18.7 Consider the regression model Yi = b1Xi + b2Wi + 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 b n

1 and b n

2 be the OLS estimators for this model. Show that

a. Whether or not Wi and ui are correlated, b n

1 ¡p b1.

b. If Wi and ui are correlated, then b n

2 is inconsistent.

c. Let b nr 1 be the OLS estimator from the regression of Y on X (the restricted regression that excludes W). Will b n

1 have a smaller asymptotic variance than b nr 1, allowing for the possibility that Wi and ui are correlated? Explain.