Exercise 7 - University of California Department of...

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University of California D. Steigerwald Department of Economics Economics 241B Exercise 7 1. Consider the population model (in deviation-from-means form) , * t t t U X Y in which * t X is latent. The observed regressor is , * t t t V X X in which t V is independent of t U . Further, we know that t V and t U are symmetrically distributed about 0. a. Detail why, in the presence of measurement error, β is not identified. b. Let med(X) be the median of n X X , , 1 and define 1 t Z if X med X t 1 t Z if X med X t . In certain texts, t Z has been advanced as a valid instrument. Is t Z a valid instrument? c. Can the assumption that t V is symmetrically distributed about 0 restore identification? If so, construct the corresponding estimator of β . 2. Consider the linear regression model (1) Y t =  X t + U t , t = 1,...,n where X t is a ( k 1 ) vector of explanatory variables, is ( k 1 ) vector of coefficients to be estimated and U t is a homoskedastic and serially uncorrelated error term with some
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