1. (40 points) Suppose the true population model is which satisfies MLR.1-MLR.5.
ID: 3365560 • Letter: 1
Question
1. (40 points) Suppose the true population model is which satisfies MLR.1-MLR.5. However, based on a random sample of wage and educ, the data analyst estimates the following model instead, In( wage) = 0+ educ + u". (a) Provide an expression of the OLS estimator * of 1 as a function of ,2, and other variables. (b) Calculate EP1] as a function of 1,2, and other variables. * (c) Derive the probability limit of as a function of 1,2, and other variables. Compare this with your answer to (b). (d) Argue that * has an asymptotic nornal distribution, and derive expressions of the mean and variance of this normal distribution.Explanation / Answer
It could be think of wages as a function of education and work experience:
Wage = f (Education,Experience).
The longer one spends on a job, the better one gets. If people are paid for their productivity, then
workers with more work experience should be more productive, and therefore, paid more. That is,
Wage/Experience> 0 ,
other things equal.
The complete relationship between wages, education, and experience can be written as
ln(Wagei) = 0 + 1Educationi + 2Experiencei + ui, ....(1)
where wages are measured in natural logs. This is a multiple regression model of wages. Because there is more than one explanatory variable, each parameter is interpreted as a partial derivative, or the change in the dependent variable for a change in the explanatory variable, holding all other variables constant. For example,
2 = ln(Wage)/( Experience) ln(Wage)/Experience Ieducation .....(2)
is the effect of experience on the log wage, holding education constant. Other ways of saying "holding experience constant" are "controlling for experience" or "accounting for the effect of experience." Because pay is measured in natural logs, 3 also can be interpreted as
2 = %Wage/Experience IEducation ....(3)
or the "return to experience" in the labor market.
If we group all workers according to their education level (less than high school, high school, some college, college graduates, and more than college), we can compare wages and work experience within education categories. This is really what multiple regression does. By looking within categories, you are holding education constant. From the univariate analysis in, we know that wages increase with education level. shows that within any given education category (i.e., reading across rows), hourly wages rise with greater work experience. This suggests 2 is positive, so that wages increase with work experience controlling for education, but also that work experience explains some of the residual variation in wages within education levels.
Likewise,
1= ln(Wage)/Education ln(Wage)/Education IExperience
is the effect of education on the log wage, holding experience constant. 2 also can be expressed as
2 = %Wage/Education IExperience , (5)
or the return to education in the labor market.
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