This paper introduces a version of empirical likelihood based on the periodogram and spectral estimating equations. This formulation handles dependent data through a data transformation (i.e., a Fourier transform) and is developed in terms of the spectral distribution rather than a time domain probability distribution. The asymptotic properties of frequency domain empirical likelihood are studied for linear time processes exhibiting both short- and long-range dependence. The method results in likelihood ratios which can be used to build nonparametric, asymptotically correct confidence regions for a class of normalized (or ratio) spectral parameters, including autocorrelations. Maximum empirical likelihood estimators are possible, as well as tests of spectral moment conditions. The methodology can be applied to several inference problems such as Whittle estimation and goodness-of-fit testing.
"A frequency domain empirical likelihood for short- and long-range dependence." Ann. Statist. 34 (6) 3019 - 3050, December 2006. https://doi.org/10.1214/009053606000000902