A local spectral approach for assessing time series model misspecification

We consider band-limited frequency-domain goodness-of-fit testing for stationary time series, without smoothing or tapering the periodogram, while taking into account the effects of parameter uncertainty (from maximum-likelihood estimation). We are principally interested in modeling short econometric time series, typically with 100 to 150 observations, for which data-driven bandwidth selection procedures for kernel-smoothed spectral density estimates are unlikely to have adequate levels. Our mathematical results take parameter uncertainty directly into account, allowing us to obtain adequate level properties at small sample sizes. The main theorems provide very general results involving joint normality for linear functionals of powers of the periodogram, while accounting for parameter uncertainty, which can be used to determine the level and power of a wide array of statistics. We discuss several applications, such as spectral peak testing and testing for the inclusion of an Unobserved Component, and illustrate our methods on a time series from the Energy Information Administration.