Serial Correlation Tests with Wavelets

This paper offers two new statistical tests for serial correlation with better power properties. The first test is concerned with wavelet-based portmanteau tests of serial correlation. The second test extends the wavelet-based tests to the residuals of a linear regression model. The wavelet approach is appealing, since it is based on the different behavior of the spectra of a white noise process and that of a weakly stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via wavelet transformation, we design tests of no serial correlation against weakly stationary alternatives. The main premise is that ratio of the high frequency variance to that of the overall variance of a white noise process is centered at 1/2 whereas the relative variance of a weakly stationary process is bounded in (0, 1). The limiting null distribution of our test is N(0,1). We demonstrate the size and power properties of our tests through Monte Carlo simulations. Our results are unifying in the sense that Durbin-Watson d test is a special case of a wavelet-based test.