TimeSeriesBook.pdf

In the example treated in section 144 the covariance

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In the example treated in Section 14.4, the covariance matrix Σ, taking the use of a constant into account and assuming 8 lags, has to be inflated by T + np +1 T = 196+4 × 8+1 196 = 1 . 168. Note also that the precision of the forecast, given Σ, diminishes with the number of parameters. 14.3 Modeling of VAR models The previous section treated the estimation of VAR models under the as- sumption that the order of the VAR, p , is known. In most cases, this as- sumption is unrealistic as the order p is unknown and must be retrieved from the data. We can proceed analogously as in the univariate case (see Sec- tion 5.1) and iteratively test the hypothesis that coefficients corresponding to the highest lag, i.e. Φ p = 0, are simultaneously equal to zero. Starting from a maximal order p max , we test the null hypothesis that Φ p max = 0 in the corresponding VAR( p max ) model. If the hypothesis is not rejected, we reduce the order by one to p max - 1 and test anew the null hypothesis Φ p max - 1 = 0 using the smaller VAR( p max - 1) model. One continues in this way until the null hypothesis is rejected. This gives, then, the appropriate order of the VAR. The different tests can be carried out either as Wald-tests (F-tests) or as likelihood-ratio tests ( χ 2 -tests) with n 2 degrees of freedom. An alternative procedure to determine the order of the VAR relies on some information criteria. As in the univariate case, the most popular ones are the Akaike (AIC), the Schwarz or Bayesian (BIC) and the Hannan-Quinn criterion (HQC). The corresponding formula are: AIC(p): ln det e Σ p + 2 pn 2 T , BIC(p): ln det e Σ p + pn 2 T ln T, HQC(p): ln det e Σ p + 2 pn 2 T ln (ln T ) ,
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264 CHAPTER 14. FORECASTING WITH VAR MODELS where e Σ p denotes the degree of freedom adjusted estimate of the covariance matrix Σ for a model of order p (see equation(13.5)). n 2 p is the number of estimated coefficients. The estimated order is then given as the minimizer of one of these criteria. In practice the Akaike’s criterion is the most popular one although it has a tendency to deliver orders which are too high. The BIC and the HQ-criterion on the other hand deliver the correct order on average, but can lead to models which suffer from the omitted variable bias when the estimated order is too low. Examples are discussed in Sections 14.4 and 15.4.5. Following L¨utkepohl (2006), Akaike’s information criterion can be ratio- nalized as follows. Take as a measure of fit the determinant of the one period approximate mean-squared errors [ MSE(1) from equation (14.8) and take as an estimate of Σ the degrees of freedom corrected version in equation (13.5). The resulting criterion is called according to Akaike (1969) the final predic- tion error (FPE): FPE( p ) = det T + np T × T T - np e Σ = T + np T - np n det e Σ . (14.9) Taking logs and using the approximations T + np T - np 1 + 2 np T and log(1 + 2 np T ) 2 np T , we arrive at AIC( p ) log FPE ( p ) .
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