(i) Let and be the intercept and slope from the regression of yi on xi, using n observations. Let c1 and c2, with c2 ( 0, be constants. Let and be the intercept and slope from the regression of c1.yi on c2.xi. Show that = (c1/c2) and = c1, thereby verifying the claims on units of measurement is Section 2.4. [To obtain , plug the scaled versions of x and y into (2.19). Then, use (2.17) for being sure to plug in the scaled x and y and the correct slope].
(ii) Now, let and be from the regression of (c1 + yi) on (c2 + xi) (with no restriction on c1 or c2). Show that / = and = / + c1 - c2.
(iii) Now, let / and / be the OLS estimates from the regression log (yi) on xi, where we must assume yi > 0 for all i. For c1 > 0, let and / be the intercept and slope from the regression of log (c1,yi) on xi. Show that and / be the intercept and slope from the regression of yi on log(c2xi). How do and / compare with the intercept and slope from the regression of yi on log (xi)?