91Ó°ÊÓ

For a sample of 600 married females, we are interested in explaining participation in market employment from exogenous characteristics in \(x_{i}\) (age, family composition, education). Let \(y_{i}=1\) if person \(i\) has a paid job and 0 otherwise. Suppose we estimate a linear regression model $$ y_{i}=x_{i}^{\prime} \beta+\varepsilon_{i} $$ by ordinary least squares. (a) Give two reasons why this is not really an appropriate model. As an alternative, we could model the participation decision by a probit model. (b) Explain the probit model. (c) Give an expression for the loglikelihood function of the probit model. (d) How would you interpret a positive \(\beta\) coefficient for education in the probit model? (e) Suppose you have a person with \(N\). What is your prediction for her labour market status \(y_{i}\) ? Why? (f) To what extent is a logit model different from a probit model? Now assume that we have a sample of women who are not working \(\left(y_{i}=0\right)\), part-time working \(\left(y_{i}=1\right)\) or full-time working \(\left(y_{i}=2\right)\). (g) Is it appropriate, in this case, to specify a linear model as \(y_{i}=x_{i}^{\prime} \beta+\varepsilon_{i} ?\) (h) What alternative model could be used instead that exploits the information contained in part-time versus full-time working? (i) How would you interpret a positive \(\beta\) coefficient for education in this latter model? (j) Would it be appropriate to pool the two outcomes \(y_{i}=1\) and \(y_{i}=2\) and estimate a binary choice model? Why or why not?