Testing for Fourth Order Autocorrelation in Quarterly Regression Equations

A test for fourth order autocorrelation in the error term of a regression equation estimated from quarterly data is described. The development draws on the finite sample results of Durbin and Watson and illustrates how their procedure for the first order case can be generalized. In the model y = X, B + u where X is a matrix of fixed regressors and u, = put-4 + Bt, an appropriate test statistic for Ho: p = 0 is the statistic d4 = {z-) 2/z2 computed from the least squares regression residuals z = y - Xb. Bounds to the significance points of d4 are tabulated. Maximum likelihood estimation methods are described; these are equally appropriate when lagged values of the dependent variable appear among the regressors, and they provide asymptotic tests for general-autoregressive error structures, as well as for the special case ut = oe1u_-1 + 04ut4 - aLa4ut-5 + et. Examples from the empirical literature are presented. THE POSSIBILITY THAT the errors in a regression equation estimated from quarterly data possess fourth order autocorrelation was considered, among other things, in a recent paper 28, and a non-parametric test was proposed. Appropriate generalized least squares estimation methods were also described. In this paper we first present a more rigorous solution to the problem of testing for fourth order autocorrelation, which utilizes the approach introduced by Durbin and Watson 6 and 7. We then describe non-linear estimation methods which simultaneously estimate the regression coefficients and the parameters of the simple fourth order or more general autoregressive error structures. The usual interpretation of the error or disturbance term in econometric models is that it represents the effect of omitted or unobservable variables on the dependent variable. The error term might thus be expected to display certain features of observed economic variables, in particular, when quarterly data are employed, seasonal variation. Equally, when seasonally unadjusted data are being employed in order that one may attempt to explain seasonal variation in the dependent variable, along with other types of variation, by means of explanatory economic or seasonal dummy variables, then the presence of non-systematic seasonal variation, or an incomplete accounting for seasonality by the regressors, will produce seasonal effects in the error term, with the possible consequence of fourth order autocorrelation. Thus we require a test for correlation not between the errors 1 Some of the results contained in this paper were reported in my paper Estimation and Tests for Quarterly Regression Equations with Autocorrelated Errors presented at the Second World Congress of the Econometric Society in Cambridge, September, 1970. I am grateful to David Hendry for comment and discussion and, in Section 3, for the use of his computer program, to Zvi Griliches and an anonymous referee for comments, to Andrew Tremayne for research assistance, and to M. I. Nadiri and Michael Parkin for supplying their data. Added in proof: After this was written an unpublished paper by H. D. Vinod entitled Generalization of the Durbin-Watson Statistic for Higher Order Autoregressive Processes was brought to my attention; this considers statistics similar to d4 for tests of higher order autocorrelation in the non