Weighted Portmanteau Tests Revisited: Detecting Heteroscedasticity, Fitting Nonlinear and Multivariate Time Series

In the 2011 SAS® Global Forum, two weighted portmanteau tests were introduced for goodness-of-fit of an Autoregressive-Moving Average (ARMA) time series process. This result is summarized and extended for use as a diagnostic tool in detecting nonlinear and variance-changing processes such as the Generalized Autoregressive Conditional Heteroscedasticity process. The efficacy of the weighting scheme is shown in simulation experiments and analysis of stock market data. The statistics are easy to implement in SAS® and source code is provided. Lastly, the versatility of this methodology is discussed for a fitted GARCH, Vector ARMA, and other time series processes. INTRODUCTION After fitting a regression model with respect to time, or modeling the trend and seasonality of a time series, a plethora of situations are known to occur in which the residual error terms of the model are not independent. This violates the underlying assumptions of most statistical testing and modeling and can lead to a multitude of problems. Modeling the serial-correlation of the errors does not address all of the issues but allows us to have better forecast (predictive models). However, much like checking the adequacy of the regression through an F-test, checking the adequacy of the fitted time-series model is of the utmost importance. Let { } be an observed time series for where is the number of observations. Suppose { } is generated by a stationary and invertible ARMA( , ) process of the form