A COMPANION TO Theoretical Econometrics
Vector autoregressions
Vector autoregressions, which were introduced to econometrics by Sims (1980), have the form:
Yt = ц t + A(L)YM + є t, (27.8)
where Yt is a n x 1 vector time series, є t is a n x 1 serially uncorrelated disturbance, A(L) is a pth order lag polynomial matrix, and pt is a n x 1 vector of deterministic terms (for example, a constant or a constant plus linear time trend). If there are no restrictions on the parameters, the parameters can be estimated asymptotically efficiently (under Gaussianity) by OLS equation by equation. Multistep forecasts can be made either by replacing the left-hand side of (27.8) by Yt+h, or by h-fold iteration of the one-step forecast.
Two important practical questions are the selection of the series to include in Yt (the choice of n) and the choice of the lag order p in the VAR(p). Given the choice of series, the order p is typically unknown. As in the univariate case, it can be estimated by information criteria. This proceeds as discussed following (27.3), except that 62 is replaced by the determinant of t (the MLE of the variance-covariance matrix of є^, and the relevant number of parameters is the total free parameters of the VAR; thus, if there are no deterministic terms, IC( p) = lndet(t) + n2pg(T). The choice of series is typically guided by economic theory, although the predictive least squares (PLS) criterion (which is similar to an information criterion) can be useful in guiding this choice, cf. Wei (1992).
The issue of whether to difference the series is further complicated in the multivariate context by the possible presence of cointegration among two or more of the n variables. The multiple time series Yt is said to be cointegrated if each element of Yt is integrated of order 1 (is I(1); that is, has an autoregressive unit root) but there are k > 1 linear combination, a' Yt, that are I(0) (that is, which do not have a unit AR root) (Engle and Granger, 1987). It has been conjectured that long-run forecasts are improved by imposing cointegration when it is present. However, even if cointegration is correctly imposed, it remains to estimate the parameters of the cointegrating vector, which are, to first-order, estimated consistently (and at the same rate) if cointegration is not imposed. If cointegration is imposed incorrectly, however, asymptotically biased forecasts with large risks can be produced. At short horizons, these issues are unimportant to first-order asymptotically. By extension of the univariate results that are known for long - horizon forecasting, one might suspect that pretesting for cointegration could improve forecast performance, at least as measured by the asymptotic risk. However, tests for cointegration have very poor finite sample performance (cf. Haug,
1996) , so it is far from clear that in practice pretesting for cointegration will improve forecast performance. Although much of the theory in this area has been worked out, work remains on assessing the practical benefits of imposing cointegration for forecasting. For additional discussions of cointegration, see Watson (1994), Hatanaka (1996) and Chapter 30 in this volume by Dolado, Gonzalo, and Marmol.
It should be noted that there are numerous subtle issues involved in the interpretation of and statistical inference for VARs. Watson (1994) surveys these issues, and two excellent advanced references on VARs and related small linear time series models are Lutkepohl (1993) and Reinsel (1993). Also, VARs provide only one framework for multivariate forecasting; for a different perspective to the construction of small linear forecasting models, see Hendry (1995).
