Alternative Tests of Independence between Stochastic Regressors and Disturbances

IN TESTING HYPOTHESES on the coefficients of a linear regression model with stochastic regressors it is well known that the usual t test and F test are applicable if the stochastic regressors are statistically independent of the disturbances [3, p. 268; 5, pp. 27-28]. Also, there are cases in which economic hypotheses can be stated in terms of the independence of stochastic regressors and disturbances, the best known examples being the current versus the permanent income hypotheses and the recursiveness hypothesis in a simultaneous equations model. Therefore, it is desirable to develop a procedure that can be used to test the hypothesis that the stochastic regressors and disturbances are independent. In this paper, we examine four alternative tests of independence between the stochastic regressors and disturbances. In the rest of this section we specify the stochastic model and state the hypotheses to be tested. In Section 2 we present two finite sample tests. In Section 3 two alternative asymptotic tests are given and asymptotic power functions of all four tests are examined. In Section 4 we give examples of applications of the test in econometrics. We consider the following linear model: