Thus, the explanatory variable and the error term in (13.5.8) are correlated, which violates the crucial assumption of the classical linear regression model that the explanatory variable is uncorrelated with the stochastic disturbance term. If this assumption is violated, it can be shown that the OLS
estimators are not only biased but also inconsistent, that is, they remain biased even if the sample size n increases indefinitely.28
28As shown in App. A, f is a consistent estimator of f if, as n increases indefinitely, the sampling distribution of f will ultimately collapse to the true f. Technically, this is stated as plim^^f = f. As noted in App. A, consistency is a large-sample property and is often used to study the behavior of an estimator when its finite or small-sample properties (e.g., unbiased-ness) cannot be determined.
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For model (13.5.8), it is shown in Appendix 13A, Section 13A.3 that plim ß = ß
where aW and aX* are variances of wi and X*, respectively, and where plim ¡3 means the probability limit of 3.
Since the term inside the brackets is expected to be less than 1 (why?), (13.5.10) shows that even if the sample size increases indefinitely, ¡> will not converge to ¡. Actually, if 3 is assumed positive, ¡3 will underestimate ¡, that is, it is biased toward zero. Of course, if there are no measurement errors in X (i.e., aw = 0), ¡3 will provide a consistent estimator of ¡.
Therefore, measurement errors pose a serious problem when they are present in the explanatory variable(s) because they make consistent estimation of the parameters impossible. Of course, as we saw, if they are present only in the dependent variable, the estimators remain unbiased and hence they are consistent too. If errors of measurement are present in the explanatory variable(s), what is the solution? The answer is not easy. At one extreme, we can assume that if aw is small compared to aX, for all practical purposes we can "assume away" the problem and proceed with the usual OLS estimation. Of course, the rub here is that we cannot readily observe or measure aw and aX* and therefore there is no way to judge their relative magnitudes.
One other suggested remedy is the use of instrumental or proxy variables that, although highly correlated with the original X variables, are un-correlated with the equation and measurement error terms (i.e., ui and wi). If such proxy variables can be found, then one can obtain a consistent estimate of ¡. But this task is much easier said than done. In practice it is not easy to find good proxies; we are often in the situation of complaining about the bad weather without being able to do much about it. Besides, it is not easy to find out if the selected instrumental variable is in fact independent of the error terms ui and wi.
In the literature there are other suggestions to solve the problem.29 But most of them are specific to the given situation and are based on restrictive assumptions. There is really no satisfactory answer to the measurement errors problem. That is why it is so crucial to measure the data as accurately as possible.
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