Multivariate Portmanteau Test For Autoregressive Models with Uncorrelated but Nonindependent Errors

Abstract

. We study the asymptotic behaviour of the least squares estimator, of the residual autocorrelations and of the Ljung–Box (or Box–Pierce) portmanteau test statistic for multiple autoregressive time series models with nonindependent innovations. Under mild assumptions, it is shown that the asymptotic distribution of the portmanteau tests is that of a weighted sum of independent chi‐squared random variables. When the innovations exhibit conditional heteroscedasticity or other forms of dependence, this asymptotic distribution can be quite different from that of models with independent and identically distributed innovations. Consequently, the usual chi‐squared distribution does not provide an adequate approximation to the distribution of the Box–Pierce goodness‐of‐fit portmanteau test in the presence of nonindependent innovations. Hence we propose a method to adjust the critical values of the portmanteau tests. Monte carlo experiments illustrate the finite sample performance of the modified portmanteau test.

An Improved Measure for Lack of Fit in Time Series Models

“…Here, we consider n = 160 and 320 with lags m = 2 and 3 and 3 and 4, respectively. The results are in Table 2 and, consistent with Francq and Raïssi (2007), the tests appear to have conservative type I error rates. Although not reported here, the statistics based off the asymptotic Table 2: Rate of rejections, out of 10,000 replications, under the null hypothesis (δ = 0) at two sample sizes n, two lags m, and three significance levels for data generated as a weak VAR (2) Lastly, in a study analogous to Figures 1 and 2, consider the possible improvement by using our recommended statistic.…”

“…The results in Francq and Raïssi (2007) demonstrate the portmanteau test for weak VAR processes can be conservative for large lags relative to the sample size. Here, we consider n = 160 and 320 with lags m = 2 and 3 and 3 and 4, respectively.…”

“…The authors are grateful to Professors Francq and Raïssi for sharing their Fortran code implementing the algorithm in Francq and Raïssi (2007).…”

“…We follow the estimation procedure in Francq and Raïssi (2007) and choose the intermediate autoregressive order, r ∈ {0, 1, 2, 3} in step 6 of their algorithm, via BIC. Here, we only consider a maximum order of 3 as we are working with smaller sample sizes.…”

Comparison of procedures for fitting the autoregressive order of a vector error correction model

“…Bauwens, Laurent and Rombouts (2006)). Taking into account the dependence of the errors is important for testing the adequacy of linear models as pointed out by Francq, Roy and Zakoïan (2005) or Francq and Raïssi (2007). In addition the study of VECM in nonstandard situations has attracted much attention in the recent years.…”