In the general case where we have a var with p

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Unformatted text preview: is cumbersome and requires a normality assumption for the disturbances. 44 Information Criteria for VAR Lag Length Selection • Multivariate versions of the information criteria are required. These can be defined as: where all notation is as above and k′ is the total number of regressors in all equations, which will be equal to g2k + g for g equations, each with k lags of the g variables, plus a constant term in each equation. The values of the information criteria are constructed for 0, 1, … lags (up to some prespecified maximum ). 45 Does the VAR Include Contemporaneous Terms? • So far, we have assumed the VAR is of the form • But what if the equations had a contemporaneous feedback term? • We can write this as • This VAR is in primitive form. 46 Primitive versus Standard Form VARs • We can take the contemporaneous terms over to the LHS and write or B yt = β0 + β1 yt-1 + ut • We can then pre-multiply both sides by B-1 to give yt = B-1β0 + B-1β1 yt-1 + B-1ut or yt = A0 + A1 yt-1 + et • This is known as a standard form VAR, which we can estimate using OLS. 47 Block Significance and Causality Tests • It is likely that, when a VAR includes many lags of variables, it will be difficult to see which sets of variables have significant effects on each dependent variable and which do not. For illustration, consider the following bivariate VAR(3): • This VAR could be written out to express the individual equations as y1t = α10 + β11 y1t −1 + β12 y2t −1 + γ 11 y1t − 2 + γ 12 y2t − 2 + δ 11 y1t −3 + δ 12 y2t −3 + u1t y2t = α 20 + β 21 y1t −1 + β 22 y2t −1 + γ 21 y1t − 2 + γ 22 y2t − 2 + δ 21 y1t −3 + δ 22 y2t −3 + u2t • We might be interested in testing the following hypotheses, and their implied restrictions on the parameter matrices: 48 Block Significance and Causality Tests (cont’d) • • • • • Each of these four joint hypotheses can be tested within the F-test framework, since each set of restrictions contains only parameters drawn from one equation. These tests could also be referred to as Granger causality tests. Granger causality tests seek to answer questions such as “Do changes in y1 cause changes in y2?” If y1 causes y2, lags of y1 should be significant in the equation for y2. If this is the case, we say that y1 “Granger-causes” y2. If y2 causes y1, lags of y2 should be significant in the equation for y1. If both sets of lags are significant, there is “bi-directional causality” 49 Assume a lag length of p X t = c1 + α1 X t −1 + α 2 X t − 2 + .....α p X t − p + β1Yt −1 + β 2Yt − 2 + ....β pYt − p + at Estimate by OLS and test for the following hypothesis H 0 : β1 = β 2 = ...... = β p = 0 (Yt does not Granger - cause X t ) H1 : any β i ≠ 0 Unrestricted sum of squared residuals ˆ RSS1 = ∑ at 2 t Restricted sum of squared residuals ˆ ˆ RSS2 = ∑ at ( RSS2 − RSS1 ) / p F= RSS1 /(T − 2 p − 1) 2 t reject if F > Fα ,( p ,T −2...
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