12 Estimation with Instruments

12 Estimation with Instruments - Economics 140A Estimation...

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Economics 140A Estimation with Instruments In our previous lecture, we discussed the problems arising from measurement error. In particular, we found that measurement error in a regressor leads to correlation between a regressor and the error. Correlation between a regressor and the error in turn leads to bias in the OLS coe¢ cient estimator. If measurement error is present, how can we correct the problem and reduce (or remove) the correlation between the regressor and the error? The answer is through the use of an additional variable termed an instrument. An instrument is a variable that theory does not suggest belongs in the regression, but that is correlated with the problem regressor. Because theory does not suggest the instrument as a regressor, the instrument is uncorrelated with the regression error. Formally, consider the two-variable regression model in deviation-from-means form Y t = t + U t : As we have seen, the OLS estimator is a method-of-moments estimator that uses the population moment E ( X t U t ) = 0 : That is, the OLS estimator is the value B such that n X t =1 X t ( Y t BX t ) = 0 : If the regressor and the error are correlated, then E ( X t U t ) 6 = 0 . As a result, the OLS estimator does not equate the population moment and the sample analog (because the sample analog is set to zero) and so is biased. An improved estimator is obtained by again equating population moments and their sample analogs. Although the regressor and the error are correlated, the regressor and the instrument Z t are uncorrelated E ( Z t U t ) = 0 : The instrumental variable estimator is the method-of-moments estimator B IV for which n X t =1 Z t ( Y t B IV X t ) = 0 : (0.1)
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From (0.1) it follows that B IV = P n t =1 Z t Y t P n t =1 Z t X t ; for which EB IV = + E n t =1 Z t U t P n t =1 Z t X t ± 6 = Despite the fact that E ( U t j Z t ) = 0 , which follows from E ( Z t U t ) = 0 , the estima- tor is biased because we cannot condition on X t when treating U t as random. (If X t and U t to be random.) The bias of B IV does diminish as the sample grows, so B IV is a consistent estimator of , while the OLSE is inconsistent. The two requirements for an instrument are clearly seen. The instrument must be uncorrelated with the
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12 Estimation with Instruments - Economics 140A Estimation...

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