Transcript Economics 310

Economics 310
Lecture 22
Limited Dependent Variables
Examples of limited dependent
variables
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Decision
not.
Decision
Decision
Decision
not.
Decision
or not.
to go to graduate school or
to get married or not.
to have a child or not.
to vote for a proposition or
to send child to private school
Modeling Decision
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This yes or no type decision leads to a
dummy variable.
The dependent variable of our model is
a dummy variable.
We will be modeling the probability
function, P(Y=1).
The statistical Model
T heobservablechoice variableis a discret erandom
variable. Choicedepends on bot h observableand
unobservable characterist ics of theindividual and
thealternatives available to theindividual.
T heprobability distribut ion of Bernoullirandom variableis
g(yi )  Pi yi (1  Pi )1 yi yi  0,1
g (1)  P ( y i  1)  Pi
g (0)  P ( y i  0)  1  Pi
E ( yi )  Pi
Var ( yi )  Pi (1  Pi )
For our case, we will makePi a funct ionof individual
characterist ics and charactistics of thechoice.
Simplest Model
Linear Probability Model
For our basic regression model
y i  E [ y i ]  ei
E [ y i ]  Pi  1   2 X i 2     k X ik
Model suffers from het eroscedast icit y.
var(ei )  Pi (1  Pi )
which variesfrom observat ion t o observat ion
and is a funct ionof t heexplanat ory variables.
Picture of LPM
yˆ
1
0
X
X0
X1
Problems of LPM
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Predictions outside 0-1 range.
Heteroscedasticity
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This can be solved and a estimated GLS
estimator developed.
Coefficient Determination has little
meaning.
Constant marginal effect.
Probit Statistical Model
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The probit model is a nonlinear (in the
probability) statistical model that achieves the
objective of relating the choice probability Pi
to explanatory factors in such a way that the
probability remains in the (0,1] interval.
Model can be developed from several
theories.
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Threshold theory
Utility theory
Probit Model
Assume we havean index (ut ilit yindex) of t heform:
I t  1   2 X i 2     k X ik
Let   ( 1
 2   k )' and x i'  (1 X i 2  X ik )
Then I t  x i 
'
Assume peoplehavea random t hreshold, such t hatan eventoccursif t heindex of
of personaland eventcharact erist ics is great er t han t het hreshold. Weassume t he
t hresholdpossess a st andardnormaldist ribut ion.
It
P (yi  1)  P ( zi  I t )  F (I t )   (2 ) 1/ 2 e


z2
2
dz
Interpreting the Probit Model
1
F(I)
0
0
I
Interpreting the Probit Model
Pi

xij
F ( x i  )
'
xij
xi 
'
 F ' ( xi  )
xij
 f ( xi  )  j
'
Maxim umchangeoccurs when I i  x i   0
Estimating Probit Parameters
We have g(yi )  Pi yi (1  Pi )1 yi yi  0,1
For n independent observations, we have
g( y1
n
n
i 1
i 1
y 2  y n )   g(yi )   Pi yi (1  Pi )1 yi
n
  F ( x i  ) yi [1  F ( x i  )]1 yi
'
'
i 1
We call this thelikelihoodfunctionand can writeit as
n
l (  )   F ( x i  ) yi [1  F ( x i  )]1 yi
'
'
i 1
we maximizeit by solving thefollowingk equations
l (  )
 ln[l (  )]
 0 j  1,  , k or
0
 j
 j
Estimating Probit Model using
LIMDEP
read; nobs=13081; nvar=5;names=1;file=wlottq07205.asc
$CREATE; COMPUTER=HESCU1A=1
$CREATE; AGE=PRTAGE
$CREATE; AGESQ=AGE*AGE
$CREATE; NONWHITE=PERACE>1
$CREATE; FEMALE=PESEX=2 $CREATE; EARNING=PTERNWA
$PROBIT; LHS=COMPUTER; RHS=ONE,AGE,AGESQ,NONWHITE,FEMALE,EARNING
$STOP $
Results of probit estimation
Computer ownership model
Variable
Coefficient Standard Error b/St.Er. P¢¦Z¦>z|
Mean of X
--------------------------------------------------------------------Index function for probability
Constant
-.1504969
.91575E-01
-1.643
.10029
AGE
.8665748E-03
.49262E-02
.176
.86036
38.79
AGESQ
-.1163434E-03
.59141E-04
-1.967
.04916
1669.
NONWHITE
-.4021405
.31576E-01 -12.736
.00000
.1499
FEMALE
.1392186
.23382E-01
5.954
.00000
.4955
EARNING
.7477787E-03
.31510E-04
23.732
.00000
573.2