On testing for serial correlation of unknown form using wavelet thresholding

Omnibus procedures for testing serial correlation are developed, using spectral density estimation and wavelet shrinkage. The asymptotic distributions of the wavelet coefficients under the null hypothesis of no serial correlation are derived. Under some general conditions on the wavelet basis, the wavelet coefficients asymptotically follow a normal distribution. Furthermore, they are asymptotically uncorrelated. Adopting a spectral approach and using results on wavelet shrinkage, new one-sided test statistics are proposed. As a spatially adaptive estimation method, wavelets can effectively detect fine features in the spectral density, such as sharp peaks and high frequency alternations. Using an appropriate thresholding parameter, shrinkage rules are applied to the empirical wavelet coefficients, resulting in a non-linear wavelet-based spectral density estimator. Consequently, the advocated approach avoids the need to select the finest scale J, since the noise in the wavelet coefficients is naturally suppressed. Simple data-dependent threshold parameters are also considered. In general, the convergence of the spectral test statistics toward their respective asymptotic distributions appears to be relatively slow. In view of that, Monte Carlo methods are investigated. In a small simulation study, several spectral test statistics are compared, with respect to level and power, including versions of these test statistics using Monte Carlo simulations.