Empirical Methods for
Microeconomic Applications
University of Lugano, Switzerland
May 27-31, 2013
William Greene
Department of Economics
Stern School of Business
1C. Extensions of Binary Choice Models
Agenda for 1C
•
•
•
•
•
•
Endogenous RHS Variables
Sample Selection
Dynamic Binary Choice Model
Bivariate Binary Choice
Simultaneous Equations
Ordered Choices
•
•
Ordered Choice Model
Application to BHPS
Endogeneity
Endogenous RHS Variable
•
U* = β’x + θh + ε
y = 1[U* > 0]
E[ε|h] ≠ 0 (h is endogenous)
•
•
•
Case 1: h is continuous
Case 2: h is binary, e.g., a treatment effect
Approaches
•
•
Parametric: Maximum Likelihood
Semiparametric (not developed here):


GMM
Various approaches for case 2
Endogenous Continuous Variable
U* = β’x + θh + ε
= ρ.
y = 1[U* > 0]
 Correlation
This is the source of the endogeneity
h = α’z
+u
E[ε|h] ≠ 0  Cov[u, ε] ≠ 0
Additional Assumptions:
(u,ε) ~ N[(0,0),(σu2, ρσu, 1)]
z
= a valid set of exogenous
variables, uncorrelated with (u,ε)
Endogenous Income in Health
Income responds to
Age, Age2, Educ, Married, Kids, Gender
0 = Not Healthy
1 = Healthy
Healthy = 0 or 1
Age, Married, Kids, Gender, Income
Determinants of Income (observed and
unobserved) also determine health
satisfaction.
Estimation by ML (Control Function)
Probit fit of y to x and h will not consistently estimate (,)
because of the correlation between h and  induced by the
correlation of u and . Using the bivariate normality,
 x  h  ( /  )u 
u

Prob( y  1| x, h)   
2


1 
Insert
ui = (hi - αz )/u and include f(h|z ) to form logL
logL=




 hi - αz i


x


h






i
i
u


(2 y  1) 
log


 i

2
1


N 


i=1 






log 1   hi - αz i  
 u 
u
 

  
  
   
 
  
  




Two Approaches to ML
(1) Full information ML. Maximize the full log likelihood
with respect to (,, u , , )
(The built in Stata routine IVPROBIT does this. It is not
an instrumental variable estimator; it is a FIML estimator.)
(2) Two step limited information ML. (Control Function)
(a) Use OLS to estimate  and u with a and s.
(b) Compute vˆi = uˆi /s = (hi  az i ) / s
 x  h  vˆ 
i
i
ˆ
ˆ  x  h  vˆ 
  log 
(c) log   i
i
i
i
2


1 

The second step is to fit a probit model for y to (x,h,vˆ) then
solve back for (,,) from (,,) and from the previously
estimated a and s. Use the delta method to compute standard errors.
FIML Estimates
---------------------------------------------------------------------Probit with Endogenous RHS Variable
Dependent variable
HEALTHY
Log likelihood function
-6464.60772
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------|Coefficients in Probit Equation for HEALTHY
Constant|
1.21760***
.06359
19.149
.0000
AGE|
-.02426***
.00081
-29.864
.0000
43.5257
MARRIED|
-.02599
.02329
-1.116
.2644
.75862
HHKIDS|
.06932***
.01890
3.668
.0002
.40273
FEMALE|
-.14180***
.01583
-8.959
.0000
.47877
INCOME|
.53778***
.14473
3.716
.0002
.35208
|Coefficients in Linear Regression for INCOME
Constant|
-.36099***
.01704
-21.180
.0000
AGE|
.02159***
.00083
26.062
.0000
43.5257
AGESQ|
-.00025***
.944134D-05
-26.569
.0000
2022.86
EDUC|
.02064***
.00039
52.729
.0000
11.3206
MARRIED|
.07783***
.00259
30.080
.0000
.75862
HHKIDS|
-.03564***
.00232
-15.332
.0000
.40273
FEMALE|
.00413**
.00203
2.033
.0420
.47877
|Standard Deviation of Regression Disturbances
Sigma(w)|
.16445***
.00026
644.874
.0000
|Correlation Between Probit and Regression Disturbances
Rho(e,w)|
-.02630
.02499
-1.052
.2926
--------+-------------------------------------------------------------
Partial Effects: Scaled Coefficients
Conditional Mean
E[ y | x, h]   (x  h)
h  z  u  z  u v where v ~ N[0,1]
E[y|x,z,v] =[x  (z  u v)]
Partial Effects. Assume z = x (just for convenience)
E[y|x,z,v]
 [x  (z  u v)](  )
x

E[y|x,z ]
 E[y|x,z,v] 
 Ev 
 (  )
[x  (z  u v)](v)dv


x
x


The integral does not have a closed form, but it can easily be simulated :
R
E[y|x,z ]
1
Est.
 (  )
[x  (z  u vr )]
x
R r 1
For variables only in x, omit  k . For variables only in z, omit k .


Endogenous Binary Variable
U* = β’x + θh + ε
Correlation = ρ.
 This is the source of the endogeneity
y
= 1[U* > 0]
h* = α’z
+u
h
= 1[h* > 0]
E[ε|h*] ≠ 0  Cov[u, ε] ≠ 0
Additional Assumptions:
(u,ε) ~ N[(0,0),(σu2, ρσu, 1)]
z
= a valid set of exogenous
variables, uncorrelated with (u,ε)
Endogenous Binary Variable
P(Y = y,H = h) = P(Y = y|H =h) x P(H=h)
This is a simple bivariate probit model.
Not a simultaneous equations model - the estimator
is FIML, not any kind of least squares.
Doctor = F(age,age2,income,female,Public)
Public = F(age,educ,income,married,kids,female)
FIML Estimates
---------------------------------------------------------------------FIML Estimates of Bivariate Probit Model
Dependent variable
DOCPUB
Log likelihood function
-25671.43905
Estimation based on N = 27326, K = 14
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------|Index
equation for DOCTOR
Constant|
.59049***
.14473
4.080
.0000
AGE|
-.05740***
.00601
-9.559
.0000
43.5257
AGESQ|
.00082***
.681660D-04
12.100
.0000
2022.86
INCOME|
.08883*
.05094
1.744
.0812
.35208
FEMALE|
.34583***
.01629
21.225
.0000
.47877
PUBLIC|
.43533***
.07357
5.917
.0000
.88571
|Index
equation for PUBLIC
Constant|
3.55054***
.07446
47.681
.0000
AGE|
.00067
.00115
.581
.5612
43.5257
EDUC|
-.16839***
.00416
-40.499
.0000
11.3206
INCOME|
-.98656***
.05171
-19.077
.0000
.35208
MARRIED|
-.00985
.02922
-.337
.7361
.75862
HHKIDS|
-.08095***
.02510
-3.225
.0013
.40273
FEMALE|
.12139***
.02231
5.442
.0000
.47877
|Disturbance correlation
RHO(1,2)|
-.17280***
.04074
-4.241
.0000
--------+-------------------------------------------------------------
Partial Effects
Conditional Mean
E[ y | x, h]   (x  h)
E[ y | x, z ]  Eh E[ y | x, h]
 Prob(h  0 | z )E[ y | x, h  0]  Prob( h  1| z )E[ y | x, h  1]
  (z ) (x)   (z ) (x  )
Partial Effects
Direct Effects
E[ y | x, z ]
   (z )(x)   (z )(x  )  
x
Indirect Effects
E[ y | x, z ]
z
  (z ) (x)  (z ) (x  )  
 (z )   (x  )   (x)  
Identification Issues
•
•
•
•
Exclusions are not needed for estimation
Identification is, in principle, by “functional form”
Researchers usually have a variable in the
treatment equation that is not in the main probit
equation “to improve identification”
A fully simultaneous model
•
•
•
y1 = f(x1,y2), y2 = f(x2,y1)
Not identified even with exclusion restrictions
(Model is “incoherent”)
Selection
A Sample Selection Model
U* = β’x
+ ε
Correlation = ρ.
y
= 1[U* > 0]
This is the source of the “selectivity:
h* = α’z
+u
h
= 1[h* > 0]
E[ε|h] ≠ 0  Cov[u, ε] ≠ 0
(y,x) are observed only when h = 1
Additional Assumptions:
(u,ε) ~ N[(0,0),(σu2, ρσu, 1)]
z
= a valid set of exogenous
variables, uncorrelated with (u,ε)
Application: Doctor,Public
3 Groups of observations: (Public=0), (Doctor=0|Public=1), (Doctor=1|Public=1)
Sample Selection
Doctor = F(age,age2,income,female,Public=1)
Public = F(age,educ,income,married,kids,female)
Sample Selection Model: Estimation
f(y1,y 2 ) = Prob[y1 = 1| y 2 =1] * Prob[y 2 =1] (y1 =1,y 2 =1)
= Prob[y1 = 0 | y 2 =1] * Prob[y 2 =1] (y1 = 0,y 2 =1)
= Prob[y 2 = 0]
(y 2 = 0)
Terms in the log likelihood :
(y1 =1,y 2 =1) Φ2 (β1 x i1,β2 x i2 ,ρ) (Bivariate normal)
(y1 = 0,y 2 =1) Φ2 (-β1 x i1,β2 x i2 ,-ρ) (Bivariate normal)
(y 2 = 0)
Φ(-β2 x i2 )
(Univariate normal)
Estimation is by full information maximum likelihood.
There is no "lambda" variable.
ML Estimates
---------------------------------------------------------------------FIML Estimates of Bivariate Probit Model
Dependent variable
DOCPUB
Log likelihood function
-23581.80697
Estimation based on N = 27326, K = 13
Selection model based on PUBLIC
Means for vars. 1- 5 are after selection.
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------|Index
equation for DOCTOR
Constant|
1.09027***
.13112
8.315
.0000
AGE|
-.06030***
.00633
-9.532
.0000
43.6996
AGESQ|
.00086***
.718153D-04
11.967
.0000
2041.87
INCOME|
.07820
.05779
1.353
.1760
.33976
FEMALE|
.34357***
.01756
19.561
.0000
.49329
|Index
equation for PUBLIC
Constant|
3.54736***
.07456
47.580
.0000
AGE|
.00080
.00116
.690
.4899
43.5257
EDUC|
-.16832***
.00416
-40.490
.0000
11.3206
INCOME|
-.98747***
.05162
-19.128
.0000
.35208
MARRIED|
-.01508
.02934
-.514
.6072
.75862
HHKIDS|
-.07777***
.02514
-3.093
.0020
.40273
FEMALE|
.12154***
.02231
5.447
.0000
.47877
|Disturbance correlation
RHO(1,2)|
-.19303***
.06763
-2.854
.0043
--------+-------------------------------------------------------------
Estimation Issues
•
This is a sample selection model applied to a
nonlinear model
•
•
•
•
There is no lambda
Estimated by FIML, not two step least squares
Estimator is a type of BIVARIATE PROBIT MODEL
The model is identified without exclusions
(again)
A
Dynamic
Model
Dynamic Models
y it  1[x it  y i,t 1  it  ui > 0]
Two similar 'effects'
Unobserved heterogeneity
State dependence = state 'persistence'
Pr(y it  1 | y i,t 1 ,..., y i0 , x it ,u]  F[ x it   y i,t 1  ui ]
How to estimate ,  , marginal effects, F(.), etc?
(1) Deal with the latent common effect
(2) Handle the lagged effects:
This encounters the initial conditions problem.
Dynamic Probit Model: A Standard Approach
(1) Conditioned on all effects, joint probability
P(y i1 , y i2 ,..., y iT | y i0 , x i ,ui )   t 1 F( x it β  y i,t 1  ui , y it )
T
(2) Unconditional density; integrate out the common effect
P(y i1 , y i2 ,..., y iT | y i0 , x i )




P(y i1 , y i2 ,..., y iT | y i0 , x i ,ui )h(ui | y i0 , x i )dui
(3) Density for heterogeneity
h(ui | y i0 , x i )  N[  y i0  x iδ, u2 ], x i = [x i1 ,x i2 ,...,x iT ], so
ui =   yi0  x iδ + wi (contains every period of x it )
(4) Reduced form
P(y i1 , y i2 ,..., y iT | y i0 , x i ) 

T

t 1
 
F(  x it β  y i,t 1  y i0  x iδ  u w i , y it )h(w i )dw i
This is a random effects model
Simplified Dynamic Model
Projecting ui on all observations expands the model enormously.
(3) Projection of heterogeneity only on group means
h(ui | y i0 , x i )  N[  y i0  x iδ, u2 ] so
ui =   y i0  x iδ + w i
(4) Reduced form
P(y i1 , y i2 ,..., y iT | y i0 , x i ) 

T

t 1
 
F(  x it β  y i,t 1  y i0  x iδ  u w i , y it )h(w i )dw i
Mundlak style correction with the initial value in the equation.
This is (again) a random effects model
A Dynamic Model for Public Insurance
Dynamic Common Effects Model
Bivariate
Model
Gross Relation Between Two Binary Variables
Cross Tabulation Suggests Presence or
Absence of a Bivariate Relationship
+-----------------------------------------------------------------+
|Cross Tabulation
|
|Row variable is DOCTOR
(Out of range 0-49:
0)
|
|Number of Rows = 2
(DOCTOR
= 0 to 1)
|
|Col variable is HOSPITAL (Out of range 0-49:
0)
|
|Number of Cols = 2
(HOSPITAL = 0 to 1)
|
+-----------------------------------------------------------------+
|
HOSPITAL
|
+--------+--------------+------+
|
| DOCTOR|
0
1| Total|
|
+--------+--------------+------+
|
|
0|
9715
420| 10135|
|
|
1| 15216
1975| 17191|
|
+--------+--------------+------+
|
|
Total| 24931
2395| 27326|
|
+-----------------------------------------------------------------+
Tetrachoric Correlation
A correlation measure for two binary variables
Can be defined implicitly
y1 * = μ1 + ε1, y1 = 1(y1* > 0)
y 2 * = μ2 + ε 2 ,y 2 = 1(y 2 * > 0)
 0   1 ρ  
 ε1 
  ~ N   , 

ε
0
ρ
1

 2
  
ρ is the tetrachoric correlation between y1 and y 2
Log Likelihood Function
for Tetrachoric Correlation
logL =  i=1logΦ2 (2yi1 -1)μ1,(2y i2 -1)μ2 ,(2y i1 -1)(2y i2 -1)ρ 
n
=  i=1logΦ2  qi1μ1,qi2μ2 ,qi1qi2ρ
n
Note : qi1 = (2y i1 -1) = -1 if y i1 = 0 and +1 if y i1 = 1.
Φ2 = Bivariate normal CDF - must be computed
using quadrature
Maximized with respect to μ1,μ2 and ρ.
Estimation
+---------------------------------------------+
| FIML Estimates of Bivariate Probit Model
|
| Maximum Likelihood Estimates
|
| Dependent variable
DOCHOS
|
| Weighting variable
None
|
| Number of observations
27326
|
| Log likelihood function
-25898.27
|
| Number of parameters
3
|
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
Index
equation for DOCTOR
Constant
.32949128
.00773326
42.607
.0000
Index
equation for HOSPITAL
Constant
-1.35539755
.01074410 -126.153
.0000
Tetrachoric Correlation between DOCTOR
and HOSPITAL
RHO(1,2)
.31105965
.01357302
22.918
.0000
A Bivariate Probit Model
•
•
•
Two Equation Probit Model
No bivariate logit – there is no
reasonable bivariate counterpart
Why fit the two equation model?
•
•
Analogy to SUR model: Efficient
Make tetrachoric correlation conditional on
covariates – i.e., residual correlation
Bivariate Probit Model
y1 * = β1x1 + ε1, y1 = 1(y1* > 0)
y 2 * = β2 x 2 + ε 2 ,y 2 = 1(y 2 * > 0)
 0   1 ρ  
 ε1 
  ~ N   , 

ε
0
ρ
1

 2
  
The variables in x 2 and x 2 may be the same or
different. There is no need for each equation to have
its 'own variable.'
ρ is the conditional tetrachoric correlation between y1 and y 2 .
(The equations can be fit one at a time. Use FIML for
(1) efficiency and (2) to get the estimate of ρ.)
Estimation of the Bivariate Probit Model
(2yi1 -1)β1 xi1,

n

logL =  i=1logΦ2 (2yi2 -1)β2 x i2 ,

(2yi1 -1)(2y i2 -1)ρ
=  i=1logΦ2  qi1β1 x i1,qi2β2 x i2 ,qi1qi2ρ
n
Note : qi1 = (2yi1 -1) = -1 if y i1 = 0 and +1 if y i1 = 1.
Φ2 = Bivariate normal CDF - must be computed
using quadrature
Maximized with respect to β1,β2 and ρ.
Parameter Estimates
---------------------------------------------------------------------FIML Estimates of Bivariate Probit Model for DOCTOR and HOSPITAL
Dependent variable
DOCHOS
Log likelihood function
-25323.63074
Estimation based on N = 27326, K = 12
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------|Index
equation for DOCTOR
Constant|
-.20664***
.05832
-3.543
.0004
AGE|
.01402***
.00074
18.948
.0000
43.5257
FEMALE|
.32453***
.01733
18.722
.0000
.47877
EDUC|
-.01438***
.00342
-4.209
.0000
11.3206
MARRIED|
.00224
.01856
.121
.9040
.75862
WORKING|
-.08356***
.01891
-4.419
.0000
.67705
|Index
equation for HOSPITAL
Constant|
-1.62738***
.05430
-29.972
.0000
AGE|
.00509***
.00100
5.075
.0000
43.5257
FEMALE|
.12143***
.02153
5.641
.0000
.47877
HHNINC|
-.03147
.05452
-.577
.5638
.35208
HHKIDS|
-.00505
.02387
-.212
.8323
.40273
|Disturbance correlation (Conditional tetrachoric correlation)
RHO(1,2)|
.29611***
.01393
21.253
.0000
---------------------------------------------------------------------| Tetrachoric Correlation between DOCTOR
and HOSPITAL
RHO(1,2)|
.31106
.01357
22.918
.0000
--------+-------------------------------------------------------------
Marginal Effects
•
What are the marginal effects
•
•
•
Possible margins?
•
•
•
•
Effect of what on what?
Two equation model, what is the conditional mean?
Derivatives of joint probability = Φ2(β1’xi1, β2’xi2,ρ)
Partials of E[yij|xij] =Φ(βj’xij) (Univariate probability)
Partials of E[yi1|xi1,xi2,yi2=1] = P(yi1,yi2=1)/Prob[yi2=1]
Note marginal effects involve both sets of regressors.
If there are common variables, there are two effects
in the derivative that are added.
Bivariate Probit Conditional Means
Prob[yi1 = 1,y i2 = 1] = Φ2 (β1xi1,β2 xi2 ,ρ)
This is not a conditional mean. For a generic x that might appear in either index function,
Prob[y i1 = 1,y i2 = 1]
= gi1β1 + gi2β2
x i
 β x - ρβx 
 βx - ρβ x 
2 i2
1 i1
2 i2
 ,gi2 = φ(β2 x i2 )Φ  1 i1

gi1 = φ(β1 x i1 )Φ 
2
2




1- ρ
1- ρ




The term in β1 is 0 if x i does not appear in x i1 and likewise for β2 .
Φ (βx ,β x ,ρ)
E[yi1 | x i1, x i2 ,yi2 = 1] = Prob[y i1 = 1| x i1, x i2 ,y i2 = 1] = 2 1 i1 2 i2
Φ(β2 x i2 )
E[yi1 | x i1, x i2 ,yi2 = 1]
Φ (β x ,β x ,ρ)φ(β x )
1
=
gi1β1 + gi2β2  - 2 1 i1 2 i2 2 2 i2 β2

x i
Φ(β2 x i2 )
[Φ(β2 x i2 )]
 gi1 
 gi2
Φ2 (β1 xi1,β2 xi2 ,ρ)φ(β2 xi2 ) 
=
β
+
 1 
 β2
2



Φ(
β
x
)
Φ(
β
x
)
[Φ(
β
x
)]

2 i2 

2 i2
2 i2

Direct Effects
Derivatives of E[y1|x1,x2,y2=1] wrt x1
+-------------------------------------------+
| Partial derivatives of E[y1|y2=1] with
|
| respect to the vector of characteristics. |
| They are computed at the means of the Xs. |
| Effect shown is total of 4 parts above.
|
| Estimate of E[y1|y2=1] = .819898
|
| Observations used for means are All Obs. |
| These are the direct marginal effects.
|
+-------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
AGE
.00382760
.00022088
17.329
.0000
43.5256898
FEMALE
.08857260
.00519658
17.044
.0000
.47877479
EDUC
-.00392413
.00093911
-4.179
.0000
11.3206310
MARRIED
.00061108
.00506488
.121
.9040
.75861817
WORKING
-.02280671
.00518908
-4.395
.0000
.67704750
HHNINC
.000000
......(Fixed Parameter).......
.35208362
HHKIDS
.000000
......(Fixed Parameter).......
.40273000
Indirect Effects
Derivatives of E[y1|x1,x2,y2=1] wrt x2
+-------------------------------------------+
| Partial derivatives of E[y1|y2=1] with
|
| respect to the vector of characteristics. |
| They are computed at the means of the Xs. |
| Effect shown is total of 4 parts above.
|
| Estimate of E[y1|y2=1] = .819898
|
| Observations used for means are All Obs. |
| These are the indirect marginal effects. |
+-------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
AGE
-.00035034
.697563D-04
-5.022
.0000
43.5256898
FEMALE
-.00835397
.00150062
-5.567
.0000
.47877479
EDUC
.000000
......(Fixed Parameter).......
11.3206310
MARRIED
.000000
......(Fixed Parameter).......
.75861817
WORKING
.000000
......(Fixed Parameter).......
.67704750
HHNINC
.00216510
.00374879
.578
.5636
.35208362
HHKIDS
.00034768
.00164160
.212
.8323
.40273000
Marginal Effects: Total Effects
Sum of Two Derivative Vectors
+-------------------------------------------+
| Partial derivatives of E[y1|y2=1] with
|
| respect to the vector of characteristics. |
| They are computed at the means of the Xs. |
| Effect shown is total of 4 parts above.
|
| Estimate of E[y1|y2=1] = .819898
|
| Observations used for means are All Obs. |
| Total effects reported = direct+indirect. |
+-------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
AGE
.00347726
.00022941
15.157
.0000
43.5256898
FEMALE
.08021863
.00535648
14.976
.0000
.47877479
EDUC
-.00392413
.00093911
-4.179
.0000
11.3206310
MARRIED
.00061108
.00506488
.121
.9040
.75861817
WORKING
-.02280671
.00518908
-4.395
.0000
.67704750
HHNINC
.00216510
.00374879
.578
.5636
.35208362
HHKIDS
.00034768
.00164160
.212
.8323
.40273000
Marginal Effects: Dummy Variables
Using Differences of Probabilities
+-----------------------------------------------------------+
| Analysis of dummy variables in the model. The effects are |
| computed using E[y1|y2=1,d=1] - E[y1|y2=1,d=0] where d is |
| the variable. Variances use the delta method. The effect |
| accounts for all appearances of the variable in the model.|
+-----------------------------------------------------------+
|Variable
Effect
Standard error
t ratio (deriv) |
+-----------------------------------------------------------+
FEMALE
.079694
.005290
15.065 (.080219)
MARRIED
.000611
.005070
.121 (.000511)
WORKING
-.022485
.005044
-4.457 (-.022807)
HHKIDS
.000348
.001641
.212 (.000348)
Computed using
difference of probabilities
Computed using scaled
coefficients
Simultaneous
Equations
A Simultaneous Equations Model
Simultaneous Equations Model
y1 * = β1x1 + γ1y 2 + ε1, y1 = 1(y1* > 0)
y 2 * = β2 x 2 + γ 2 y1 + ε 2 ,y 2 = 1(y 2 * > 0)
 0   1 ρ  
 ε1 
  ~ N   , 

ε
0
ρ
1

 2
  
This model is not identified. (Not estimable.
The computer can compute 'estimates' but
they have no meaning.)
bivariate probit;lhs=doctor,hospital
;rh1=one,age,educ,married,female,hospital
;rh2=one,age,educ,married,female,doctor$
Error
809: Fully simultaneous BVP model is not identified
Fully Simultaneous ‘Model’
(Obtained by bypassing internal control)
---------------------------------------------------------------------FIML Estimates of Bivariate Probit Model
Dependent variable
DOCHOS
Log likelihood function
-20318.69455
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------|Index
equation for DOCTOR
Constant|
-.46741***
.06726
-6.949
.0000
AGE|
.01124***
.00084
13.353
.0000
43.5257
FEMALE|
.27070***
.01961
13.807
.0000
.47877
EDUC|
-.00025
.00376
-.067
.9463
11.3206
MARRIED|
-.00212
.02114
-.100
.9201
.75862
WORKING|
-.00362
.02212
-.164
.8701
.67705
HOSPITAL|
2.04295***
.30031
6.803
.0000
.08765
|Index
equation for HOSPITAL
Constant|
-1.58437***
.08367
-18.936
.0000
AGE|
-.01115***
.00165
-6.755
.0000
43.5257
FEMALE|
-.26881***
.03966
-6.778
.0000
.47877
HHNINC|
.00421
.08006
.053
.9581
.35208
HHKIDS|
-.00050
.03559
-.014
.9888
.40273
DOCTOR|
2.04479***
.09133
22.389
.0000
.62911
|Disturbance correlation
RHO(1,2)|
-.99996***
.00048
********
.0000
--------+-------------------------------------------------------------
A Latent Simultaneous Equations Model
Simultaneous Equations Model in the latent variables
y1 * = β1 x1 + γ1y 2* + ε1, y1 = 1(y1* > 0)
y 2 * = β2 x 2 + γ 2 y1* + ε 2 , y 2 = 1(y 2 * > 0)
 0   1 ρ  
 ε1 
 ε  ~ N  0  ,  ρ 1  

 2
  
Note the underlying (latent) structural variables in
each equation, not the observed binary variables.
This model is identified. It is hard to interpret. It can
be consistently estimated by two step methods.
(Analyzed in Amemiya (1979) and Maddala (1983).)
A Recursive Simultaneous Equations Model
Recursive Simultaneous Equations Model
y1 * = β1x1 +
ε1, y1 = 1(y1* > 0)
y 2 * = β2 x 2 + γ 2 y1 + ε 2 ,y 2 = 1(y 2 * > 0)
 0   1 ρ  
 ε1 
~
N
  , 
 

ε
0
ρ
1

 2
  
This model is identified. It can be consistently and efficiently
estimated by full information maximum likelihood. Treated as
a bivariate probit model, ignoring the simultaneity.
Ordered
Choices
Ordered Discrete Outcomes
•
•
E.g.: Taste test, credit rating, course grade,
preference scale
Underlying random preferences:
•
•
•
•
•
Existence of an underlying continuous preference scale
Mapping to observed choices
Strength of preferences is reflected in the discrete
outcome
Censoring and discrete measurement
The nature of ordered data
Bond
Ratings
Health Satisfaction (HSAT)
Self administered survey: Health Satisfaction (0 – 10)
Continuous Preference Scale
Modeling Ordered Choices
•
Random Utility (allowing a panel data
setting)
Uit =  + ’xit + it
= ait +
•
•
it
Observe outcome j if utility is in region j
Probability of outcome = probability of cell
Pr[Yit=j] = Prob[Yit < j] - Prob[Yit < j-1]
= F(j – ait) - F(j-1 – ait)
Ordered Probability Model
y*  βx  , we assume x contains a constant term
y  0 if y* 0
y = 1 if 0 < y*  1
y = 2 if 1 < y*  2
y = 3 if 2 < y*  3
...
y = J if  J-1 < y*   J
In general : y = j if  j-1 < y*   j , j = 0,1,...,J
-1  , o  0,  J  ,  j-1   j , j = 1,...,J-1
Combined Outcomes for Health Satisfaction
Probabilities for Ordered Choices
Prob[y=j]=Prob[ j-1  y*   j ]
= Prob[ j-1  βx     j ]
= Prob[βx     j ]  Prob[βx     j1 ]
= Prob[   j  βx ]  Prob[   j1  βx ]
= F[ j  βx ]  F[ j1  βx]
where F[] is the CDF of .
Probabilities for Ordered Choices
μ1 =1.1479
μ2 =2.5478
μ3 =3.0564
Coefficients
 What are the coefficients in the ordered probit model?
There is no conditional mean function.
Prob[y=j|x ]
 [f( j1  β'x)  f( j  β'x)] k
x k
Magnitude depends on the scale factor and the coefficient.
Sign depends on the densities at the two points!
 What does it mean that a coefficient is "significant?"
Effects of 8 More Years of Education
An Ordered Probability
Model for Health Satisfaction
+---------------------------------------------+
| Ordered Probability Model
|
| Dependent variable
HSAT
|
| Number of observations
27326
|
| Underlying probabilities based on Normal
|
|
Cell frequencies for outcomes
|
| Y Count Freq Y Count Freq Y Count Freq
|
| 0
447 .016 1
255 .009 2
642 .023
|
| 3 1173 .042 4 1390 .050 5 4233 .154
|
| 6 2530 .092 7 4231 .154 8 6172 .225
|
| 9 3061 .112 10 3192 .116
|
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Index function for probability
Constant
2.61335825
.04658496
56.099
.0000
FEMALE
-.05840486
.01259442
-4.637
.0000
.47877479
EDUC
.03390552
.00284332
11.925
.0000
11.3206310
AGE
-.01997327
.00059487
-33.576
.0000
43.5256898
HHNINC
.25914964
.03631951
7.135
.0000
.35208362
HHKIDS
.06314906
.01350176
4.677
.0000
.40273000
Threshold parameters for index
Mu(1)
.19352076
.01002714
19.300
.0000
Mu(2)
.49955053
.01087525
45.935
.0000
Mu(3)
.83593441
.00990420
84.402
.0000
Mu(4)
1.10524187
.00908506
121.655
.0000
Mu(5)
1.66256620
.00801113
207.532
.0000
Mu(6)
1.92729096
.00774122
248.965
.0000
Mu(7)
2.33879408
.00777041
300.987
.0000
Mu(8)
2.99432165
.00851090
351.822
.0000
Mu(9)
3.45366015
.01017554
339.408
.0000
Ordered Probit Partial Effects
Partial effects at means of the data
Average Partial Effect of HHNINC
Predictions of the Model:Kids
+----------------------------------------------+
|Variable
Mean Std.Dev. Minimum Maximum |
+----------------------------------------------+
|Stratum is KIDS = 0.000. Nobs.= 2782.000
|
+--------+-------------------------------------+
|P0
| .059586 .028182 .009561 .125545 |
|P1
| .268398 .063415 .106526 .374712 |
|P2
| .489603 .024370 .419003 .515906 |
|P3
| .101163 .030157 .052589 .181065 |
|P4
| .081250 .041250 .028152 .237842 |
+----------------------------------------------+
|Stratum is KIDS = 1.000. Nobs.= 1701.000
|
+--------+-------------------------------------+
|P0
| .036392 .013926 .010954 .105794 |
|P1
| .217619 .039662 .115439 .354036 |
|P2
| .509830 .009048 .443130 .515906 |
|P3
| .125049 .019454 .061673 .176725 |
|P4
| .111111 .030413 .035368 .222307 |
+----------------------------------------------+
|All 4483 observations in current sample
|
+--------+-------------------------------------+
|P0
| .050786 .026325 .009561 .125545 |
|P1
| .249130 .060821 .106526 .374712 |
|P2
| .497278 .022269 .419003 .515906 |
|P3
| .110226 .029021 .052589 .181065 |
|P4
| .092580 .040207 .028152 .237842 |
+----------------------------------------------+
This is a restricted
model with outcomes
collapsed into 5 cells.
Fit Measures
•
•
•
•
There is no single “dependent variable” to explain.
There is no sum of squares or other measure of
“variation” to explain.
Predictions of the model relate to a set of J+1
probabilities, not a single variable.
How to explain fit?
•
•
•
Based on the underlying regression
Based on the likelihood function
Based on prediction of the outcome variable
Log Likelihood Based Fit Measures
Fit Measure Based on Counting Predictions
This model always
predicts the same cell.
A Somewhat Better Fit
Different Normalizations
•
NLOGIT
•
•
•
•
Y = 0,1,…,J, U* = α + β’x + ε
One overall constant term, α
J-1 “thresholds;” μ-1 = -∞, μ0 = 0, μ1,… μJ-1, μJ = + ∞
Stata
•
•
•
Y = 1,…,J+1, U* = β’x + ε
No overall constant, α=0
J “cutpoints;” μ0 = -∞, μ1,… μJ, μJ+1 = + ∞
α̂
μˆ j
αˆ
μˆ j - αˆ
Generalizing the Ordered Probit
with Heterogeneous Thresholds
Index = βxi
Threshold parameters
Standard model : μ-1 = -, μ0 = 0, μ j > μ j-1 > 0, μJ = +
Preference scale and thresholds are homogeneous
A generalized model (Pudney and Shields, JAE, 2000)
μij = α j + γ j zi
Note the identification problem. If zik is also in xi (same variable) then
μij - βxi = α j + γzik - βzik +... E.g.,
( j + 1Age +  2Married) - (  + 1Age  2Sex )
 ( j +  2Married) - ( + (1  1 ) Age  2Sex )
No longer clear if the variable is in x or z (or both)
Differential Item Functioning
People in this
country are
optimistic –
they report
this value as
‘very good.’
People in this
country are
pessimistic –
they report
this same
value as ‘fair’
Panel Data
•
Fixed Effects
•
•
•
The usual incidental parameters problem
Partitioning Prob(yit > j|xit) produces estimable
binomial logit models. (Find a way to combine
multiple estimates of the same β.
Random Effects
•
Standard application
Incidental Parameters Problem
Table 9.1 Monte Carlo Analysis of the Bias of the MLE in Fixed
Effects Discrete Choice Models (Means of empirical sampling
distributions, N = 1,000 individuals, R = 200 replications)
Random Effects
Dynamic Ordered Probit Model
Model for Self Assessed Health
•
British Household Panel Survey (BHPS)
•
•
•
•
•
Waves 1-8, 1991-1998
Self assessed health on 0,1,2,3,4 scale
Sociological and demographic covariates
Dynamics – inertia in reporting of top scale
Dynamic ordered probit model
•
•
Balanced panel – analyze dynamics
Unbalanced panel – examine attrition
Dynamic Ordered Probit Model
Latent Regression - Random Utility
h *it = xit +  H i ,t 1 + i + it
xit = relevant covariates and control variables
It would not be
appropriate to include
hi,t-1 itself in the model
as this is a label, not a
measure
H i ,t 1 = 0/1 indicators of reported health status in previous period
Hi ,t 1 ( j ) = 1[Individual i reported h it  j in previous period], j=0,...,4
Ordered Choice Observation Mechanism
h it = j if  j 1 < h*it   j , j = 0,1,2,3,4
Ordered Probit Model - it ~ N[0,1]
Random Effects with Mundlak Correction and Initial Conditions
i =  0  1H i ,1 + 2 xi + u i , u i ~ N[0,2 ]
Testing for Attrition Bias
Three dummy variables added to full model with unbalanced panel
suggest presence of attrition effects.
Attrition Model with IP Weights
Assumes (1) Prob(attrition|all data) = Prob(attrition|selected variables) (ignorability)
(2) Attrition is an ‘absorbing state.’ No reentry.
Obviously not true for the GSOEP data above.
Can deal with point (2) by isolating a subsample of those present at wave 1 and the
monotonically shrinking subsample as the waves progress.
Inverse Probability Weighting
Panel is based on those present at WAVE 1, N1 individuals
Attrition is an absorbing state. No reentry, so N1  N2  ...  N8.
Sample is restricted at each wave to individuals who were present at
the previous wave.
d it = 1[Individual is present at wave t].
d i1 = 1  i, dit  0  d i ,t 1  0.
xi1  covariates observed for all i at entry that relate to likelihood of
being present at subsequent waves.
(health problems, disability, psychological well being, self employment,
unemployment, maternity leave, student, caring for family member, ...)
Probit model for dit  1[xi1  wit ], t = 2,...,8. ˆ it  fitted probability.
t
Assuming attrition decisions are independent, Pˆit   s 1 ˆ is
ˆ  d it
Inverse probability weight W
it
P̂it
Weighted log likelihood
logLW   i 1  t 1 log Lit (No common effects.)
N
8
Estimated Partial Effects by Model
Partial Effect for a Category
These are 4 dummy variables for state in the previous period. Using
first differences, the 0.234 estimated for SAHEX means transition from
EXCELLENT in the previous period to GOOD in the previous period,
where GOOD is the omitted category. Likewise for the other 3 previous
state variables. The margin from ‘POOR’ to ‘GOOD’ was not interesting
in the paper. The better margin would have been from EXCELLENT to
POOR, which would have (EX,POOR) change from (1,0) to (0,1).