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 gn i = 1X1iX2i = 0. Show that b n

1 = gn i = 1X1iYi > gn i = 1X21 i.

d. Suppose that gn i = 1X1iX2i  0. Derive an expression for b n

1 as a function of the data 1Yi, X1i, X2i2, 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 gn i = 11X1i - X121X2i - X22 = 0. Show that b

n 1 = gn i = 11X1i - X121Yi - Y 2 > gn i = 11X1i - X12 2. How does this compare to the OLS estimator of b1 from the regression that omits X2?