On wavelet-based testing for serial correlation of unknown form using Fan’s adaptive Neyman method

Test procedures for serial correlation of unknown form with wavelet methods are investigated. A new test statistic is motivated using a canonical multivariate normal hypothesis testing model. It relies on empirical wavelet coefficients of a wavelet-based spectral density estimator. The choice of the Haar wavelet function is advocated, since evidence demonstrates that the choice of the wavelet function is not critical. Under the null hypothesis of no serial correlation, the asymptotic distribution of a vector of empirical wavelet coefficients is derived, which is asymptotically a multivariate normal distribution. A test statistic is proposed based on that asymptotic result, which presents the serious advantage to be completely data-driven or adaptive, avoiding the selection of any smoothing parameters. Furthermore, under a suitable class of fixed alternatives, the wavelet-based method is consistent against serial correlation of unknown form. The test statistic is expected to exhibit good power properties when the true spectral density displays significant spatial inhomogeneity, such as seasonal or business cycle periodicities. However, the convergence of the test statistic towards its asymptotic distribution is relatively slow. Thus, Monte Carlo methods based on random samples are suggested to determine the corresponding critical values. In a simulation study, the new methodology is compared with several test statistics, with respect to their exact levels and powers. The robustness properties of the spectral methods based on Monte Carlo critical values are also investigated empirically, when the error terms are weak white noises.