6.11 (Requires calculus) Consider the regression model Yi = b1X1i + b2X2i + ui for i = 1,

c, n. (Notice that there is no constant term in the regression.)

Following analysis like that used in Appendix (4.2):

a. Specify the least squares function that is minimized by OLS.

b. Compute the partial derivatives of the objective function with respect to b1 and b2.

c. Suppose that gni

= 1X1iX2i = 0. Show that b n

1 = gni

= 1X1iYi > gni

= 1X21 i.

d. Suppose that gni

= 1X1iX2i  0. Derive an expression for b n

1 as a function of the data (Yi, X1i, X2i), i = 1,

c, n.

e. Suppose that the model includes an intercept: Yi = b0 + b1X1i +

b2X2i + ui. Show that the least squares estimators satisfy b n

0 =

Y - b n

1X1 - b n

2X2.

f. As in (e), suppose that the model contains an intercept. Also suppose that gni

= 1(X1i - X1)(X2i - X2) = 0. Show that b n

1 =

gni

= 1(X1i - X1)(Yi - Y )>gni

= 1(X1i - X1)2. How does this compare to the OLS estimator of b1 from the regression that omits X2?