This paper aims at developing omnibus procedures for testing for serial correlation using spectral density estimation and wavelet shrinkage. We derive the asymptotic distributions of the wavelet coefficients under the null hypothesis of no serial correlation. 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 of Fan (1996), 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. Using proposals of Fan (1996), 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, which correspond essentially to parametric bootstrap test procedures. In a small simulation study, the following test statistics are compared, with respect to level and power: the new one-sided test statistics using wavelet thresholding, the one-sided wavelet-based test statistics of Lee and Hong (2001), the one-sided kernel-based test statistics of Hong (1996), and the bootstrapped versions of these spectral test statistics.
Published September 2009 , 34 pages