1. Lecture 15 Tobit model for corner solution
Research Method
Lecture 15
Tobit model for corner solution

2. The corner solution responses
Corner solution responses example
1 Amount of charitable donation: Many people do not donate. Thus a significant fraction of the data has zero value.
2. Hours worked by married women: Many married women do not work. Thus, a significant fraction of the data has zero hours worked.

3. Tobit model is used to model such situations.

4. The model
For the explanatory purpose, I use one explanatory variable model. This can be extended, off course, to multiple variable cases.

5. Consider to estimate the effect of education x on the married women’s hours worked y.
In tobit model, we start with a latent variable y*, which is only partially observed by the researcher:
y*=β0+β1x+u and u~N(0,σ2)
If y* is positive, then y* is equal to the actual hours worked: y. But if y* is negative, then the actual hours worked, y, is equal to zero. We also assume u is normally distributed.

6. The model can be conveniently written as: yi*=β0+β1xi+ui …………………..(1)
such that
yi=yi* if yi*>0
yi=0 if yi*≤0
and
ui~N(0,σ2)
We introduced i-subscript to denote ith observation. We assume that equation (1) satisfies the Classic Linear Model assumptions.
The actual hours worked.

7. Graphical illustration
y*, y
When y* is negative, actual hours worked is zero.
Educ

8. The variable, y*, can be negative, but if it is negative, then the actual hours worked is equal to zero.
In this way, the Tobit model deals with the fact that, many women do not work, thus,the hours worked is zero many women .

9. The estimation procedure
The estimation procedure is by maximum likelihood estimation.
If the hours worked is positive (i.e., for the women who are working), yi*=yi, thus
ui= yi- β0+β1xi
Thus, the likelihood function for a working woman is given by the hight of the density function:

10. If the actual hours worked is zero (i. e
If the actual hours worked is zero (i.e., for women who are not working), we only know that y*≤0. Thus, the likelihood contribution is the probability that y*≤0, which is given by:

11. To summarize,
Let Di be a dummy variable that takes 1 if yi>0. Then, the above likelihood contribution can be written as.

12. The likelihood function, L, is obtained by multiplying all the likelihood contributions of all the observations.
The values of β0,β1 and σ that maximize the likelihood function are the Tobit estimators of the parameters.
In actual computation, you maximize Log(L).

13. Exercise
Using Mroz.dta, estimate the hours worked equation for married women using Tobit model. Included in the model, nwifeinc, educ, exper, expersq, age kidslt6, kidsge6.

14. Answer

15. The partial effects (marginal effects)
As can be seen, estimated parameters βj measures the effect of xj on y*.
But in corner solution, we are interested in the effect of xj on actual hours worked y.
In the next few slides, I will show how to compute the effect of an increase in explanatory variable on the expected value of y.

16. Note that the expectation of y given x is given by
E(y|x)=P(y>0)E(y|y>0,x)
+P(y=0)E(0|y=0,x)
=P(y>0)E(y|y>0,x) …………..(1)
Now, let me compute E(y|y>0,x).
zero

17. Now, we use the fact that if v is a standard normal variable and c is a constant, then
In our case, c=-(β0+β1x). (Note that the expectation is also conditioned on x, so you can treat x as a constant.). Thus, we have

18. Now, let me compute P(y>0|x)
This term is called the inverse Mill’s ratio, and denoted by λ(.)
Now, let me compute P(y>0|x)

19. By plugging (2) and (3) into (1), we have

20. From the above computation, you can see that there can be two ways to compute the partial effect.
The effect of x on hours worked for those who are working.
The overall effect of x on hours worked.

21. As can be seen, both partial effects depends on x
As can be seen, both partial effects depends on x. Therefore, they are different among different observations in the data.
However, we need to know the overall effect rather than the effect for specific person in the data.
As was the case in the Probit models or logit models, there are two ways to compute the ‘overall partial effect’.

22. The first is the Partial Effect at Average (PEA)
The first is the Partial Effect at Average (PEA). You simply plug the average value of x in the partial effect formula. This is automatically computed by STATA.
The second is the Average Partial Effect. You compute the partial effect for each individual in the data, then take the average.

23. Exercise
Using Mroz.dta, estimate the hours worked equation for married women using Tobit model. Included in the model, nwifeinc, educ, exper, expersq, age kidslt6, kidsge6.
1. Compute the effect of education on hours worked for those who are currently working:
2. Compute the effect of education on hours worked for the entire observations:

24. Partial effect at average for working women: Computing manually.

25. Partial effect at average for working women: Compute automatically.

26. Partial effect at average for all the obs: Compute manually.
Partial effect at average for all the obs: Compute automatically