Trends and Breaks in Cointegrated VAR Models


 Cecily Harrison
 3 years ago
 Views:
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, N0033 Oslo, Norway. Homepage:
2 2
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
Chapter 6: Multivariate Cointegration Analysis
Chapter 6: Multivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie VI. Multivariate Cointegration
More informationEstimation and Inference in Cointegration Models Economics 582
Estimation and Inference in Cointegration Models Economics 582 Eric Zivot May 17, 2012 Tests for Cointegration Let the ( 1) vector Y be (1). Recall, Y is cointegrated with 0 cointegrating vectors if there
More informationChapter 4: Vector Autoregressive Models
Chapter 4: Vector Autoregressive Models 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie IV.1 Vector Autoregressive Models (VAR)...
More informationNormalization and Mixed Degrees of Integration in Cointegrated Time Series Systems
Normalization and Mixed Degrees of Integration in Cointegrated Time Series Systems Robert J. Rossana Department of Economics, 04 F/AB, Wayne State University, Detroit MI 480 EMail: r.j.rossana@wayne.edu
More informationThe VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium
More informationSYSTEMS OF REGRESSION EQUATIONS
SYSTEMS OF REGRESSION EQUATIONS 1. MULTIPLE EQUATIONS y nt = x nt n + u nt, n = 1,...,N, t = 1,...,T, x nt is 1 k, and n is k 1. This is a version of the standard regression model where the observations
More informationChapter 5: Bivariate Cointegration Analysis
Chapter 5: Bivariate Cointegration Analysis 1 Contents: Lehrstuhl für Department Empirische of Wirtschaftsforschung Empirical Research and und Econometrics Ökonometrie V. Bivariate Cointegration Analysis...
More informationThe Feeble Link between Exchange Rates and Fundamentals: Can We Blame the Discount Factor?
LUCIO SARNO ELVIRA SOJLI The Feeble Link between Exchange Rates and Fundamentals: Can We Blame the Discount Factor? Recent research demonstrates that the welldocumented feeble link between exchange rates
More informationTesting The Quantity Theory of Money in Greece: A Note
ERC Working Paper in Economic 03/10 November 2003 Testing The Quantity Theory of Money in Greece: A Note Erdal Özmen Department of Economics Middle East Technical University Ankara 06531, Turkey ozmen@metu.edu.tr
More informationExplaining Cointegration Analysis: Part II
Explaining Cointegration Analysis: Part II David F. Hendry and Katarina Juselius Nuffield College, Oxford, OX1 1NF. Department of Economics, University of Copenhagen, Denmark Abstract We describe the concept
More informationThe link between unemployment and inflation using Johansen s. cointegration approach and vector error correction modelling.
Proceedings 59th ISI World Statistics Congress, 2530 August 2013, Hong Kong (Session CPS102) p.4340 The link between unemployment and inflation using Johansen s cointegration approach and vector error
More informationThe NominaltoReal Transformation
The NominaltoReal Transformation Hans Christian Kongsted PhD course at Sandbjerg, May 2 Based on: Testing the nominaltoreal transformation, J. Econometrics (25) Nominaltoreal transformation: Reduce
More informationThe EngleGranger representation theorem
The EngleGranger representation theorem Reference note to lecture 10 in ECON 5101/9101, Time Series Econometrics Ragnar Nymoen March 29 2011 1 Introduction The GrangerEngle representation theorem is
More informationOn the long run relationship between gold and silver prices A note
Global Finance Journal 12 (2001) 299 303 On the long run relationship between gold and silver prices A note C. Ciner* Northeastern University College of Business Administration, Boston, MA 021155000,
More informationFinancial Integration of Stock Markets in the Gulf: A Multivariate Cointegration Analysis
INTERNATIONAL JOURNAL OF BUSINESS, 8(3), 2003 ISSN:10834346 Financial Integration of Stock Markets in the Gulf: A Multivariate Cointegration Analysis Aqil Mohd. Hadi Hassan Department of Economics, College
More informationState Space Time Series Analysis
State Space Time Series Analysis p. 1 State Space Time Series Analysis Siem Jan Koopman http://staff.feweb.vu.nl/koopman Department of Econometrics VU University Amsterdam Tinbergen Institute 2011 State
More informationChapter 5: The Cointegrated VAR model
Chapter 5: The Cointegrated VAR model Katarina Juselius July 1, 2012 Katarina Juselius () Chapter 5: The Cointegrated VAR model July 1, 2012 1 / 41 An intuitive interpretation of the Pi matrix Consider
More informationInternet Appendix to Stock Market Liquidity and the Business Cycle
Internet Appendix to Stock Market Liquidity and the Business Cycle Randi Næs, Johannes A. Skjeltorp and Bernt Arne Ødegaard This Internet appendix contains additional material to the paper Stock Market
More informationChapter 1. Vector autoregressions. 1.1 VARs and the identi cation problem
Chapter Vector autoregressions We begin by taking a look at the data of macroeconomics. A way to summarize the dynamics of macroeconomic data is to make use of vector autoregressions. VAR models have become
More informationTEMPORAL CAUSAL RELATIONSHIP BETWEEN STOCK MARKET CAPITALIZATION, TRADE OPENNESS AND REAL GDP: EVIDENCE FROM THAILAND
I J A B E R, Vol. 13, No. 4, (2015): 15251534 TEMPORAL CAUSAL RELATIONSHIP BETWEEN STOCK MARKET CAPITALIZATION, TRADE OPENNESS AND REAL GDP: EVIDENCE FROM THAILAND Komain Jiranyakul * Abstract: This study
More informationINDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition)
INDIRECT INFERENCE (prepared for: The New Palgrave Dictionary of Economics, Second Edition) Abstract Indirect inference is a simulationbased method for estimating the parameters of economic models. Its
More informationOverview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model
Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written
More informationAre the US current account deficits really sustainable? National University of Ireland, Galway
Provided by the author(s) and NUI Galway in accordance with publisher policies. Please cite the published version when available. Title Are the US current account deficits really sustainable? Author(s)
More informationTHE EFFECTS OF BANKING CREDIT ON THE HOUSE PRICE
THE EFFECTS OF BANKING CREDIT ON THE HOUSE PRICE * Adibeh Savari 1, Yaser Borvayeh 2 1 MA Student, Department of Economics, Science and Research Branch, Islamic Azad University, Khuzestan, Iran 2 MA Student,
More informationChapter 6. Econometrics. 6.1 Introduction. 6.2 Univariate techniques Transforming data
Chapter 6 Econometrics 6.1 Introduction We re going to use a few tools to characterize the time series properties of macro variables. Today, we will take a relatively atheoretical approach to this task,
More informationC: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)}
C: LEVEL 800 {MASTERS OF ECONOMICS( ECONOMETRICS)} 1. EES 800: Econometrics I Simple linear regression and correlation analysis. Specification and estimation of a regression model. Interpretation of regression
More information1 Short Introduction to Time Series
ECONOMICS 7344, Spring 202 Bent E. Sørensen January 24, 202 Short Introduction to Time Series A time series is a collection of stochastic variables x,.., x t,.., x T indexed by an integer value t. The
More informationJim Gatheral Scholarship Report. Training in Cointegrated VAR Modeling at the. University of Copenhagen, Denmark
Jim Gatheral Scholarship Report Training in Cointegrated VAR Modeling at the University of Copenhagen, Denmark Xuxin Mao Department of Economics, the University of Glasgow x.mao.1@research.gla.ac.uk December
More information1 Teaching notes on GMM 1.
Bent E. Sørensen January 23, 2007 1 Teaching notes on GMM 1. Generalized Method of Moment (GMM) estimation is one of two developments in econometrics in the 80ies that revolutionized empirical work in
More informationVector Time Series Model Representations and Analysis with XploRe
01 Vector Time Series Model Representations and Analysis with plore Julius Mungo CASE  Center for Applied Statistics and Economics HumboldtUniversität zu Berlin mungo@wiwi.huberlin.de plore MulTi Motivation
More informationDEPARTMENT OF ECONOMICS CREDITOR PROTECTION AND BANKING SYSTEM DEVELOPMENT IN INDIA
DEPARTMENT OF ECONOMICS CREDITOR PROTECTION AND BANKING SYSTEM DEVELOPMENT IN INDIA Simon Deakin, University of Cambridge, UK Panicos Demetriades, University of Leicester, UK Gregory James, University
More informationLeast Squares Estimation
Least Squares Estimation SARA A VAN DE GEER Volume 2, pp 1041 1045 in Encyclopedia of Statistics in Behavioral Science ISBN13: 9780470860809 ISBN10: 0470860804 Editors Brian S Everitt & David
More informationNonStationary Time Series, Cointegration and Spurious Regression
Econometrics 2 Fall 25 NonStationary Time Series, Cointegration and Spurious Regression Heino Bohn Nielsen 1of32 Motivation: Regression with NonStationarity What happens to the properties of OLS if variables
More informationNonStationary Time Series andunitroottests
Econometrics 2 Fall 2005 NonStationary Time Series andunitroottests Heino Bohn Nielsen 1of25 Introduction Many economic time series are trending. Important to distinguish between two important cases:
More informationA Trading Strategy Based on the LeadLag Relationship of Spot and Futures Prices of the S&P 500
A Trading Strategy Based on the LeadLag Relationship of Spot and Futures Prices of the S&P 500 FE8827 Quantitative Trading Strategies 2010/11 MiniTerm 5 Nanyang Technological University Submitted By:
More informationTime Series Analysis III
Lecture 12: Time Series Analysis III MIT 18.S096 Dr. Kempthorne Fall 2013 MIT 18.S096 Time Series Analysis III 1 Outline Time Series Analysis III 1 Time Series Analysis III MIT 18.S096 Time Series Analysis
More informationForeign Direct Investment and Economic Growth in Sri Lanka
Sri Lankan Journal of Agricultural Economics. Vol. 6, No. 1, 2004 Foreign Direct Investment and Economic Growth in Sri Lanka N. Balamurali and C. Bogahawatte * ABSTRACT This paper examines the relationship
More informationIs the Forward Exchange Rate a Useful Indicator of the Future Exchange Rate?
Is the Forward Exchange Rate a Useful Indicator of the Future Exchange Rate? Emily Polito, Trinity College In the past two decades, there have been many empirical studies both in support of and opposing
More informationIs the Basis of the Stock Index Futures Markets Nonlinear?
University of Wollongong Research Online Applied Statistics Education and Research Collaboration (ASEARC)  Conference Papers Faculty of Engineering and Information Sciences 2011 Is the Basis of the Stock
More informationThe LongRun Relation Between The Personal Savings Rate And Consumer Sentiment
The LongRun Relation Between The Personal Savings Rate And Consumer Sentiment Bradley T. Ewing 1 and James E. Payne 2 This study examined the long run relationship between the personal savings rate and
More informationTime Series Analysis
Time Series Analysis Identifying possible ARIMA models Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos
More informationCausality between Government Expenditure and National Income: Evidence from Sudan. Ebaidalla Mahjoub Ebaidalla 1
Journal of Economic Cooperation and Development, 34, 4 (2013), 6176 Causality between Government Expenditure and National Income: Evidence from Sudan Ebaidalla Mahjoub Ebaidalla 1 This study aims to determine
More informationTime Series Analysis
Time Series Analysis Forecasting with ARIMA models Andrés M. Alonso Carolina GarcíaMartos Universidad Carlos III de Madrid Universidad Politécnica de Madrid June July, 2012 Alonso and GarcíaMartos (UC3MUPM)
More informationEMPIRICAL INVESTIGATION AND MODELING OF THE RELATIONSHIP BETWEEN GAS PRICE AND CRUDE OIL AND ELECTRICITY PRICES
Page 119 EMPIRICAL INVESTIGATION AND MODELING OF THE RELATIONSHIP BETWEEN GAS PRICE AND CRUDE OIL AND ELECTRICITY PRICES Morsheda Hassan, Wiley College Raja Nassar, Louisiana Tech University ABSTRACT Crude
More informationUnivariate and Multivariate Methods PEARSON. Addison Wesley
Time Series Analysis Univariate and Multivariate Methods SECOND EDITION William W. S. Wei Department of Statistics The Fox School of Business and Management Temple University PEARSON Addison Wesley Boston
More informationCOINTEGRATION AND CAUSAL RELATIONSHIP AMONG CRUDE PRICE, DOMESTIC GOLD PRICE AND FINANCIAL VARIABLES AN EVIDENCE OF BSE AND NSE *
Journal of Contemporary Issues in Business Research ISSN 23058277 (Online), 2013, Vol. 2, No. 1, 110. Copyright of the Academic Journals JCIBR All rights reserved. COINTEGRATION AND CAUSAL RELATIONSHIP
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationThe Effect of Infrastructure on Long Run Economic Growth
November, 2004 The Effect of Infrastructure on Long Run Economic Growth David Canning Harvard University and Peter Pedroni * Williams College 
More informationThe University of Chicago, Booth School of Business Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay Solutions to Final Exam
The University of Chicago, Booth School of Business Business 41202, Spring Quarter 2013, Mr. Ruey S. Tsay Solutions to Final Exam Problem A: (40 points) Answer briefly the following questions. 1. Describe
More informationSales forecasting # 2
Sales forecasting # 2 Arthur Charpentier arthur.charpentier@univrennes1.fr 1 Agenda Qualitative and quantitative methods, a very general introduction Series decomposition Short versus long term forecasting
More informationWorking Papers. Cointegration Based Trading Strategy For Soft Commodities Market. Piotr Arendarski Łukasz Postek. No. 2/2012 (68)
Working Papers No. 2/2012 (68) Piotr Arendarski Łukasz Postek Cointegration Based Trading Strategy For Soft Commodities Market Warsaw 2012 Cointegration Based Trading Strategy For Soft Commodities Market
More informationCointegration. Basic Ideas and Key results. Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board
Cointegration Basic Ideas and Key results Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board Summer School in Financial Mathematics Faculty of Mathematics & Physics University of Ljubljana
More informationREASSESSMENT OF SUSTAINABILITY OF CURRENT ACCOUNT DEFICIT IN INDIA
SouthEastern Europe Journal of Economics 1 (2012) 6779 REASSESSMENT OF SUSTAINABILITY OF CURRENT ACCOUNT DEFICIT IN INDIA AVIRAL KUMAR TIWARI * ICFAI University, Tripura Abstract In this study, we examined
More informationIS THERE A LONGRUN RELATIONSHIP
7. IS THERE A LONGRUN RELATIONSHIP BETWEEN TAXATION AND GROWTH: THE CASE OF TURKEY Salih Turan KATIRCIOGLU Abstract This paper empirically investigates longrun equilibrium relationship between economic
More informationChapter 12: Time Series Models
Chapter 12: Time Series Models In this chapter: 1. Estimating ad hoc distributed lag & Koyck distributed lag models (UE 12.1.3) 2. Testing for serial correlation in Koyck distributed lag models (UE 12.2.2)
More informationThe information content of lagged equity and bond yields
Economics Letters 68 (2000) 179 184 www.elsevier.com/ locate/ econbase The information content of lagged equity and bond yields Richard D.F. Harris *, Rene SanchezValle School of Business and Economics,
More informationIntroduction to Matrix Algebra
Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra  1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary
More informationComovements of NAFTA trade, FDI and stock markets
Comovements of NAFTA trade, FDI and stock markets Paweł Folfas, Ph. D. Warsaw School of Economics Abstract The paper scrutinizes the causal relationship between performance of American, Canadian and Mexican
More informationFULLY MODIFIED OLS FOR HETEROGENEOUS COINTEGRATED PANELS
FULLY MODIFIED OLS FOR HEEROGENEOUS COINEGRAED PANELS Peter Pedroni ABSRAC his chapter uses fully modified OLS principles to develop new methods for estimating and testing hypotheses for cointegrating
More informationRelationship between Commodity Prices and Exchange Rate in Light of Global Financial Crisis: Evidence from Australia
Relationship between Commodity Prices and Exchange Rate in Light of Global Financial Crisis: Evidence from Australia Omar K. M. R. Bashar and Sarkar Humayun Kabir Abstract This study seeks to identify
More informationGovernment bond market linkages: evidence from Europe
Applied Financial Economics, 2005, 15, 599 610 Government bond market linkages: evidence from Europe Jian Yang Department of Accounting, Finance & MIS, Prairie View A&M University, Prairie View, TX 77446,
More informationNCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )
Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates
More informationCorporate Defaults and Large Macroeconomic Shocks
Corporate Defaults and Large Macroeconomic Shocks Mathias Drehmann Bank of England Andrew Patton London School of Economics and Bank of England Steffen Sorensen Bank of England The presentation expresses
More informationANALYSIS OF EUROPEAN, AMERICAN AND JAPANESE GOVERNMENT BOND YIELDS
Applied Time Series Analysis ANALYSIS OF EUROPEAN, AMERICAN AND JAPANESE GOVERNMENT BOND YIELDS Stationarity, cointegration, Granger causality Aleksandra Falkowska and Piotr Lewicki TABLE OF CONTENTS 1.
More informationEnergy consumption and GDP: causality relationship in G7 countries and emerging markets
Ž. Energy Economics 25 2003 33 37 Energy consumption and GDP: causality relationship in G7 countries and emerging markets Ugur Soytas a,, Ramazan Sari b a Middle East Technical Uni ersity, Department
More informationCentre for Central Banking Studies
Centre for Central Banking Studies Technical Handbook No. 4 Applied Bayesian econometrics for central bankers Andrew Blake and Haroon Mumtaz CCBS Technical Handbook No. 4 Applied Bayesian econometrics
More informationWhy the saving rate has been falling in Japan
MPRA Munich Personal RePEc Archive Why the saving rate has been falling in Japan Yoshiaki Azuma and Takeo Nakao January 2009 Online at http://mpra.ub.unimuenchen.de/62581/ MPRA Paper No. 62581, posted
More informationStatistical Machine Learning
Statistical Machine Learning UoC Stats 37700, Winter quarter Lecture 4: classical linear and quadratic discriminants. 1 / 25 Linear separation For two classes in R d : simple idea: separate the classes
More informationLecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay
Lecture 4: Seasonal Time Series, Trend Analysis & Component Model Bus 41910, Time Series Analysis, Mr. R. Tsay Business cycle plays an important role in economics. In time series analysis, business cycle
More informationy t by left multiplication with 1 (L) as y t = 1 (L) t =ª(L) t 2.5 Variance decomposition and innovation accounting Consider the VAR(p) model where
. Variance decomposition and innovation accounting Consider the VAR(p) model where (L)y t = t, (L) =I m L L p L p is the lag polynomial of order p with m m coe±cient matrices i, i =,...p. Provided that
More informationAdaptive DemandForecasting Approach based on Principal Components Timeseries an application of datamining technique to detection of market movement
Adaptive DemandForecasting Approach based on Principal Components Timeseries an application of datamining technique to detection of market movement Toshio Sugihara Abstract In this study, an adaptive
More informationFORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits
Technical Paper Series Congressional Budget Office Washington, DC FORECASTING DEPOSIT GROWTH: Forecasting BIF and SAIF Assessable and Insured Deposits Albert D. Metz Microeconomic and Financial Studies
More information7 Time series analysis
7 Time series analysis In Chapters 16, 17, 33 36 in Zuur, Ieno and Smith (2007), various time series techniques are discussed. Applying these methods in Brodgar is straightforward, and most choices are
More informationBias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes
Bias in the Estimation of Mean Reversion in ContinuousTime Lévy Processes Yong Bao a, Aman Ullah b, Yun Wang c, and Jun Yu d a Purdue University, IN, USA b University of California, Riverside, CA, USA
More informationIntroduction to General and Generalized Linear Models
Introduction to General and Generalized Linear Models General Linear Models  part I Henrik Madsen Poul Thyregod Informatics and Mathematical Modelling Technical University of Denmark DK2800 Kgs. Lyngby
More informationSTOCK MARKET VOLATILITY AND REGIME SHIFTS IN RETURNS
STOCK MARKET VOLATILITY AND REGIME SHIFTS IN RETURNS ChiaShang James Chu Department of Economics, MC 0253 University of Southern California Los Angles, CA 90089 Gary J. Santoni and Tung Liu Department
More informationReal Exchange Rates and Real Interest Differentials: The Case of a Transitional Economy  Cambodia
Department of Economics Issn 14415429 Discussion paper 08/10 Real Exchange Rates and Real Interest Differentials: The Case of a Transitional Economy  Cambodia Tuck Cheong Tang 1 Abstract: This study
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationPITFALLS IN TIME SERIES ANALYSIS. Cliff Hurvich Stern School, NYU
PITFALLS IN TIME SERIES ANALYSIS Cliff Hurvich Stern School, NYU The t Test If x 1,..., x n are independent and identically distributed with mean 0, and n is not too small, then t = x 0 s n has a standard
More informationSimple Linear Regression Inference
Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation
More informationTesting for Granger causality between stock prices and economic growth
MPRA Munich Personal RePEc Archive Testing for Granger causality between stock prices and economic growth Pasquale Foresti 2006 Online at http://mpra.ub.unimuenchen.de/2962/ MPRA Paper No. 2962, posted
More informationTrend and Seasonal Components
Chapter 2 Trend and Seasonal Components If the plot of a TS reveals an increase of the seasonal and noise fluctuations with the level of the process then some transformation may be necessary before doing
More information1 Cointegration. ECONOMICS 266, Spring, 1997 Bent E. Sørensen March 1, 2005
ECONOMICS 266, Spring, 1997 Bent E. Sørensen March 1, 2005 1 Cointegration. The survey by Campbell and Perron (1991) is a very good supplement to this chapter  for further study read Watson s survey for
More informationThe pricevolume relationship of the Malaysian Stock Index futures market
The pricevolume relationship of the Malaysian Stock Index futures market ABSTRACT Carl B. McGowan, Jr. Norfolk State University Junaina Muhammad University Putra Malaysia The objective of this study is
More informationijcrb.com INTERDISCIPLINARY JOURNAL OF CONTEMPORARY RESEARCH IN BUSINESS AUGUST 2014 VOL 6, NO 4
RELATIONSHIP AND CAUSALITY BETWEEN INTEREST RATE AND INFLATION RATE CASE OF JORDAN Dr. Mahmoud A. Jaradat Saleh A. AIHhosban Al albayt University, Jordan ABSTRACT This study attempts to examine and study
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution
More informationPrincipal Components Analysis (PCA)
Principal Components Analysis (PCA) Janette Walde janette.walde@uibk.ac.at Department of Statistics University of Innsbruck Outline I Introduction Idea of PCA Principle of the Method Decomposing an Association
More informationMarketing Mix Modelling and Big Data P. M Cain
1) Introduction Marketing Mix Modelling and Big Data P. M Cain Big data is generally defined in terms of the volume and variety of structured and unstructured information. Whereas structured data is stored
More informationVI. Real Business Cycles Models
VI. Real Business Cycles Models Introduction Business cycle research studies the causes and consequences of the recurrent expansions and contractions in aggregate economic activity that occur in most industrialized
More informationChapter 9: Univariate Time Series Analysis
Chapter 9: Univariate Time Series Analysis In the last chapter we discussed models with only lags of explanatory variables. These can be misleading if: 1. The dependent variable Y t depends on lags of
More informationMachine Learning in Statistical Arbitrage
Machine Learning in Statistical Arbitrage Xing Fu, Avinash Patra December 11, 2009 Abstract We apply machine learning methods to obtain an index arbitrage strategy. In particular, we employ linear regression
More informationTHE IMPACT OF EXCHANGE RATE VOLATILITY ON BRAZILIAN MANUFACTURED EXPORTS
THE IMPACT OF EXCHANGE RATE VOLATILITY ON BRAZILIAN MANUFACTURED EXPORTS ANTONIO AGUIRRE UFMG / Department of Economics CEPE (Centre for Research in International Economics) Rua Curitiba, 832 Belo Horizonte
More informationStock prices and exchange rates in Sri Lanka: some empirical evidence
Guneratne B. Wickremasinghe (Australia) Stock prices and exchange rates in Sri Lanka: some empirical evidence Abstract This paper examines the relationship between stock prices and exchange rates in Sri
More informationTHE EFFECT OF MONETARY GROWTH VARIABILITY ON THE INDONESIAN CAPITAL MARKET
116 THE EFFECT OF MONETARY GROWTH VARIABILITY ON THE INDONESIAN CAPITAL MARKET D. Agus Harjito, Bany Ariffin Amin Nordin, Ahmad Raflis Che Omar Abstract Over the years studies to ascertain the relationship
More informationChapter 5. Analysis of Multiple Time Series. 5.1 Vector Autoregressions
Chapter 5 Analysis of Multiple Time Series Note: The primary references for these notes are chapters 5 and 6 in Enders (2004). An alternative, but more technical treatment can be found in chapters 1011
More informationTHE RELATIONSHIP BETWEEN TRADE OPENNESS AND INVESTMENT IN SYRIA
THE RELATIONSHIP BETWEEN TRADE OPENNESS AND INVESTMENT IN SYRIA Adel Shakeeb MOHSEN * ABSTRACT This study attempts to investigate the effect of trade openness, GDP and population on the investment in Syria
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationModule 6: Introduction to Time Series Forecasting
Using Statistical Data to Make Decisions Module 6: Introduction to Time Series Forecasting Titus Awokuse and Tom Ilvento, University of Delaware, College of Agriculture and Natural Resources, Food and
More informationEXPORT INSTABILITY, INVESTMENT AND ECONOMIC GROWTH IN ASIAN COUNTRIES: A TIME SERIES ANALYSIS
ECONOMIC GROWTH CENTER YALE UNIVERSITY P.O. Box 208269 27 Hillhouse Avenue New Haven, Connecticut 065208269 CENTER DISCUSSION PAPER NO. 799 EXPORT INSTABILITY, INVESTMENT AND ECONOMIC GROWTH IN ASIAN
More informationEconometrics Modeling and systems estimation
André K. Anundsen and Claudia Foroni, (Norges Bank), and Ragnar Nymoen (Department of Economics) ECON 4160 Econometrics Modeling and systems estimation TEACHING PLAN Autumn 2015 Lectures and computer classes:
More information