Using ATS to Turn Time Series Estimation and Model Diagnostics into Fast Regression Estimation and Model Diagnostics

The Average Transform Smooth (ATS) statistical methods [McRae, Mallows, and Cleveland], are applied to measurements of a non-gaussian random variable to make them close to gaussian. This gaussianization makes use of the well known concept of variance stabilizing transformation, but takes it further by first averaging blocks of r measurements, transforming next, and then smoothing. The smoothing can be nonparametric, or can be the fitting of a parametric model. The gaussianization makes analysis simpler and more effective.

In this work ATS is applied to the periodogram of a stationary parametric time series, and makes use of the periodogram large sample properties given the true power spectrum [Brillinger], to develop a new approach to parametric time series model estimation and model diagnostics. The ATS results and the theory are reformulated as a regression model, PPS-REG, involving true power spectrum and the periodogram. PPS-REG has attractive properties: iid gaussian error terms with mean 0 and a known variance; accurate estimation; much faster estimation than the classical maximum likelihood when the time series is large; enables the use of the very powerful classical regression model diagnostics; bases the diagnostics on the power spectrum, adding substantially to the standard use of the autocovariance function for diagnosing the fits of models specified in the time domain.