Consider the relatively unlikely event that the correlation between x1i and x2i is zero.

(a) Prove that, if this is true, the bias in equation (11.6) is zero.

(b) Prove that, if this is true, the variance of b1 in equation (12.14) is equal to the variance of b in equation (5.50).

(c) What should we conclude about the importance of including x2i in our regression under these circumstances, if our objective is to estimate β1?

(d) If, instead, our objective is to estimate β2, what are the consequences of omitting x1i from our regression?

(e) If our objective is to estimate both β1 and β2 under these circumstances, what, if any, are the differences between estimating them with the regression of equation (11.12) or estimating β1 with the auxiliary regression of equation (11.2) and β2 with the auxiliary regression of equation (11.59)?

(f↜渀) Is the comparison in part e the same if CORR(x1i, x2i) ≠ 0? Why or why€not?