1. Re -scaling variables (8 points ): la ) (4 points ) Show the rescaling effect Let B. (hat ) and B, (hat ) be the OLS estimates from the regression of Y, on X,, i = 1,2, ..., n. For nonzero constants c, and c,, show that the OLS intercept and slope from the regression of coY, on c,X,, are given by cop. (hat ) and (c/c)) B , (hat ). [Hint : Start with the OLS formula for B, (hat ). Use the properties of the covariance and the variance: cov(AX, BY) = A B cov(X, Y), and var(AX)=A? var(X). To get the result for B. (hat ) use the fact that the OLS regression line goes through the mean (the one obtained from the first moment equation ).] Now apply the rescaling to the following problem. Using the data GPA2 on 4137 American college students , the following equation was estimated by OLS : colgpa = 1.392 - 0.0135 hsperc + 0.00148 sat where colgpa is measured on a four -point scale , hsperc is the percentile in the high - school graduating class , and sat is the combined match and verbal scores on the student achievement test . Variable obs Mean Std . Dev . Min Max colgpa 4137 2. 652686 . 6586347 hsperc 4137 19.23707 16.56873 1666667 92 sat 4137 1030. 331 139 . 4014 470 1540