🤖 AI Summary
This study addresses the problem of quantifying the goodness-of-fit of moving average MA(q) models to the spectral density of stationary processes by proposing a spectral-domain coefficient of determination. This coefficient extends, for the first time, the classical notion of the coefficient of determination into the framework of spectral analysis to measure how closely an MA(q) model approximates the true spectral density. Constructed via periodogram-based estimation, the proposed coefficient is shown to possess asymptotic normality under rigorous derivation, enabling the development of both a model order selection criterion and a goodness-of-fit test specifically tailored for MA(q) models. The approach adaptively identifies the minimal order q that achieves a pre-specified accuracy level, offering a method that is theoretically sound and practically useful.
📝 Abstract
We develop a spectral based coefficient of determination to measure how well the spectral density of a stationary process is represented by the class of MA($q$) models. Using periodogram-based estimators, we establish asymptotic normality, derive tests for the MA($q$) hypothesis, and construct procedures for determining the smallest order $q$ achieving a prescribed approximation quality.