BibTeX
@MISC{_chapter8,
author = {},
title = {Chapter 8 VECTOR AUTOREGRESSION TECHNIQUES},
year = {}
}
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Abstract
This chapter discusses econometric techniques for vector autoregressions (VAR). In most cases, the variables in VAR are assumed to be stationary. 1 Let yt be an n-dimensional vector stochastic process that is covariance stationary. Because yt is covariance stationary, it has a Wold representation: (8.1) yt = µ + ɛt + Ψ1ɛt−1 + Ψ2ɛt−2 + · · · = µ + Ψ(L)ɛt, where Ψ(L) = In + ∑ ∞ s=1 ΨsL s and L is the lag operator. Assuming that Ψ(L) is invertible, yt has a VAR representation. Assuming that the VAR representation is of order p: (8.2) A(L)yt = δɛ + ɛt, 1 A VAR model may include nonstationary variables. Chapter 15 treats the case where some of the variables in VAR are difference stationary and cointegrated, in terms of terminology introduced later. When the difference stationary variables are not cointegrated, we can take the first difference to make them stationary for VAR. 124 8.1. OLS ESTIMATION 125 where (8.3) δɛ = Ψ(1) −1 µ = A(1)µ, p∑ AiL i, A(L) = Ψ(L) −1 = In − i=1 ɛt = yt − Ê(yt|yt−1, yt−2, yt−3, · · ·) and (8.4) E(ɛtɛ ′ t) = Σɛ. Here Ê(·|yt−1, yt−2, yt−3, · · · ) is defined to be the linear projection operator onto the linear space spanned by a constant (say, 1) and yt−1, yt−2, yt−3, · · ·. In virtually all applications, Σɛ is not diagonal. However, the Seemingly Unrelated Regression Estimator (SUR) coincides with the OLS estimator for (8.2) because the regressors are identical for all regressions when OLS is applied to each row of (8.2). 8.1 OLS Estimation The VAR (8.2) gives a system of regression equations. It may appear that the SUR estimator should be used to estimate these equations because the error terms are contemporaneously correlated. However, the OLS and SUR estimators coincide, because the regressors are the same for all equations. Hence, we can estimate each equation by OLS. It is often convenient to have a matrix expression to write the OLS estimators for the VAR system. For this purpose, rewrite (8.2) by staking it from t = 1, · · · , T after transpose: (8.5)
Keyphrases
vector autoregression technique sur estimator var representation ols estimator ols estimation linear projection operator var model yt yt difference stationary variable wold representation linear space seemingly unrelated regression estimator n-dimensional vector stochastic process econometric technique nonstationary variable error term regression equation lag operator var system vector autoregressions first difference matrix expression
