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On Comparing Several Spectral Densities

We investigated the problem of testing equality among spectral densities of several independent stationary processes. Our main methodological contribution is the introduction of a novel semiparametric log-linear model that links all of the spectral densities under consideration. This model is motivated by the asymptotic properties of the periodogram ordinates and specifies that the logarithmic ratio of G − 1 spectral density functions with respect to the Gth is linear in some parameters. Then the problem of testing equality of several spectral density functions is reduced to a parametric problem. Under this assumption, the large-sample theory of the maximum likelihood estimator was studied, and it was found that the estimator is asymptotically normal even when the model is misspecified. The development of the asymptotic theory is based on a new contrast function that might be useful for other spectral domain time series problems. The results are applicable to a variety of models, including linear and nonlinear processes. Simulations and data analysis support further the theoretical findings.