Trends and Breaks in Cointegrated VAR Models


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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, N0033 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 Longrun 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 CobbDouglas 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 longrun 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 ndimensional 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 BeveridgeNelson 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, nonlinear 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. Longrun properties in cointegrated VAR models Longrun 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 reunification. 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 scaleelasticity 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 nonexperts. 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 closedform 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 cobreaking 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, Likelihoodbased 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 2528, 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 overidentified restrictions can be tested against economic theory. A lot of work has been accomplished in estimating longrun cointegration relationships in economics. However, other parameters in a cointegrated VAR model also have longrun 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 ndimensional vector of nonstationary 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 premultiply 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, premultiply (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, premultiplying with β gives (1.5. To find (1.6, rearrange (1.4 to αµ = Γγ η, premultiply 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 1974Q31987Q3, 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 Â = ( ] ] [ β unr [ β 1 ( ] [ unr [ β b ] unr is used to derive α 1 = α unr Â 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: 1974Q31987Q3. 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 OsterwaldLenum (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 λmax 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 λmax 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 OsterwaldLenum (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 tvalues. 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 pvalue [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 (tvalue 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 pvalue 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 superconsistent, 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
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