Problem Set 2
This problem set covers panel data models - including Fixed Effects and Random Effects, maximum likelihood estimation, and interpreting output from nonlinear estimation.
1. You are interested in studying the responsiveness of labor supply to wages. The original data is from Jim Ziliak (1997) ”Efficient Estimation With Panel Data when Instruments are Predetermined: An Empirical Comparison of Moment-Condition Estimators” Journal of Business and Economic Statistics, 15, 419-431 MOM.dat was downloaded from the JBES website and is available to you on Blackboard. MOM.dat has data on 532 men from 1979 to 1988. Data are space-delimited ordered by person with a separate line for each year. The panel is balanced so there will be 5320 observations in total. id is a unique identifier assigned to each man in the sample (1-532). The observations are ordered: id 1 1979, id 1 1980, ..., id 1 1988, id 2 1979, id 2 1980, ... There are 8 variables: lnhr lnwg kids ageh agesq disab id year
yit = ci + xi β + uit
yit = lnhrit =man i’s ln(hours worked) at time t
ci =person-specific unobserved desire to work that is time-invariant
x1it = lnwgit =man i’s ln(wage) at time t,
x2it = kidsit =man i’s number of kids at time t, x3it = agehit =man i’s age at time t,
x4it = agesqit=man i’s age squared at time t,
x5it = disabit =indicator equal to 1 if man i is disabled at time t.
(a) Interpret β1.
(b) Estimate βˆ by pooled ordinary least squares. Report default standard errors and heteroskedasticity-robust standard errors (clustering across time for a given individual).
(c) Interpret βˆ1,OLS .
(d) Re-estimate βˆ by pooled ordinary least squares, including year dummies. Test the null hypothesis that the year effects on men’s hours worked are jointly zero, at the 5% significance level. How has βˆ1,OLS changed?
(e) Under what conditions are βˆ1,OLS consistent estimators for β1? Do you think those conditions hold?
(f) Estimate βˆ using the within estimator (fixed effects model). Report default standard errors and panel-robust standard errors.
(g) Under what conditions are conditions hold? βˆ1,F E consistent estimators for β1? Do you think those
(h) Estimate βˆ using Feasible Generalized Least Squares (random effects model).
(i) Under what conditions are conditions hold? βˆ1,RE consistent estimators for β1? Do you think those
(j) Are the estimated coefficients on lnwg, βˆ1, similar across models?
(k) Is there a systematic difference between the default standard errors (assuming ho- moskedasticity and no serial correlation) and the panel-robust standard errors?
(l) Perform. the Hausman test of the difference between the fixed and random effects estimate of β. What do you conclude at the 5% significance level.
(m) Given your arguments about which estimators are likely consistent and your estimates of β1 given this sample, what would you surmise about whether the labor supply curve is upward sloping?
2. Let grad be an indicator variable for whether a student-athlete graduates from a large uni- versity within 5 years of starting. Using data from 420 student-athletes, the following logit estimates were obtained:
ˆ P [grad = 1|hsGPA, SAT, Study] = G(−1.17+.24hsGPA+.00058SAT +.073Study); G(z) = 1 + eZ/eZ
(a) Holding hsGPA fixed at 3 and SAT fixed at 1200, compute the estimated difference in the graduation probability for someone who spent 10 hours per week in study hall (Study=10) and someone who spent 5 hours per week (Study=5).
(b) Compute the marginal effect of an extra hour of study for someone who hsGPA=3, SAT=1200, and Study=10.
3. Read in the Charity data set (on the blackboard site). The data set has 4,268 observations on people who have responded to past requests by donating at least once. Where the variables are
y1i = respondi =indicator if person i responded with a donation(gift) in the most recent request
y2i = gifti =amount of i’s gift in response to the most recent request, x1i = resplasti =indicator if person i responded to the most recent mailing, x2i = weekslasti =number of weeks since i’s last response,
x3i = proprespi =i’s response rate to mailings,
x4i = mailsyeari=number of mailings i receives per year,
x5i = giftlasti =amount of i’s most recent gift,
x6i = avggifti =mean amount of i’s past gifts.
You are interested in the effect of the number of mailings per year on the response probability for donating, so that you can asses the cost and benefit of increasing mailings. You (partially) specify the model
P [y1 = 1|X] = G(β0 + β1X1 + β2X2 + β3X3 + β4X4 + β5X5 + β6X6) + c
(a) Report the ordinary least squares estimates of the Linear Probability Model using heteroskedasticty-robust standard errors.
(b) Interpret the coefficient estimate on mailsyear, βˆ4.
(c) Report the Probit estimates of β along with the log likelihood.
(d) Write down the implied G(·) for the Probit model.
(e) Which, if any, of the Probit coefficient estimates are statistically significant?
(f) Calculate the Marginal effect of mailings per year in two ways: averaged over the sample and evaluated at the sample average.
(g) Report the Logit estimates of β along with the log likelihood.
(h) Calculate the Marginal effect of mailings per year in two ways: averaged over the sample and evaluated at the sample average.
(i) Compare the three binary response models on the basis of statistical significance of βˆ4.
(j) Compare the three binary response models on the basis of estimated Marginal Effects.
(k) Compare the logit and probit binary response models on the basis of log-likelihood.