Lesson 17: Vector AutoRegressive Models

Transcription

1 Dipartimento di Ingegneria e Scienze dell Informazione e Matematica Università dell Aquila, umberto.triacca@ec.univaq.it

2 Vector AutoRegressive models The extension of ARMA models into a multivariate framework leads to Vector AutoRegressive (VAR) models Vector Moving Average (VMA) models Vector AutoregRegressive Moving Average (VARMA) models

3 Vector AutoRegressive models The Vector AutoRegressive (VAR) models, made famous in Chris Sims s paper Macroeconomics and Reality, Econometrica, 1980, are one of the most applied models in the empirical economics.

4 Vector AutoRegressive models Let be a K-variate random process. { yt = (y 1t,..., y Kt ) ; t Z }

5 Vector AutoRegressive models We say that the process {y t ; t Z} follows a vector autoregressive model of order p, denoted VAR(p) if y t = ν + A 1 y t A p y t p + u t, t Z p is a positive integer, A i are fixed (K K) coefficient matrices, ν = (ν 1,..., ν K ) is a fixed (K 1) vector of intercept terms, u t = (u 1t,..., u Kt ) is ak-dimensional white noise with covariance matrix Σ u. The covariance matrix Σ u is assumed to be nonsingular.

6 Example: Bivariate VAR(1) or [ y1t y 2t ] = [ ν1 ν 2 ] [ a11 a + 12 a 21 a 22 ] [ y1t 1 y 2t 1 y 1t = ν 1 + a 11 y 1t 1 + a 12 y 2t 1 + u 1t ] [ u1t + u 2t ] y 2t = ν 1 + a 21 y 1t 1 + a 22 y 2t 1 + u 2t where cov(u 1t, u 2s ) = σ 12 for t = s, 0 otherwise

7 Advantages of VAR models 1 They are easy to estimate. 2 They have good forecasting capabilities. 3 The researcher does not need to specify which variables are endogenous or exogeneous. All are endogenous. 4 In a VAR system is very easy to test for Granger non-causality.

8 Problems with VARs 1 So many parameters. If there are K equations, one for each K variables and p lags of each of the variables in each equation, (K + pk 2 ) parameters will have to be estimated. 2 VARs are a-theoretical, since they use little economic theory. Thus VARs cannot used to obtain economic policy prescriptions.

9 Estimation In this lesson, the estimation of a vector autoregressive model is discussed.

10 Estimation Consider the VAR(1) y 1t = ν 1 + a 11 y 1t 1 + a 12 y 2t 1 + u 1t y 2t = ν 1 + a 21 y 1t 1 + a 22 y 2t 1 + u 2t where cov(u 1t, u 2s ) = σ 12 for t = s, 0 otherwise.

11 Estimation The model corresponds to 2 regressions with different dependent variables and identical explanatory variables. We could estimate this model using the ordinary least squares (OLS) estimator computed separately from each equations.

12 Estimation It is assumed that a time series of the y variables is available. In addition, a presample value is assumed to be available. y 1 = [y 11, y 21 ],..., y T = [y 1T, y 2T ] y 0 = [y 10, y 20 ]

13 Estimation Consider the first equation y 1t = ν 1 + a 11 y 10 + a 12 y 2t 1 + u 1t ; t = 1,..., T y 11 = ν 1 + a 11 y 10 + a 12 y 20 + u 11 y 12 = ν 1 + a 11 y 11 + a 12 y 21 + u 12. y 1T = ν 1 + a 11 y 1T 1 + a 12 y 2T 1 + u 1T

14 Estimation We define y 1 = [y 11,..., y 1T ], 1 y 1,0 y 2,0 1 y 1,1 y 2,1 X 1 =... 1 y 1,T 1 y 2,T 1, π 1 = [ν 1, a 11, a 12 ], u 1 = [u 11,..., u 1T ],

15 Thus y 1 = Xπ 1 + u, The OLS estimator π 1 is given by ˆπ 1 = (X X) 1 X y 1

16 Estimation Consider the second equation y 2t = ν 2 + a 21 y 10 + a 22 y 2t 1 + u 2t ; t = 1,..., T y 21 = ν 2 + a 21 y 10 + a 22 y 20 + u 21 y 22 = ν 2 + a 21 y 11 + a 22 y 21 + u 22. y 2T = ν 2 + a 21 y 1T 1 + a 22 y 2T 1 + u 2T

17 Estimation We define y 2 = [y 21,..., y 2T ], π 2 = [ν 2, a 21, a 22 ], u 2 = [u 21,..., u 2T ],

18 Estimation Thus y 2 = Xπ 2 + u 2, The OLS estimator π 2 is given by ˆπ 2 = (X X) 1 X y 2

19 OLS estimators ˆπ 1 = (X X) 1 X y 1 π 1 = [ν 1, a 11, a 12 ] ˆπ 2 = (X X) 1 X y 2 π 2 = [ν 2, a 21, a 22 ]

20 OLS Estimation Are the OLS estimators for π 2 and π 2 efficient estimators?

21 Seemingly Unrelated Regressions Estimation We remember that cov(u 1t, u 2t ) = σ 12 0 When contemporaneous correlation exists, it may be more efficient to estimate all equations jointly, rather than to estimate each one separately using least squares. The appropriate joint estimation technique is the GLS estimation

22 Seemingly Unrelated Regressions Equations Let as consider the following set of two equations y 1t = β 10 + β 11 x 1t + β 12 z 1t + u 1t y 2t = β 10 + β 11 x 2t + β 12 z 2t + u 2t

23 Seemingly Unrelated Regressions Equations The two equation can be written in the usual matrix algebra notation as y 1 = X 1 β 1 + u 1 y 2 = X 2 β 2 + u 2

24 Seemingly Unrelated Regressions Equations where y i = [y i1,..., y it ] i = 1, 2 1 x i,1 z i,1 1 x i,2 z i,2 X i =... 1 x i,t z i,t i = 1, 2, β i = [β i0, β i1, β i2 ], i = 1, 2, and u i = [u i1,..., u it ], i = 1, 2,

25 Seemingly Unrelated Regressions Equations Further, we assume that 1 E[u it ] = 0 i = 1, 2 t = 1, 2...T 2 var(u it ) = σi 2 i = 1, 2 t = 1, 2...T ; 3 E[u it u jt ] = σ ij i, j = 1, 2; 4 E[u it u js ] = 0 for t s and i, j = 1, 2;

26 Seemingly Unrelated Regressions Equations Now we write the two equations as the folllowing super model [ ] [ ] [ ] [ ] y1 X1 0 β1 u1 = + 0 X 2 β 2 u 2 or y 2 y = Xβ + u The first point to note is that it can be shown that least squares applied to this system is identical to applying least squares separately to each of the two equations.

27 Seemingly Unrelated Regressions Equations Note that the covariance matrix of the joint disturbace vector u is given by [ ] [ ] σ 2 Φ = 1 I σ 12 I σ 2 σ 12 I σ2 2I = 1 σ 12 σ 12 σ2 2 I T = Σ I T

28 Seemingly Unrelated Regressions Equations The disturbance covariance matrix Φ is of dimension (2T 2T ).An important point to note is that Φ cannot be written as scalar multiplied by a 2T -dimensional identity matrix. Thus the best linear unbiased estimator for β is given by the generalized least squares estimator ˆβ GLS = (X Φ 1 X) 1 X Φ 1 y = [X (Σ 1 I T )X] 1 X(Σ 1 I T )y It has lower variance than the least squares estimators for β because it takes into account the contemporaneous correlation between the disturbances in different equation.

29 Seemingly Unrelated Regressions Equations There are two conditions under the which least squares is identical to generalized least squares. 1 The first is when all contemporaneus covariances are zero, σ 12 = 0. 2 The second is when the explanatory variables in each equation are identical, X 1 = X 2

30 Estimation of A VAR model The use of generalized least squares estimator does not lead to a gain in efficiency when each equation contains the same explanatory variables. Thus the autoregressive cooefficients of our VAR(1) model can be estimated, without loss of estimation efficiency, by ordinary least squares. This in turn is equivalent to estimating each equation separately by OLS

31 Estimation of A VAR model The (2 2) unknown covariance matrix Σ may be consistent estimated by ˆΣ whose elements are ˆσ ij = û iûj T 2p 1 for i, j = 1, 2 where û i = y i Xˆπ i

32 Estimation of A VAR model In general, the autoregressive cooefficients of a K-dimensional VAR(p) model y t = ν + A 1 y t A p y t p + u t can be estimated, without loss of estimation efficiency, by ordinary least squares.in the first equation, we have to run the regression y 1t on y 1t 1,..., y Kt 1,..., y 1t p,..., y Kt p, in the second equation, we regress and so on. y 12 on y 1t 1,..., y Kt 1,..., y 1t p,..., y Kt p,

33 Conclusion Since the VAR(p) model is just a Seemingly Unrelated Regression (SUR) model where each equation has the same explanatory variables, each equation may be estimated separately by ordinary least squares without losing efficiency relative to generalized least squares.