Violations of the Classical Assumptions
In this chapter, we relax the assumptions made in Chapter 3 one by one and study the effect of that on the OLS estimator. In case the OLS estimator is no longer a viable estimator, we derive an alternative estimator and propose some tests that will allow us to check whether this assumption is violated.
Violation of assumption 1 implies that the mean of the disturbances is no longer zero. Two cases are considered:
Case 1: E(щ) = л = 0
The disturbances have a common mean which is not zero. In this case, one can subtract л from the Mj’s and get new disturbances u* = ui — л which have zero mean and satisfy all the other assumptions imposed on the uj’s. Having subtracted л from щ we add it to the constant a leaving the regression equation intact:
Yi = a* + pXi + u* i = 1, 2,…,n (5.1)
where a* = a + л. It is clear that only a* and в can be estimated, and not a nor л. In other words, one cannot retrieve a and л from an estimate of a* without additional assumptions or further information, see problem 10. With this reparameterization, equation (5.1) satisfies the four classical assumptions, and therefore OLS gives the BLUE estimators of a* and в. Hence, a constant non-zero mean for the disturbances affects only the intercept estimate but not the slope. Fortunately, in most economic applications, it is the slope coefficients that are of interest and not the intercept.
Case 2: E(ui) = лі
The disturbances have a mean which varies with every observation. In this case, one can transform the regression equation as in (5.1) by adding and subtracting лі. The problem, however, is that a* = a + лі now varies with each observation, and hence we have more parameters than observations. In fact, there are n intercepts and one slope to be estimated with n observations. Unless we have repeated observations like in panel data, see Chapter 12 or we have some prior information on these a*, we cannot estimate this model.
B. H. Baltagi, Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-20059-5_5, 95
© Springer-Verlag Berlin Heidelberg 2011