Question

Suppose you want to estimate the model

$Y = \beta _{0} + \beta _{1}X + U$

but, unfortunately, you don't observe Y. Instead, a mismeasured version of Y is available, $\widetilde{Y}$ . (e.g. people may not report their true earnings, but their incomes with some error).

We assume that $\widetilde{Y}$ = Y + V where V is a measurement error which satisfies

Cov (Y,V ) = 0

Cov (U,V ) = 0:

Given that Y is not observed, you go ahead an estimate the regression

$\widetilde{Y}$ = 0 + 1X + $\widetilde{U}$

(a) How would the precision of the estimator be affected? Hint: Assume homoskedasticity and look at the (homoskedasticity-only) standard error formula for $\widehat{\beta _{_{1}}}$ .

(b) Suppose now the measurement error is correlated with X , Cov (X,V ) $\neq$ 0. (E.g., people report their wrong incomes in such a way that correlates with their educational level (rich people may understate their income, while poor may overstate it). Also, assume E (V) = 0. Prove that OLS $\widehat{\beta _{_{1}}}$ converges in probability to

$\beta _{_{1}}+ \frac{Cov(X,V)}{Var(X)}$

Answer 1