Econometric Take Home Exam

Econometric Final          

Chang Shen N15973701

Question 1

(a)

[OLS]

    The coefficient for educ in ‘Printout for OLS’ is -0.1220755. The coefficient for educ in ‘Printout for Probit’ is -0.0281153.

    The first number comes from a Linear Probability Model. The interpretation of the slope coefficient of educ, is that if the years of education of the head of the household increases by 1, the probability that there is someone in this household made at least one visit to a doctor in the last month will decrease by 12.2% percent.

[Probit]

The later one comes from a Probit Model. First we construct an index, named Z.        

The probability of the y conditioned on all regressors is a function of the regressors only through the index z.

    Pr (y = 1 | x) = F(Z) and Z =   Here y is docvis and x denotes all regressors.[pic 1]

So in this case the coefficient of educ is actually the coefficient with respect to the index, not y. It is the coefficient from Z= . The interpretation of it, is that if the years of education of the head of the household increases by 1, Z the index, which is statistic value in standard normal probability function, will decrease by -0.0281153.    [pic 2]

If we want to know how much the probability will change with respect to educ, we need to know the marginal effect.

(b)

    If we denote the standard normal density as . The marginal effect of a probit model is calculated by:   where Z=[pic 3][pic 4][pic 5]

    To estimate the marginal effect you need to select a value for x in order to estimate a value for Z.

    So the difference in the ‘Printout for Margins(1)’ and ‘Printout for Margins(2)’ lies in how to “average” the marginal effect.

    Margins(1) estimates Z for every observation in sample and its marginal effect. Then take the average of all the marginal effects for each observation.

    Margins(2) calculate the mean of each variable, and then put these mean values into the function to calculate the margins at mean.

    And as we can see from the printout, the average marginal effect for all educ in the sample is -0.104003, while the marginal effect at the mean value of educ is -0.0105967. Theses two numbers are different but very close.

(c)

    I put 1.hhkids=0,hhninc=40,age=32,educ=15,1.married=0 into the probit model, times their coefficients respectively and I can calculated that Z = 0.1435162. Next I calculated the cumulative probability of standard normal function F(Z) and get the probability to be 0.55705874.