+1 and Φ = ( In, Γ 1, Γ 2,..., Γ p 1. The 3 It is straightforward to include intercepts in (1.8.

25 1.3. The linear switching algorithm 25 log likelihood function (apart form a constant becomes log L ( α, β, γ, Γ 1, Γ 2,..., Γ p 1, s, Ω = T 2 log Ω ( T [ (ΦZt Γγ αβ Xt 1 sd t Ω 1 ( ΦZ t Γγ αβ Xt 1 sd ] t. t=1 The maximization problem is to maximize (1.10 under the restrictions (1.7 - (1.9. In Theorem 1 we derive the conditional maximum likelihood estimators for this maximization problem. It turns out to be convenient also to use a log likelihood function where we condition on the growth rates. For a given set of growth rates satisfying (1.9 we can define Z0t (ψ = X t γ, Z1t = X t 1, Z 2t (ψ = vec ( X t 1 γ, X t 2 γ,..., X t p+1 γ, D t and Θ = ( Γ 1, Γ 2,..., Γ p 1, s. The log likelihood function conditioned on the growth rates is log L ( α, β, Γ 1, Γ 2,..., Γ p 1, s, Ω; γ = T 2 log Ω ( T [ (Z 0t αβ Z1t ΘZ 2t Ω 1 ( Z0t αβ Z1t ] ΘZ 2t. t=1 Furthermore, we define and M ij S ij (ψ = T 1 T (ψ = T 1 T t=1 t=1 Zit Z jt, i, j = 0, 1, 2, (1.12 Rit R jt, i, j = 0, 1, (1.13 where R 0t (ψ and R 1t (ψ are the residuals we obtain by regressing Z 0t (ψ and Z 1t on Z 2t (ψ respectively. Finally, is the Kronecker product. Theorem (The conditional maximum likelihood estimators The conditional maximum likelihood estimators for Θ, φ, ϕ, Ω and ψ in (1.10 under the restrictions (1.7 - (1.9

26 26 Chapter 1. Growth Rates are given by Θ (ψ, φ, ϕ = M 02 (M 22 1 αβ M 12 (M 22 1, (1.14 φ (ψ, ϕ, Ω = ϕ (ψ, φ, Ω = [ ( H β H β H β ] 1 α Ω 1 α S11 ( [(α Ω 1 I n+1 vecs10 α Ω 1 α S11 h β ], (1.15 [ ] 1 [ ( H α (Ω 1 β S11 β H α H α Ω 1 β ] vecs10,(1.16 Ω (ψ, φ, ϕ = S 00 αβ S 10 S 01 β α + αβ S 11 β α, (1.17 ψ (φ, ϕ, Θ, Ω = [H γγ Ω 1 ΓH γ ] 1 [ H γγ Ω 1 ( ΦZ αβ X ] sd Γh γ, (1.18 where Z = T 1 T t=1 Z t, X = T 1 Z 1t = T 1 X t 1 and D = T 1 D t. See the appendix for the proof. The term sd in (1.18 equals to zero if we have the same number of observations for each season in the calendar year (and D includes no impulse dummies. If, however, we have an estimation period with more observation from some seasons than from others, this term will generally not equal zero. In the example below the estimation period is 1974Q3-1987Q3, which means that we have one more observation from the third quarter than the others. Note that β X = β X µ in (1.18. The first part is the average cointegration mean in the estimation period, and µ is the system cointegration mean. These will not generally be equal. We now suggest the following estimation procedure: The maximum likelihood estimators of ψ, φ, ϕ, Θ and Ω may be obtained by the following iterative procedure, starting from a set of initial values {ψ 0, φ 0, ϕ 0, Θ 0, Ω 0 } : I III V ψ j = ψ ( φ j 1, ϕ j 1, Θ j 1, Ω j 1 ϕ j = ϕ ( ψ j, φ j, Ω j 1 Ω j = Ω ( ψ j, φ j, ϕ j II IV φ j = φ ( ψ j, ϕ j 1, Ω j 1 Θ j = Θ ( ψ j, φ j, ϕ j j = 1, 2,... The iterative procedure needs a set of starting values. In fact, it only needs starting values for the free growth rates parameters (ψ, the cointegration vectors (φ and the

27 1.3. The linear switching algorithm 27 loading parameters (ϕ since starting values for the other parameters (Θ and Ω can be calculated by (1.14 and (1.17. It may be tempting to use the relations in Theorem with unrestricted parameters to compute starting values for ψ, φ and ϕ too. However, this is not a good idea when there are more than one cointegration vector. The unrestricted estimator of β is only unique up to a rotation which spans the same space. When restrictions are imposed on β these restrictions may lead to a rotation of this space. To take account of this, we use the method described in Doornik (1995. Let vec b = H β φ + h β = H β (H ( β vec β unr h β + h β, where the subscript unr indicates the parameters are revealed by the unrestricted cointegrated VAR model. Define [ ] as dropping those rows which have no restrictions in them; if this yields less than r rows, then add rows back in, so that the [ ] matrix is q r, with q r. Then the least square estimator  = ( ] ] [ β unr [ β 1 ( ] [ unr [ β b ] unr is used to derive α 1 = α unr  1. Now the loading matrix α is consistent with the restricted β, and we can use the relations in theorem 1 to calculate starting values for ψ, φ and ϕ; φ 0 = φ ( γ unr, α 1, Ω unr, ϕ 0 = ϕ ( γ unr, φ 0, Ω unr, ψ 0 = ψ ( φ 0, ϕ 0, Θ unr, Ω unr. As discussed in Johansen (1991, the distribution of β is mixed normal (i.e. the variance matrix is stochastic, provided that identifying restrictions are imposed on α and β. The discussion there also indicates that inference on β may be done as if α were known, and vice versa. Following this result, we compute the variance of β as V (vecβ = T [ ( ] 1 (H T k β TH β α Ω 1 α S11 H β H β, (1.19 where the term inside the square brackets is (the negative of the double derivative of (1.11 with respect to φ. The scale factor T/ (T k (where k is the integer part of

28 28 Chapter 1. Growth Rates the ratio between the freely estimated parameters in the system and the number of the dependent variables in the system is used to control for degrees of freedom, see Doornik (1995. The standard deviations of vecβ are the square roots of the diagonal elements in (1.19. The distribution of α is normal, and the variance is In (1.20 we use V (vecα = T [ ( ] 1 (H α TH α Ω 1 β S T 11 β H α H k α. (1.20 S 11 = T 1 T R 1t R 1t, t=1 where R 1t are the residuals we obtain by regressing Z 1t = X t 1 on Z 2t = ( 1, X t 1,. X t 2,..., X t p+1, D t The reason for using β S 11 β in (1.20 instead of β S11 β (which we would obtain if we used the double derivatives of (1.11 with respect to ψ is to take account for the covariance between the intercepts in the cointegration relations and the other coefficients outside the cointegration vectors. The expression in (1.20 is used to compute the variance of α in the standard literature, see e.g. Johansen (1991. We compute this variance of γ conditional on Γ = I n p 1 i=1 Γ i as V (γ = T [ ( ] 1 (H γ TH γ Γ Ω 1 Γ H γ H T k γ, (1.21 which is the observed information matrix for γ (adjusted for degrees of freedom. The distribution of γ is normal, see Johansen (1996, Theorem 13.6 and Application: Danish money demand To illustrate the estimation method we use data for money demand in Denmark. This is the data used by Johansen and Juselius (1990 to illustrate how one can restrict the intercepts to lie in the cointegration space. Restricting the intercepts to lie in the cointegration space implies restricting the variables in the system not to grow over time. This might be realistic for the bond rate (i b and the deposit rate (i d, but not for (the logs of real money (m2 and real income (y. The data are plotted in Figure 1.1. Centered seasonal dummies are included in the empirical analysis. We use the same estimation period as Johansen and Juselius (1990: 1974Q3-1987Q3. In the VEqCM 2 lags are included. In contrast to Johansen and Juselius (1990 we include the

29 1.4. Application: Danish money demand m2 6.1 y i b 0.12 i d Figure 1.1: Plot of real money (m2, real income (y, the bond rate (i b and the deposit rate (i d. Source: Johansen and Juselius (1990. intercepts unrestricted. 4 Table 1.1 includes two columns of critical values for each of the two tests. The first row (labelled 95% contains the standard critical values in a system with the intercepts unrestricted. In the second row (labelled 95% the critical values for the case where the true model has no deterministic trends are reported, see Osterwald-Lenum (1992. The latter set of critical values is reported since we cannot reject the hypothesis that there is no growth in the system. The rank test indicates that there is one or zero cointegration vectors in the data. The λ-max test supports one cointegration vector at a five per cent significance level and the trace test supports one cointegration vector at a 10 per cent level (independent of which of the two tables of critical values we use. 5 We continue the analysis by assuming that there is one cointegration vector among the variables. Estimating the system with one cointegration vector yields the following equilibrium relation: m2 = 1.04 (0.14 y 5.22 (0.56 ib (1.10 id 6.02 (0.87 ( The results are obtained by combining PcFiml 9.2 (see Doornik and Hendry, 2001 and Ox 2.1 (see Doornik, In Johansen and Juselius (1990 the λ-max test is significant at five per cent. However, their trace test is not significant even at the 10 per cent level (though very close to be so.

30 30 Chapter 1. Growth Rates Table 1.1: Money demand: Cointegration rank H 0 : rank = r λ λ max 95% 95% trace 95% 95% r = r r r One asterisk denote significance at the 5 per cent level. The columns labelled 95% contain the standard critical values, and in the columns labelled 95% the critical values for the case where the true model has no deterministic trends are reported. The critical values are taken from Osterwald-Lenum (1992. The difference between the bond rate and the deposit rate can be interpreted as the cost of holding money. We therefore restrict the money demand to be homogenous of degree zero in the two interest rates. In addition, we restrict the income elasticity to equal unity, i.e. β = H β φ + h β = ( b + µ This yields m ( y ( i b ( i d ( t = ( ( ( (0.015 m Γ 1 y i b i d ( ( m2 y i b i d 6.19 (0.53 (0.04 t 1 t 1 + ŝd t + ε t. (1.23 The results indicate a positive growth in money and income. In annual terms these growth rates are 3.3 and 1.5 per cent respectively. The results also indicate a negative growth in the interest rates; a 0.5 percentage points annual decrease in the bond rate and a 0.2 percentage points decrease annually in the deposit rate. However, most of the growth rates parameters are insignificant (measured with their t-values. We now impose the restriction that there is no underlying growth in the two inter-

31 1.4. Application: Danish money demand 31 Table 1.2: Money demand: Likelihood ratio test of reductions Equation log L log Ω p-value [d. f.] ( ( [2] ( [4] ( [6] est rates. These restrictions imply R γ = ( In addition we have the restriction β γ = 0. With the restriction we have imposed on the cointegration vector, this restriction involves (1, 1, b, b γ = 0. Therefore, the total set of restrictions on γ can be written as ( β R γ γ = 1 1 b b γ 1 γ 2 γ 3 γ 4 0 = 0. 0 The restrictions can also be expressed as γ = H γ ψ = where ψ is a scalar ψ, We see that the restrictions imposed on the growth rates imply that real money and real income grow at the same rate, i.e. γ 1 = γ 2. Imposing the restrictions on the growth rates we get the following results: 6 In our example H γ is independent of b, which means we do not have to update H γ for each iteration. Generally, however, H γ will change when the unrestricted parameters in β changes, and H γ must therefore be updated for each iteration.

32 32 Chapter 1. Growth Rates Table 1.3: Cointegration coefficient estimates for different restrictions on α and γ α = (,,, α = (,, 0, 0 γ = (,,, b = 5.907, µ = b = 5.808, µ = (0.531 (0.037 (0.560 (0.038 γ = (,, 0, 0 b = 5.889, µ = b = 5.805, µ = (0.523 (0.037 (0.559 (0.039 γ = (0, 0, 0, 0 b = 5.884, µ = b = 5.811, µ = (0.523 (0.038 (0.560 (0.040 Asterisk denote that the parameter is unrestricted. m ( y ( i b i d t = ( ( ( (0.015 m Γ 1 y i b i d ( ( m2 y i b i d 6.21 (0.52 (0.04 t 1 t 1 + ŝd t + ε t (1.24 The common estimated growth rate for money and income corresponds to an annual growth rate of 1.6 per cent. From the estimated model we see that the two interest rates may be weakly exogenous. Imposing weak exogeneity yields equation (1.25. m ( y ( i b i d t (0.053 ( = ( (0.059 m2 y i b i d 6.20 (0.56 ( t 1 0 m Γ 1 y i b + ŝd t + ε t (1.25 i d Table 1.2 shows that none of the restrictions imposed are rejected. (The unrestricted system (1.22 is always the alternative hypothesis. From equation (1.25 we see that the growth rates for money and income is probably not significant (t-value of 1.3. Imposing the restriction that there is no growth in the system (the restrictions imposed t 1

33 1.5. Conclusions and suggestions for further work 33 by Johansen and Juselius, 1990, we get a log likelihood value of and a corresponding p-value of 0.27 (with 7 degrees of freedom. We therefore can not reject that all the growth rates equals zero. In Table 1.3 we see how the estimates of the parameters in the restricted cointegrated vector β = (1, 1, b, b, µ change with different restrictions on the loading parameters (α and growth rates (γ. From the table we see that the restrictions on the loading parameters change the estimates of the cointegration vector (and particularly b more than restrictions on the growth rates do. There can be two reasons for this result. First, the restrictions on the loading vector are more binding, as can be seen from the relatively large drop in the log likelihood value as the restrictions of weak exogeneity are imposed. More binding restrictions will normally change the other parameters more. Second, the multiplicative relationship between α and β may lead to that restrictions on α will be more important than restrictions on γ with respect to the cointegration vector. 1.5 Conclusions and suggestions for further work Sometimes it is relevant to estimate and restrict growth rates and cointegration mean levels in VAR models. These parameters may have economic interpretations, and in particular restrictions on the growth rates are interesting to test. We show that this can be achieved by using an iterative procedure. When restricting growth rates and cointegration mean levels, the degrees of freedom increase. If these restrictions are valid, the estimates of the other parameters in the system will be more precise. On the other hand, the parameters in the cointegration vectors are super-consistent, and the gain may not be large. In the Danish data we see that the estimates hardly change by including these restrictions. However, this may be so as these restrictions are barely binding. More research will be needed in order to learn how important restrictions on growth rates and cointegration mean levels are for the estimates of the cointegration vectors. The method presented here can easily be extended to models including more deterministic variables. A deterministic trend is often included in the cointegration vectors. When a trend is included, the growth rates are no longer orthogonal to the cointegration vectors. However, if ρ is the vector of trend coefficients in the cointegration vectors, β γ = ρ will capture the restrictions between the coefficients. Sometimes we also want to include step dummies in the system. If included, these will pick up changes in the growth rates as well as changes in the cointegration mean levels. However, we may also want to know how the step dummy influences the

34 34 Chapter 1. Growth Rates system: Does the step dummy change the cointegration mean levels, or the growth rates only? Do all growth rates change when the step dummy is included? Such questions can be answered by applying the same method to the step dummy as for the intercepts. Take testing of purchasing power parity as an example: In many countries the inflation rate was higher in the 1980s than in the 1990s. If we test for purchasing power parity, we may include a shift dummy to take account of the shift in the growth rate for prices. However, the shift dummy may pick up shift in the real exchange rate as well as shift in the growth rates. Utilizing the estimation procedure presented here we can test whether the shift in the real exchange rate is significant or not. References Boswijk, H. P. (1995, Identifibility of cointegrated systems, Discussion Paper ti , Tinberger Institute, University of Amsterdam. Clements, M. P. and D. F. Hendry (1999, Forecasting Non-Stationary Economic Time Series, The MIT Press. Doornik, J. A. (1995, Testing general restrictions on the cointegrating space. Nuffield College, Oxford OX1 INF, UK. Doornik, J. A. (2001, Object-Oriented Matrix Programming using Ox, London: Timberlake Consultants Press. Doornik, J. A. and D. F. Hendry (2001, Modelling Dynamic Systems Using PcGive, Volume II, Timberlake Consumltats ltd. London. Engle, R. F. and C. W. J. Granger (1987, Co-integration and error correction: Representing, estimating and testing, Econometrica 55, Hansen, H. and K. Juselius (1994, Manual to Cointegration Analysis of Time Series CATS in RATS, Institute of Economics, University of Cobenhagen. Johansen, S. (1988, Statistical analysis of cointegration vectors, Journal of Economic Dynamics and Control 12, Johansen, S. (1991, Estimating and hypothesis testing of cointegration vectors in gaussian vector autoregressive models, Econometrica 59,

35 References 35 Johansen, S. (1996, Likelihood-based Inference in Cointegrated Vector Autoregressive Models, 2nd printing. Oxford: Oxford University Press. Johansen, S. and K. Juselius (1990, Maximum likelihood estimation and inference on cointegration - with application to the demand of money, Oxford Bulletin of Economics and Statistics 52, Osterwald-Lenum, M. (1992, A note with quantiles of the asymptotic distribution of the ML cointegration rank test statistics, Oxford Bulletin of Economics and Statistics 54,

36 36 Chapter 1. Growth Rates 1.A Appendix Proof of Theorem 1. To prove the theorem we use trab = trba = (veca vecb and vec (AXB = (B A vecx = ((A B vecx, where tr is the trace operator. Deriving equation (1.14 is straightforward, see e.g. Johansen (1996, p. 90. Equations (1.15 and (1.16 (see Boswijk, 1995, Theorem 2: The derivatives of (1.11 with respect to φ and ϕ (under restrictions (1.8 and (1.7 are log L φ log L ϕ ] = TH β [S vec 10 Ω 1 α S11 β α Ω 1 α, (1.a [ = TH αvec β S10 Ω 1 β S11 β α Ω 1]. (1.b Setting (1.a equal zero, substituting (1.7 in (1.a and solving for φ yields (1.15. Similarly, setting (1.b equal to zero and using (1.b and (1.8 leads to (1.16. Equation (1.17: Solving (1.11 with respect to Ω and using log Ω / Ω = Ω 1 = Ω 1 together with (1.12 and (1.13 leads to (1.17. Equation (1.18: The derivative of (1.10 with respect to ψ (under the restriction (1.9 is log L ψ [( = TH γvec Z Φ X β α D s γ Γ ] Ω 1 Γ Setting (1.c equal to zero and using (1.9 lead to (1.18. (1.c

37 Chapter 2 Identifying Structural Breaks in Cointegrated VAR Models 37

38 38 Chapter 2. Structural Breaks Abstract The paper describes a procedure for decomposing the deterministic terms in cointegrated VAR models into growth rate parameters and cointegration mean level parameters. These parameters express long-run properties of the model. For example, the growth rate parameters tell us how much to expect (unconditionally the variables in the system to grow from one period to the next, representing the underlying (steady state growth in the variables. The procedure can be used for analysing structural breaks when the deterministic terms include shift dummies and broken trends. By decomposing the coefficients into interpretable components, different types of structural breaks can be identified. Both shifts in intercepts and shifts in growth rates, or combinations of these, can be tested for. The ability to distinguish between different types of structural breaks makes the procedure superior compared to alternative procedures. Furthermore, the procedure utilizes the information more efficiently than alternative procedures. Finally, interpretable coefficients of different types of structural breaks can be identified. Keywords: Johansen procedure, cointegrated VAR, structural breaks, growth rates, cointegration mean levels. JEL classification: C32, C51, C52. Acknowledgments: Thanks to Søren Johansen, Ragnar Nymoen, Terje Skjerpen and Anders Rygh Swensen for valuable comments on various versions of the paper. An earlier version of the paper was presented at the Econometric Society European Meeting (ESEM conference in Madrid (August 20-24, 2004.

39 2.1. Introduction Introduction In analysing a dynamic econometric model we are often interested in identifying and testing long-run properties. The cointegrating vectors are examples of long run relationships between different variables. However, also the underlying growth rates (i.e. steady state growth rates can be identified in cointegrated vector autoregressive (VAR models. Hungnes (2002 shows how these growth rates can be estimated within a full information maximum likelihood framework, as well as how to test for restrictions on these growth rates. The growth rates tell us how much to expect (unconditionally the variables in the system to grow from one period to the next. When the system is used for forecasting, the vector of growth rates is very important in providing good forecasts. In fact, as the forecasting horizon approaches infinity, the forecast will be determined by this vector only, see Clements and Hendry (1999, pp Structural breaks often imply changes in the growth rates of the variables. With many and frequent structural breaks in time series integrated of order 1, it will normally be best to estimate the system as if the variables in it were integrated of order 2. With less frequent structural breaks identification is possible. Structural breaks have been discussed intensively in the context of univariate autoregressive time series. Perron (1989 suggests three models: Model A, a crash model, with change in intercept but where the slope of the linear trend is unchanged; Model B, a changing growth model, allows a change in the slope of trend function without any sudden change in the level at the time of the break; and Model C, where both intercept and slope are changed at the time of the break. Johansen et al. (2000 present a generalization of model C in a multivariate framework, and allow for testing hypotheses corresponding to Model A. In this paper all deterministic terms in a cointegrated VAR model are decomposed into interpretable components. The corresponding coefficients describe the long run (steady state growth rates in the variables, and possibly shifts in level and growth rates (the latter depending on the type of deterministic variables that are included in the system. Combined with the coefficients of the cointegrating vectors, they also describe levels and trends (and possibly shifts in these in the cointegrating vectors. The decomposition therefore allows us to test all three types of structural breaks suggested in Perron (1989. The paper presents a model C for a multivariate framework where we allow for testing hypotheses corresponding to both A and B. In addition the method presented here makes it possible to identify the growth rates and the size of the different types of shifts.

40 40 Chapter 2. Structural Breaks Johansen et al. (2000 show how the traditional cointegration analysis can be used in order to identify some types of structural breaks. They show that within their framework they can identify (and test for shifts in the trends in the cointegrating vectors, but not in the levels of the cointegrating vectors. In order to use traditional cointegration analysis one needs to disregard observations following immediately after structural breaks by including impulse dummies. The number of impulse dummies after the break corresponds to the number of lags in the system, and the inclusion of these dummies implies a reduction in the effective sample. An alternative could be to use a two step approach, where the coefficients of the deterministic part are estimated in the first step and a traditional cointegration analysis could be conducted on the de-trended time series in a second step. Saikkonen and Lütkepohl (2000 suggest such a two-step approach. However, they only consider testing the cointegration rank, and do not consider how to impose restrictions on the system in order to test for different types of structural breaks. The estimation approach suggested here therefore has three important advantages compared to the alternatives. First, it allows testing for all the different types of structural breaks. Second, it utilizes all the information better by not disregarding observations after a break point. Third, it identifies interpretable coefficients of the different types of structural shifts. On the other hand, a disadvantage with the approach suggested here, is that it involves a more complicated maximizing problem. However, a program, GRaM (see Hungnes, 2005, has been developed to estimate these systems. The estimation approach is illustrated by applying two data sets. In the first illustration the same data set as in Johansen et al. (2000 is used, and it is shown that the approach in the present paper can handle more types of breaks. In the second illustration a money demand system for Germany covering the period of the (re- unification is analysed. The data set is used to test for different types of structural breaks. The estimation approach in the present paper does not consider identification of the cointegrating rank. To determine the cointegrating rank in data series with structural breaks, the procedures in Johansen et al. (2000 or Saikkonen and Lütkepohl (2000 can be applied. Only situations where the break points are known are considered in this paper. Lütkepohl et al. (2004 suggest an approach for identifying the break point and cointegrating rank if the break points are unknown (but the number of breaks is known. We do not consider structural breaks that change the number of cointegrating relationsships. Neither do we consider general changes in the coefficients in the cointegrating relationships (but only for those corresponding to deterministic variables. For such structural breaks, see Hansen (2003.

41 2.2. Model formulation 41 The paper is organized as follows: In Section 2.2 the model is formulated. Section 2.3 presents the estimation problem. In Section 2.4 the estimation procedure is used to identify structural breaks on two different data sets. Section 2.5 concludes and describes other situations where this procedure can be informative. Throughout the paper we define the orthogonal complement of the full column rank matrix A as A such that A A = 0 and (A, A has full rank. (The orthogonal complement of a nonsingular matrix is 0, and the orthogonal complement of a zero matrix is an identity matrix of a suitable dimension. Furthermore, for a matrix A with dimension n m (m n, we define A = A (A A Model formulation Conventional formulation of cointegrated VAR Let Y t be an n-dimensional vector of variables that are integrated of order one at most. α and β are matrices of dimension n r (where r is the number of cointegrating vectors and β Y t is an r 1 vector where all elements are I(0. Furthermore, Γ i (i = 1, 2,..., p 1 are n n matrices of coefficients, where p is the number of lags. is the difference operator. D t is a vector of deterministic variables. The errors ε t are assumed to be Gaussian white noise (ε t NID (0, Ω. Y t = α ( β p 1 Y t 1 + Γ i Y t i + δdt + ε t, t = 1, 2,..., T. (2.1 i=1 It is common to distinguish between deterministic variables that are restricted to lie in the cointegration space and those which are not. Let δdt = δ 0 D0,t + δ 1D1,t, where includes the deterministic variables restricted to lie in the cointegrating space (i.e D 0,t such that δ 0 = αα δ 0 or equivalently α δ 0 = 0. Disregarding different types of dummies (such as impulse dummies, shift dummies and seasonal dummies, ( the most common two specifications for these deterministic variables are D0,t, D 1,t = (1, ( (i.e. restricted constant, excluding a linear drift in Y t, labelled H c and D0,t, D 1,t = (t, 1 (i.e. restricted linear trend, excluding a quadratic trend in Y t, labelled H l. If, in practice there are trends in the data H l is recommended, and in systems without trends H c is recommended. Let us assume that the process in (2.1 is generated by hypothesis H l. The system grows at the unconditional rate E [ Y t ] = γ with long run (cointegration mean levels

42 42 Chapter 2. Structural Breaks E [ β (Y t γ ] = µ. We can re-parameterize the system as Y t γ = α ( β Y t 1 µ ρ (t 1 + p 1 Γ i ( Y t i γ + ε t, (2.2 i=1 where ρ β γ (2.3 is the vector of trend coefficients in the cointegrating vectors. For the system to be stable, the following restriction must hold: Condition Assume that n r of the roots of the characteristic polynomial A (z = (1 z I n αβ z p 1 Γ i (1 z z i i=1 are equal to 1 and the remaining roots are outside the complex unit circle Alternative formulation of cointegrated VAR Here we will present the cointegrated system slightly differently from Equation (2.1. There are two reasons for changing the representation. First, it will be easier to interpret. Second, it will be easier to formulate structural breaks in the system. Let D t be a vector of q deterministic variables, such as trend and seasonally dummies. The system can then be written as Y t γ D t = α ( β (Y t 1 γd t 1 µ + where γ is now an n q matrix of coefficients. p 1 Γ i [ Y t i γ D t i ] + ε t, (2.4 i=1 If D t = t, the system in (2.4 is equal to the system in (2.2. This is the case with linear trend in the variables, i.e. H l. In the case where there are no trends in the variables, D t vanishes from (2.4, and the system can be written as Y t = α ( β Y t 1 µ + p 1 Γ i Y t i + ε t. (2.5 i=1 In either case there is a one-to-one correspondence between the system written in the conventional way, as in (2.1, and in the alternative way, as in (2.4 or (2.5. If the

43 2.2. Model formulation 43 system in (2.1 is estimated with e.g. coefficients of (2.4. ( D0,t, D 1,t = (t, 1, we can always identify the Also, when seasonal dummies are included, (2.1 and (2.4 are statistically equivalent. Generally, however, when other deterministic variables are included in D t there is no such one-to-one relationship between the formulations in (2.1 and (2.4. An alternative way to write the system, is to write the system where the deterministic components are removed. Let Y d be defined as Y with the deterministic components removed, i.e. Y d t = Y t γd t with D t as the vector of deterministic variables and γ as the corresponding matrix of coefficients. Hence, the system can alternatively be written as Y d t = α (β p 1 Yt 1 d µ + i=1 Γ i Y d t i + ε t. (2.6 We have the following theorem: Theorem (Granger s representation theorem with deterministic variables Under Condition 2.2.1, Y t in (2.4 has the moving average representation Y t = C t i=1 ε i + ι + γd t + B t, (2.7 where C = β ( α Γβ 1 α with Γ = I n p 1 i=1 Γ i. The process B t is stationary with zero expectation. The level coefficients ι depends on initial values in such a way that µ = β ι. (2.8 Proof. By using Yt d = Y t γd t (i.e. the system in (2.6, the proof follows from the proof of Theorem 4.2. in Johansen (1996. By formulating the system as in (2.4 we achieve that the representation of the process in (2.7 is valid in the whole sample. This is an advantage of this approach compared to Johansen et al. (2000 where such a representation does not exist in the periods after a break. (This is the reason why Johansen et al. (2000 exclude these observation points by including impulse dummies.

44 44 Chapter 2. Structural Breaks Structural breaks If there are structural breaks in the time series, there might be both level shifts and trend shifts. A shift dummy picks up the level shift in a time series. A shift dummy is a dummy equal to zero up till a specified period and unity afterwards. Both the shift dummy and the corresponding broken trend are included in the vector of deterministic variables, D t. A broken trend picks up the trend shift. The broken trend is constructed as the accumulated value of the corresponding shift dummy. Therefore, the accumulated shift dummy has no level shift. The advantage of defining the broken trend without a shift in the levels, is that it will be much easier to identify the different types of structural breaks. We first consider different types of structural breaks in the cointegration space. The coefficient matrix ρ = β γ contains information about these breaks. If the (vector of coefficients in ρ corresponding to the shift dummy are significantly different from zero, this implies a change of the intercepts in the cointegrating space. Similarly, significant coefficients of the accumulated shift dummy (i.e. the broken trend implies a shift in the slope of the trend in the cointegrating vectors. Therefore, according to the definitions in Perron (1989, the cointegration space follows a crash model (Model A if the coefficients of the shift dummy are significant whereas the coefficients of the broken trend are insignificant. On the other hand, the cointegrating vectors behave as a changing growth model (Model B if the coefficients of the shift dummy are insignificant whereas the coefficients of the broken trend are significant. If the coefficients of both the level-shift and trend-shift dummies are significant, the cointegrating space behaves as Model C. The different types of structural breaks in the time series can be identified in similar ways by examining the coefficient matrix γ. The time series follow a crash model (A if the corresponding coefficients of the shift dummy are significant whereas the corresponding coefficients of the broken trend are insignificant. Correspondingly, the time series follow a changing growth model (B if the corresponding coefficients of the shift dummy are insignificant whereas the corresponding coefficients for the broken trend are significant. If the time series follow a crash model (A, it is impossible to construct a linear relationship of the time series of type B. Therefore, the coefficients of the broken trend in the cointegrating space must be zero as well. Similarly, if the time series follow a changing growth model (B none of the cointegrating vectors can be of type A. However, the reverse implication does not apply. If there are no trend or level shifts in the cointegrating vectors, this does not imply that there are no shifts in the

45 2.3. Estimation 45 time series. The time series may still have trend and level shifts. If so, we say that the cointegrating vectors also are co-breaking vectors. The concept of co-breaking was introduced by Hendry and Mizon (1998. If deterministic breaks in a system of equations can be removed by taking linear combinations of the variables, the variables are said to co-break. Not many analysis on co-breaking have been done. An important reason is that one needs at least as many breaks as variables in the system. If not, there will always exist at least one linear combination of the variables where the deterministic breaks can be removed. Hendry and Mizon (1998 label such situations spurious co-breaking. Here we do not test if there are any linear combination that remove the deterministic breaks. We only test if the cointegrating vectors also are co-breaking vectors. To be precise, let sp (B denote the co-breaking space (as sp ( β denotes the cointegrating space. The hypothesis we are testing is if sp ( β sp (B. 2.3 Estimation Restrictions There are different methods for imposing restrictions on the cointegrating vectors. Here we will only consider restrictions on the cointegrating space. Let β = ( β, µ and Xt = (X t, 1, such that restrictions on the cointegration mean levels can also be imposed. These restrictions on the cointegrating space can be written as R β β = 0, (2.9 where each column in R β represents a restriction on β. Equivalently, we can write β = H β φ β, (2.10 where H β = ( R β. We may refer to φ β as the free parameters in β under the imposed restrictions. However, this is not entirely correct. The cointegration space, and therefore φ β, is only unique up to a normalization and rotation of the cointegration space. We may therefore introduce the normalization φ β = (I r, φ b. Next, we look at how to impose restrictions on the n q matrix γ (where q is the number of deterministic variables. The restrictions we want to impose on γ can be written as R γγ = 0. (2.11

46 46 Chapter 2. Structural Breaks These restrictions can be restrictions on both level and trend shifts. The restrictions on γ can alternatively be written as γ = H γ φ γ, (2.12 where H γ = (R γ and φ γ are the free parameters in γ. As discussed above, restrictions on γ imply restrictions on ρ. However, if we want to test for different types of structural breaks in the cointegrating space, we have to impose restrictions on ρ directly. Let these restrictions be written as R ρρ = 0 R [ ρ γ Jβ ] = 0, (2.13 where J = (I n, 0 n 1. Since ρ = β γ = β Jγ, restrictions on ρ therefore imply restrictions on the product of β and γ. 1 (These restrictions may be transformed into restrictions on β or γ, see Section The estimation problem Next, we consider how to estimate the system. First, suppose we knew φ b and φ γ (and therefore β( and γ. Then the remaining ( coefficients could ( be estimated by applying OLS. Let l α (φ b, φ γ, H β φ β, H γ φ γ, Γ 1 φ b, φ γ,..., Γ p 1 φ b, φ γ, Ω (φ b, φ γ be the corresponding log-likelihood value. Problem The maximum likelihood estimates of β, γ and ρ can be derived from (2.10, (2.12 and ρ = β γ respectively, where φ γ and φ b are given by the solution of the following maximization problem { ( max l α (φ φ γ,φ b, φ γ b, H β φ β, H γ φ γ, Γ 1 ( φ b, φ γ,..., Γ p 1 ( φ b, φ γ subject to R ρ [H γ φ γ JH β φ b ] } = 0., Ω (φ b, φ γ The estimation problem described above is solved using GRaM (an acronym for Growth Rates and cointegration Means, which is programmed in Ox Professional 3.3 (see Doornik (2001. Since it utilizes OxPack, the program is interactive and easy to use. 2 1 If there are restrictions on ρ, then φ b and φ γ are not the free parameters in β and γ. This is because the restrictions imposed on ρ imply restrictions between φ b and φ γ. 2 The latest version is downloadable from my homepage The program requires Ox Professional 3.3 (or later versions, since it applies the function MaxSQP. MaxSQP implements a sequential quadratic programming technique to maximize a non-linear function subject to non-linear constraints.

47 2.3. Estimation Alternative formulations of the estimation problem To apply Problem we must use an algorithm that allows for maximizing under restrictions. However, many maximizing algorithms, such as BFGS (Broyden, Fletcher, Goldfarb and Shanno and SA (simulated annealing, do not allow for such restrictions. An alternative is to transform the restrictions on ρ into restrictions on β or γ. (The restrictions might have to be updated for each iteration. 3 If rank ( R β + rank ( Rρ n r (i.e. the total number of restrictions on β and ρ do not exceed the number of variables minus the number of cointegrating vectors, the maximization problem could be simplified. Suppose we know φ γ (and therefore γ, we could construct the variables Y d t = Y t γd t (i.e. de-trended variables in (2.6, and estimate the remaining coefficients as suggested in e.g. Johansen (1996, Chap , where the restrictions imposed on β now are the joint set of R β and R ργ. The joint restrictions could be written as ( R β R ργ J β = 0, and the maximization problem could be written as { ( max l α (φ ( ( ( } φ γ, β φ γ, H γ φ γ, Γ 1 φ γ,..., Γ p 1 φ γ, Ω (φ γ, (2.14 γ where H γ = ( R β, J γr ρ. This alternative formulation can be used in most of the empirical applications in this paper. If there are many restrictions on β and γ, i.e. rank ( R β + rank ( Rρ > n r, the restrictions on ρ can be transformed into restrictions on γ. The joint restrictions on γ is then 4 ( I n R γ β R ρ vecγ = 0. By applying this form of the restriction on γ, the maximizing problem is equal to that in Problem 2.3.1, but without the constraint (since the constraint is already imposed on γ. (In this situation we use H γ = ( I R γ, β R ρ with vecγ = H γ φ γ and H β = (R γ with β = H β φ β. 3 If BFGS or SA is chosen as maximizing algorithm in GRaM, the program identifies the appropriate formulation based on the number of restrictions. (If SQP is chosen, GRaM applies Problem Here, is the operator for the Kronecker product and veca indicates that all columns in A are stacked in one row vector.

48 48 Chapter 2. Structural Breaks Distribution of the likelihood tests The distribution of most of the likelihood ratio (LR tests that apply are shown in the literature to be χ 2 -distributed. The LR test for the restrictions on β, as formulated in (2.9, is known to be asymptotically χ 2 -distributed with r rank ( R β degrees of freedom, see e.g. (Johansen, 1996, Section Johansen et al. (2000 show that (at least a subset of the restrictions on γ, as formulated in (2.11, are asymptotically χ 2 - distributed with n rank (R γ degrees of freedom. 5 The restrictions on β and γ are independent, such that the total numbers of degrees of freedom is just the sum (i.e. r rank ( R β + n rank (Rγ. Restrictions on ρ can be reformulated into restrictions on β or γ; therefore if restrictions on β and γ can be tested based on a χ 2 -distribution, restrictions on ρ can be tested based on the same distribution as well. The appropriate degrees of freedom can also be found by transforming these restrictions into restrictions on β or γ. 2.4 Empirical illustrations Uncovered interest parity and the Italian/German exchange rate Johansen et al. (2000 apply their method to analyse the uncovered interest parity (UIP hypothesis between Germany and Italy. We analyse the same data using our method. The data used in the analysis are first differences of log consumer price indices for Italy and Germany ( pt I, pd t ; the first difference of log nominal exchange rate between Italian Lira and German Mark ( e t+1 representing the rational expectation to future exchange rates; and nominal interest rates on long-term treasury bonds in both countries (it I, id t. The data, Y t = ( pt I, pd t, e t+1, it I, id t, is plotted in the left part of Figure Johansen et al. (2000 introduce two break points; in 1980q1 and 1992q3. The former corresponds to the creation of the EMS (but is also supposed to capture the oil price shock and the modification on the US monetary policy, and the latter corresponds to the exit of Italy from the EMS and the reunification of Germany. The vector of deterministic variables is therefore given by D t = (t, D1980q1 t, D1992q3 t, cum (D1980q1 t, cum (D1992q3 t. Here D1980q1 t is a step dummy equal to zero before 1980q1 and unity from 1980q1. D1992q3 t is defined similarly, and cum (D1980q1 t and cum (D1992q3 t are the variables for the corresponding broken trends. 5 Johansen et al. (2000 only consider shifts in the trend. 6 The data are available at Source: Johansen et al. (2000.

49 2.4. Empirical illustrations p I p D i I ( i D + e i I i D 0.02 ( i D p D e ( i I p I ( i D p D Figure 2.1: The time series in the left part: inflation in Italy ( p I and Germany ( p D ; interest rates in Italy (i I and Germany (i D ; and log difference of LIT/DM exchange rate ( e. In the right part the three components we expect to be stationary are plotted. In Figure 2.1 we see that there are trends in inflation and interest rates. Therefore, we use the model H l. The estimation period is 1973q4-1995q4, and the number of lags is set to 2 (p = 2. The cointegration rank test results are reported in Table 2.1. The significance probabilities are computed according to Johansen et al. (2000. They are based on an approximation using a Gamma distribution (suggested by Doornik, 1998 in the presence of structural breaks. The reported significance probabilities suggest a rank of two. Johansen et al. (2000, analysing the same data set but handling the breaks differently, find support of three cointegrating vectors. To be able to compare our results with theirs, we continue the analysis with three cointegrating vectors. 7 Following Johansen et al. (2000, we suggest the three stationary linear combinations reported below, cf. the right part of Figure 2.1. They correspond to the UIP 7 However, the probability values are only asymptotically valid. Johansen (2002 shows that there can be significant discrepancies between the true critical values and the asymptotical values.

50 50 Chapter 2. Structural Breaks Table 2.1: Cointegration rank test Rank: loglik Hypothesis Trace p-value r = r = r r = r r = r r = r r = r = Table 2.2: Restrictions on the cointegration space Test loglik LR d.f. p-value No restrictions Restrictions on coint. space hypothesis, the German real interest rate, and the real interest rate differential: y 1t = it I (it D + e t+1, ( y 2t = it D pt D, ( ( y 3t = it I pt I it D pt D These suggested stationary components imply that the cointegration space should span the space of the matrix reported below: β sp The restrictions imposed on the cointegration space when the vector of variables is.. augmented with an intercept (i.e. β = ( β, µ is therefore R β = ( : : 0 where the left 2 5 matrix (which involve the restrictions on β is orthogonal to the cointegration space, and the right vector with zeros corresponds to that we have not imposed any restrictions on the intercept in the cointegration space. The restrictions implied by the suggested cointegration space are not accepted at a 5 per cent level, see Table 2.2. In the following we test different types of breaks both with and without the suggested restrictions on the cointegrated space. In the upper part of Table 2.3 different test of breaks in trends and levels in the,

51 2.4. Empirical illustrations 51 Table 2.3: Test results for structural breaks in cointegrating vectors Test loglik LR d.f. p-value No restrictions No trend in first period No trend in second period No trend in third period No trend-break from 1st to 2nd No trend-break from 2nd to 3rd No level-break from 1st to 2nd No level-break from 2nd to 3rd Test loglik LR d.f. p-value Restricted cointegration space No trend in first period No trend in second period No trend in third period cointegration space are tested when the restrictions on the cointegration space are not imposed. Some of the tests are repeated in the bottom part of Table 2.3 when the cointegration restrictions are imposed. The first three hypotheses in Table 2.3 correspond to the hypotheses in Table 9 in Johansen et al. (2000. For each of the three subperiods (1973q4-1979q4, 1980q1-1992q2, 1992q3-1995q4 the hypothesis of no trend is rejected at a 5 per cent level. The reported significance probabilities are somewhat smaller than the corresponding values reported in Johansen et al. (2000. The discrepancies stem from the fact that they include impulse dummies in two periods after the breaks. 8 In the next two lines of Table 2.3 hypotheses of no break between the different periods are tested. Both hypotheses are rejected. These hypotheses could also be tested by the method in Johansen et al. (2000. However, tests of level-breaks can not be tested by the method suggested in Johansen et al. (2000. The test results reported in Table 2.3 show that such hypotheses are rejected, using the procedure introduced in the current paper. The three first restrictions are re-tested when the restrictions on the cointegration space are imposed, see the last part of Table Note that, at this stage of the 8 In our framework we can produce test results that are approximately equal to those obtained in Johansen et al. (2000 by extending the vector of deterministic variables with the impulse dummies (I1980q2 t, I1980q3 t, I1992q4 t, I1993q1 t. 9 We analyse the system also with these restrictions on the cointegrating space imposed even though they were rejected (see Table 2.2, because they were included in the analysis of Johansen et al. (2000. They found that these restrictions in the cointegration space could not be rejected when combined with additional restrictions on the trends in the cointegrating space. 10 Here rank ( R β = 2 and rank ( Rρ = 1. Therefore; rank ( Rβ + rank ( Rρ = 3 n r = 2, and the

52 52 Chapter 2. Structural Breaks Table 2.4: Test results for structural breaks in variables Test loglik LR d.f. p-value No restrictions No trend in first period No trend in second period No trend in third period Test loglik LR d.f. p-value Restricted cointegration space No trend in first period No trend in second period No trend in third period testing, the restrictions that there are no trends in the cointegration space are rejected for the first and third period. For the second period, however, the restrictions are now far from being rejected. This is consistent with the impression obtained when looking at the right part of Figure 2.1: in the second period (1980q1-1992q2 there are no trends in the suggested stationary relationships. We can also test hypotheses about breaks in the trends of the variables. In Table 2.4 results from these tests are reported. Again we test both with and without the restricted cointegration space. From the upper part of Table 2.4 we see that the hypotheses that there is no trend is rejected for all the three periods, when no restrictions are imposed on the cointegration space. When restrictions are imposed on the cointegration space, however, we can not reject the hypothesis that there is no trend in the variables in the second period Money demand in unified Germany The unification of Germany in 1990 lead to dramatic shifts in German time series. Lütkepohl and Wolters (1998 and Saikkonen and Lütkepohl (2000 use a data set covering the unification to estimate a model for money demand in Germany. They use four variables; (log of real money M3 (m, (log of real GNP (y, an opportunity cost of money (r and inflation (Dp. The opportunity cost of money is defined as the difference between a long term interest rate and the own rate on M3. The data run from 1975q1 to 1996q4. 11 The official date of unification is October 3, However, the monetary unification took place July 1, In the data series for money and income there is a significant shift in level from alternative formulation of the maximizing problem in (2.14 can not be used. Therefore, one of the two alternative formulations must be used. 11 The data are available at: ftp:// /pub/econometrics/germanm3.zip. Source: Lütkepohl and Wolters (1998.

53 2.4. Empirical illustrations 53 m y r p Figure 2.2: The money stock (m, (real income/gdp (y, real interest rate (r and inflation quarter/quarter ( p in Germany (West Germany until 1990q2 and unified Germany thereafter. 1990q3, cf. Figure 2.2. For the opportunity cost of money there is no obvious shift in level or trend. To take account for the shift in level for money and income we include a shift dummy in our empirical analysis. This shift dummy is zero until 1990q2 and one thereafter. In order to test for the different types of structural shifts suggested by Perron (1989 we also include a broken trend, defined as the cumulate of the shift dummy. Due to significant seasonal pattern, especially in income and inflation, we also include (centered seasonal dummies. The cointegration rank test results are reported in Table 2.5, where the significance probabilities are computed according to Johansen et al. (2000. The reported values suggest a rank of two, which is the same cointegration rank as Saikkonen and Lütkepohl (2000 found. Table 2.6 reports the test for homogeneity between money and income in the two cointegrating vectors. Homogeneity is clearly rejected. Next, we test the different types of structural breaks, both in the cointegrating space and for the variables. The test of no trend-break in the cointegrating space can not be rejected, since the restriction implies only a slight reduction in the log-likelihood (cf. Table 2.7. However, the hypothesis of no level-break is rejected at the conventional

54 54 Chapter 2. Structural Breaks Table 2.5: Cointegration rank test Rank: loglik Hypothesis Trace p-value r = r 4 r = r r = r r = r r = r = Table 2.6: Test of restrictions on the cointegrating space Test loglik LR d.f. p-value No restrictions Restrictions on cointgration space levels of significance (since the significance probability is only 0.8 per cent. The joint hypothesis, implying no breaks in the cointegration space has a significance probability of 2.2 per cent. If we apply a 5 per cent significance level we reject these restrictions, but with a 1 per cent significance level it can formally not be rejected. Also for the variables we can not reject the hypotheses of no trend-break, but reject the hypotheses of no level-break. Not surprisingly, also the joint hypothesis of no structural breaks is clearly rejected. In Table 2.7 also the results of a joint combined test of no breaks in the cointegration space and no trend-break in the variables are reported. This has a significance probability of 4 per cent, and whether we reject the hypothesis or not depends on the critical level we choose to use. Table 2.7: Tests of structural breaks Test loglik LR d.f. p-value No restrictions No trend-break in cointgrating space No level-break in cointgrating space No break in cointegrating space No trend-break in variables No level-break in variables No break in variables No break in cointegrating space & no trend-break in variables

55 2.5. Conclusions Conclusions In this paper we have shown how to decompose deterministic parts in cointegrated VAR models into interpretable long run counterparts. The inference procedure includes hypothesis testing on the corresponding coefficients. The procedure is applied on two data sets in which structural breaks are an important feature, and we show how to test for different types of structural breaks in these two concrete settings. However, it is important to note that this method can be used for analysing other types of problems than structural breaks. A decomposition of the deterministic terms into interpretable counterparts will often be important even though there are no structural breaks. For example, Hungnes (2002 used a simplified version of the procedure to estimate a cointegrated VAR model where some variables were allowed to grow and some were not. As mentioned in the introduction, the growth rates are important in forecasting, and identifying the growth rates is important in order to judge the forecasting ability of a model. Another application for the procedure is to test for zero-mean convergence. Zeromean convergence implies that the difference between two variables (say; output per capita in two countries converges to a stationary series with zero mean. To test if the mean is zero, the deterministic terms must be decomposed. References Clements, M. P. and D. F. Hendry (1999, Forecasting Non-Stationary Economic Time Series, The MIT Press. Doornik, J. A. (1998, Approximations to the asymptotic distribution of cointegration tests, Journal of Economic Surveys 12, Reprinted in M. McAleer and L. Oxley (1999: Practical Issues in Cointegration Analysis. Oxford: Blackwell Publishing. Doornik, J. A. (2001, Object-Oriented Matrix Programming using Ox, London: Timberlake Consultants Press. Hansen, P. R. (2003, Structural changes in the cointegrated vector autoregressive model, Journal of Econometrics 114, Hendry, D. F. and G. E. Mizon (1998, Exogeneity, causality, and co-breaking in economic policy analysis of a small econometric model of money in the UK, Empirical Economics 23,

56 56 Chapter 2. Structural Breaks Hungnes, H. (2002, Restricting growth rates in cointegrated VAR models, Revised version of Discussion Papers 309, Statistics Norway. (Downloadable at Hungnes, H. (2005, Identifying the deterministic components in GRaM for Ox Professional - user manual and documentation. Johansen, S. (1996, Likelihood-based Inference in Cointegrated Vector Autoregressive Models, 2nd printing. Oxford: Oxford University Press. Johansen, S. (2002, A small sample correction for the test of cointegration rank in the vector autoregressive model, Econometrica 70, Johansen, S., R. Mosconi and B. Nielsen (2000, Cointegration analysis in the presence of structural breaks in the deterministic trend, Econometrics Journal 3, Lütkepohl, H. and J. Wolters (1998, A money demand system for German M3, Empirical Economics 23, Lütkepohl, H., P. Saikkonen and C. Trenkler (2004, Testing for the cointegrating rank of a VAR process with level shift at unknown time, Econometrica 72, Perron, P. (1989, The great crash, the oil price shock, and the unit root hypothesis, Econometrica 57, Saikkonen, P. and H. Lütkepohl (2000, Testing for the cointegrating rank of a VAR process with structural shifts, Journal of Business and Economic Statistics 18,

57 Chapter 3 A Demand System for Input Factors when there are Technological Changes in Production 57

58 58 Chapter 3. Factor Demand Abstract In a system with n input factors there are n 1 independent cost shares (or budget shares. An often-used approach in estimating factor demand systems is to (implicitly or explicitly assume that there is a (independent cointegrating relationship for each of the n 1 independent cost shares. However, due to technological changes there might not be as many cointegrating relationships as there are (independent cost shares. The paper presents a flexible demand system that allows for both factor neutral technological changes as well as technological changes that affect the relative use of the different factors. The empirical tests indicate that there are fewer cointegrating relationships than usually implied by using conventional estimation approaches. This result is consistent with technological changes. We argue that since such unexplained technological changes are likely to affect input factor decisions, a demand system that allows for such changes should be chosen. Keywords: Factor demand, technological changes, growth rates. JEL classification: C32, C52, D24. Acknowledgments: Thanks to Søren Johansen, Bent Nielsen, Ragnar Nymoen, Terje Skjerpen and Anders Rygh Swensen for valuable comments on an earlier version of this paper.

59 3.1. Introduction Introduction Technological improvements might change the coefficients in the production function. It is therefore too restrictive to assume that the production technology is fixed in the period the data set spans. Even to allow for a stochastic technological progress, is a too restrictive formulation given the pervasive technological change. The present paper presents a more flexible production function where most of the coefficients are allowed to change during the estimation period. We allow for both factor neutral technological changes as well as technological changes that change the relative use of the different input factors. Only the elasticities of scale and substitution are assumed to be constant in the estimation period. The procedure takes into account that the data series for production and input factors are non-stationary. Therefore, a cointegrating vector autoregressive framework is used. However, the system is reformulated in order to make it possible to identify and impose restrictions on the growth rates of the different input factors. This reformulation is suggested by Hungnes (2002, 2005a and the system is estimated by GRaM for Ox, see Hungnes (2005b. A technological change in the production might have a permanent effect on the use of the different input factors. However, if there is no drift in the distribution parameters in the production function, technological changes will only have a temporary effect on the growth of the input factors. In the long run, the only reason for differences in the underlying growth in the input factors are changes in the relative factor prices. Corrected for the effect of changes in relative prices, the growth rates for the different input factors should be the same. In the present paper a factor demand system is estimated. The system includes as many as 7 input factors, and output. It is therefore a relative large system. In the empirical analysis we use gross investment as a proxy of the service flow of capital, and the investment price as the associated price. The paper is organized as follows. In Section 3.2 we present the theoretical model and show the implication of the model on the expected growth of the different input factors. In Section 3.3 the results from the empirical analysis are presented. Section 3.4 concludes. 3.2 Theoretical model In the presentation of the theoretical model we will only consider long term properties. For simplicity, departures from and adjustments back towards these long run

60 60 Chapter 3. Factor Demand relationships are ignored in this section. (However, the empirical analysis in the next section will allow for such temporary effects. Linear factor demand systems are often modelled based upon a production function where only factor neutral progress is allowed. However, technological improvements might not be factor neutral. Many of the coefficients in the production function may therefore change over time. 1 In this paper we will only assume that two parameters do not change over time. Both the elasticity of substitution (σ and the elasticity of scale (κ are assumed unaltered by the technological progress. Based on a CES (constant elasticity of substitution production technology the demand for input factor i conditioned on the production level x can be derived. The factor demand will depend on the price of the input factor p i relatively to the price of other input factors (represented by a weighted factor price p A. The demand function can be written as 2 v i,t = σ ln δ i,t σ (p i,t p A,t 1 κ θ t + 1 κ x t. (3.1 In (3.1 δ 1,t,..., δ n,t are time-varying distribution parameters; δ i,t > 0 ( i, t, n j=1 δ j,t = 1 ( t. (With a Cobb-Douglas technology these distribution parameters express the optimal factor cost ratios. The time-dependence of the δ s is interpreted as picking up factor-biased technological changes. 3 The latent stochastic variable θ t represents the factor neutral technology level. Generally, the expression of the weighted factor price, p A, is rather complicated. However, if σ = 1 (i.e. with Cobb-Douglas production function it is simply the weighted average of the different input factors, where the weight is equal to the optimal cost share. To calculate weighted factor prices, p A, the observable cost shares in each time period are used. By calculating the aggregated factor price by observed cost shares, the aggregated factor price also becomes observable. The factor demand function can be rewritten as v i,t + p i,t p A,t = σ ln δ i,t + (1 σ (p i,t p A,t 1 κ θ t + 1 κ x t, (3.1 The expression on the left hand side of (3.1 is the real cost of factor i or factor i 1 Non-linear factor demand systems can also allow for other technological changes. However, these are normally only allowed to be deterministic. In the present paper, the technological changes are allowed to be stochastic. 2 See Appendix 3.A for how the factor demand functions are derived. 3 However, parameter instability may also stem from other reasons, such as aggregation (over firms effects.

61 3.2. Theoretical model 61 adjusted for relative factor prices. This expression will be treated as one variable, and we will refer to it as factor i (i.e. neglecting adjusted for relative factor prices. If we assume σ = 1 the number of (effective variables in the analysis is reduced. The factor neutral technological level is assumed to follow a stochastic trend where γ θ is the drift parameter: θ t = θ 0 + γ θ t + t ɛ θ,s, (3.2 s=1 where the error term sequence ɛ θ,t (t = 1,..., t = T is independent identically distributed stochastic variables with a zero mean. If ɛ θ,t = 0 t, (3.3 simplifies to a deterministic factor neutral process. The relative weights of the different factors are also allowed to follow stochastic processes. 4 ln δ i,t = ln δ i,0 + t ɛ i,s, (3.3 s=1 where the error term sequence ɛ i,t (t = 1,..., t = T is independent identically distributed stochastic variables with a zero mean. There are no drift parameters in (3.3. The absence of a drift implies that we do not expect some input factors to be more important (and others to be less important over time. In order to write our system in matrix notation we need some definitions: Let I n be the identity matrix of dimension n. Furthermore, let 1 n m be a unity matrix of dimension n m, i.e. a matrix where all elements are unity. Similarly, let 0 n m be a zero matrix of dimension n m, i.e. a matrix of zeros. Taking account of (3.2 and (3.3 included, the system in (3.1 can be written in matrix form as v t + p t p ( A,t θ0 = σ ln δ 0 κ 1 n 1 + ((1 σ I n (p t p A,t ( 1 ( + κ 1 γθ t ( n 1 x t κ 1 n 1 t + σɛ s 1 κ ɛ θs, (3.4 s=1 where v t = (v 1,t,..., v n,t, p t = (p 1,t,..., p n,t, p A,t = p A,t 1 n 1, δ 0 = (δ 1,0,..., δ n,0, ɛ t = (ɛ 1,t,..., ɛ n,t and ɛ θt = ɛ θ,t 1 n 1. 4 Due to the fact that the relative weights sum to unity, the errors in (3.2 are not independent. However, this non-linear relationship between the errors is not important in/for the present analysis.

62 62 Chapter 3. Factor Demand If σ = 1, i.e. with Cobb-Douglas production function, this can be written as v t + p t p A,t = σ ln δ 0 + ( θ0 κ 1 n 1 ( 1 ( κ 1 γθ n 1 x t κ 1 n 1 t + t ( ɛ s 1 κ ɛ θs, (3.5 s=1 Equations (3.4 and (3.5 include unobservable stochastic components. Below it is investigated how to remove these components, depending on the type of technological progress Factor demand relationships Before we present the general case, two special cases will be presented. First, consider the case where the factor neutral technological progress is deterministic and there are no changes of the distribution parameters in the production function: Case Deterministic factor neutral technological progress Defined as: ɛ θ,t = 0, t and ɛ i,t = 0, i, t. Then v t + p t p ( A,t θ0 = σ ln δ 0 κ 1 n 1 + ((1 σ I n (p t p A,t ( 1 ( + κ 1 γθ n 1 x t κ 1 n 1 t, (3.6 expresses n relationships among observable variables. If there are no unobserved stochastic components in the production function, the factor demand equation for each input factor depends only on observable variables. The demand for an input factor depends on its relative price, the level of production, and a deterministic trend that represents the factor neutral technological progress. Another special case is when there is a stochastic factor neutral drift in production, but no stochastic components in the distribution parameters (equation (3.3. Case Stochastic factor neutral technological progress Defined as: ɛ i,s = 0, i, t. Then it is possible to remove the stochastic components in (3.4 by pre-multiply by the matrix B = (I n 1, 1 (n 1 1 (or any matrix spanning the same

63 3.2. Theoretical model 63 space, which yields B [v t + p t p A,t ] = B (σ ln δ 0 + B ((1 σ I n (p t p A,t. (3.7 Equation (3.7 expresses n 1 relationships among the observable variables where some terms have disappeared because B 1 n 1 = 0. (Note that B p A,t = 0, so this term could also be cancelled out. Equation (3.7 expresses relative demand for the input factors: The demand for input factor i (i = 1, 2,..., n 1 will increase relatively to factor n if the price for factor i decreases relatively to the price of factor n, i.e. v i v n = constant σ (p i p n. (We can of course normalize on another input factor than factor n. In this case, none of the relationships of observable variables include the level of production, see (3.7. Proposition In both Case and Case the expression of the cost share does not involve any unobserved stochastic processes. The log of the cost share for input factor i is given by ln n exp (p i,t v i,t n j=1 exp ( = σ ln δ i + (1 σ δ j ln δ j + (1 σ (p i,t p A,t. (3.8 p j,t v j,t j=1 Otherwise, the expression for at least one of the cost ratios will involve stochastic processes. Proof. See Appendix 3.B for proof. Note that if the elasticity of substitution is unity, i.e. Cobb-Douglas technology, δ i is the cost ratio for input factor i. According to Proposition 3.2.1, the expression for each cost ratio involves only observable variables if the distribution parameters, δ 1,..., δ n, do not follow a stochastic process. This is a common assumption when modelling factor demand. Also other common approaches imply that the cost ratios can be expressed by observable variables only. For example, with a translog cost function, the corresponding cost ratios are functions of the different factor prices (plus, possibly, a deterministic trend. In this paper we want to allow for both factor neutral and non-neutral technological progress. Even then, it is possible to find linear combinations of the equations in (3.6 such that the unobserved stochastic components cancel out. Such linear combination will exist if there are linear relationships between the stochastic errors. Assume that the errors can be written as ɛ t 1 κ ɛ θt = Aν t, (3.9

64 64 Chapter 3. Factor Demand where A is an n (n r matrix with full column rank, and ν t is a vector (with n r elements of errors describing the common trends in the demand system. To see how these common trends can be removed, we need to define the orthogonal complement of a matrix. Let the n r matrix B be the orthogonal complement to A, i.e. B = A. The orthogonal complement of the full column rank matrix A is written as A with properties such that A A = 0 and (A, A has full rank. (The orthogonal complement of a non-singular matrix is 0, and the orthogonal complement of a zero matrix is an identity matrix with a suitable dimension. The matrix B is not unique. The matrix ˇB = B Q, where Q is a non-singular matrix (of dimension r r, will also be a representation of the orthogonal complement to A. Then ˇB and B are said to span the same space, and we write this as sp ˇB = spb. For our purpose, the non-uniqueness of B does not represent a problem since we only require that the space spanned by B is unique. The highest number of independent linear combination of factor demand functions where the common trends are removed is given by r. These linear combinations can be derived by pre-multiply (3.9 with the r n matrix B = A, such that B (ɛ s 1 κ ɛ θs = 0. (3.10 The system becomes B [v t + p t p A,t ] ( = B (σ ln δ 0 B θ0 κ 1 n 1 + B ((1 σ I n (p t p A,t ( + B 1 ( κ 1 n 1 x t B γ θ κ 1 n 1 t (3.11 Case and Case are special cases of (3.11: If r = n, i.e. B has full rank, one can use B = I n, and the system coincides with Case If r = n 1 and B sp ( I n 1, 1 (n 1 1, the system coincides with Case Growth and growth rates The theoretical model presented above involves information on the growth of the different input factors. First, take the difference of (3.4. Second, take the expectation. (The second step is not necessary with deterministic factor neutral technological progress, i.e. Case We then get the expected growth rate for each input factor.

65 3.3. Empirical analysis 65 Let be the difference operator and E t the expectation operator. (The subscript indicates that the expectation is formed at the beginning of period t. E t [v t + p t p A,t ] = ((1 σ I n E t (p t p A,t γ θ κ 1 n 1 + (since E t ( t s=1 ɛ s = Et ɛ t = 0 and E t ( t s=1 ɛ θs = Et ɛ θt = 0. ( 1 κ 1 n 1 E t x t (3.12 From equation (3.12 we see that the expected growth of the different input factors (adjusted for changes in relative input prices depends on expected productivity growth (E t θ t = γ θ, expected production growth (E t x t, and the scale elasticity (κ. If σ = 1, the expected growth of the real input factors also depends on the expected changes in relative input prices. It also follows from (3.12 that the expected growth in all the factors are equal, conditioned that the relative input prices do not change. (This stems from the assumption that no factors become more important over time, i.e. that there is no drift in ( Empirical analysis The theoretical background presented in the previous section addresses long run properties only. One would not expect the demand for the different input factors to be on its (long-run equilibrium level in each period. However, one would expect the use of the input factors to be adjusted back towards its equilibrium level if they differ from this level. To analyse the factor demand system a cointegrated VAR model is applied. Following Hungnes (2002, the deterministic parts are decomposed into interpretable counterparts. However, we extend this approach by also decomposing the effect of exogenous variables into interpretable effects on the endogenous variables. We recall the following definitions: Strict stationarity is defined as a stochastic process whose joint distribution of observations is not a function of time, i.e. the joint distribution of (X t1, X t2,..., X tk is the same as the distribution of ( X t1 +h, X t2 +h,..., X tk +h. Weak stationarity (or covariance stationarity describes a process where the first two moments are not functions of time. A stochastic process is called I(d if it is weakly stationary after differencing d times, but not weakly stationary after differencing d 1 times. We are modelling a system with the n input factors plus production. Let Y be a vector of these variables, which are assumed to be non-stationary variables integrated

66 66 Chapter 3. Factor Demand of order one (I(1 at most. Furthermore, let Z t be a vector of q deterministic and exogenous variables. The coefficient matrix γ is of dimension n q. The coefficient matrices α and β are of dimension n r (where r is the number of cointegrating vectors and - as will be shown below - corresponds to the rank of B in the previous section and β (Y t γz t comprises r cointegrating I(0 relations. Furthermore, Γ i (i = 1, 2,..., p 1 are n n matrices of coefficients, where p denotes the number of lags (in levels. The errors ε t are for simplicity assumed to be Gaussian white noise, ε t N (0, Ω. Y t γ Z t = α ( β (Y t 1 γz t 1 µ + p 1 Γ i ( Y t i γ Z t 1 + ε t (3.13 i=1 As for the matrix B, only the space spanned by β is identifiable. Condition Assume that n r of the roots of the characteristic polynomial A (z = (1 z I n αβ z p 1 Γ i (1 z z i i=1 are equal to 1 and the remaining roots are outside the complex unit circle. Under Condition 3.3.1, the system grows at the unconditional rate E t [ Y t ] = γ Z t with long run (cointegrating mean levels E t [ β (Y t γz t ] = µ. Under Condition 3.3.1, Y t has the (moving-average representation Y t = ι + γz t + C t i=1 ε i + Λ t, (3.14 where C = β ( α Γβ 1 α with Γ = I n p 1 i=1 Γ i, and the vector process Λ t is stationary with expectation zero. The level vector ι depends on initial values in such a way that β ι = µ. Note the similarity between (3.14 and (3.4: Y t corresponds to v t + p t p A,t ; ι to (σ ln δ 0 ( θ0 κ 1 n 1 ; γz t to ((1 σ I n (p t p A,t + C t i=1 ε i to σ t s=1 ɛ s 1 κ t s=1 ɛ θs. ( 1κ 1 n 1 x t ( γ θ κ 1 n 1 t; and Λ t only captures short run dynamics, and was therefore ignored in the theoretical section above. Since (3.14 is just another representation of (3.13, this shows that

67 3.3. Empirical analysis 67 our empirical system is suitable for estimating and testing the hypotheses/restrictions from/of the theoretical part. Proposition has some important empirical implications. Let c i be the log of the cost share for factor i. If σ = 1 then c i = ln δ i, and the cost share will be (weakly stationary if δ i is (weakly stationary. Generally (i.e. when σ > 0 c i will cointegrate with p i p A with cointegrating vector (1, (1 σ if σ ln δ i + (1 σ n j=1 δ j ln δ j follows a stationary process. We will assume that this expression is stationary if processes of the δ s are stationary. 5 Therefore, if the empirical process of all the distribution parameters (δ 1,t,..., δ n,t are (strictly stationary, there will be a cointegrating relationship between each cost share and the relative factor prices. Since the cost shares sum to unity, there will be n 1 independent cointegrating relationships between the cost shares and the relative factor prices. Therefore, if the process of the distribution parameters are strictly stationary, there will be (at least n 1 cointegrating relationships in our analysis. (This still holds even if we are not modelling the cost shares explicitly. If, in addition, the process of the (factor neutral technological process is (weakly trend stationary, there will be an additional independent cointegrating relationship in our system. Therefore, if both θ and the δ s follow stationary processes (i.e. Case 3.2.1, there will be one independent cointegrating relationship for each input factor and the cointegrating space is described by (3.6. And if the δ s are stationary and θ follows a non-stationary I(1 process (i.e. Case 3.2.2, there are n 1 independent cointegrating relationships and the cointegrating space is described by (3.7. If not all of the distribution parameters follow stationary processes, there will not be a cointegrating relationship for each cost share. And, at least if θ does not follow a trend stationary process, the number of independent cointegrating relationships will be less than n Distribution of cointegrating rank test From the comparison of (3.14 and (3.4 we noted that the vector Z t must include production (x t. In the cointegrating rank test we therefore condition on the production level. Even though we condition on exogenous variables in the cointegrating rank test we can apply standard critical values. Therefore, we do not use critical values for partial systems. 6 5 A non-linear function of weakly stationary processes is not necessary weakly stationary. However, a non-linear function of strictly stationary processes is strictly stationary. 6 Thanks to Bent Nielsen and Søren Johansen for pointing out that with the formulation of the model chosen here the normal critical values should be used.

68 68 Chapter 3. Factor Demand Let Z be partitioned into exogenous and deterministic variables, i.e. Z t = (X t, D t, where X t is q X 1 and D t is q D 1. Similarly, partition γ as (γ X, γ D and ρ as (ρ X, ρ D with appropriate dimensions. Furthermore, to simplify, we only include one lag here, i.e. p = 1. The system in (3.13 can be written as Y t = γ Z t + α ( β Y t 1 ρz t 1 µ + ε t, (3.15 where ρ = β γ (i.e. ρ X = β γ X and ρ D = β γ D. To write the partial system we define ρ = (ρ X, ρ D. The conventional partial formulation of a partial system can be written as 7 Y t = γ Z t + α ( β Y t 1 ρ Z t 1 µ + u t, (3.16 where ρ D = β γ D (which implies ρ D = ρ D 8 and no restrictions between the coefficients matrixes γ X, β and ρ X. The partial system can be rewritten as Y t = γ Z t + α ( β (Y t 1 γz t 1 (ρ X ρ X X t 1 µ + u t. (3.17 Writing the partial system as in (3.17 it is easy to see why this system not only depends on the common trends from the errors (n r, but also on the number of exogenous variables (q X. The reason why we must adjust for exogenous variables when determining the cointegrating rank in such partial systems, is that the number of coefficients in the expression ρ X ρ X depends on the cointegrating rank as well. It is the presence of the term (ρ X ρ X X t 1 that makes the normal critical tables for determining the cointegrating rank invalid. However, since we implicitly impose the restriction ρ X = ρ X, this term vanishes, and we can apply normal critical values for the cointegrating rank tests. To support the claim that normal critical values can be used to determine the rank in (3.13, independent of how many exogenous variables we include, we simulate different quantiles of the distribution of the cointegrating rank test. These simulated quantiles are reported in the right part of Table 3.1. When there are no exogenous variables, the formualtions in (3.15 and (3.16 are identical. Therefore, the simulated quantiles should be close to the asymptotic critical values. Comparing the first column in the right part of Table 3.1 with the corresponding row in the left part of Table 3.1 show that these quantiles are approximately equal. 7 See Harbo et al. (1998 on conditional systems. 8 To be able to rewrite the partial system like this, D t can only include trend and seasonal dummies, see Hungnes (2005a. With other deterministic variables included, (3.16 is only an approximation. However, the approximation will not affect the asymptotically critical values.

69 3.3. Empirical analysis 69 Table 3.1: Asymptotic and simulated quantiles for systems with D t = t Partial system, as (3.16 Representation here, as (3.15 q X = 0 quantlie q X = 1 q X = 2 q X = 3 q X = 0 q X = 1 q X = 2 q X = 3 n r In the left part: Asymptotic quantiles (of Trace test for systems with D t = t in (3.16, based on a Gamma-distribution as suggested in Doornik (1998, 2003 for different number of exogenous variables (q X. The reported quantiles are taken from Tables 4 and 13 in Doornik (2003; the first column (q X = 0 is taken from the former and the next three columns (q X = 1, 2, 3 are taken from the latter. In the right part: Simulated quantiles (of Trace test for H l (i.e. with D t = t in (3.15. The number of observations is 500 and number of replications is 1000 for each combination of (n r, q X, n r = 1, 2, 3, 4 & q X = 0, 1, 2, 3. However, the simulated quantiles for systems with one ore more exogenous variables differ from the asymptotic quantiles reported in the left part of the table. We see that the asymptotic quantiles increase with the number of exogenous variables, the simulated quantiles do not. Since the simulated quantiles seem to be unaffected by including exogenous variables in (3.15, this indicates that we can apply the conventional asymptotical quantiles for closed systems in the cointegrating rank test Factor demand with Cobb-Douglas Technology (σ = 1 To achieve unbiased estimates of the elasticity of substitution, relative factor prices must be (weakly exogenous (with respect to the parameter of interest, see Richard (1980. If price changes occur due to changes in demand, the estimate of the elasticity of substitution may be (downward biased. In the present analysis it is assumed that the substitution elasticity is unity, i.e. σ = 1. Then our analysis can be based on a Cobb-Douglas production function. This implies that we do not include the relative prices in the vector of exogenous variables. In the analysis 7 different input factors are used, see Table 3.2. Among these are two

70 70 Chapter 3. Factor Demand Table 3.2: Input factors and corresponding prices Notation Input factor price L Labour p L, wage E Electricity p E, price index F Fuel p F, price index M Other materials p M, price index J b Buildings p J,b, price index J te Transport equipment p J,te, price index J m Machinery p J,m, price index energy inputs (electricity and fuel and three different types of real capital (buildings, transport equipment and machinery. For the three real capital inputs we use the investment cost as a proxy for the cost of using that particular real capital factor. Alternatively one could use the real capital stocks multiplied by the user cost of the input factors. The latter has theoretical advantages, but might be more complicated to use, from a practical point of view, since the user cost is not directly observable and because there can be large measurement problems for the real capital time series. The vector of endogenous variables are therefore defined as follows: Y t = v L + p L p A v E + p E p A v F + p F p A v M + p M p A v J,b + p J,b p A v J,te + p J,te p A v J,m + p J,m p A t The vector of exogenous variables are defined as Z t = [ In the vector Z we could also have included other variables, such as seasonal dummies. However, seasonal dummies are not included in this analysis. x t t Our system in (3.13 combined with the choices of Y and Z implies that we indirectly impose a common factor restriction between the variables in Y and the production x. However, inclusion of lagged differences of production (i.e. x t and/or lagged differences of endogenous variables (i.e. Y t in Z would remove this implied restric- ].

71 3.3. Empirical analysis Labour Fuel Electricity Other materials Buildings Machinery Transport equipment Production Figure 3.1: The data series used in the estimation. The time series for the input data are defined as v i + p i p A where i = L, E, F, M, Jb, Jt, Jm. Production x (log-transformed is reported in the lower right part of the table. tion. To limit the number of parameters to be estimated, we do not include lagged differences (of Y in Z in this analysis. Due to the flexibility in the description of technological changes we believe that the theory can be applied to model the factor demand system in all industry sectors. In the empirical illustration below we apply national accounts data from the Norwegian Building and construction industry. Figure 3.1 displays the data series used in the estimation. As can be seen from the plots, there are cycles in many of the time series. In Figure 3.2 the cost shares are plotted. If the δ s follows stationary processes, the cost shares should follow stationary processes as well (when σ = 1.When σ = 1 these cost shares should be stationary around a fixed level when the δ s follows stationary processes. The graphs in Figure 3.2 indicate that this is not the case. By comparing (3.14 and (3.5 we see that the theoretical part implies restrictions on the matrix of coefficients γ. Since, in the absence of changes in (relative factor prices, the use of each input factor should grow at the same rate and react similarly to

72 72 Chapter 3. Factor Demand 0.40 Labour Electricity Fuel Other materials Buildings Transport equipment Machinery Figure 3.2: The cost shares for the different input factors. changes in production, this coefficient matrix must have the form γ = ( 1 κ 1 7 1, γ θ κ (3.18 The matrix β in (3.13 corresponds to the matrix B in the theoretical part, i.e. sp (β = sp (B. Therefore, it follows that these two matrixes must have the same rank, i.e. rank (β = rank (B. Remark 1 (Case In addition to that γ has the form given in (3.18, Case implies that β has full rank. Remark 2 (Case In addition to that γ has the form given in (3.18, Case implies (. that rank (β = n 1 and furthermore that β sp I n 1, 1 (n 1 1 Table 3.3 reports the tests of the rank of β. The reported probability values are based on a Gamma distribution, as suggested by Doornik (1998. According to these significance probabilities the hypotheses of a rank equal to 2 or higher are not rejected, whereas hypotheses of a rank equal to 1 or 0 are rejected. Therefore, the cointegrating rank test indicates a rank of 2, so the rank is neither full nor equal to n 1. Hence, the number of independent cointegrating relationships is consistent with technological changes.

73 3.3. Empirical analysis 73 Table 3.3: Cointegrating rank test likelihood LR (vs r=7 p-value r= r= r= r= r= r= r= r= The estimation period is 1980q1-2002q4 (i.e. T=88. The p-values are calculated based on a Gamma distribution as suggested by Doornik (1998. In the estimation 2 lags are used, i.e. p = 2. The estimation results are obtained by using GRaM 0.99, see Hungnes (2005b. Table 3.4: Likelihood ratio tests likelihood LR d.f. p-value κ γ θ No restr. (r= Scale-elasticity No drift % + Scale-el. = % + No techn % Results from testing different restrictions on γ. LR denotes the likelihood ratio; d.f. the degrees of freedom; and p-value the significance value. The elasticity of scale (κ and technological progress (γ θ are reported (when possible under different restrictions. γ θ is reported in per cent in annual terms. The estimation results are obtained by using GRaM 0.99, see Hungnes (2005b. In Table 3.4 the results of different hypotheses on γ (tested against the system with no restrictions on γ are reported. The first test is labelled Scale-elasticity. Here we restrict all the elements in the first column of γ to be equal, a restriction which is not rejected. The reciprocal of the estimated value is the estimated κ, i.e. the scale-elasticity. The estimated scale elasticity is close to unity. The next test, No drift, tests the hypothesis that there is no deterministic trend in the cost ratios. This involves restricting all elements in the second column of γ to be equal. (The sign + indicates that this is an additional test, i.e. we restrict both columns in γ. This hypothesis is also not rejected. When both these restrictions are imposed, the technological growth is identified. According to our estimation results the technological growth is about 0.2 per cent in annual terms. Two additional hypotheses are also tested. The first of these hypotheses is if the scale elasticity is unity. This hypothesis is not rejected, and we conclude that the scale elasticity is not significantly different from unity. The second hypothesis is if the tech-

74 74 Chapter 3. Factor Demand Table 3.5: Cointegrating rank test (without trend likelihood LR (vs r=8 p-value r= r= r= r= r= r= r= r= See Table 3.3 nological growth is zero. This hypothesis is also not rejected. Hence the technological growth is not significantly positive. The non-rejection of the test of no technological growth implies that there are no significant trends in the data series. However, in Table 3.3 we have assumed a significant trend in the cointegrating rank test. To take account of the fact that the trend is insignificant, we test the cointegrating rank without a trend. The results of this cointegrating rank test are reported in Table 3.5. Also when not including the trend, the hypotheses of a rank equal to 2 or higher are rejected, whereas the hypotheses of a rank equal to 1 or 0 are rejected. Therefore, also the cointegrating rank test without a trend included indicates a rank of Factor demand with CES Technology The results in Section indicate that there are less cointegrating relationships than there are (independent cost shares. However, this result might stem from our a priori value of the elasticity of substitution. In order to test robustness of the realized rank finding, we conduct cointegrating rank tests for different (imposed substitutionelasticities. To do this, we re-define the vector of endogenous variables to and keep Z t unchanged. Y = v L + σ (p L p A v E + σ (p E p A v F + σ (p F p A v M + σ (p M p A v J,b + σ ( p J,b p A v J,te + σ (p J,te p A v J,m + σ (p J,m p A,

75 3.3. Empirical analysis 75 Table 3.6: Cointegrating rank tests for different values of σ Cointegrating rank test with trend σ = 0 σ = 0.1 σ = 0.5 σ = 1 σ = 2 likel. p-val likel. p-val likel. p-val likel. p-val likel. p-val r= r= r= r= r= r= r= r= Cointegrating rank test without trend σ = 0 σ = 0.1 σ = 0.5 σ = 1 σ = 2 likel. p-val likel. p-val likel. p-val likel. p-val likel. p-val r= r= r= r= r= r= r= r= Testing the significance of the trend (cointegrating rank restricted to 2 σ = 0 σ = 0.1 σ = 0.5 σ = 1 σ = 2 LR p-val LR p-val LR p-val LR p-val LR p-val r= Table 3.6 reports the results of the cointegrating rank tests for different values of the substitution-elasticity (σ. Two things should be noted: First, for each rank the likelihood values are highest for σ = 0 or σ = 0.1. This indicates that if we had estimated the substitution-elasticity, the estimate would have been close to zero. Second, for all the chosen values of σ in Table 3.6, the cointegrating rank test indicates a rank of 2, confirming that the rank is substantially less than the number of (independent cost shares. Similarly test of restrictions on γ as those presented in Table 3.4 are reported in Appendix 3.C for different values of σ.

76 76 Chapter 3. Factor Demand 3.4 Conclusions The present paper suggests an approach for estimating factor demand systems with technological changes. The approach allows for both factor neutral technological progress as well as technological changes that change the relative use of the different input factors. The estimation approach makes it possible to impose the restriction that the expected growth in all input factors are equal. This corresponds to assuming that all changes in relative use of input factors not explained by changes in relative input prices, are unpredictable based on the given information set. By applying the estimation approach elaborated here we can estimate the scaleelasticity and the expected technological growth. The identification of these properties is important for detecting if the system has reliable long-run properties (such as growth rates, especially if one aims to use the system for forecasting. In the present paper the focus is on estimating a factor demand system. The reduced cointegrating rank is consistent with technological changes. Conventional estimation approaches might undermine the extent of technological changes. The approach presented here can also be used to analyse consumer demand (because tastes might change. In a system with 9 consumer groups, Raknerud et al. (2003 show that there are only 6 cointegrating vectors. The reduced number of rank (compared to the number of goods indicates that there have been changes in the utility function. These changes in the utility function may stem from changes in tastes. However the lack of a stable utility function may be the result of changes in income distribution. Nevertheless, independent of the reason, it may be important to allow for such instability when estimating the demand system. References Doornik, J. A. (1998, Approximations to the asymptotic distribution of cointegration tests, Journal of Economic Surveys 12, Reprinted in M. McAleer and L. Oxley (1999: Practical Issues in Cointegration Analysis. Oxford: Blackwell Publishing. Doornik, J. A. (2003, Asymptotic tables for cointegration tests based on the Gammadistribution approximation. Accompanying note to Doornik (1998. (Downloadable at Harbo, I., S. Johansen, B. Nielsen and A. Rahbek (1998, Asymptotic inference on coin-

77 References 77 tegrating rank in partial systems, Journal of Business and Economic Statistics 16, Hungnes, H. (2002, Restricting growth rates in cointegrated VAR models, Revised version of Discussion Papers 309, Statistics Norway. (Downloadable at Hungnes, H. (2005a, Identifying structural breaks in cointegrated VAR models, Discussion Papers 422, Statistics Norway. (Downloadable at Hungnes, H. (2005b, Identifying the deterministic components in GRaM for Ox Professional - user manual and documentation. Raknerud, A., T. Skjerpen and A. R. Swensen (2003, A linear demand system within a seemingly unrelated time series equation framework, Discussion Papers 345, Statistics Norway. Richard, J.-F. (1980, Models with several regimes and changes in exogeneity, Review of Economic Studies 47, 1 20.

78 78 Chapter 3. Factor Demand 3.A Deriving the factor demand equations In this appendix we show how to derive the factor demand equations. Here, untransformed variables are used, not log transformed variables as in the main part of the paper. The level variables used are X = exp{x} for production, V i = exp{x i } for input factor i, P i = exp{p i } for the price of input factor i, and P A = exp{p A } expressing the aggregated input factor price. For simplicity, the time subscript is dropped. The production function, with substitution elasticity σ is ( Θ X = n j=1 δ jv σ 1 σ Θ ( n j=1 Vδ j j j σ σ 1 κ for σ > 0, σ = 1 κ for σ = 1 where i δ i = 1 and Θ = exp{θ }. Furthermore, define, ( n Θ = Θ δ δ j j j=1 κσ, where Θ = exp{θ}. When σ = 1 the cost minimizing problem is given by { n min P j V j s.t. X σ 1 V 1,...,V n j=1 which yields the first order conditions σ κ 1 = (Θ σ 1 σ 1 κ n j=1 } δ j V σ 1 σ j, P i λ (Θ σ 1 σ κ 1 σ 1 σ δ iv 1 σ i = 0, (i = 1,..., i = n where λ is the Lagrange multiplier. Solving these equations for the input factors yield V i = λ (Θ σ 1 σ κ 1 σ 1 σ δ i P i σ, (i = 1,..., i = n. These expressions can be used to insert for the input factors in the transformed prod-

79 3.A. Appendix A 79 uct function X σ 1 σ κ 1 = (Θ σ 1 σ 1 κ n δ j j=1 ( = (Θ σ 1 κ σ 1 λ σ 1 σ λ (Θ σ 1 σ κ 1 σ 1 σ δ j P j σ ( σ 1 n δ σ j P1 σ j j=1. σ 1 σ Solving for the Lagrange multiplier yields λ = (Θ 1 κ = (Θ 1 κ ( X σ 1 σ ( n δ δ j j j=1 κ σ ( σ 1 σ 1 σ σ 1 X 1 σκ PA, ( 1 n δ σ j P1 σ j j=1 1 1 σ where we have defined the weighted factor price as P A = ( n δ δ j j j=1 ( n δ σ j P1 σ j j=1 1 1 σ. (3-a Inserting the expression for the Lagrange parameter into the expression for the factor demand yields V i = [ ( (Θ κ 1 n j=1 δδ j j ] 1 σ σ 1 X σκ 1 P A (Θ σ 1 P i σ κ 1 σ 1 σ δ i σ = δ i (Θ [( n δ κδ j j j=1 σ ] 1 σκ X 1 σκ P A P i σ ( σ = δi σ Θ κ 1 1 PA X κ. P i This is the conditional demand function for the input factor when assuming a CES production function. It can be shown (following the same approach as above that this expression also applies when the substitution elasticity equals unity, i.e. with Cobb- Douglas technology. The estimation could have been conducted as in the paper, but by simply substitute v i + p i p A for i = L, E, F, M, Jb, Jt, Jm with v i + σ (p i p A.

80 80 Chapter 3. Factor Demand The expression for the weighted factor price in (3-a is only valid when σ = 1. Here we will show the expression for the aggregated factor price when σ = 1. Taking logs of (3-a yield ln (P A = ( n δ j ln ( ln δ j + j=1 n j=1 δσ j P1 σ j 1 σ Both the nominator and the denominator in the last part of the expression above approach zero when σ 1. Therefore we apply L-Hopital s rule: 9 ( ln lim σ 1 n j=1 δσ j P1 σ j 1 σ = lim σ 1 = lim σ 1 = lim σ 1 ( σ [ σ ( [ σ ln n j=1 δσ j P1 σ j σ (1 σ n j=1 δσ j P1 σ j n j=1 δ σ j = lim P1 σ j σ 1 = = ] [ n j=1 δσ j P1 σ j ] 1 1 ( n j=1 exp [ ] ] σ ln δ j + (σ 1 ln P j n j=1 δσ j P1 σ j [ ] σ ln δj ln P j n j=1 δσ j P1 σ j n [ ] δ j ln δj ln P j j=1 n δ j ln δ j + j=1 n ( ln j=1 P δ j j This yields P A = n j=1 Pδ j j ( P A = n j=1 δδ j j when σ = 1, or generally ( n j=1 δσ j P1 σ j 1 1 σ for σ > 0, σ = 1 n j=1 Pδ j j for σ = 1. 3.B Proof of Proposition Proof. From Appendix 3.A we have the expression for the demand of input factor i: ( σ V i = δi σ Θ κ 1 1 PA X κ P i 9 Thanks to Terje Skjerpen for showing how to prove that the expression converges to n j=1 Pδ j j when σ = 1.

81 3.C. Appendix C 81 Therefore, the cost of factor i is and the cost ratio is P i V i = δ σ i Θ 1 κ X 1 κ P σ A P 1 σ i, P i V i n j=0 P jv j = = = Θ κ 1 X κ 1 PA σδσ i( P1 σ i Θ κ 1 X κ 1 PA σ n j=0 ( [ δ σ i P1 σ i n j=0 δσ i P1 σ i ( n j=1 δδ j j n j=1 δδ j j ( δ σ i P1 σ i 1 σ δ σ i P 1 σ i n j=0 δσ i P1 σ i 1 1 σ ] 1 σ. Applying that the expression in the square brackets is equal to P A yields ( C CES Production Function - Empirical results In Section 3.3 we only tested the restictions on γ in the Cobb-Douglas case (i.e. σ = 1. Here we reproduce these tests for different values of the substitution-elasticity (i.e. σ. As can be seen from Table 7, the estimate of the scale-elasticity is close to unity and the estimate of the technological growth is close to zero for all choices of σ. However, for σ = 2 the estimate of the technological growth is significantly positive.

82 82 Chapter 3. Factor Demand Table 7: Likelihood ratio tests for different choices of σ σ = 0 likelihood LR d.f. p-val κ γ θ No restr. (r = Scale-elasticity No drift % + Scale-el. = % + No techn % σ = 0 likelihood LR d.f. p-val κ γ θ No restr. (r = Scale-elasticity No drift % + Scale-el. = % + No techn % σ = 0 likelihood LR d.f. p-val κ γ θ No restr. (r = Scale-elasticity No drift % + Scale-el. = % + No techn % σ = 0 likelihood LR d.f. p-val κ γ θ No restr. (r = Scale-elasticity No drift % + Scale-el. = % + No techn % Hypotheses testing on γ for different values of σ. For σ = 1 see Table 3.4.

83 Appendix A Identifying the Deterministic Components in Cointegrated VAR Models using GRaM for Ox Professional - User Manual and Documentation 83

84 84 Chapter A. GRaM Abstract GRaM (Growth Rates and cointegration Means is a program for estimating long-run properties in cointegrated VAR models. As other program packages for cointegration analysis, the cointegrating vectors and their corresponding adjustment parameters can be estimated. In addition, the program decomposes the parameters for the deterministic terms into interpretable counterparts such as growth rate parameters and cointegration mean level parameters. These parameters express long-run properties of the model. For example, the growth rate parameters tell us how much to expect (unconditionally the variables in the system to grow from one period to the next, and will therefore be especially important to identify if the system is to be used for forecasting. GRaM also allows for different types of hypotheses testing. Both linear tests on the cointegrating space and on the growth rates are allowed for, i.e. we can test for both the long-run relationship between the time series and the long-run growth in the different time series. This document describes the program. A tutorial shows how to use the program.

85 A.1. Introduction 85 A.1 Introduction A.1.1 System requirements Requirements: - Windows - Ox Professional Simulated Annealing (Ox package A.1.2 Installation You need Ox Professional, (c J. A. Doornik, version 3.3. (or newer in order to run the program. (Please note that you cannot use the console version of Ox. If you do not have Ox Professional you can buy if from the following address: If you have purchased Ox Professional 3.0, 3.1 or 3.2, and have a valid licence code, then you can download the upgrade to version 3.3 from this address: oxpro.html It is important that you upgrade Ox Professional to the version 3.3 (or newer. This is because the program applies a function (MaxSQP that is not implemented in earlier versions of Ox. You must also run Windows to run GRaM for Ox. (I apologize for this... The reason is that the program use the OxPack interface in Ox Professional, and this runs (unfortunately only under Windows. 1 Finally, you need to include the MaxSA package in the directory /Ox/Packages/. (The MaxSA package provides an implementation of Simulated Annealing, see Goffe 1 In the next version of Ox, Ox Professional 4.x, OxPack will probably run under any platform (and not just Windows. GRaM will therefore probably also work on all platforms.

86 86 Chapter A. GRaM et al. (1994 for a documentation of the algorithm. You can download this package from Charles Bos home page: cbos/software/maxsa.html The next step is to download GRaM. You can download it from the following page: Unzip the files and save them in the directory /Ox/Packages/. To run the program start OxPack, which is a part of Ox Professional. In OxPack, choose Add/Remove Package.. under the Package menu. Choose Browse to find gram.oxo. (If you followed the instruction above, you will find it in the directory /Ox/Packages/. Finally, press the button Add. Now you can choose GRaM in the package menu. And that is it! A.1.3 User license and citation GRaM (Growth Rates and cointegration Means 0.99 is written by Håvard Hungnes. It is free to use under the following conditions: 1. Its use in published research is acknowledged. Please cite this document in your list of references. 2. (Disclaimer The code is distributed as is, with no warranties as to fitness for any purpose. Use it at your own risk. Ox Professional 3.3 (or newer is required to use the package, see Installation. Ox should be cited whenever it is used. For example you could say in the text: the results are generated using Ox version 3.30 (see Doornik, 2001 and then give the references: Doornik (2001, Object-Oriented Matrix Programming Using Ox, 4th ed. London: Timberlake Consultants Press.

87 A.2. Theoretical introduction 87 A.2 Theoretical introduction A.2.1 Introduction In analysing a dynamic econometric model we are often interested in identifying and testing long-run properties. The cointegrating vectors are examples of long run relationships between different variables. However, also the underlying growth rates (i.e. steady state growth rates can be identified in cointegrated vector autoregressive (VAR models. The growth rates tell us how much to expect (unconditionally the variables in the system to grow from one period to the next. When the system is used for forecasting, the vector of growth rates is very important in providing good forecasts. In fact, as the forecasting horizon approaches infinity, the forecast will be determined by this vector only. GRaM decomposes all deterministic terms in a cointegrated VAR model into interpretable components. The corresponding coefficients describe the long run (steady state growth rates for the variables, and possibly shifts in level and growth rates (the latter depending on the type of deterministic variables that are included in the system. Combined with the coefficients for the cointegrating vectors, they also describe level and trends (and possibly shifts in these in the cointegrating vectors. Examples of applications: Hungnes (2002; cointegrated VAR model where some variables were allowed to grow and some were not. Hungnes (2005; cointegrated VAR with structural breaks. Another application for the procedure is to test for zero-mean convergence. Zeromean convergence implies that the difference between two variables (say; output per capita in two countries converges to a stationary series with zero mean. To test if the mean is zero, the deterministic terms must be decomposed.

88 88 Chapter A. GRaM Throughout the paper we define the orthogonal complement of the full column rank matrix A as A such that A A = 0 and (A, A has full rank. (The orthogonal complement of a nonsingular matrix is 0, and the orthogonal complement of a zero matrix is an identity matrix of a suitable dimension.furthermore, for a matrix A with dimension n m (m n with full column rank, we define A = A (A A 1. A.2.2 Model formulation and Granger s representation theorem Let Y t be an n-dimensional vector of variables that are integrated of order one at most. α and β are matrices of dimension n r (where r is the number of cointegrating vectors and β Y t comprises r cointegrating I(0 relations. Furthermore, Γ i (i = 1, 2,..., p 1 are n n matrices of coefficients, where p is the number of lags. is the difference operator. D t is a vector of deterministic variables. The errors ε t are assumed to be Gaussian white noise (ε t NID (0, Ω. Y t = α ( β p 1 Y t 1 + Γ i Y t i + δdt + ε t, t = 1, 2,..., T. i=1 (A.1 It is common to distinguish between deterministic variables that are restricted to lie in the cointegration space and those which are not. Let δdt = δ 0 D0,t + δ 1D1,t, where includes the deterministic variables restricted to lie in the cointegrating space (i.e D 0,t such that δ 0 = αα δ 0 or equivalently α δ 0 = 0. Disregarding different types of dummies (such as impulse dummies, shift dummies and seasonal dummies, ( the most common two specifications for these deterministic variables are D0,t, D 1,t = (1, ( (i.e. restricted constant, excluding a linear drift in Y t, labelled H c and D0,t, D 1,t = (t, 1 (i.e. restricted linear trend, excluding a quadratic trend in Y t, labelled H l. If there are trends in the data H l is recommended, and in systems without trends H c is recommended. Let us assume that the process in (A.1 is generated by hypothesis H l. The system grows at the unconditional rate E [ Y t ] = γ with long run (cointegration mean levels E [ β (Y t γ ] = µ. We can re-parameterize the system as where Y t γ = α ( β Y t 1 µ ρ (t 1 + ρ β γ p 1 Γ i ( Y t i γ + ε t, i=1 (A.2 (A.3 is the vector of the trend coefficients in the cointegrating vectors. For the system to be

89 A.2. Theoretical introduction 89 stable, the following restriction must hold: Condition A.2.1 Assume that n r of the roots of the characteristic polynomial A (z = (1 z I n αβ z p 1 Γ i (1 z z i i=1 are equal to 1 and the remaining roots are outside the complex unit circle. In GRaM we formulate the cointegrated system slightly differently than in (A.1. We do this in order to make the deterministic components easier to interpret, such as in (A.2. Let D t be a vector of q deterministic variables, such as trend and seasonally dummies. The system can then be written as Y t γ D t = α ( β (Y t 1 γd t 1 µ + p 1 Γ i [ Y t i γ D t i ] + ε t, i=1 (A.4 where γ is now an n q matrix of coefficients. If D t = t, the system in (A.4 is equal to the system in (A.2. This is the case with linear trend in the variables, i.e. H l. In the case where there are no trends in the variables, D t vanishes from (A.4, and the system can be written as Y t = α ( β Y t 1 µ + p 1 Γ i Y t i + ε t. i=1 (A.5 In both these cases there is a one-to-one correspondence between the system written in the normal way, as in (A.1, and( in the alternative way, as in (A.4 or (A.5. If the system is estimated as in (A.1 with D0,t, D 1,t = (t, 1, we can always identify the coefficients in (A.4. Also when seasonal dummies are included, (A.1 and (A.4 are two alternative ways of writing the same system. Generally, however, there is no such one-to-one relationship between the formulations in (A.1 and (A.4. An alternative way to write the system, is to write the system where the deterministic components are removed. Let Y d be defined as Y with the deterministic components removed, i.e. Y d t = Y t γd t with D t as the vector of deterministic variables and γ as the corresponding matrix of

90 90 Chapter A. GRaM coefficients. Hence, the system can alternatively be written as Y d t = α (β p 1 Yt 1 d µ + i=1 Γ i Y d t i + ε t. (A.6 Now we have the following theorem: Theorem A.2.1 (Granger s representation theorem with deterministic variables Under Condition A.2.1, Y t in (A.4 has the moving average representation Y t = C t i=1 ε i + ι + γd t + B t, (A.7 where C = β ( α Γβ 1 α with Γ = I n p 1 i=1 Γ i. The process B t is stationary with zero expectation. The level coefficients ι depends on initial values in such a way that µ = β ι. (A.8 Proof. By using Yt d = Y t γd t (i.e. the system in (A.6, the proof follows from the proof of Theorem 4.2. in Johansen (1996. The formulation of the system as in (A.2 involves that the representation of the process in (A.7 is valid in the whole sample and independent of the type of deterministic variables included in D t.

91 A.3. GRaM tutorial 91 A.3 GRaM tutorial A.3.1 Introduction To start GRaM you need to start OxPack and choose GRaM under Packages. (If you can not find GRaM under this menu you have not installed GRaM properly. See Section A.1.2 on page 85 on how to install GRaM. In order to follow this tutorial you need to load the data file luetkepohl.in7. (You can find this file /Ox/Packages/Gram/Data/. Lütkepohl and Wolters (1998 and Saikkonen and Lütkepohl (2000 use a data set covering the unification to estimate a model for money demand in Germany. 2 First we look at the data. Choose Graphics... under the Tools menu in GiveWin. (Alternatively you could click on the graphics button or press Alt+G. Highlight the following data series: m, y, r and Dp. Then press << and choose Apply separate actual values plots in the top left part of the window. Now you should get a graph which looks something like Figure A.1 on the next page. The four variables are; (log of real money M3 (m, (log of real GNP (y, an opportunity cost of money (r and inflation (Dp. The opportunity cost of money is defined as the difference between long term interest rate and the own rate on M3. In the data series for money and income there is a significant shift in the level from 1990q3. This corresponds to the monetary (re- unification of Germany, which took place July 1, For the opportunity cost of money and the inflation rate there is no obvious shift in level or trend. To take account for the shift in level for money and income, a shift dummy is constructed. This shift dummy, D1990q3, is zero until 1990q2 and unity thereafter. A 2 The data are available at: ftp:// /pub/econometrics/germanm3.zip. Source: Lütkepohl and Wolters ( The official date of unification is October 3, However, the monetary unification took place about three months earlier.

92 92 Chapter A. GRaM m y r p Figure A.1: The money stock (m, (real income/gdp (y, real interest rate (r and inflation quarter/quarter ( p in Germany (West Germany until 1990q2 and unified Germany thereafter. Figure A.2: GRaM in OxPack corresponding broken trend, defined as the cumulate of the shift dummy, is also included to allow for testing shifts in the growth rates, see Hungnes (2005. There is (possibly a seasonal pattern in the income and inflation rate series. Therefore, some seasonal dummies are constructed; CS1, CS2 and CS3. A.3.2 Formulation Now, let us turn to the estimation of this data set. If you have started GRaM, OxPack will look something like Figure A.2. Choose Formulate... under Model (or click on the highlighted Formulate button, or press

93 A.3. GRaM tutorial 93 Figure A.3: Data selection CTL+Y. You will then get to the Data selection menu. In the Data selection menu you choose the data series to be included in the model. First choose the number of lags; here 2 lags. Then choose the variables to be included (both endogenous and deterministic variables; here all variables in the Database. Finally, choose which variables to be endogenous and which to be deterministic. GRaM automatically suggests the first variable only to be endogenous, and the remaining being deterministic. Here we want the first four variables (money, income, opportunity cost of money, and the inflation rate to be endogenous. Therefore, highlight variables from y to Dp in the Model, and click the button Endogenous. 4 The Data selection menu should now look something like Figure A.3. The next menu is the Estimate model menu, see Figure A.4 on the following page. Here you can choose different types of simulation algorithms, and you can adjust the estimation period. The different types of simulation algorithms will be explained in more details later. Now, use Maximum likelihood (MaxSQP. In the General restrictions menu (Figure A.5 on the next page you choose the number cointegrating vectors, and the number of restrictions on the different coefficient matrixes. Choose 2 for Cointegrating rank, and zero for all restrictions. (In the next subsections you will learn about imposing restrictions. After you click on OK on the General restrictions menu, the program starts to 4 You do not have to highlight the deterministic variables and choose Deterministic ; if a variable is not chosen to be Endogenous, GRaM will assume it is Deterministic.

94 94 Chapter A. GRaM Figure A.4: Estimate Model Figure A.5: General restrictions simulate in order to try to find the maximum likelihood value and the corresponding estimates. The program will (probably write some intermediate results. After a few seconds (depending on how fast your computer is, you will get the final result, see Figure A.6 on the facing page. The first few lines report intermediate results from the simulation. We will ignore this part of the output here.

95 A.3. GRaM tutorial 95 Ox version 3.30 (Windows (C J.A. Doornik, GRaM package version 0.99, (c H. Hungnes, 2004, object created on GRaM( 1 Estimating luetkepohl.in7 using M_MAXSQP STEP 0 : Starting values STEP 1 : Simulation ('C.I. space restrictions' it0 f= e2=6.209e+004 step=1 it100 f= e2= step=1 it106 f= e2=2.054e-007 step=1 Strong convergence STEP 3 : Switching Sw0: f = > Sw1: f = > gamma' m y r Dp D1990q e CS CS CS trend e e-005 cumd1990q st.d. D1990q CS CS CS trend cumd1990q alpha m y r Dp st.d beta m y r Dp Constant rho' D1990q CS CS CS trend cumd1990q Eigenvalues of companion matrix real imag modulus log-likelihood T/2log Omega no. of observations 85 no. of parameters 54 rank of long-run matrix 2 no. long-run restr. 0 cointegration space is not identified SQP using analytical derivatives (eps1=1e-005; eps2=0.0005: Strong convergence Figure A.6: GiveWin output

96 96 Chapter A. GRaM The next lines reports the estimated coefficient matrixes, with the corresponding standard deviations. First, γ (gamma is reported (with standard deviations, followed by α (alpha, β (beta and ρ (rho. (The standard deviation for β can not be computed when no restrictions are imposed. The eigenvalues of the companion matrix are also reported. These should be either on or inside the unit circle, i.e. the modulus should be unity or less. (The number of unit roots should equal the number of endogenous variable minus the cointegrating rank, see Condition A.2.1 on page 89. These eigenvalues are reported to evaluate the reliability of the results; if one or more of the eigenvalues are outside the unit circle results are not reliable (and the corresponding estimated system is explosive. Finally, the likelihood value and some other information about the system are reported. Also the results from the simulation algorithm are reported. The output here is similarly to the output in Multiple-Equation Dynamic Modelling... package in Pc- Give, see Doornik and Hendry (2001. Now, consider the estimated γ in Figure A.6 on the preceding page. The second last row of the estimated γ, labelled trend, reports the trend in the different variables in the system. Since both money and income are measured in logs, the corresponding coefficients can be interpreted as growth rates. The growth rate for money is therefore 0.53 per cent each quarter. This corresponds to an annual growth rate of The estimated quarterly growth rate for income is 1.37 per cent, which corresponds to an annual growth rate of 5.61 per cent. The coefficients for the trend in the variables for opportunity cost and inflation rate are approximately zero, implying no growth in these variables. (Based on the t- values, i.e. the ratio between the estimated coefficient and its corresponding standard deviation, the null hypothesis of each of these two growth rates to be zero would not be rejected. In the last line of the estimated γ in Figure A.6 on the page before the corresponding coefficients for cumd1990q3 are reported. Since cumd1990q3 is defined as the cumulated value of the step dummy D1990q3, this variable is zero until 1990q2, and increases by one unit each quarter thereafter. The corresponding coefficients therefore picks up the shift in the trend in the different variables. Therefore, the quarterly growth rate for money is 1.12 per cent after 1990q2 (i.e The reported coefficient for the trend is therefore the trend (or growth rate in the variable before 1990q3. 5 The annual growth rate can be calculated as γ a = ( 1 + γ q 4 1, where γa is the annual growth rate and γ q is the quarterly growth rate. However, when the growth rate is close to zero, the simplification γ a 4 γ q works well.

97 A.3. GRaM tutorial 97 The first line of the estimated γ in Figure A.6 on page 95 reports the level shift in the different variables. As expected, the shift is large for both money and income, and the shift is much less for the opportunity cost and the inflation rate. From the reported γ we can also see how much seasonality there is in the different variables. The reported coefficients indicate that there is most seasonality in income and inflation, which we also guessed based on the graphs of the series. A.3.3 Imposing restrictions on γ Restrictions on all variables In the previous subsection γ was estimated without restrictions. But we may also want to impose restrictions on γ (or some other matrixes of coefficients. One interesting hypothesis is to test if there is a significant shift in the trend (or the growth rate in the variables from 1990q3. This implies testing if the coefficients in the last row of the estimated γ in Figure A.6 on page 95 are zero. In GRaM restrictions on γ are imposed by applying restriction matrixes. The restrictions we want to impose on γ is written as R γ γ = 0. (A.9 The hypothesis that the coefficients for the trend shift are zero can be tested with the restriction matrix R = (0, 0, 0, 0, 0, 1, since γ m,d1990q3 γ y,d1990q3 γ r,d1990q3 γ Dp,D1990q3 γ m,cs0 γ y,cs0 γ r,cs0 γ Dp,CS0 ( γ m,cs1 γ y,cs1 γ r,cs1 γ Dp,CS γ m,cs0 γ y,cs2 γ r,cs2 γ Dp,CS2 γ m,trend γ y,trend γ r,trend γ Dp,trend = ( γ m,cumd90q3 γ y,cumd90q3 γ r,cumd90q3 γ Dp,cumD90q3 implies γ i,cumd1990q3 = 0, i = {m, y, r, Dp}. Since you have already formulated the model, you do not have to do that again. Therefore, choose Estimate... under Model (or click on the highlighted Estimate button, or press CTL+L. You will then get to the Estimate Model menu. Press OK. In the General restrictions menu change the number of restrictions on γ in the second last line to 1, see Figure A.7 on the following page. (We choose 1 because we

98 98 Chapter A. GRaM Figure A.7: General restrictions (2 only impose one restriction on the γ matrix. Press OK. Now you come to the menu Restrictions on gamma. For each restrictions, there are two choices. The first choice is whether you want the restrictions to be restrictions on all the (endogenous variables in your system, or on only one of them. If you press on the underlined all, a roll-down menu appears. Here you can chose between all, m, y, r, and Dp. We want to impose the restriction on all the variables in our system, and we therefore choose all (which is the default. The second choice is the restriction you want to impose. To make the specification of the restriction vector simpler, GRaM generates a row of equally many zeros as elements in the restriction vector. In addition, in the upper part of the menu, a row of the corresponding deterministic variables are reported. This should make it easier to to construct the restriction vectors. We want to restrict the coefficients for cumd1990q3 to be zero for all the variables in the system. According to the line in the upper part of the menu, the variable cumd1990q3 corresponds to the last element in the restriction vector. Therefore, we change the last element in the restriction vector to unity (or any non-zero number, see Figure A.8 on the next page. Press OK, and GRaM starts to simulate. After a few seconds, the results are reported to the screen. The last part of the print-out should look something like Figure A.9 on the facing page. In the system we have estimated 4 independent coefficient restrictions were imposed, i.e. one restriction for each of the four endogenous variables. This is also re-

99 A.3. GRaM tutorial 99 Figure A.8: Restrictions on γ log-likelihood T/2log Omega no. of observations 85 no. of parameters 54 rank of long-run matrix 2 no. long-run restr. 4 cointegration space is not identified SQP using analytical derivatives (eps1=1e-005; eps2=0.0005: Strong convergence Figure A.9: GiveWin output (part of ported in the output (no. long-run restr.: 4. Since this system is a restricted version of the previous, the log likelihood value is less ( vs We can apply a likelihood ratio test to test the (null hypothesis that there is no trend-break in the (endogenous variables in the system. The likelihood ratio is twice the difference between the likelihood values, i.e. 2 ( = To find the corresponding significance probability, we choose Progress... under Model. In the Progress menu, just choose OK. The program reports information about the general and restricted system. It also reports an χ 2 test of the restriction. (We assume this ratio is χ 2 distributed, see Section A.4.8 on page 120. The reported probability value is 0.381, and with this probability value you would not reject the null hypothesis (that there is no shift in the trend - or growth rates - in the variables. Note that we would have got the same estimation results if we had deleted cum1990q3 as a deterministic variable in our system, and estimated that system without restrictions.

100 100 Chapter A. GRaM Restriction on one variable only Above we imposed the restriction that there were no trend-breaks in any of the endogenous variables. Here we will show how to impose the additional restrictions that there is no trend in the opportunity cost or in the inflation rate. In the General restrictions menu we therefore choose 3 restrictions on gamma in the second last line. In GRaM restrictions can also be imposed on one vector in γ. If we want to impose the restriction that there is no trend in the opportunity cost, this can be written as ( γ r,d1990q3 γ r,cs0 γ r,cs1 γ r,cs2 γ r,trend γ r,cumd90q3 ( = In the Restrictions on gamma menu we shall now formulate 3 restrictions. The first restriction is the same as above. Since GRaM remembers the restrictions, we do not have to change the first restriction. The second restriction is to restrict the trend for the opportunity cost to be zero. Change therefore all to r in the line Restriction 2 on... The restriction vector should read R = (0, 0, 0, 0, 1, 0. Similarly, for the third restriction we change all to Dp and replaces the second last zero with unity, see Figure A.10 on the facing page. In the output GRaM reports that there are now 6 independent coefficient restrictions, and the likelihood is now Comparing it with the unrestricted system yields χ 2 = 4.9[0.56], or comparing it with the system with restrictions only on the broken trend yields χ 2 = 0.7[0.70]. A.3.4 Imposing other (coefficient restrictions Restrictions on γ Not all interesting restrictions on γ can be imposed by the formulation above. For example, in the system we look at, we might want to test if the growth rates for money and income are equal. To allow for restrictions between the coefficients for the

101 A.3. GRaM tutorial 101 Figure A.10: Restrictions on γ (2 deterministic variables in γ, GRaM also allows for restrictions of the form 6 R γγ = 0, (A.10 if the restriction is to hold for all deterministic variables, or R γ,i γ i = 0, (A.11 if it refers to only one deterministic variable. (Here γ i is the i th (column vector in γ. The restriction that the growth rates for money and income are equal, can therefore be imposed by the formulation which implies γ m,trend (A.11. ( γ m,trend γ y,trend γ r,trend γ Dp,trend ( = 0 = γ y,trend. This is an example of restrictions of the form in To impose this restriction, go to the General restrictions menu. Now, change the number of restrictions on γ in the last line to 1 (and keep the 3 restrictions on γ. After 6 In (A.10 the restrictions are imposed on γ, whereas in (A.9 the restrictions are imposed on the transformed of γ, i.e. on γ.,

102 102 Chapter A. GRaM you click on OK, you get to the menu for Restrictions on gamma. Do not change these restrictions, but press OK. Now you get to the menu Restrictions on gamma. The restriction we want to impose is on the trend variable only. Therefore, change all to trend. The restriction vector we want to impose on the trend is R = (1, 1, 0, 0. Therefore, change the first element in the row of zeros to 1 and the second to -1. The restriction that the growth rates for money and income are equal is now imposed, and you can let GRaM simulate by clicking OK. The resulting likelihood value is , and the likelihood ratio of this additional restriction is with a significance probability of 0.00 (with one degree of freedom. Therefore, the hypothesis of equal growth rates for money and income is rejected. Restrictions on ρ Restrictions on γ involve restrictions on the variables in the system. Restrictions on ρ involve restrictions on the cointegrating vectors in the system. We will now consider restrictions on the deterministic variables in the cointegrating vectors. The test we will impose, is if there is a significant shift in the trend (or the growth rates in the two cointegrating vectors from 1990q3. The restrictions are imposed similarly to the restrictions on γ, that is of the form R ρρ = 0 (A.12 if the restriction is to be imposed on all cointegrating vectors. 7 The hypothesis that the coefficients for the trend shift are zero in both cointegrating vectors can be tested with the restriction matrix R ρ = (0, 0, 0, 0, 0, 1, since ( ρ CI1,D1990q3 ρ CI2,D1990q3 ρ CI1,CS0 ρ CI2,CS0 ρ CI1,CS1 ρ CI2,CS1 ρ CI1,CS0 ρ CI2,CS2 ρ CI1,trend ρ CI2,trend ( = 0 0 ρ CI1,cumD90q3 ρ CI2,cumD90q3 implies γ i,cumd1990q3 = 0, i = {CI1, CI2}. (Here CI1 refers to the first cointegrating vector and CI2 refers to the second cointegrating vector. 7 In the present version of GRaM it is not possible to impose restrictions on the individual cointegrating vectors.

103 A.3. GRaM tutorial 103 In the General restrictions choose 1 in the row for Restrictions on rho, and choose 0 for the restrictions on γ and γ. (I.e. we are only considering restrictions on ρ. In the Restrictions on rho menu, keep all after Restriction 1 on.., since we will impose the restriction on all (or both, since there are only two cointegrating vectors cointegrating vectors. 8 Next, change the last 0 to 1, so the restriction vector becomes R ρ = (0, 0, 0, 0, 0, 1. Press OK. The likelihood value with this restriction is , which yields χ 2 = 1.23[0.54]. The restriction is therefore not rejected. This we would expect, since the restriction of no trend break in any of the variables (which is a stronger restriction was not rejected either. We can test for co-breaking when we test restrictions on ρ. The concept of cobreaking was introduced by Hendry and Mizon (1998. If deterministic breaks in a system of equation can be removed by taking linear combination of the system variables, the variables are said to co-break. With GRaM we can test if the co-breaking vectors coincide with the cointegrating vectors. For example, the restriction of no level shift in the variables (R γ = (0, 0, 0, 0, 0, 1 is rejected in our system. However, if the restriction of no level shift in the two cointegrating vectors (R ρ = (0, 0, 0, 0, 0, 1 is not rejected, the (space spanned by the cointegrating vectors equals the (space spanned by the co-breaking vectors. Then we can say that the cointegrating space also represents a co-breaking space. Restrictions on β As other programs, GRaM allows for restrictions among the endogenous variables in the cointegrating space, β. In addition, GRaM allows for restrictions on the intercept of the cointegrating vectors, i.e. µ in (A.2 on page 88. These coefficients (µ may be interpreted as the cointegration mean levels, at least if ρ = 0. The coefficient restrictions on β and µ may be formalized as R β β = c β, (A.13 where β = (β, µ. In (A.13 we have included a level variable, c β, which makes it possible to normalise the cointegrating vectors. 8 In the present version all is the only possible choice, see footnote 7.

104 104 Chapter A. GRaM 3.00 Fitted m Residual: m Detrended: m Fitted y Residual: y Detrended: y Fitted r Residual: r Detrended: r Fitted Dp Residual: Dp Detrended: Dp Figure A.11: Graphical output A.3.5 Graphical output With GRaM you can also get some graphical output of your estimation results. Estimate the system with cointregation rank 2 and without restrictions. Choose Graphical Analysis under the Test menu. You will then get a graphical output similarly to Figure A.11. In the first column the true time series are plotted together with one period predictions ( fitted. In the second column the residuals are plotted. In the last column the de-trended variables are plotted. The de-trended variables are defined by Y det t = Y t γ D t, (A.14 i.e. where the deterministic components are removed from the series. In these detrended data series there should be no deterministics such as trend and seasonality. (If there are, this indicates that you have not included all necessary deterministic variables in the system. From the graph we see that the trending is removed in the series for money and income. And in the graph for de-trended inflation a large degree of the variation is removed, which indicates that much of the variation in inflation were seasonal related.

105 A.3. GRaM tutorial 105 A.3.6 Using GRaM together with PcGive and PcGets In GRaM you can estimate the long run properties of your system, and construct the corresponding de-trended variables, see (A.14. These de-trended variables can be saved in the database and then be used in other programs such as PcGive or PcGets. To save the de-trended variables choose Store in database... under the Test menu, and cross off for De-trended variables. Then choose names for your de-trended variables. (GRaM suggests a prefix Det before the original labels of your data series. A.3.7 The Batch editor All estimation in GRaM can also be achieved through batch commands in GiveWin. Here only the new commands necessary to apply with GRaM are documented. For other batch commands, please look up in the GiveWin documentation. rank(rank : Sets the cointegration rank. Must appear before restrictions and estimate. restrictions(label,... : Impose restrictions. label must be one of the following; beta (if the restrictions shall be imposed on β, rhot (if the restrictions shall be imposed on ρ, gammat (if the restrictions shall be imposed on γ or gamma (if the restrictions shall be imposed on γ. The next numbers represent the restriction (or restrictions. The first number after the label expresses if the restriction is to be imposed on the whole matrix of coefficients or on only one of the vectors. The number 0 represents a restriction on the whole matrix, otherwise the number corresponds to the vector the restriction is to be imposed on. The next numbers correspond to the restriction vector as they are imposed in GRaM. As an example, consider the batch code when estimating the German Money data set with no trend break for money and income and no trend for the opportunity cost of money and for the inflation rate, i.e. as the restrictions as in Figure A.10 on page 101. Click the batch icon in GiveWin. Then the batch code in Figure A.12 on the next page appears. We now consider the restrictions command in the batch code. The label is gammat implies that the restrictions are imposed on γ. The first number is 0, implying that the following restriction shall be imposed on all columns in γ. The next numbers, 0, 0, 0, 0, 0, 1, corresponds to the restriction vector. In the next restriction, the number 3 tells us that the next restriction shall be imposed on the third column in γ, whereas the next numbers, 0, 0, 0, 0, 1, 0 is the restriction vector. Finally, the numbers in the final restriction imply that a similar restriction is also imposed in the forth column in γ. startingvalues(istart: Method for providing starting values, see Section A.4.4

106 106 Chapter A. GRaM module("oxpack"; package("gram"; usedata("luetkepohl.in7"; system { Y = m, y, r, Dp; D = m_1, m_2, y_1, y_2, r_1, r_2, Dp_1, Dp_2, D1990q3, D1990q3_1, D1990q3_2, CS1, CS1_1, CS1_2, CS2, CS2_1, CS2_2, CS3, CS3_1, CS3_2, trend, trend_1, trend_2, cumd1990q3, cumd1990q3_1, cumd1990q3_2; } rank(2; restrictions( gammat, 0, 0, 0, 0, 0, 0, 1; restrictions( gammat, 3, 0, 0, 0, 0, 1, 0; restrictions( gammat, 4, 0, 0, 0, 0, 1, 0; estimate("m_maxsqp", 1975, 4, 1996, 4; Figure A.12: Batch code for a description of the different alternatives. Use 0 for Zeros (or almost zero; use 1 for As for rank = 0; use 2 for OLS - Static regression (gamma restrictions imposed; use 3 for OLS - Static regression (rho restrictions imposed; or 4 for JMN. switching(cswitch, iswitch, dswitch: These commands are used to adjust settings in the switching algorithm. cswitch; the maximum number of iteratins (default is iswitch; print every iswitch th iteration, default is 100, 0 corresponds to not writing intermediate results. dswitch; is the convergence criteria (default is 1.0e-15. The algorithm stops if the increase in likelihood from one iteration to the next is less than this value. MaxControl(mxIter, iprint, bcompact and MaxControlEps(dEps1, deps2: These commands can be used to adjust the settings of the iteration parameters for MaxSQP and MaxBFGS. mxiter is the maximum number of iteration; default is 1000, use -1 to leave the current value unchanged. iprint; print every iprint th iteration, default is 100, 0 corresponds to not writing intermediate results, use -1 to leave unchanged. bcompact; if 1 uses compact form for iteration results. deps1 and deps2 changes the convergence criteria, see Doornik (2001. MaxSAControl(mxEval, iprint, MaxSAControlEps(dEps, ineps and MaxSAControlStep (ins, int, drt, vm, VC: These commands adjust settings for MaxSA, see the corresponding documentation. Note that only one element (and not vectors can be chosen for vtm and vc. Fianlly, MaxControlInitTemp(adT is used to adjust the initial temperature. In GRaM the chosen initial temperature is much smaller than the default in the MaxSA package, and results in a warning when MaxSA is used. The choice of initial temperature in GRaM that small is done in order to prevent explosive solutions and to increase the speed until the maximising algorithm converges.

107 A.4. Technical documentation 107 A.4 Technical documentation A.4.1 The estimation problem without restrictions In this section conditional estimators for γ and β (respectively are presented. Furthermore, the estimation problem is formulated. First, consider the estimator for γ. By defining Yt = (Y t, 1 and β = ( β, µ the system can be written as Y t p 1 Γ i Y t i i=1 = αβ Yt 1 (I + n, Γ1 αβ, Γ2 (Ip+1,..., Γ p γ D t D t 1 : D t p + ε t (A.15 where Γ 1 = Γ 1 + I n, Γ i = Γ i Γ i 1 (i = 2, 3,..., p 1 and Γ p = Γ p 1. Define Z t = vec ( Y t,..., Y t p+1, D v t = vec ( ( D t,..., D t p, Φ = In, Γ 1,..., Γ p 1, ( and Φ = I n, Γ1 αβ, Γ2,..., Γ p. The cointegrated VAR becomes ΦZ t = αβ Y t 1 + Φ ( I p+1 γ D v t + ε t. (A.16 Define S DD = T 1 D v t D v t, S ZD = T 1 Z t D v t, S YD = T 1 Y t 1 Dv t, and similarly for S ZZ and S YZ = S ZY. Furthermore, implicitly define the matrix M

108 108 Chapter A. GRaM with dimension q (p n qn by 9 vec ( I p+1 γ = Mvecγ. (A.18 Theorem A.4.1 With given estimates of α, β, Γ 1,..., Γ p 1 and Ω, the conditional estimator for γ becomes vecγ ( α, β, Γ 1,..., Γ p 1, Ω = and corresponding variance matrix [ ( M S DD Φ Ω 1 Φ ] 1 M [ M vec (Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD ], V (vecγ = T 1 [ M ( S DD Φ Ω 1 Φ M] 1. Proof. To prove the theorem we use tr (AB = tr (BA = (veca vecb and vec (AXB = (B A vecx where tr is the trace operator. The log-likelihood function (when a constant is disregarded can be written as where log L = T 2 trω 1 ΦS ZZ Φ + trω 1 αβ S YY β α trω 1 ΦS ZY β α trω 1 αβ S YZ Φ +N + trω 1 Φ ( I p+1 γ S DD ( Ip+1 γ Φ + N N = trω 1 ΦS ZD ( Ip+1 γ Φ + trω 1 αβ S YD ( Ip+1 γ Φ = tr ( I p+1 γ ( Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD = ( vec ( I p+1 γ ( vec Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD = ( vecγ M vec (Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD., (A.19 9 The explicit form of this matrix is given as M = [(( I p+1 K q,p+1 ( vecip+1 I q In ] Kq,n, (A.17 where K q,n is the commutation ( matrix, see e.g. Magnus and Neudecker (1988, pp This is defined as K q,n = q i=1 n j=1 J i,j J i,j, where J i,j is an q n matrix with all elements equal to zero except the (i, j th element, which is unity (see e.g. Lütkepohl, 1996, p. 116.

109 A.4. Technical documentation 109 The derivative of (A.19 with respect to vecγ is log L vecγ (A.20 [ = TM vec (Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD Φ Ω 1 Φ ( I p+1 γ ] S DD. Disregarding the T, the last part of the right hand side of (A.20 can be written as M vec (Φ Ω 1 Φ ( I p γ S DD and combining (A.20 and (A.21 yields ( = M S DD Φ Ω 1 Φ vec ( I p γ [ ( = M S DD Φ Ω 1 Φ ] M vecγ, (A.21 log L [ ] vecγ = T M vec (Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD [ ( T M S DD Φ Ω 1 Φ ] M vecγ. (A.22 Setting this equal to zero and solving for vecγ yields the estimator, and taking the derivative of (A.22 with respect to vecγ yields the inverted negative of the variance matrix. Let Z 0,t (γ = Y t γ D t, Z 1,t (γ = [ (Y t 1 γd t 1, 1 ] and Z 2,t (γ = vec ( Y t 1 γd t 1,..., Y t p+1 γd t p+1. The system becomes Z 0,t = αβ Z 1,t + ΘZ 2,t + ε, (A.23 where Θ = ( Γ 1,..., Γ p 1. An estimator for β in the equation above could be found by applying reduced rank regression, see e.g. Anderson (1951 or Johansen (1996. However, here only a conditional estimator for β is considered. The equation above can be estimated by considering the concentrated log-likelihood function, which up to a constant term is given by logl = T 2 trω 1 ( S 00 S 01 β α αβ S 10 + αβ S 11 β α, (A.24 where S ij = T 1 ( Z i Z i Z i Z 2 [ Z 2 Z 2 ] 1 Z 2 Z j, i, j = 0, 1, (A.25 with Z 0, Z 1 and Z 2 the data matrices of Z 0,t, Z 1,t and Z 2,t, respectively.

110 110 Chapter A. GRaM Theorem A.4.2 The conditional estimator for β is β = [α Ω 1 α S 11 ] 1 [α Ω 1 I n+1 ] vecs 10. Proof. The derivative of (A.34 (with respect to vecβ is log L [ ] vecβ = Tvec S 10 Ω 1 α S 11 β α Ω 1 α (A.26 Setting this equal to zero, and solving for β yields the (conditional estimator in the theorem. Now, consider how to estimate the system. First, suppose β and γ were known. Then the remaining ( coefficients could be estimated by applying ordinary least square in (A.4. Let l α (β, γ, β, γ, Γ 1 (β, γ,..., Γ p 1 (β, γ, Ω (β, γ be the corresponding log-likelihood value. Problem A.4.1 The maximum likelihood estimates for β and γ are the coefficient matrices that solve the following maximization problem { ( } max β,γ l α (β, γ, β, γ, Γ 1 (β, γ,..., Γ p 1 (β, γ, Ω (β, γ. The solution to this problem must imply that (the empirical counterparts to (A.20 and (A.26 are equal to zero, i.e. that the first order conditions are satisfied. Problem A.4.1 describes how GRaM considers the estimation problem. The program tries to maximize the log-likelihood, and applies the two first order conditions (A.20 and (A.26 to increase the speed and probability to reach the maximum. A.4.2 The estimation problem with restrictions In this section the estimation problem with restrictions is described. On the γ matrix all linear restrictions among the elements are allowed. Therefore, the restrictions on γ can be written as R v γ vecγ = 0. (A.27a The restrictions on γ can alternatively be written as vecγ = H v γ φ v γ, (A.27b

111 A.4. Technical documentation 111 ( where Hγ v = R v γ. In the present version of GRaM only restrictions on the space spanned by the cointegration space are allowed. Define β = ( β, µ and X t = (X t, 1, such that restrictions on the cointegration mean levels also can be imposed. The restrictions on β can be written as 10 R m β β = 0. (A.28a Equivalently, we can write β = Hβ m φm β, (A.28b where Hβ (R m = m β. We may also want to impose restrictions on ρ. Let these restrictions be written as R m ρ ρ = 0, which can be transformed into restrictions on β as [ ] R m ρ γ J β = 0, where J = (I n, 0 n 1, or into restrictions on γ as [ ] β R m β vecγ = 0. (A.29a (A.29b (A.29c Now, consider how to estimate the system. First, suppose β and γ were known. Then the remaining coefficients ( ( could be estimated by applying ordinary least square on (A.4 on page 89. Let l α φ m β, φv γ, Hβ m φm β, Hv γ φ v γ, Γ 1 (φ m β, φv γ,..., Γ p 1 (φ m β, φv γ, ( Ω φ m β, φv γ be the corresponding log-likelihood value. Problem A.4.2 (General problem with matrix restrictions The maximum likelihood estimates for β, γ and ρ can be derived from (A.27b, (A.28b and ρ = β γ respectively, where φ m β and φv γ are given by the solution of the following maximization problem: { ( ( max l α φ m φ m β,φv β, φv γ γ, Hβ m φm β, Hv γ φ v γ, Γ 1 (φ m β, φv γ,..., Γ p 1 (φ m β, φv γ [ ] } subject to φ m β Hm β J R m ρ Hγφ v v γ = 0. (, Ω φ m β, φv γ The solution must imply that (the empirical counterparts to the following first order con- 10 GRaM also allows for normalization restrictions of the type R m β β = c β 1 1 r. However, when maximizing the likelihood, the normalization is ignored. The estimates of β and α are adjusted for these normalization after the likelihood is maximized.

112 112 Chapter A. GRaM ditions holds: H v γ M [ vec ( Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD Φ Ω 1 Φ ( I p+1 γ ] [( ] S DD λ β J R m ρ Hγ v = 0 [ ] vec Hβ m S 10Ω 1 α + Hβ m S 11β α Ω 1 α [ ] λ I r R v ρ γ JHβ m R m ρ where λ is the vector of Lagrange multipliers. (A.30a = 0 (A.30b ( γ β = 0 (A.30c To apply Solution A.4.2 we need an algorithm that allows restrictions. Alternatively, we could transform the restrictions on ρ to restrictions on γ or β. Remark A Restrictions on ρ can be transformed to restrictions on γ by applying (A.29c. The joint set of restrictions imposed on γ is therefore ( R v γ β R m ρ vecγ = 0, (A.31 and we can apply Solution A.4.2 (without the restriction with the modification that H v γ is the orthogonal compliment to the transposed matrix in the parenthesis in (A.31. Remark A Restrictions on ρ can be transformed to restrictions on β by applying (A.29b. The joint set of restrictions imposed on β is therefore ( R m β R m ρ γ J β = 0. (A.32 where J = (I n, 0 n 1. The system can now be maximized by applying Solution A.4.2 (without the constraint where Hβ m parenthesis in (A.32. is the orthogonal compliment to the transposed matrix in the A.4.3 Conditional Reduced Rank Regression Here it will be shown how the remaining coefficients may be estimated under the restrictions on β and ρ, given γ. When γ is known, possible restrictions on ρ must be transformed into restrictions on β. Therefore, the total set of restrictions are given as

113 A.4. Technical documentation 113 in (A.32. The H matrix in (A.28b is therefore defined as the orthogonal compliment of the transposed parentheses in (A.32. Let Z 0,t (γ = Y t γ D t, Z 1,t (γ = [ (Y t 1 γd t 1, 1 ] and Z2,t (γ = vec ( Y t 1 γd t 1,..., Y t p+1 γd t p+1. The system becomes Z 0,t = αβ Z 1,t + ΘZ 2,t + ε, (A.33 where Θ = ( Γ 1,..., Γ p 1. An estimator for β in the equation above could be found by applying reduced rank regression, see e.g. Anderson (1951 or Johansen (1996. However, here only a conditional estimator for β is considered. The equation above can be estimated by considering the concentrated log-likelihood function, which up to a constant term is given by logl = T 2 trω 1 ( S 00 S 01 β α αβ S 10 + αβ S 11 β α, (A.34 where S ij = T 1 ( Z i Z i Z i Z 2 [ Z 2 Z 2 ] 1 Z 2 Z j, i, j = 0, 1, (A.35 with Z 0, Z 1 and Z 2 the data matrices of Z 0,t, Z 1,t and Z 2,t, respectively. As shown in Johansen (1996, p. 91 the estimator for α and Ω (conditioned on β and the supposed known γ are given by α = S 01 β ( β S 11 β 1, Ω = S 00 S 01 β ( β S 11 β 1 β S 10. (A.36 (A.37 With the restrictions β = H β φ β imposed, minimizing the determinant of Ω corresponds to solving the following eigenvalue problem (see Johansen (1996, pp and Theorem 7.2 on p. 107 λ H β S 11H β H β S 10S 1 00 S 01H β = 0, (A.38 for the r highest eigenvalues 1 > λ 1 >... > λ r with the corresponding eigenvectors (v 1,..., v r. Then we use β = H β φ β with φ β = (v 1,..., v r as the estimator.

114 114 Chapter A. GRaM A.4.4 Starting values In GRaM you can choose between different methods for computing starting values. The default method is descried in Section A.4.4. Some, but not all, coefficient restrictions are taken into account when the starting values are chosen. All restrictions imposed on γ are taken into account. This means that we are really considering starting values of φ v γ, and the corresponding starting values for γ is therefore given by (A.27b. Also the restrictions on β as given in (A.28a are taken into account. Restrictions on ρ are not taken into account when starting values are computed if the chosen simulation algorithm is MaxSQP. However, if MaxBFGS or MaxSA are used as starting values, the restrictions on ρ are transformed into restrictions on the cointegration space (if possible. Therefore the joint set of restrictions that are taken into account when the starting values for β are calculated are given by (A.28b, where H m β is given by the orthogonal compliment to the transformed matrix in the parenthesis in (A.32. Zeros (or almost zero Here, the starting values of the elements in the matrix γ are zero or close to zero. 11 (If restrictions are imposed on γ, the elements in φ v γ are set equal to (or close to zero, and (A.27b is used to calculate the corresponding starting values for γ. Based on these starting values for γ, GRaM apply (A.14 on page 104 to de-trend the data series. With these de-trended variables, an estimate for β is identified by normal reduced rank. Restrictions on β formed as in (A.28b are imposed when estimating starting values for β. Depending on the simulation algorithm that is used, also restrictions via ρ may be imposed. As for rank = 0 Here GRaM calculates starting values in two steps. In the first step the starting values to γ are set equal to (or close to, se above zero (but no starting values for β are calculated. In the second step GRaM use the MaxBFGS algorithm to simulate for new starting values for γ in the system where cointegration rank equal to zero is imposed. 11 Sometimes it can be problematic to use starting values of the elements in γ equal to zero. When there are restrictions on ρ, these restrictions are allays fulfilled when γ consists of zeros only, see (A.29a. Therefore, when γ equals zero restrictions on ρ will not involve any restrictions on β. However, when the simulation algorithm tries to change value of the elements in γ slightly, the restrictions on ρ will imply restrictions on β. The algorithm may therefore not move from the situation where all the elements in γ equals zero. This problem is solved by choosing starting values not equal (but close to zero.

115 A.4. Technical documentation 115 (Therefore, restrictions on β and ρ are not taken into account. However, the restrictions on γ are imposed. Based on the starting values for γ from the first step, starting values for β are computed by reduced rank. Here the restrictions on the cointegration space (i.e. equation (A.28b or the joint set of restrictions on β and ρ given by Remark A are imposed, depending on which simulation algorithm is used. (The former is used when MaxSQP is chosen and the latter is used when MasBFGS or MaxSA is chosen. OLS - Static regression (gamma restrictions imposed Here the starting values of γ are found by estimating Y t = constant + γ D t + u t (A.39 when the restrictions given by (A.27a are imposed on γ. Only a first round estimate is found, where the identity matrix is used as a proxy for the covariance matrix of the errors. OLS - Static regression (rho restrictions imposed Also here starting values of γ are found by estimating (A.39, but here the restrictions on γ are combined with restrictions on the space of ρ, i.e. ( R v γ I n R m ρ vecγ = 0. (A.40 Since we are imposing such strong restrictions on γ, we only need to impose the restrictions on the cointegration space (see equation (A.28b independent of which simulation algorithm we use. The restrictions implied by (A.29a are already imposed on γ. JMN/SL - Johansen, Mosconi, Nilsen and Saikkonen, Lütkepohl The system in (A.4 on page 89 can be rewritten as y t = α ( [ β p 1 p 1 Y t 1 µ ρd t 1 + Γ i Y t i + γ D t i=1 i=1 Γ i γ D t i ] + ε t (A.41 Use β = H m β φm β, γ = H m γ φ m γ, and ρ = H m ρ φ m ρ, where Hm β = (R m β, Hm γ =

116 116 Chapter A. GRaM ( ( ( R m γ, and Hm ρ = R m ρ, R m γ. Note that the special construction of Hρ m secures ( ( that all restrictions on γ also are imposed on ρ, i.e. span Hρ m span Hγ m. 12 With these restrictions the system can be rewritten as ( Y t = α φ m β, φm [ Hm β Y t 1] ρ Hρ m D t 1 1 Y t 1. + ( Y t p+1 Γ 1,..., Γ p 1, Ψ 0, Ψ 1,..., Ψ p 1 H γ D t H γ D t 1. H γ D t p ε t, with some restrictions on the coefficient matrices Ψ i (which are ignored when computing the starting values. When the coefficient restrictions in the Ψ s are ignored, the system above may include linear dependent variables. Therefore, let [...] 0 only include linearly independent variables (i.e. exclude variables that are linearly dependent. Furthermore, in [...] 1, exclude all variables that are linearly dependent of other variables in [...] 0 or [...] 1. Now α, β, Γ i (i = 1,..., p 1 and Ω can be estimated by applying reduced rank technics, see e.g. Johansen (1996, pp Based on these estimates, γ can be estimated by 13 vecγ ( α, β, Γ 1,..., Γ p 1, Ω [ = Hγ v H v ( γ M S DD Φ Ω 1 Φ 1 MHγ] v (A.42 ] (Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD, [ H v γ M vec where the same notation as in Theorem A.4.1 on page 108 is used. The proof is similar to the proof of Theorem A.4.1. If MaxSQF is chosen as the maximizing algorithm, the estimates of β and γ achieved by the procedure above is used as starting values. If MaxBFGS or MaxSA is used as 12 See Johansen et al. (2000, Section 4 for a discussion on the problems when this span restriction is not fulfilled. 13 To use the equation below one needs an estimate of β (and not only β. The present version of GRaM use ˆµ = t=1 T ˆβ Y t 1. However, ˆµ = t=1 T ˆβ Y t 1 ˆφ m [ ρ Hρ m D t 1 will probably be a more ]1 precise estimate.

117 A.4. Technical documentation 117 maximizing algorithm, β is estimated again where also the restrictions on ρ are imposed. A.4.5 Simulation If MaxSQP is used as the simulation algorithm the log-likelihood function formulated in Problem A.4.2 is maximized, and the first order conditions (A.30a - (A.30c are used to making the algorithm analytical. 14 If MasBFGS or MaxSA is used as the simulation algorithm, GRaM uses a modification of Problem A.4.2 described in the remarks. Remark A is used if possible (i.e. when there are not so many restrictions on β and ρ that the orthogonal complement to the parenthesis (A.32 is not defined, Remark A otherwise. A.4.6 Switching In Section A.4.1 it was shown that the system could be written as ΦZ t = αβ Y t 1 + Φ ( I p+1 γ D v t + ε t, (A.43 where Z t = vec ( Y t,..., Y t p+1, D v t = vec ( ( D t,..., D t p, Φ = In, Γ 1,..., Γ p 1, ( and Φ = I n, Γ1 αβ, Γ2,..., Γ p. Alternatively, the system could be written as Z 0,t = αβ Z 1,t + ΘZ 2,t + ε, (A.44 where Z 0,t (γ = Y t γ D t, Z 1,t (γ = [ (Y t 1 γd t 1, 1 ], Z2,t (γ = vec ( Y t 1 γd t 1,..., Y t p+1 γd t p+1 and Θ = ( Γ 1,..., Γ p 1. Furthermore, define S DD = T 1 D v t D v t, S ZD = T 1 Z t D v t, S YD = T 1 Y t 1 Dv t, 14 The restrictions (A.30a and (A.30b are divided into two parts: One part that corresponds to the maximizing problem without restrictions on ρ, and one part based on the restrictions R m ( ρ β γ = 0, see description of MaxSQP in Ox (Doornik, 2001.

118 118 Chapter A. GRaM (and similarly for S ZZ and S YZ = S ZY, M ij = T 1 TZ it Z jt, i, j = 0, 1, 2, t=1 S ij = M ij M i2 M22 1 M 2j, i, j = 0, 1, and implicitly define the matrix M with dimension q (p n qn by 15 vec ( I p+1 γ = Mvecγ. Theorem A.4.3 The conditional maximum likelihood estimators for β, α, Θ ( Γ 1,..., Γ p 1, γ and Ω under the restrictions (A.27a, (A.28a and (A.29a are given by β = H β φ β (A.45 α = S 01 β ( β S 11 β 1 (A.46 Θ = M 02 (M 22 1 αβ M 12 (M 22 1 (A.47 vecγ = Hγ [H ( v γm S DD Φ Ω 1 Φ ] 1 MH γ [ ] H γm vec (Φ Ω 1 ΦS ZD Φ Ω 1 αβ S YD (A.48 Ω = S 00 αβ S 10 S 01 β α + αβ S 11 β α (A.49 where φ β is the solution of the eigenvalue problem in Section A.4.3, and or or H β = with J = (I n, 0 n 1. H β = H β = ( ( ( R m β and H γ = R v γ, β R m ρ ( ( (R m β, J γr m ρ and H γ = R v γ ( ( ( (R m β, J γr m ρ and H γ = R v γ, β R m ρ (I (II (III Proof. The equations (A.45, (A.46 and (A.49 are shown in Section A.4.3; for (A.47 see Johansen (1996, Section 6.1 and (A.48 see Theorem A.4.1 on page See footnote 9 on page 108 for the explicit form of M.

119 A.4. Technical documentation 119 From (I - (III it can be seen that the restrictions on ρ are imposed on either β or γ, or on both. The advantage of using (III in an algorithm is that is secures that the likelihood value increases for each iteration. However, it might reduce the parameter space the algorithm can search. The advantage of using (I or (II in an algorithm is that it seeks a broader parameter space than if (III is used. However, it might lead to a decrease in the likelihood value. GRaM uses the following algorithm: Algorithm A.4.1 Estimates for β, α, Θ ( Γ 1,..., Γ p 1, γ and Ω under the restrictions (A.27a, (A.28a and (A.29a may be obtained by the following iterative procedure, starting from a set of initial values {β, α, Θ, γ, Ω}: β i = β (ˆα i 1, ˆγ i 1, ˆΩ i 1 ; H β α i = α ( ˆβ i, ˆγ i 1, ˆΩ i 1 Θ i = Θ ( ˆβ i, ˆγ i 1, ˆΩ i 1 vecγ i = γ ( ˆβ i, ˆα i, ˆΘ i, ˆΩ i 1 ; H γ Ω i = Ω ( ˆβ i, ˆα i, ˆγ i i = 1, 2,... Use (I. However, if an iteration leads to a decreased likelihood, use the estimates from the previous iteration and use (III in this iteration. In the next iteration, use (I, unless this leads to a decreased likelihood. (And so forth... Continue until convergence. A.4.7 Standard errors GRaM computes standard errors. The standard errors for γ are based on the conditional variance matrix for γ. With given estimates of α, β, Γ 1,..., Γ p 1 and Ω, the conditional variance matrix for γ under the restriction (A.27a becomes [ V (vecγ ( = T 1 H H M S DD Φ Ω 1 Φ 1 MH] H, where S DD is defined elsewhere. The standard errors for vec (γ are the square root of the diagonal elements in this matrix.

120 120 Chapter A. GRaM The conditional variance matrix for β is shown in Doornik (1995 to be given by V (vecβ ( [ ( ( = T 1 I r Hβ m I r Hβ (α ] m Ω 1 α S 11 I r Hβ m 1 ( I r Hβ m, (with S 11 defined elsewhere and the standard errors for vecβ are given by the square root of the diagonal elements in this matrix. Since ρ = β γ, the variance of ρ depends on the variances of both β and γ. It is therefore simpler to calculate the variance of ρ conditioned of β or γ. Note that ρ = H ρ H ρρ + R ρ R ρρ where H = H (H H 1 with H ρ = ( R ρ. Since R ρρ = 0 and ( ρ = β γ, we have ρ = H ρ H ργ β. Vectorizing this yields vecρ = I r H ρ H ργ vecβ. Then the variance of ρ becomes ( V (vecρ = I r H ρ H ργ ( V (vecβ I r γ H ρ H ρ if we condition on γ. The standard errors for vec (ρ are the square root of the diagonal elements in this matrix. Note that the standard errors of β and ρ are only computed if appropriate normalization restrictions are imposed on the cointegration space. A.4.8 Distribution for the likelihood tests When GRaM tests restrictions, it applies the χ 2 -distribution. For restrictions on β it has been shown in e.g. Boswijk (1995 or Boswijk and Doornik (2004 that this is the correct distribution. Boswijk and Doornik (2004 also shows that restrictions on β is χ 2 in the H c case, (i.e D t is empty, or at least does not include a trend. In the H l case µ is not included in the cointegration vectors when estimating cointegrated VAR models in its normal form (A.1, and therefore similar results are not available. For restrictions on γ, Johansen et al. (2000 show that (at least a subset of restrictions of the form R γ are χ 2 -distributed. Restrictions on ρ can be reformulated into restrictions on β or γ; so if restrictions on β and γ can be tested based on a χ 2 -distribution, restrictions on ρ can be tested based on the same distribution as well. In order to apply the χ 2 test one needs to know the degrees of freedom in the test, i.e. how many independent ( restrictions that are imposed. The number of restrictions on γ is given by rank R v γ. If there are no restrictions on β there are (n + 1 r r independent elements i β. If restrictions are imposed, Boswijk (1995 shows that the number of independent ele-

121 A.4. Technical documentation 121 ments in β is given by rank [( I r β Hβ ]. If both restrictions on β and γ are considered; H v β = (R v β, ( I r J γ R v ρ The number of independent restrictions imposed on β and ρ is therefore given by (n + 1 r r rank [( I r β Hβ ]. The degrees of freedom used in the χ 2 test is the sum of restrictions on γ and the restrictions on β and ρ, i.e. the degrees of freedom are given by (n + 1 r r rank [( I r β ] ( Hβ + rank R v γ..

122 122 Chapter A. GRaM References Anderson, T. W. (1951, Estimating linear restrictions on regression coefficients for multivariate normal distributions, Annals of Mathematical Statistics 22, Boswijk, H. P. (1995, Identifibility of cointegrated systems, Discussion Paper ti , Tinberger Institute, University of Amsterdam. Boswijk, H. P. and J. A. Doornik (2004, Identifying, estimating and testing restricted cointegrated systems: An overview, Statistica Neerlandica 58(4, Doornik, J. A. (1995, Testing general restrictions on the cointegrating space. Nuffield College, Oxford OX1 INF, UK. Doornik, J. A. (2001, Object-Oriented Matrix Programming using Ox, London: Timberlake Consultants Press. Doornik, J. A. and D. F. Hendry (2001, Modelling Dynamic Systems Using PcGive, Volume II, Timberlake Consumltats ltd. London. Goffe, W. L., G. D. Ferrier and J. Rogers (1994, Global optimalization of statistical functions with simulated annealing, Journal of Econometrics 60, Hendry, D. F. and G. E. Mizon (1998, Exogeneity, causality, and co-breaking in economic policy analysis of a small econometric model of money in the UK, Empirical Economics 23, Hungnes, H. (2002, Restricting growth rates in cointegrated VAR models, Revised version of Discussion Papers 309, Statistics Norway. (Downloadable at Hungnes, H. (2005, Identifying structural breaks in cointegrated VAR models, Discussion Papers 422, Statistics Norway. (Downloadable at

123 References 123 Johansen, S. (1996, Likelihood-based Inference in Cointegrated Vector Autoregressive Models, 2nd printing. Oxford: Oxford University Press. Johansen, S., R. Mosconi and B. Nielsen (2000, Cointegration analysis in the presence of structural breaks in the deterministic trend, Econometrics Journal 3, Lütkepohl, H. and J. Wolters (1998, A money demand system for German M3, Empirical Economics 23, Lütkepohl, H. (1996, Handbook of Matrices, New York: John Wiley & Sons. Magnus, J. R. and H. Neudecker (1988, Matrix Differential Calculus wih Applications in Statistics and Econometrics, New York: John Wiley & Sons. Saikkonen, P. and H. Lütkepohl (2000, Testing for the cointegrating rank of a VAR process with structural shifts, Journal of Business and Economic Statistics 18,

124 Håvard Hungnes Trends and Breaks in Cointegrated VAR Models In analysing a dynamic econometric model we are often interested in identifying and testing long-run properties. The cointegrating vectors are examples of long run relationships between different variables. However, also the underlying growth rates (i.e. steady state growth rates can be identified in cointegrated vector autoregressive (VAR models. The growth rates tell us how much to expect (unconditionally the variables in the system to grow from one period to the next. When the system is used for forecasting, the vector of growth rates is very important in providing good forecasts. In fact, as the forecasting horizon approaches infinity, the forecast will be determined by this vector only. In this thesis I show how the deterministic terms in a cointegrated VAR model can be decomposed into interpretable components. The corresponding coefficients describe the long run (steady state growth rates for the variables, and possibly shifts in level and growth rates (the latter depending on the type of deterministic variables that are included in the system. Combined with the coefficients for the cointegrating vectors, they also describe level and trends (and possibly shifts in these in the cointegrating vectors. The thesis consists of three papers plus a documentation of GRaM. GRaM is an Ox program for estimating the systems in the thesis. The program is menu driven and easy to use, and can be downloaded from my homepage