13 Error Location and Scale Variation

13 Error Location and Scale Variation - Economics 140A...

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Economics 140A Error: Location and Scale Variation We turn now to the remaining classic assumptions that refer to the error and are statistical in meaning. Today we focus on the assumption that the error has mean zero (location) and constant variance (scale). Error Location We assume that EU t = 0 . As the error term is unobservable, there is no way to test the assumption. Further, because the error term includes all model misspeci°cations, it is unlikely that the mean of the error is zero. It seems we must investigate the consequences of a non-zero error mean. Let EU t = ° and suppose that ° 6 = 0 . Then the population model Y t = ± 1 + ± 2 X t + U t = ( ± 1 + ° ) + ± 2 X t + ( U t ° ° ) : The two expressions on the right are observationally equivalent. Because the error is unobserved, the mean of the error and the intercept are not separately identi°ed. As a result, it is often di¢ cult to interpret the intercept. (In addition, the intercept often captures behavior well o/ the support of the data, when X t = 0 , and we do not believe the extrapolation of our model to this case.) The consequence of a non-zero error mean is simply to alter the intercept, none of the other coe¢ cients are a/ected. If, however, we have omitted an intercept, then the slope coe¢ cients are a/ected. Consider a two-variable model of costs in which the intercept captures °xed cost. If we know that °xed costs are zero, it is tempting to suppress the intercept. Yet suppressing an intercept is dangerous, the remaining coe¢ cient estimators are unbiased only if the mean of the error actually is zero. To motivate, consider a case in which, due to mismeasurement of the dependent variable, the mean of the error term is not zero. From theory we believe that the intercept should be omitted from the regression as we are con°dent the sample mean of Y t and of X t is zero for all samples. Thus we exclude an intercept and do not remove sample means from the dependent variable. With systematic mismeasurement in the dependent variable, as would be the case if Y t ° Y ° t tended to be positive, the mean of U t will not be zero. A nonzero mean for U t is absorbed in the intercept, but there is no intercept in this regression! As a result, the
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regression line is inappropriately forced through the origin and bias will result. If EU t 6 = 0 , then EB = ± + P n t =1 X t EU t P n t =1 X 2 t : A positive error mean leads to a positive bias, if the regressor is also positive in mean. A negative error mean leads to a negative bias, if the regressor is positive in mean. The signs change if the regressor is negative in mean.
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