Introductory Econometrics 4th Edition by Jeffrey Wooldridge

Exercise 6

Let 0 and 1 be the intercept and slope from the regression of y i on x i , using n observations. Let c 1 and c 2 , with c 2 0, be constants. Let 0 and 1 be the intercept and slope from the regression of c 1 y i on c 2 x i. Show that 1 = (c 1 /c 2 ) 0 and 0 = c 1 0 , thereby verifying the claims on units of measurement in Section 2.4. [Hint: To obtain 1 , plug the scaled versions of x and y into (2.19). Then, use (2.17) for 0 , being sure to plug in the scaled x and y and the correct slope.]
(ii) Now, let 0 and 1 be from the regression of (c 1 + y i ) on (c 2 + x i ) (with no restriction on c 1 or c 2 ). Show that l = 1 and 0 = 0 + c 1 - c 2 1.
(iii) Now, let 0 and 1 be the OLS estimates from the regression log(y i ) on x i , where we must assume y i. 0 for all i. For c 1 0, let 0 and 1 be the intercept and slope from the regression of log(c 1 y i ) on x i. Show that .
(iv) Now, assuming that x. 0 for all i, let 0 and 1 be the intercept and slope from the regression of y. on log(c 2 x i ). How do 0 and 1 compare with the intercept and slope from the regression of y i on log(x i )