Shapes of Stationary Autocovariances

This note introduces shape orderings for stationary time series autocorrelation and partial autocorrelation functions and explores some of their convergence rate ramifications. The shapes explored include decreasing hazard rate and new better than used, orderings that are familiar from stochastic processes settings. Time series models where these shapes arise are presented. The shapes are used to obtain explicit geometric convergence rates for mean squared errors of one-step-ahead forecasts.