The Dual Frequency Spectral Density Function of Locally Periodic Stationary Processes With an Application to Testing for Correlation Between Different Frequency Bands of a Time Series

Abstract

Harmonizable processes are a class of nonstationary time series, that are characterized by their dependence between different frequencies of a time series. The covariance between two frequencies is the dual frequency spectral density, an object analogous to the spectral density function. Local stationarity is another popular form of nonstationarity, though thus far, little attention has been paid to the dual frequency spectral density of a locally stationary process. The focus of this paper is on the dual frequency spectral density of local stationary time series and locally periodic stationary time series, its natural extension. We show that there are some subtle but important differences between the dual frequency spectral density of an almost periodic stationary process and a locally periodic stationary time series. Estimation of the dual frequency spectral density is typically done by smoothing the dual frequency periodogram. We study the sampling properties of this estimator under the assumption of locally periodic stationarity. In particular, we obtain a Gaussian approximation for the smoothed dual frequency periodogram over a group of frequencies, allowing for the number of frequency lags to grow with sample size. These results are used to test for correlation between different frequency bands in the time series. The variance of the smooth dual frequency periodogram is quite complex. However, by identifying which covariances are the most pertinent we propose a nonparametric method for consistently estimating the variance. This is necessary for constructing confidence intervals or testing aspects of the dual frequency spectral density. Simulations are given to illustrate our results.