Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 14

Suppose that you are interested in estimating the ceteris paribus relationship between y and x1. For this purpose, you can collect data on two control variables, x2 and x3. (For concreteness, you might think of y as final exam score, x1 as class attendance, x2 as GPA up through the previous semester, and x3 as SAT or ACT score.) Let be the simple regression estimate from y on x1 and let be the multiple regression estimate from y on x1,x2,x3.
(i) If x1 is highly correlated with x2 and x3 in the sample, and x2 and x3 have large partial effects on y, would you expect and to be similar or very different Explain.
(ii) If x1 is almost uncorrelated with x2 and x3, but x2 and x3 are highly correlated, will and tend to be similar or very different Explain.
(iii) If x1 is highly correlated with x2 and x3, and x2 and x3 have small partial effects on y, would you expect se or se to be smaller Explain.
(iv) If x1 is almost uncorrelated with x2 and x3, x2 and x3 have large partial effects on y, and x2 and x3 are highly correlated, would you expect se or se to be smaller Explain.