hws06 - Economics 140A Answers to Exercise 6 1. (Fall 1991...

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Economics 140A Answers to Exercise 6 1. (Fall 1991 Final Examination) a) If Y t replaces Y t , then we have added measurement error in the dependent variable. From our earliest discussions of the composition of U t , we determined that estimators are generally unbiased if U t consists only of measurement error in the dependent variable. The logic is that mismeasurement of the dependent variable does not lead to any relation between the regressors and the error. With no such relation, the OLS estimator B is unbiased. To understand the e/ects The most common approach is to remove the sample means from the data, thus the dependent variable in use would be Y t = Y t Y n . For such a depen- dent variable, increases in the measurement error correspond to increases in the variance of U t . As the variance of U t increases, so too does the variance of B and we are less likely to reject any null hypothesis concerning . Mathematically Y t = t + ( U t + V Y t ) ; where the regression error is now U t + V Y t . For the OLS estimator E ( B j X ) = E n t =1 X t Y t P n t =1 X 2 t ± ± ± ± X ² = + P n t =1 X t E ( U t + V Y t j X ) P n t =1 X 2 t ; where X = f X t g n t =1 . The last term equals zero because X t is uncorrelated with both U t and V Y t . b) If X t replaces X t , then we have added measurement error to the regressor. The measurement error in the regressor is added to U t and this compound error is correlated with X t , so B is biased. Mathematically Y t = t + ( U t Xt ) ; where the regression error is now U t Xt . For the OLS estimator EB = + E P n t =1 X t ( U t Xt ) P n t =1 X 2 t = P n t =1 ( X t V Xt ) P n t =1 X 2 t : Although the second term in the last equality on the right cannot be easily calcu- lated, as both the numerator and denominator contain random variables, the re- gressor X t is negatively correlated with the component of the error term Xt .
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This note was uploaded on 09/04/2011 for the course ECON 140a taught by Professor Staff during the Fall '08 term at UCSB.

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hws06 - Economics 140A Answers to Exercise 6 1. (Fall 1991...

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