9.2.2 Choice of Lag Length

9.2.2 Choice of Lag Length

The standard approach for choosing the lag length for a VAR is to treat this as a model selection problem and use information criteria. Popular methods in the literature on VARs are the BIC and AIC. As we have seen, SUR estimation simplifies to OLS estimation equation by equation in a system with the same variables appearing in each regression. This explains why it is common practice to search over different values for $\mathbf{p}$ and hence exclude the same regressors from each equation. Here the search is not over all possible models but instead involves a sequence of models corresponding to each lag length, i.e., either setting all elements of $A_P$ to 0 or leaving all of them unrestricted. With n variables, $\mathbf{p}$ lags, and T observations, and assuming that a constant term is included in each equation of the VAR, the BIC and AIC information criteria take the forms

$$\mathrm{BIC}(p)=\ln |{\hat{\Sigma }}_p|+n(np+1)\frac{\ln (T)}{T},\text{\hspace{1em}(9.10)}$$

$$\mathrm{AIC}(p)=\ln |{\hat{\Sigma }}_p|+n(np+1)\frac{2}{T},\text{\hspace{1em}(9.11)}$$

where $\begin{array}{r}{\hat{\Sigma }}_p=T^{-1}{\sum }_{t=t}^T{\hat{u}}_t{\hat{u}}_t^{\prime }\end{array}$ . In each case the objective is to identify the model (indexed by $p)$ that minimizes the information criterion. In practice the search is sometimes not conducted over all possible values of $\mathbf{p}$ but restricted to include a minimum value of $\mathbf{p}$ that reflects the periodicity of the data. For example a minimum of four lags is sometimes chosen when the data are quarterly. Such restrictions reflect seasonality patterns believed to be present in the data. Moreover, it is common practice not to allow for gaps in the lags, i.e., all lags up to the pth order are included as opposed to, for example, dropping lag $\mathbf{p}-1$

9.2.3 Granger Causality Tests

Each variable predicts every other variable in the general VAR. However, in VARs that include many variables, it is quite likely that some individual variables are not particularly useful for forecasting all the other variables. Granger (1969a) used a variable’s predictive content and notions of causality to develop a definition of causality that depends on the conditional distribution of the predicted variable. Consider the conditional density of a random variable ${\gamma }_{it}$ given its own past history and that of a set of other random variables, $L_{{\hat{P}}_{3t}}(y_{it}|y_{1t-1},\dots ,y_{1t-p},\dots ,y_{nt-1},\dots ,y_{nt-p})$ . Suppose that this density is independent of the past history of the r th random variable, $y_{rt},i.e.$

$$\begin{array}{r}{p}_{y_{it}}(y_{it}|{\cup }_{j=1}^n{y}_{jt-k},k=1,\dots ,p)=p_{y_{it}}(y_{it}|{\cup }_{j\ne r}{y}_{jt-k},k=1,\dots ,p),\end{array}\text{\hspace{1em}(9.12)}$$

so the ith density does not depend on $y_{rt-1},\dots ,y_{rt-p}.$ When this holds, ${\mathbf{y}}_{\mathbf{rt}}$ is said not to Granger-cause ${\gamma }_{it}$ in the universe of variables $y_{1t},\dots ,y_{nt}$ . If such a statement were true, the past of ${\mathbf{y}}_{\mathbf{rt}}$ would not be useful for forecasting ${\gamma }_{it}$ one step ahead regardless of the loss function. The proviso “in the universe of variables $y_{1t},\dots ,y_{nt}$ is important. It is entirely possible that a finding of Granger causality can be overturned by adding more variables to the system $-y_{rt}$ may simply have nonzero coefficients because of omitted variable bias.

In practice, the condition in (9.12) is not what is usually examined to see whether a variable adds value from a predictive perspective. Part of the problem is that testing independence for the conditional density is not straightforward. Instead, it is common to test for lack of linear predictability rather than independence, and thus focus on the weaker notion that the lags $y_{rt-1},\dots ,y_{rt-p}$ do not enter the equation for ${\gamma }_{it}$ for the linear VAR. This is a test for exclusion in a $\mathrm{VAR}.$ and is thus simple to undertake. Consider the bivariate VAR,

$$\begin{array}{r}\left(\genfrac{}{}{0ex}{}{y_{1t}}{y_{2t}}\right)=\left(\genfrac{}{}{0ex}{}{a_{11}(L)a_{12}(L)}{a_{21}(L)}\right)\left(\genfrac{}{}{0ex}{}{y_{1t-1}}{y_{2t-1}}\right)+\left(\genfrac{}{}{0ex}{}{u_{1t}}{u_{2t}}\right).\end{array}\text{\hspace{1em}(9.13)}$$

The test is simply that $a_{12}(L)=0$ for ${\gamma }_{1t}$ not to be Granger-caused by ${\gamma }_{2t}$ . This more limited notion of predictability is really a statement that past information in ${\mathbf{y}}_{2t}$ is not useful in predicting the mean of ${\gamma }_{1t}$ and so this concept is most relevant under MSE loss.

Under MSE loss, Granger causality tests examine whether all lags of some variable can be removed from the forecasting equation. If lags of a variable are constrained to have zero coefficients, the same variables will no longer appear in each equation and hence OLS estimation equation by equation is no longer asymptotically efficient. In such cases either full information maximum likelihood (FIML) or minimum distance methods can be used to estimate the coefficients. For further discussion, see Hamilton (1994, chapter 11).