Testing Adequacy of ARMA Models using a Weighted Portmanteau Test on the Residual Autocorrelations

In examining the adequacy of a statistical model, an analysis of the residuals is often performed. This includes anything from performing a residual analysis in a simple linear regression to utilizing one of the portmanteau tests in time-series analysis. When modeling an autoregressive-moving average time series we typically use the Ljung-Box statistic on the residuals to see if our fitted model is adequate. In this paper we introduce two new statistics that are weighted variations of the common Ljung-Box and, the less-common, Monti statistics. A brief simulation study demonstrates that the new statistics are more powerful than the commonly used Ljung-Box statistic. The new statistics are easy to implement in SAS® and source code is provided. INTRODUCTION A plethora of situations are known to occur in which the errors in a regression model are not independent. This violates the underlying assumptions of regression and can lead to a multitude of problems. Modeling the errors via a time series analysis does not address all of the issues but allows us to have better 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 a time series for where is the number of observations. Suppose is generated by a stationary and invertible ARMA( , ) process of the form where are white-noise residuals. A model of this form is typically fitted with the autoregressive and moving average parameters, and respectively, estimated by their maximum likelihood or conditional least squares counterparts, and . After we have fit the model for a given and , testing for the adequacy of the fitted model follows. Most diagnostic goodness-of-fit tests are based on the residual autocorrelation coefficients provided by