Model Specification and Model Checking
Unrestricted VAR models usually involve a substantial number of parameters which in turn results in rather imprecise estimators. Therefore, it is desirable to impose restrictions that reduce the dimensionality of the parameter space. Such restrictions may be based on economic theory or other nonsample information and on statistical procedures. Of course, for structural models nonsample information is required for imposing identifying constraints. On top of that there may be further overidentifying constraints on the basis of a priori knowledge.
Tests are common statistical procedures for detecting possible restrictions. For example, t-ratios and F-tests are available for this purpose. These tests retain their usual asymptotic properties if they are applied to the short-run parameters in a VECM whereas problems may arise in the levels VAR representation as explained in the previous section. A particular set of restrictions where such problems occur is discussed in more detail in Section 5.2. In case of doubt it may be preferable to work on the VECM form. This form also makes it easy to test
restrictions on the cointegration vectors (see Chapter 30 by Dolado, Gonzalo, and Marmol in this volume).
Because the cointegrating rank r is usually unknown when the choice of p is made, it is useful to focus on the VAR form (32.1) at this stage. Various model selection criteria are available that can be used in this context. In practice, it is not uncommon to start from a model with some prespecified maximum lag length, say pmax, and apply tests sequentially, eliminating one or more variables in each step until a relatively parsimonious representation with significant parameter estimates has been found. Instead of sequential tests one may alternatively choose the lag length or determine exclusion restrictions by model selection procedures. For example, for determining the VAR order, the general approach is to fit VAR(m) models with orders m = 0,…, pmax and choose an estimator of the order p which minimizes a criterion such as
AIC(m) = log det(£„(m)) + mK2,
(see Akaike, 1974);
proposed by Hannan and Quinn (1979); or
SC(m) = log det(£„(m)) + mK2
due to Schwarz (1978). Here det() denotes the determinant, log is the natural logarithm and £„(m) = Т^^йй is the residual covariance matrix estimator for a model of order m. The term log det(£„(m)) measures the fit of a model with order m. Since there is no correction for degrees of freedom in the covariance matrix estimator the log determinant decreases (or at least does not increase) when m increases. Note that the sample size is assumed to be held constant and, hence, the number of presample values set aside for estimation is determined by the maximum order pmax. The last terms in the criteria penalize large VAR orders. In each case the estimator p of p is chosen to be the order which minimizes the desired criterion so that the two terms in the sum on the right-hand sides are balanced optimally.
The AIC criterion asymptotically overestimates the order with positive probability whereas the last two criteria estimate the order consistently (plim p = p or p ^ p a. s.) under quite general conditions, if the actual DGP has a finite VAR order and the maximum order pmax is larger than the true order. These results not only hold for I(0) processes but also for I(1) processes with cointegrated variables (Paulsen, 1984). Denoting the orders selected by the three criteria by p(AIC), p(HQ), and p(SC), respectively, the following relations hold even in small samples of fixed size T > 16 (see Lutkepohl, 1991, chs 4 and 11): p(SC) < p(HQ) < p(AIC).
Model selection criteria may also be used for identifying single coefficients that may be replaced by zero or other exclusion restrictions. After a model has been set up, a series of checks may be employed to confirm the model’s adequacy. Some such checks will be mentioned briefly in a subsequent section. Before that issue is taken up, procedures for specifying the cointegrating rank will be reviewed.