(Requires calculus) Consider the regression model Yi, = 1X1i + 2X2i + ui for i = 1,...,...

(Requires calculus) Consider the regression model
Yi, = β1X1i + β2X2i + ui
for i = 1,..., 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 bi and b2.
(c) Suppose

(d) Suppose ˆ‘ni=1 XtiX2i ‰  0. Derive an expression for 1 as a function of the data (Yi,X1i,X2i), i = 1,..., n.
(e) Suppose that the model includes an intercept:

How does this compare to the OLS estimator of β1 from the regression that omits X2?

Transcribed Image Text:

E-XX2 = 0. Show that B1 = EX1,Y/EX %3D %3D