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Suppose a researcher tried to examine the effect of household income on car ownership.

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Question 1

Suppose a researcher tried to examine the effect of household income on car ownership. Car ownership is a binary variable for which y = 1 if a household owns a car or more, and =0 otherwise. A logit model was estimated and the regression results are presented as follows:

Independent variables Coefficient estimate Standard error

Constant 0.48 0.066

Income 0.20 0.089

Children 0.01 0.004

No. of observations 3200

LR Chi-square (df = 2) 23.16

 

Where Income is the household annual income in thousand dollars, and Children is the number of children in the household. The data were obtained from a household survey which covered 3,200 households. 

(a) Write down the regression model if you are going to estimate the same relationship between the car ownership and the independent variables using linear probability model. 

(b) By applying the estimates obtained from the logit model, write down in the expression for the predicted probability of car ownership. 

(c) Suppose the average household income is $2,500 and the average number of children per household is 3. Calculate the partial effect of Children at the average when the number of children increases from 2 to 3. 

(d) Interpret the likelihood ratio statistic (i.e., the LR Chi-square).

 

Question 2

Use the dataset nls77a.dta for this question. The dataset is a national longitudinal survey which contains the wage and demographic information between 1977 and 1988 of women aged 23-35 in 1977 (the variable of the id of each woman is “idcode” and the year of each wave of survey is “year”). 

A model for the wage rates of women with individual fixed effects is as follows: 

 

where ln_w is the natural log of wage of individual i in year t, age is the age in years, union is a dummy variable which equals 1 if the individual joined the union and equals 0 otherwise, ttl_exp is the total years of work experience, tenure is the years of tenure, ai is the individual fixed effects, and uit is the idiosyncratic error. 

(a) Estimate equation (1) using fixed effects (FE) and report the results. 

(b) How many individuals are used in the FE estimation? How many waves were included in the panel data? How many total observations would be used if each individual had data on all variables for all the waves? 

(c) Would you choose dummy variable regression method to estimate equation (1)? Why? 

(d) Interpret the coefficient on union and comment on its significance. 

(e) Estimate equation (1) using random effects (RE) and report the results. 

(f) Conduct the Hausman test to compare the FE model with the RE model and report the test results. Based on the test results, would you recommend to use fixed-effects or random-effects estimation? Why?

 

Question 3

The data set 401KSUBS.DTA contains a binary variable for eligibility in a 401(k) plan (e401k), and several other variables. 401(k) plan is a pension plan available in U.S. and we want to examine the probability of the individual to be eligible to the 401(k) pension plan. Consider the following model:

 

(a) What fraction of individuals in the sample are eligible for participation in a 401(k) plan? 

(b) Should you use the heteroskedasticity-robust standard errors to estimate the LPM? Why? Estimate the linear probability model (LPM) of equation (1) and report the estimated coefficients with (appropriate) standard errors and other important statistics. 

(c) Using the results in part (b), interpret the estimate of 𝛽4 and comment on its significance. 

(d) Estimate equation (1) by probit and logit also. Report the estimated coefficients with standard errors and other important statistics from probit and logit. 

(e) Why are the coefficient estimates from LPM, probit and logit different from each other? 

(f) Calculate the average partial effects of family size (f size) for f size = 1,2,3,…,8 using the results from the logit model. Interpret the average partial effect of family size at f size =4. Why the average partial effect of family size is not a constant?

 

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