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The CEF error

Exercise 3.23 The data matrix is (y; X) with X = [X1; X2] ; and consider the transformed
regressor matrix Z = [X1; X2 X1] : Suppose you do a least-squares regression of y on X; and a
least-squares regression of y on Z: Let b
2 and e
2 denote the residual variance estimates from the
two regressions. Give a formula relating b
2 and e
2
? (Explain your reasoning.)
Exercise 3.24 Use the data set from Section 3.22 and the sub-sample used for equation (3.50)
(see Section 3.25) for data construction)
(a) Estimate equation (3.50) and compute the equation R2 and sum of squared errors.
(b) Re-estimate the slope on education using the residual regression approach. Regress log(Wage)
on experience and its square, regress education on experience and its square, and the residuals
on the residuals. Report the estimates from this Önal regression, along with the equation R2
and sum of squared errors. Does the slope coe¢ cient equal the value in (3.50)? Explain.
(c) Are the R2 and sum-of-squared errors from parts (a) and (b) equal? Explain.
Exercise 3.25 Estimate equation (3.50) as in part (a) of the previous question. Let ebi be the
OLS residual, ybi the predicted value from the regression, x1i be education and x2i be experience.
Numerically calculate the following:
(a) Pn
i=1 ebi
(b) Pn
i=1 x1iebi
(c) Pn
i=1 x2iebi
(d) Pn
i=1 x
2
1i
ebi
(e) Pn
i=1 x
2
2i
ebi
(f) Pn
i=1 ybieb
CHAPTER 3. THE ALGEBRA OF LEAST SQUARES 103
(g) Pn
i=1 eb
2
i
Are these calculations consistent with the theoretical properties of OLS? Explain.
Exercise 3.26 Use the data set from Section 3.22.
(a) Estimate a log wage regression for the subsample of white male Hispanics. In addition to
education, experience, and its square, include a set of binary variables for regions and marital
status. For regions, you create dummy variables for Northeast, South and West so that
Midwest is the excluded group. For marital status, create variables for married, widowed or
divorced, and separated, so that single (never married) is the excluded group.
(b) Repeat this estimation using a di§erent econometric package. Compare your results. Do they
agree?