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16 Estimation with Instruments

16 Estimation with Instruments - Economics 241B Estimation...

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Economics 241B Estimation with Instruments Measurement Error a variable. At some level, every variable is measured with error. For example, if we measure a person±s height as 6 feet 2 inches, it is unlikely that they are exactly 6 feet 2 inches tall. It is simply that they are closer to 6 feet 2 inches than to any other easily measured height. How does mismeasurement of variables a/ect estimation of a linear regression model? If a variable is mismeasured, then the resulting error is a component of the regression error. Consider &rst the dependent variable. In our earliest discussion of the regres- sion error, we said that the best possible outcome is for the regression error to arise from random mismeasurement of the dependent variable. Mismeasurement of the dependent variable does not lead to correlation between the regressors and the error and induces no bias in our OLS estimators. In fact, measurement error of the dependent variable a/ects only the variance of the regression error. Of course, we would like the error to have as small a variance as possible, so we would like measurement error in the dependent variable to be small. Mismeasurement of the independent variable is another matter. In general, mismeasurement of the independent variable will lead to correlation between the regressor and the error and so lead to bias in the OLS coe¢ cient estimators. As all regressors may have some measurement error, we treat the problem only when we feel the measurement error in the regressors is large enough relative to the approximation error made in the model under study. For example, survey data is often subject to such large errors that it is wise to treat the regressors as though measured with error. Formally, let the population model (in deviation-from-means form) be Y t = t + U t : The regressor is measured with error, so that we observe X t , where X t = X t + V t and V t is measurement error. The estimated model is Y t = t + [ U t + ( X t X t )] = t + [ U t t ] ;
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where the error is the term in [ ] . Because V t is a component of X t , the regressor and error are correlated in the estimated model. For the OLS estimator B = + P n t =1 X t [ U t ± t ] P n t =1 X 2 t : Because X t is uncorrelated with U t , we have EB = 1 ± E ±P n t =1 X t V t P n t =1 X 2 t ²³ : It is di¢ cult to proceed further with exact results, as one cannot condition on X t while examining V t . Rather, we have as an approximation EB ² 1 ± C ( X t ;V t ) V ( X t ) ³ ; where V ( X t ) = V ( X t ) + V ( V t ) and C ( X t ;V t ) = V ( V t ) . Thus EB ² V ( X t ) V ( X t ) + V ( V t ) ³ : Measurement error in the regressor biases the estimator toward zero and the degree of the bias depends upon the variance of the measurement error. The result is intuitive, as the measurement error in the regressor becomes more pronounced, the e/ect of the regressor is obscured and the estimated coe¢ cient is biased toward
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