Testing for uncorrelated errors in ARMA models: non‐standard Andrews‐Ploberger tests

Abstract

A problem of interest in economic and finance applications is testing whether ARMA (Autoregressive moving average) errors are uncorrelated under weak assumptions, namely assumptions where the errors are neither iid nor a martingale difference. In this paper, non-standard versions of the tests of serial correlation introduced by Andrews and Ploberger (1996, hereafter AP) are proposed for diagnostic checking of ARMA errors. The original AP tests are designed for the case where the observed time series is generated by ARMA(1,1) models under the alternative and use asymptotic critical values computed by AP. The non-standard testing procedure uses AP statistics calculated from residuals and critical values based on asymptotic distribution theory derived under weak assumptions. The motivation for modifying the original AP tests is that they have attractive properties for the case for which they were originally designed: They are consistent against all non-white noise alternatives and have good all-round power against non-seasonal alternatives compared to several widely used tests in the literature, including those of Box and Pierce (1970, hereafter BP) and Ljung and Box (1978, hereafter LB) tests. A further advantage of the AP tests is that there is no need to specify a cutoff lag-length as is necessary for the BP and LB tests. We compare the non-standard AP tests with the non-standard BP and LB tests proposed by Francq et al. (2005), the tests of Hong and Lee (2007), and the tests using standardized residuals proposed by Chen (2008). In Monte Carlo experiments using ARMA models with GARCH (Generalised autoregressive conditional heteroskedasticity), EGARCH (Exponential GARCH) and non-MDS (Martingale difference sequence) innovations, the non-standard AP tests generally have better power than the other tests we consider. This suggests that the power advantage of the original AP tests extends to the more general framework considered in this paper.