Frontiers in Econometrics by Paul Zarembka

By Paul Zarembka

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The small sample behavior of the W statistic was examined by Monte Carlo means in the same work in which the previous two tests were initially examined; see Ramsey and Gilbert (1969). Consequently, it was possible to examine the statistical dependence of the W statistic with the other statistics. Under Theil residuals it would appear that W and M are not independently distributed, but do appear to be so under OLS residuals. As predicted, the W statistic had considerable power against the alternative given in Equation (2e), in which the alternative (true) hypothesis is that the distribution is central chi-square with one degree of freedom.

With respect to each test, the question asked is in a sense whether the model as formulated is “consistent” with the data with respect to certain specified characteristics. Thus, the main feature common to these procedures is that 12 The interested reader is referred to the Discussion following the 1970 paper for further details on this point. 42 JAMES В. RAMSEY they are all constructed to test for the presence of one or more important characteristics, rather than being, in some sense, summary measures of fit.

The errors listed above were discussed in terms of Equations (2a) to (2d) in Section I. The distributional properties of OLS residuals are well known and the effects of specification errors were discussed in the first section, so little needs to be added to the discussion. The dissatisfaction with OLS residuals in the classical normal linear regression model is that under the null hypothesis of no specifica­ tion error, the residual vector has a singular distribution with a nonscalar covariance matrix whose elements are functions of the regressors.

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