Suppose that you are interested in estimating the ceteris paribus relationship between y and x,. 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, x, as class attendance, x2 as GPA up through the previous semester, and x3 as SAT or ACT score.) Let B, be the simple regression estimate from y on x and let be the multiple regression estimate fro m y on xi, X2, X3. (a) If xi is highly correlated with x2 and xj in the sample, and x2 and xs have large partial effects on y, would you expect A and to be similar or very different? Explain.
(b) 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.
(c) If x1 is highly correlated with x2 and x3, and x2 and x3 have small partial effects on y, would you expect selß1] or selB1] to be smaller? Explain.
(d) If x1 is almost uncorrelated with x2 and x3, x2 and x3 have large partial effects on y, and %y and xi are highly correlated, would you expect selß] or selßi] to be smaller? Explain.