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Consistent Testing for Serial Correlation of Unknown Form

Yongmiao Hong

Abstract

This paper proposes three classes of consistent one-sided tests for serial correlation of unknown form for the residual from a linear dynamic regression model that includes both lagged dependent variables and exogenous variables. The tests are obtained by comparing a kernel-based normalized spectral density estimator and the null normalized spectral density estimator and the null normalized spectral density, using a quadratic norm, the Hellinger metric, and the Kullback-Leibler information criterion respectively. Under the null hypothesis of no serial correlation, the three classes of new test statistics are asymptotically $N(0, 1)$ and equivalent. The null distributions are obtained without having to specify any alternative model. Unlike some conventional tests for serial correlation, the null distributions of our tests remain invariant when the regressors include lagged dependent variables. Under a suitable class of local alternatives, the three classes of the new tests are asymptotically equally efficient. Under global alternatives, however, their relative efficiencies depend on the relative magnitudes of the three divergence measures. Our approach provides an interpretation for Box and Pierce's (1970) test, which can be viewed as a quadratic norm based test using a truncated periodogram. Many kernels deliver tests with better power than Box and Pierce's test or the truncated kernel based test. A simulation study shows that the new tests have good power against an AR(1) process and a fractionally integrated process. In particular, they have better power than the Lagrange multiplier tests of Breusch (1978) and Godfrey (1978) as well as the portmanteau tests of Box and Pierce (1970) and Ljung and Box (1978). The cross-validation procedure of Beltrao and Bloomfield (1987) and Robinson (1991a) works reasonably well in determining the smoothing parameter of the kernel spectral estimator and is recommended for use in practice.