The power of the durbin watson test when the errors are par(1)

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Abstract
The Durbin Watson, DW, test for first order autocorrelation in regression residuals is among the most widely applied tests in time series analysis and econometrics. A significant test statistic indicates possible mis-specification of the underlying model as well as warning of the invalidity of traditional tests of parameter restrictions. It is important that properties of such a widespread test are well understood by users. We contribute by considering the power of the DW test where the regression errors can be described by a periodic autoregressive process of order 1, PAR(1). Increasingly, the accurate specification of seasonality is used to improve in-sample fit and out of sample forecasting efficiency in time series and econometric models. It is recognised that many seasonal data can be modelled using a periodic specification whereby parameters of the model vary with the season of the year. Seasonality in time series data can be described explicitly in the model, or captured in the error process. A PAR process for the errors is a useful modelling tool and a realistic description of the errors in a seasonal model. Considering the PAR(1) process, we show such errors display both (seasonal) autocorrelation and heteroscedasticity, even if the original data is homoscedastic. However, there is no reason to suppose the implications of PAR(1) errors should be limited to, or indeed similar to, the implications of heteroscedasticity. Thus we derive the power function of the DW test for given regressor data when a periodic process describes the errors. This power function is compared to the power function of the DW test when the errors display no periodicity. Significant differences are found. The DW test is robust if the data set includes seasonal dummy variables. Otherwise the power function is seriously distorted as the error process approaches a random walk.