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# IV - Economics 245A Instruments Linear Models Consider the...

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Economics 245A Instruments Linear Models Consider the scalar regression model in deviation-from-means form Y t = t + U t : One can always consider the problem of endogeneity ( Cov ( X t ;U t ) 6 = 0 ) to be one of a mismeasured regressor. If we only had the correct measure of the regressor, then OLS could be used. It may be helpful to think of the measurement error as falling into two cases. Case 1 - The mismeasurement in X record the correct value at each observation, without any systematic tendency to misrecord the value based on the value of Y or of other regressors. For case 1, statistical assumptions can restore identi±cation. components are correlated with observed components. In such cases, instruments are needed to restore identi±cation. Statistical Measurement Error As discussed above, the classic treatment of measurement error allows statis- tical assumptions to restore identi±cation. Consider ±rst Y t = t + U t ; in which X t is measured without error. We have C ( X t ;Y t ) = ( X t ) : Both C ( X t ;Y t ) and V ( X t ) can be consistently estimated from their sample mo- ments, hence the method-of-moments (OLS) estimator is B = P X t Y t P X 2 t : Now suppose X t is measured with error X t = X t + W t :

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We assume that W t is not correlated with U t nor with X t . The observed regression model is Y t = t + ( U t t ) : We have C ( X t ;Y t ) = ( X t ) ( W t ) : Because the variance of the measurement error cannot be consistently estimated from the observed data, the OLS estimator is not consistent C ( X t ;Y t ) V ( X t ) = 1 V ( W t ) V ( X t ) ± : In essence, the parameter error. If we assume that the measurement error is symmetric around zero, then we can construct a consistent MoM estimator. Note E ² X 2 t Y t ³ = ² X 3 t ³ ² W 3 t ³ : Hence the assumption that E ( W 3 t ) = 0 implies = E ( X 2 t Y t ) E ( X 3 t ) . The resulting consistent MoM estimator is ~ B = P X 2 t Y t P X 3 t : For the more common case, in which the regressor is considered endogenous, the simple assumption of symmetric measurement error will not work (and may not make sense). In essence, the ±measurement error"
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IV - Economics 245A Instruments Linear Models Consider the...

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