Econometrica: Jul, 1972, Volume 40, Issue 4

Testing for Fourth Order Autocorrelation in Quarterly Regression Equations

https://doi.org/0012-9682(197207)40:4<617:TFFOAI>2.0.CO;2-E
p. 617-636

Kenneth F. Wallis

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 ut = put-4 + E?t, an appropriate test statistic for H0: @? = 0 is the statistic d4 = {@X(zt - Zt-4)2}/@Xzt2 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 = @?1u1-1 + @?4ut-4 - @?1@?4u-5 + Et. Examples from the empirical literature are presented.