1. Using a probit model and data from the 2008 March Current Population Survey, I estimated a probit model of the determinants of pension coverage. Three specifications were estimated. The first included all workers, and the second and third estimated the regressions separately for women and men. The results are in table 4 below. Table 4. Probit model of union membership. Probit coefficients are presented below. Data from March 2008 Current Population Survey. All workers Female Workers Male Workers age 0.0111*** 0.0115*** 0.0122*** (10.8) (7.59) (8.52) Education dummies (0-8 years omitted) 9-11 0.130 0.0318 0.209 (1.04) (0.16) (1.31) 12 0.442*** 0.231 0.601*** (4.08) (1.29) (4.38) 13-15 0.488*** 0.292 0.677*** (4.50) (1.64) (4.92) 16+ 0.558*** 0.642*** 0.443*** (5.17) (3.64) (3.22) Race dummies (white omitted) black 0.135*** 0.155** 0.136** (3.02) (2.52) (2.06) other -0.0174-0.0364 0.0100 (-0.33) (-0.47) (0.14) Constant -2.096*** -2.088*** -2.164*** (-18.1) (-11.1) (-14.6) Observations 14468 7256 7212 log-likelihood -5397-2536 -2802 a. Using the specfication for all workers, compute the probability of union membership for a white 30 year old college graduate. Give a brief description of how you derived your answer. b. Using the specification for all workers, compute the marginal effect of an additional year of age on the probability of union membership for the person described in (a). Give a brief description of how you derived your answer. c. Test the null hypothesis that the coefficients for the union membership model are identical for men and women. Provide a test statistic, indicate its distribution, and whether you reject at the null at the.05 level of significance. Give a brief description of how you generated your test statistic. d. Suppose that you are interested in knowing what female union coverage would be if, holding their characteristics constant, union membership was determined the same way that it is for men. Explain how you would use STATA to generate this prediction. (i.e. describe the
regressions you would estimate and the subsequent values that you could generate). e. For the pooled model (entire sample), the variance-covariance matrix for the coefficient estimates is presented below (eddum2-eddum5 are the education dummies with eddum5 being for the most educated group): age eddum2 eddum3 eddum4 eddum5 black other _cons age 1.060e-06 eddum2 9.868e-06.01581586 eddum3 1.530e-06.01115643.01175391 eddum4 2.951e-06.011168.01114927.0117637 eddum5 8.045e-07.01114305.0111396.01114073.0116542 black 1.796e-06 -.00007456 -.00008872 -.00005284.00001212.00199489 other 3.126e-06.00010774.00016031.00016461.00010783.0002186.00282613 _cons -.00004829 -.01158482 -.01121414 -.01128138 -.01118275 -.00025676 -.00049336.01338765 Describe how you would construct a Wald test of the hypothesis that the coefficients on black and other race are jointly equal to zero. Describe the distribution of the test statistic and under what conditions would reject the null hypothesis at the.05 level of significance.
2. Knapp and Seakes (1994) 1 examine the factors that influence whether a student defaults on a college student loan. Below is a summary of 4 probit models that they estimated where the dependent variable is one if the student defaults on the loan and is zero otherwise. Family status is a dummy that equals one if the student is from a two-parent family; graduation is a dummy that equals one if the person finished his/her degree. The other variable names are sufficiently descriptive for your task here. Notice that at the bottom of the table there is a row indicating whether "school dummies" are included. This indicates whether the specification includes a set of dummies indicating which of 26 different schools the student attended (25 dummies are included, one omitted). Also, K is the number of explanatory variables in the specification, and log(l) is the log of the likelihood function. a. Using specification (1), estimate the probability of default on a student loan for a person who has graduated, comes from a one-parent family (family status=0), whose race is not black, and whose parent's income is $50,000 (note: parent's income is measured in $1000s of dollars in this problem --- so $50,000 is recorded as 50 in the data). Show the basic steps involved in your calculation. 1 Laura Greene Knapp and Terry G. Seaks. "An Analysis of the Probabilty of Default on Federally Guaranteed Student Loans." Review of Economics and Statistics, August 1992.
b. For the person described in (a), what is the marginal effect of an extra $1000 of parental income on the probability of default? Show the basic steps involved in your calculation. c. Based on the information provided, construct a test-statistic for the null hypothesis that there is no difference in the probabilty of default across the 26 schools that students in the sample attended. Show how you construct your statistic, describe its distribution, and whether you would reject the null at the.05 level.
3. I estimated a multinomial logit model of employment behavior using data from the 2006 Current Population Survey. The three possible outcomes for a person are employed (outcome=1), unemployed (outcome=2) and out of the labor force (outcome=3). The coefficients for outcomes 2 and 3 are presented below. The coefficients for outcome 1 are normalized to zero. VARIABLES unemployed Out of labor force female 0.0575 0.677 (2.647) (74.59) age -0.129-0.305 (-33.56) (-211.3) Age-squared 0.00122 0.00379 (25.35) (225.6) # of kids aged 0-5 0.00907 0.181 (0.490) (22.78) # of kids aged 6-17 0.0711 0.199 (6.557) (42.17) Constant -0.309 3.711 (-4.467) (132.5) Observations 325458 325458 a. compute the probability that a 40 year old male with no kids is i. employed ii. unemployed b. After estimating the above multinomial logit model, I executed the following stata commands and received the output listed below:
. mfx, predict(p outcome(2)) Marginal effects after mlogit y = Pr(emp==2) (predict, p outcome(2)) =.02700302. mfx, predict(p outcome(2)) variable dy/dx Std. Err. z female* -.0048006.00056-8.50 age -.0005028.0001-4.94 age2-3.61e-06.00000-2.94 #kids<5 -.0014705.00048-3.05 #kids 6-17 -.0000104.00028-0.04 (*) dy/dx is for discrete change of dummy variablefrom 0 to 1 (*) dy/dx is for discrete change of dummy variable from 0 to 1. mfx, predict(p outcome(3)) Marginal effects after mlogit y = Pr(emp==3) (predict, p outcome(3)) =.34939412 variable dy/dx Std. Err. z female*.1515857.00198 76.51 age -.0681668.00034-198.50 age2.0008495.00000 209.74 #kids<5.0410849.0018 22.88 #kids 6-17.0446071.00106 41.91 Use the above results to compute the effect of having an additional child under the age of 5 on the probability that a person is employed. Show how you derived your estimate. d. Suppose you wish to test that children have different effects on employment behavior of men and women. Explain how you could test this hypothesis. Define the variables you would construct, the model(s) you would estimate, how you would construct your test statistic, the distribution of test statistics, and how you would decide whether to reject the null hypothesis.
4. Starting with data from the Health and Retirement Study on workers who were over age 51 in 1992 and working full-time, labor force behavior was recorded every two years between 1992 and 1998. For each person who was working full-time in a given survey, data from the subsequent survey was used to record whether they made a transition from full-time work to (i) full-time work; (ii) part-time work; (iii) retirement. These transitions are recorded respectively as FTFT, FTPT, FTRET. A multinomial logit model was used to estimate how various characteristics affected the probability of each transition. The coefficients for FTFT are normalized to zero. Table 3. Multinomial logit model of labor force transitions. FTPT FTRET coefficient t-statistic coefficient t-statistic covered by only a DB plan -0.47-4.42 0.45 6.00 covered by only a DC plan -0.41-3.63-0.18-2.00 covered by both a DB and DC plan -0.47-1.90 0.35 2.13 Age dummies (under 55 omitted) 55-60 0.36 3.22 0.38 4.34 61-62 1.12 8.21 1.76 17.97 63-65 1.47 8.76 1.93 15.96 65 and over 1.40 2.51 1.86 4.69 Education dummies (<12 yea omitted) 12 years -0.02-0.18-0.25-2.92 13-15 years 0.05 0.35-0.29-2.84 16 years or more 0.12 0.84-0.33-3.15 Male -0.38-4.18-0.17-2.55 Intercept -2.52-17.87-2.22-20.32 a. Compute the probability that a female with 12 years of education who is age 58 and has no pension plan coverage will: i.. continue in full-time employment ii. switch to part-time work iii. retire. Give a brief explanation of how you calculated your answers.. b. Compute the effect of switching to a DB plan on the probability of retiring for the person in (a). c. Federal regulations prohibit (or penalize) workers from withdrawing funds from their pension plan prior to retirement. Some policy makers have pointed out that this may make workers more likely to switch directly from full-time work to retirement and reduce the chance that they "phase" into retirement with a spell of parttime work intervening. i. Based on the information provided above, how does coverage by only a DB plan affect the chance that the worker in (a) makes each of the three possible transitions? Provide a numerical answer and provide a brief description of how you derived your answer. ii. Does your answer to (i) suggest that DB coverage does or does not "impede" phased retirement? Explain. d. Because workers can access assets in defined contribution plans after age 59, DC plans may have a different effect before and after age 59. How can you test this hypothesis? Explain the models you would estimate, how you would construct the appropriate test statistic, the distribution of the test statistic (including degrees of freedom) and the conditions under which you would reject the null.