Probability Models

Probability Models

Please use the “WOMEN13” data attached to answer the questions below.  Data include the following variables:

D = 1 if a woman is looking for a job, 0 otherwise

M = 1 if a woman is married, 0 otherwise

S = Number of years of schooling for each woman included in the analysis

1) First estimate a linear probability model (LPM) that regresses whether a woman is looking for a job (D) on whether she is married (M) and years of schooling (S).  Are the results as expected?  Interpret the results with emphasis on a) whether coefficients are statistically significant, b) marginal effects, and c) model fit.

2) Next,  estimate a “probit” model using the same data.  Are the results as expected?  Interpret the results with emphasis on a) whether coefficients are statistically significant, b) marginal effects, c) and 3) model fit.

3)  Finally,  estimate a “logit” model using the same data.  Are the results as expected?  Interpret the results with emphasis on a) whether coefficients are statistically significant, b) marginal effects, c) odds ratios, and d) model fit.   

Solution 

Question 1

Please use the “WOMEN13” data attached to answer the questions below.

Data include the following variables:

D = 1 if a woman is looking for a job, 0 otherwise

M = 1 if a woman is married, 0 otherwise

S = Number of years of schooling for each woman included in the analysis

1) First estimate a linear probability model (LPM) that regresses whether a woman is looking for a job (D) on whether she is married (M) and years of schooling (S).  Are the results as expected?  Interpret the results with emphasis on a) whether coefficients are statistically significant, b) marginal effects, and c) model fit.

Stata code:

use WOMEN13.dta, clear

* label variables

labvar D “looking for a job”

labvar M “married”

labvar S “Number of years of schooling”

* Linear Probability Model

reg D M S, robust

Stata output:

Interpretation:

According to the above LPM, the outcome variable “looking for a job” significantly decreases for married women, and increases with the number of years of schooling. If the dataset includes only unemployed women, the results are consistent with expectations. Single women cannot rely on partner’s income;therefore, they usually need a source of income, and are more likely to search it on the labor market. Better educated women are more likely to search for a job when they lack one, and to convert their human capital in material resources.

All effects are significant at p<.05. The model explains 36% of total variation.

The interpretation of the coefficients is difficult given the outcome is dichotomous and no logit model is used, but if treating the outcome as the probability to search for a job, one may say married women are with -0.38 less likely to search for a job as compared to unmarried ones; also, the likelihood to search for a job increases with .09 for each year of schooling. In other words, it takes about 4.5 years of extra-schooling to have the same impact as being unmarried.

2) Next,  estimate a “probit” model using the same data.  Are the results as expected?  Interpret the results with emphasis on a) whether coefficients are statistically significant, b) marginal effects, c) and 3) model fit.

Stata code:

*probit model

probit D M S, nolog

margins, dydx(*)

Stata output:

Interpretation:

“Looking for a job” significantly decreases for married women, and increases with the number of years of schooling. If the dataset includes only unemployed women, the results are consistent with expectations. Single women cannot rely on partner’s income; therefore, they usually need a source of income, and are more likely to search it on the labor market. Better educated women are more likely to search for a job when they lack one, and to convert their human capital in material resources.

All effects are significant at p<.05. The model explains 34% of total variation.

The interpretation of the coefficients is done studying the marginal effects. Unlike LPM, which reports marginal effects as b-coefficients, for probit one needs to compute them in post-estimation. In our case, it turns out that married women have a probability to search for job with -.36 lower as compared to unmarried. One year of schooling increases the probability to search for job with .10.

3)  Finally,  estimate a “logit” model using the same data.  Are the results as expected?  Interpret the results with emphasis on a) whether coefficients are statistically significant, b) marginal effects, c) odds ratios, and d) model fit.

Stata code:

* logit Model

logit D M S, nolog or robust

margins, dydx(*)

Stata output:

Interpretation:

As in the other two models, all coefficients are significant at p<.05 (this time even at p<.01). Married women are less likely to search for job, and education increases likelihood to search for employment. The model explains 34% of the variance in probability to search for job.

The odd-ratios are interpretable. Being married imply the odds to search for a job .08 of being unmarried. Each year of schooling increase the odds of searching for job by a factor of 1.99.

Again, when looking at marginal effects, we get a decrease of -.38 in the probability to search for job if married as compared to unmarried. Every additional school year increases the probability to look for a job, controlling for marital status, with .10.

One may observe that the estimates for marginal effects are very close in all three models.