Model Diagnostic Checks

Chapter Contents

The VARMAX Procedure

Model Diagnostic Checks

Multivariate Model Diagnostic Checks

Information Criterion
Various model selection criteria (normalized by T) can be used to choose the appropriate model. The following list includes the Akaike Information Criterion (AIC), the corrected Akaike Information Criterion (AICC), the Final Prediction Error criterion (FPE), the Hannan-Quinn Criterion (HQC), and the Schwarz Bayesian Criterion (SBC), also referred to as BIC.
${AIC} &=& \log(|\tilde{\Sigma}|) + 2r/T \ {AICC}&=& \log(|\tilde{\Sigma}|) +... ...{\Sigma}|) + 2r\log(\log(T))/T \ {SBC} &=& \log(|\tilde{\Sigma}|) + r\log(T)/T$
where r denotes the number of parameters estimated and $\tilde{\Sigma}$ is the maximum likelihood estimate of $\Sigma$ .
An example of the output was displayed in Figure 4.4.
Portmanteau Statistic Q_s
Let $C_{\epsilon}(l)$ be the residual cross-covariance matrices and $\hat \rho_{\epsilon}(l)$ be the residual cross-correlation matrices as
$C_{\epsilon}(l)=T^{-1} \sum_{t=1}^{T-l} {{\epsilon}}_t {{\epsilon}}_{t+l}'$
and
$\hat \rho_{\epsilon}(l)=\hat V_{\epsilon}^{-1/2} C_{\epsilon}(l) \hat V_{\epsilon}^{-1/2} {\rm and} \hat \rho_{\epsilon}(-l)=\hat \rho_{\epsilon}(l)'$
where $\hat V_{\epsilon}={\rm Diag}(\hat \sigma^2_{11}, ... , \hat \sigma^2_{kk} )$ and $\hat \sigma^2_{ii}$ are the diagonal elements of $\hat \Sigma$ .The multivariate portmanteau test defined in Hosking (1980) is
$Q_s=T^2 \sum_{l=1}^s (T-l)^{-1} {\rm tr} \{ \hat \rho_{\epsilon}(l)\Sigma^{-1} \hat \rho_{\epsilon}(-l)\Sigma^{-1} \}$
The statistic Q_s has approximately the chi-square distribution with k²(s-p) degrees of freedom. An example of the output was displayed in Figure 4.6.

Univariate Model Diagnostic Checks

There are various ways to perform diagnostic checks for a univariate model. For details, see the chapter on the ARIMA or AUTOREG procedure. An example of the output was displayed in Figure 4.7 and Figure 4.8.

Durbin-Watson (DW) statistics: The test statistics are computed from the residuals of the autoregressive model with order 1.
F tests for autoregressive conditional heteroscedastic (ARCH) disturbances: These test statistics are computed from the residuals of the ARCH(1) model.
F tests for AR disturbance: These test statistics are computed from the residuals of the univariate AR(1), AR(2), AR(3), and AR(4) models.
Jarque-Bera normality test: This test is helpful in determining whether the model residuals represent a white noise process.

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