Introduction to Econometrics 3rd Edition by James Stock, James Stock

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(Requires calculus) Consider the regression model
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 b 1 and b 2.
c. Suppose . Show that .
d. Suppose . Derive an expression for ß 1 as a function of the data ( Y i X₁i X₂i ) , i = 1,..., n.
e. Suppose that the model includes an intercept: . Show that the least squares estimators satisfy .
f. As in (e), suppose that the model contains an intercept. Also suppose that . Show that . How does this compare to the OLS estimator of ß x from the regression that omits X₂