Tobit Model

Tobit analysis from Mcdonald and Moffitt (1980)

1. Motivation
TOBIT MODEL
ojapar Tobit Model

2. Motivation
Motivation
Figure 1: X and Y Data
1
1
Source:https://www.bauer.uh.eduojapar Tobit Model

3. Motivation
Motivation
Figure 2: Censored Data
2
2
Source:https://www.bauer.uh.eduojapar Tobit Model

4. Motivation
Motivation
Examples
hours of work often have the same clustering
dividends paid by a company remain zero until earnings
reach some threshold value
demand for consumption goods often have values
clustered at zero
ojapar Tobit Model

5. Motivation
Introduction
Tobit model
model devised by Tobin (1958)
assumes that the dependent variable has a number of its
values clustered around a limiting value
uses all observations to estimate a regression line
coecients are called beta coecients
ojapar Tobit Model

6. Motivation
Introduction
Tobit Model can determine both
1
change in the probability of being above the limit
2
change in the value of the dependent variable if it is
already above the limit
ojapar Tobit Model

7. Motivation
Objectives
1
To quantify the decomposition of the total change in Y
2
To apply the decomposition to some journal articles that
used Tobit analysis
3
To emphasize the importance of Tobit model
ojapar Tobit Model

8. Motivation
The Model
Stochastic model underlying Tobit analysis
Yt=

Xt+tif Xt+t>0
0 if Xt+t0
for t= 1;2;3; :::;N(1)
where
N = number of observations
Ytis the dependent variable
Xtis a vector of independent variables
is a vector of unknown coecients
tare error terms that are NID (normal(0,
2
)
and independently distributed)
ojapar Tobit Model

9. Motivation
The Model
Assumption:
There is an underlying stochastic indexY

t=Xt+twhich
is observed only when it is positive. ThusY

tqualies as a
latent variable.
ojapar Tobit Model

10. Motivation
The Model
Expected value of Y (Tobin(1958)):
E[Y] =XF(z) +f(z)
where
z =
X

F(z) = cumulative distribution function (CDF) (Normal)
f(z) = probability density function (pdf) (Normal(0,1))
ojapar Tobit Model

11. Motivation
The Model
Expected value conditional upon being above the
limit
3
:
E[Y

] =E(yjy>0)
=E(yj >X)
=X+
f(z)
F(z)
(3)
3
Tobin(1958); Amemiya(1973)ojapar Tobit Model

12. Motivation
The Model
Relationship of:
E[Y] = expected value of all observations
E[Y

] = expected value conditional upon being above the
limit
F(z) = the probability of being above the limit
E[Y] =F(z)E[Y

] (4)
ojapar Tobit Model

13. Motivation
The Model
The decomposition considers the eect of a change
in thei
th
variable of X on Y.
E[Y]
Xi
=
F(z)E[Y

]
Xi
+
E[Y

]F(z)
Xi
(5)
Total change in E[Y] =change in E[Y

]weighted by F(z) +
change in F(z)weighted by E[Y

]
ojapar Tobit Model

14. Motivation
The Model
Computation of the quantity of the decomposition
4
:
andare estimated via Maximum Likelihood Estimation (MLE) using large
sample size
Xcan be evaluated at X =X
F(z) can be obtained from statistical tables
E[Y

] can be calculated from equation (3)
The two partial derivatives are
F(z)
Xi
=i

f(z)


(6)
E[Y

]
Xi
=i

1
zf(z)
F(z)

f(z)
2
F(z)
2

(7)
using
F'(z) = f(z)
f'(z) = -zf(z)for a unit normal density.
4
McDonald and Mott (1980)ojapar Tobit Model

15. Motivation
The Model
The eect of a change inXionY

isiif X is
innite.
E[Y

]
Xi
=i

1
zf(z)
F(z)

f(z)
2
F(z)
2

At innite X, F(z) = 1 and f(z) = 0.
ojapar Tobit Model

16. Motivation
The Model
Total Eect:
Substituting equation (6) and (7) to equation (5),
we will have
E[Y]
Xi
=F(z)i
ojapar Tobit Model

17. Motivation
The Model
The fraction of total eect due to the eect above
the limit:

1
zf(z)
F(z)

f(z)
2
F(z)
2

This is the information emphasized in the decomposition.
This is the fraction by which theBicoecients must be
adjusted to obtain correct regression eects for observations
above the limit.
ojapar Tobit Model

18. Motivation
Example from Literature
Source: McDonald and Mott (1980)
ojapar Tobit Model

19. Motivation
Summary
For empirical application of the model:
andmust be estimated by MLE
large sample will be used for estimation of parameters of
the model
beta coecients must be multiplied by
[1zf(z)=F(z)f(z)
2
=F(z)
2
] to obtain the correct
regression eects for observations above the limit
ojapar Tobit Model

20. Motivation
Tobit model extensions
limit value - may use an upper limit, a nonzero limit,
dierent limits for each observation, unobserved
stochastic limit, two limits (upper and lower)
simultaneous equation models
the decomposition applied to market disequilibrium
ojapar Tobit Model

21. Motivation
References
Mcdonald, J. F., and Mott, R. A. (1980)
The Uses of Tobit Analysis
Rev. Econ. Stat.62, 318-321.
ojapar Tobit Model

22. Motivation
Thank You!
ojapar Tobit Model