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2 o since x x e thatiseisapartofxsotypically cov

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Unformatted text preview: the unobserved true variable, and X as the observed mismeasured variable, then the measurement error can be written as e = X − X *. Without loss of generality, assume E(e) = 0. • Assume the true model is Y = α + β X * + u with all the required • assumptions holding true. Really estimating Y = α + β X * + u = α + β ( X − e) + u = α + β X + (u − β e). • The effect of measurement error on OLS estimates depends on our assumption about the correlation between e and X. 2 o Since X = X * + e , that is, e is a part of X, so typically Cov( X , e) ≠ 0, and OLS is biased in this case. Cov( X , e) = E ( Xe) - 0 = E ( X *e) + E (e 2 ) If Cov( X * , e) = 0 , i.e., the classical measurement error, then Cov( X , e) = E ( X *e) + E (e 2 ) = 0 + σ 2 ≠ 0. o In rare cases if Cov( X , e) = 0 , then OLS remains unbiased, and variances tend to be larger. Sample selection bias Selection based on X • If we are interested in estimating the population relationship. In general, we need a random sample, because only a random sample keeps the population relationship. A nonrandom sample may lead to biased estimates. (figure) • If we instead are only interested in estimating the relationship for the particular selected sample we have, then it is ok to choose the sample based on the X variable. What you estimate is valid for the...
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