Hypothesis Testing in Time Series via the Empirical Characteristic Function: A Generalized Spectral Density Approach

Abstract The standardized spectral density completely describes serial dependence of a Gaussian process. For non-Gaussian processes, however, it may become an inappropriate analytic tool, because it misses the nonlinear processes with zero autocorrelation. By generalizing the concept of the standardized spectral density, I propose a new spectral tool suitable for both linear and nonlinear time series analysis. The generalized spectral density is indexed by frequency and a pair of auxiliary parameters. It is well defined for both continuous and discrete random variables, and requires no moment condition. Introduction of the auxiliary parameters renders the spectrum able to capture all pairwise dependencies, including those with zero autocorrelation. The standardized spectral density can be derived by properly differentiating the generalized spectral density with respect to the auxiliary parameters at the origin. The consistency of a class of Parzen's kernel-type estimators for the generalized spectral density is established, and their optimal convergence rates are derived using the integrated mean squared error criterion. A data-dependent asymptotically optimal bandwidth (or lag order) is introduced. The kernel estimators and their derivatives are applied to construct a class of asymptotically one-sided N(0, 1) tests for generic serial dependence and hypotheses on various specific aspects of serial dependence. The latter include serial uncorrelatedness, martingale, conditional homoscedasticity, conditional symmetry, and conditional homokurtosis. All of the proposed tests, which include Hong's spectral density test for serial correlation, can be derived from a unified framework. An empirical application to Deutschemark exchange rates highlights the approach.