We consider tests for lack of fit in ARMA models with nonindependent innovations. In this framework, the standard Box-Pierce and Ljung-Box portmanteau tests can perform poorly. Specifically, the usual text book formulas for asymptotic distributions are based on strong assumptions and should not be applied without careful consideration. In this article we derive the asymptotic covariance matrix ρ m of a vector of autocorrelations for residuals of ARMA models under weak assumptions on the noise. The asymptotic distribution of the portmanteau statistics follows. A consistent estimator of ρ m , and a modification of the portmanteau tests are proposed. This allows us to construct valid asymptotic significance limits for the residual autocorrelations, and (asymptotically) valid goodness-of-fit tests, when the underlying noise process is assumed to be noncorrelated rather than independent or a martingale difference. A set of Monte Carlo experiments, and an application to the Standard & Poor 500 returns, illustrate the practical relevance of our theoretical results.
In many situations, we want to verify the existence of a relationship between multivariate time series. In this paper, we generalize the procedure developed by Haugh (1976) for univariate time series in order to test the hypothesis of noncorrelation between two multivariate stationary ARMA series. The test statistics are based on residual cross‐correlation matrices. Under the null hypothesis of noncorrelation, we show that an arbitrary vector of residual cross‐correlations asymptotically follows the same distribution as the corresponding vector of cross‐correlations between the two innovation series. From this result, it follows that the test statistics considered are asymptotically distributed as chi‐square random variables. Two test procedures are described. The first one is based on the residual cross‐correlation matrix at a particular lag, whilst the second one is based on a portmanteau type statistic that generalizes Haugh's statistic. We also discuss how the procedures for testing noncorrelation can be adapted to determine the directions of causality in the sense of Granger (1969) between the two series. An advantage of the proposed procedures is that their application does not require the estimation of a global model for the two series. The finite‐sample properties of the statistics introduced were studied by simulation under the null hypothesis. It led to modified statistics whose upper quantiles are much better approximated by those of the corresponding chi‐square distribution. Finally, the procedures developed are applied to two different sets of economic data.
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