9.2 CLASSICAL ESTIMATION OF VARS

9.2 CLASSICAL ESTIMATION OF VARS

After determining the n variables to include in the VAR, we must decide how many lags to use. Rather than keeping the same number of lags for all variables, refinements that eliminate some variables in some of the equations can be used. Classical estimation also requires settling on an estimation technique for the model. We first assume that the lag length, p, has been chosen and consider parameter estimation before we address the choice of model specification.

9.2.1 Estimation of VAR(p) Models

From an econometric perspective VARs are seemingly unrelated regressions (SUR) whose equations are related through the covariances of the residual terms (Zellner, 1962). In its unconstrained form where each equation includes the same lags of each of the same regressors (and the same exogenous variables as well), SUR estimation simplifies to be numerically identical to OLS estimation equation by equation, which therefore offers the easiest estimation method.2 Conversely, for constrained VARs with different regressors in each equation, typically SUR methods will result in more efficient estimates.

In sufficiently large samples and under conventional assumptions, the centered and standardized least squares estimators for the coefficients $\beta$ in (9.3) will be distributed according to

$$T^{1/2}(\hat{\beta }-{\beta }_0)\stackrel{a}{\sim }\mathrm{N}(0,\Sigma \otimes (Z^{\prime }Z{)}^{-1}),\text{\hspace{1em}(9.8)}$$

where ${\beta }_0$ is the true parameter value. Standard errors for each regression are computed using the OLS estimates equation by equation. Tests of cross-equation restrictions must, however, account for the full covariance matrix of the residuals, -.

While OLS estimation is asymptotically efficient in the unconstrained case, there are many other estimators with the same first-order asymptotic properties. Hence, asymptotic efficiency alone does not eliminate interest in alternative estimators. Moreover, other properties familiar from OLS estimation may fail. For example, consider the OLS estimator ${\hat{\beta }}_i=(Z^{\prime }Z{)}^{-1}{Z}^{\prime }{y}_i$ from the i th column of (9.3), $y_i=$ $Z{\beta }_i+u_i$ . The bias of ${\hat{\mathbf{\beta }}}_i$ is given by

$$\begin{array}{c}E\left[{\hat{\beta }}_i-{\beta }_i\right]=E\left[(Z^{\prime }Z{)}^{-1}{Z}^{\prime }{u}_i\right]\\ =E\left[E\left[(Z^{\prime }Z{)}^{-1}{Z}^{\prime }{u}_i|Z\right]\right].\end{array}\text{\hspace{1em}(9.9)}$$

When $E\left[u_i|Z\right]=0$ , this bias is 0 and the OLS estimator is unbiased. However it is impossible for this to hold here, since the shocks $u_{is}$ affect ${\gamma }_{jt}$ y for $t>s$ , i.e., current shocks affect future values of the regressors. Since u determines the path of y and hence appears inside $Z,$ in general $E\left[u_i|Z\right]\ne 0$ and the OLS estimator is not unbiased.3

The claim that OLS is asymptotically efficient may also not be very comforting from a forecasting perspective. It only means that in large samples the distribution of the parameters is well approximated by a normal distribution with a variance– covariance matrix that is as small as possible. In practice, however, we often do not have a great amount of data at hand and need to estimate many parameters. In such situations it is not obvious that the number of degrees of freedom is large enough to ensure that the asymptotic approximation is reasonable. If the asymptotic approximation is not reasonable, efficiency calculations based on it are clearly not relevant. Alternatively, a wide range of analytical and bootstrap methods for bias correction have been suggested; see, e.g., Bauer, Rudebusch, and Wu (2012) for an application to the term structure of interest rates.

After determining which variables to include in the VAR, we need to choose the lag length of the VAR. Often the choice uses the methods for model selection discussed in chapter 6 which readily apply to this problem. Beyond this, some researchers also employ Granger causality tests to remove individual variables from some of the equations. The idea is to remove predictors from the model if there is insufficient evidence that such predictors are useful.