Question: Consider the regression model, Yi = 1Xi + 2Wi + ui, where for simplicity the intercept is omitted and all variables are assumed to have

Consider the regression model, Yi = Î²1Xi + Î²2Wi + ui, where for simplicity the intercept is omitted and all variables are assumed to have a mean of zero. Suppose that Xi is distributed independently of (Wi, ui) but Wi and ui might be correlated and let Î²1 and Î²2 be the OLS estimators for this model. Show that
a. Whether or not Wi and ui are correlated,

b. If Wi and ui are correlated, then Î²2 is inconsistent.
c. Let Î²1r be the OLS estimator from the regression of Y on X (the restricted regression that excludes W). Provide conditions under which Î²1 has a smaller asymptotic variance than Î²1r, allowing for the possibility that Wi and ui are correlated.

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