Wavelet‐Based Tests for Comparing Two Time Series with Unequal Lengths

Test procedures for assessing whether two stationary and independent time series with unequal lengths have the same spectral density (or same auto‐covariance function) are investigated. A new test statistic is proposed based on the wavelet transform. It relies on empirical wavelet coefficients of the logarithm of two spectral densities' ratio. Under the null hypothesis that two spectral densities are the same, the asymptotic normal distribution of the empirical wavelet coeffcients is derived. Furthermore, these empirical wavelet coefficients are asymptotically uncorrelated. A test statistic is proposed based on these results. The performance of the new test statistic is compared to several recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed test is very comparable to the current test statistics in most cases. The main advantage of our proposed test statistic is that it is constructed very simply and is easy to implement.