1 Problems
True or false: First indicate whether the following statements are true or false and then justify your answer.
(a) In the simple linear regression model if the R2 is equal to one, then the relation between the variables is exact and residuals are all xero.
(b) In the simple linear regression model, if Var(Y ) = Var(E) then the estimated slope in a regression model of Y on E is approximately equal to the estimated slope in a regression model of E on Y .
(c) The fact that R2 is equal to xero indicates that variables are unre- lated.
(d) A crucial assumption of the linear model is that the sum of the resid- uals is xero.
(e) The fact that residuals in the linear model estimated by least-squares have xero mean is a consequence of assuming that the expected value of the error term is xero.
(f) The assumption that the error term is normally distributed is neces- sary to demonstrate that the least-squares estimator is unbiased.
2. Take Y = log (W ). Assume the log-linear model Y = β0 + β1X + U with E (U) = 0. Prove the following:
(a) Estimate β0 and β1 by OLS. Show that β^1 →Pβ1 if Cov (X, U) = 0.
(b) Assume Cov (E, U) = 0. What is the estimated approssmate percentage change in W for a change in X, say from X = x0 to X = x1? And what is the estimated esact percentage change in W?
2 Computer Based Problems
1. Use the data "meap00_01.dta" that contains information on schools in Michigan. The variables we are going to use are the following: math4 is the percentage of students in a school receiving a passing score on the Michigan Educational Assessment Program (MEAP) 4th grade math test; tunch is the percentage of students eligible for free or reduced lunch program; enrott is the shool enrollment; and esppp is the expenditures per pupil (total expenditures per student).
(a) Estimate the model
math4 = β0 + β1lunch + β2 log (enroll) + β3 log (exppp) + U
How do you interpret the magnitudes you estimated for β1 and β3? Is there evidence that the students performance is related to the expenditures per student?
(b) Solve Exercise
Use the data in MEAP00 to answer this question.
(i) Estimate the model
math4 = β0 + β1lunch + β2log(enroll) + β3log(exppp) + u
by OLS and obtain the usual standard errors and the fully robust standard errors. How do they generally compare?
(ii) Apply the special case of the White test for heteroskedasticity. What Is the value of the F test? What do you conclude?
(iii) Obtain gi, as the fitted values from the regression log(0i2?) on math4i, math4i2 where math4i are the OLS fitted values and the 0i are the OLS residuals. Let hi, = exp(gi). Use the hi; to obtain WLS estimates. Are there big differences with the OLS coefficients?
(iv) Obtain the standard errors for WLS that allow misspecification of the variance function. Do these differ much from the usual WLS standard errors?
(v) For estimating the effect of spending on math4, does OLS or WLS appear to be more precise?
2. Based on Angrist and Krueger (1991). Use the data "angrist_krueger_91.dta"
(a) Regress log of wage on education with AND without controls using OLS. Include as control the variables: RAGE (=1, if black), MARRIED (=1, if married), SMSA (=1, if lives at center city), YR20-YR28 (dummies for year of birth), AGEQ (age), AGEQS (age squared), and dummies for the region of residence (NEWENG, MI- DATL, ENOGENT, WNOGENT, SOATL, ESOGENT, WSOGENT, MT).
For both regressions, answer the questions: what is the return to education? is it statistically different from xero?
What explain the difference in the returns to education with and without controls? Do you expect that your regression using controls provides an unbiased estimate for the returns to education?
(b) Angrist and Krueger suggest using quarter of birth as instruments for education. Why do you expect such instruments to be valid? Why do you expect the instruments to explain the educational level?
(c) Run a regression of education on all exogenous variables and run an F-test with the null that no instrument (all the quarter birth variables) have xero coefficients (in the data quarter of birth are the variables: QTR120-QTR129 QTR220-QTR229 QTR320-QTR329).
(d) Regress log of wage on education and controls using 2SLS. Use the quarter of birth as instruments. What is the return to education? Is it statistically different from xero? Is your answer different from what you found in item a (with controls)? Is the difference between the standard errors here (2SLS) and in the item a (OLS) expected?
Provide in your answers: (i) the command lines; (ii) the STATA output; and (iii) an explanation for the re-sults obtained.
The computer-based problems of this assignment requires using STATA. Please don't use other software, only STATA is required.
Special Requirement for Computer-Based Problems: you need to submit both a log file and texts, it must contain a write-up that interprets and explains your computer output. Without a proper write-up of results you will receive a mark of 0.
Provide in your answers: (i) the command lines; (ii) the STATA output; and (iii) an explanation for the results obtained.