Suppose that you are interested in estimating the ceteris paribus relationship between y and xi. 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 is highly correlated with x2 and x3 in the sample, and x2 and x3 have large partial effects on y would you expect and / 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.