Research Method

1
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 womens hours worked y.
• In tobit model, we start with a latent variable
  y, which is only partially observed by the
  researcher
• yß0ß1xu and uN(0,s2)
• 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
• yiß0ß1xiui ..(1)
• such that
• yiyi if yigt0
• yi0 if yi0
• and
• uiN(0,s2)
• 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), yiyi, thus
• ui yi- ß0ß1xi
• 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., for
  women who are not working), we only know that
  y0. Thus, the likelihood contribution is the
  probability that y0, which is given by

11

• To summarize,
• Let Di be a dummy variable that takes 1 if yigt0.
  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 ß0,ß1 and s 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 ßj 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(yx)P(ygt0)E(yygt0,x)
• P(y0)E(0y0,x)
• P(ygt0)E(yygt0,x) ..(1)
• Now, let me compute E(yygt0,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-(ß0ß1x). (Note that the
  expectation is also conditioned on x, so you can
  treat x as a constant.). Thus, we have

18
This term is called the inverse Mills ratio, and
denoted by ?(.)
Now, let me compute P(ygt0x)
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. 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).
  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