Discrete Choice Modeling
William Greene
Stern School of Business
New York University
Lab Sessions
Lab Session 2
Analyzing Binary
Choice Data
Data Set: Load PANELPROBIT.LPJ
Fit Basic Models
Partial Effects for Interactions
Prob[ y  1| x]  [  1 x  2 z  3 x 2  4 xz ]
 [ A]
Partial Effects?
P
 [ A](1  23 x  4 z )
x
P
 [ A](2  4 x)
z
Compute without extensive additional computation of
extra variables, etc.
Partial Effects

Build the interactions into the model
statement
PROBIT ; Lhs = Doctor
; Rhs = one,age,educ,age^2,age*educ $

Built in computation for partial effects
PARTIALS ; Effects:
Age & Educ = 8(2)20
; Plot(ci) $
Average Partial Effects
--------------------------------------------------------------------Partial Effects Analysis for Probit Probability Function
--------------------------------------------------------------------Partial effects on function with respect to AGE
Partial effects are computed by average over sample observations
Partial effects for continuous variable by differentiation
Partial effect is computed as derivative
= df(.)/dx
--------------------------------------------------------------------df/dAGE
Partial
Standard
(Delta method)
Effect
Error
|t| 95% Confidence Interval
--------------------------------------------------------------------Partial effect
.00441
.00059
7.47
.00325
.00557
EDUC
= 8.00
.00485
.00101
4.80
.00287
.00683
EDUC
= 10.00
.00463
.00068
6.80
.00329
.00596
EDUC
= 12.00
.00439
.00061
7.18
.00319
.00558
EDUC
= 14.00
.00412
.00091
4.53
.00234
.00591
EDUC
= 16.00
.00384
.00138
2.78
.00113
.00655
EDUC
= 18.00
.00354
.00192
1.84
-.00023
.00731
EDUC
= 20.00
.00322
.00250
1.29
-.00168
.00813
Useful Plot
More Elaborate Partial Effects

PROBIT ; Lhs = Doctor
; Rhs = one,age,educ,age^2,age*educ,
female,female*educ,income $

PARTIAL ; Effects: income
@ female = 0,1
? Do for each subsample
| educ = 12,16,20 ? Set 3 fixed values
& age = 20(10)50 ? APE for each setting
Constructed Partial Effects
Predictions
List and keep predictions
Add ; List ; Prob = PFIT
to the probit or logit command
(Tip: Do not use ;LIST with large samples!)
Sample ; 1-100 $
PROBIT ; Lhs=ip ; Rhs=x1
; List ; Prob=Pfit $
DSTAT ; Rhs = IP,PFIT $
Testing a Hypothesis – Wald Test
Wald Statistic
-1 ˆ
ˆ
ˆ
Wald = (β - 0)[Est.Var(β - 0)] (β - 0)
SAMPLE ; All $
PROBIT ; Lhs = IP ; RHS = Sectors,X1 $
MATRIX ; b1 = b(1:3) ; v1 = Varb(1:3,1:3) $
MATRIX ; List ; Waldstat = b1'<V1>b1 $
CALC
; List ; CStar = CTb(.95,3) $
Testing a Hypothesis – LM Test
Lagrange Multiplier Test
LM = g(βˆ 0 )[Est.Hessian0 ]-1g(βˆ 0 )
βˆ 0 = MLE with restrictions imposed
Hessian is computed at βˆ 0 .
PROBIT ; LHS = IP ; RHS = X1 $
PROBIT ; LHS = IP ; RHS = X1,Sectors
; Start = b,0,0,0 ; MAXIT = 0 $
Results of an LM test
Maximum iterations reached. Exit iterations with status=1.
Maxit = 0. Computing LM statistic at starting values.
No iterations computed and no parameter update done.
+---------------------------------------------+
| Binomial Probit Model
|
| Dependent variable
IP
|
| Number of observations
6350
|
| Iterations completed
1
|
Note: Wald
| LM Stat. at start values
163.8261
|
equaled 163.236.
| LM statistic kept as scalar
LMSTAT
|
| Log likelihood function
-4228.350
|
| Restricted log likelihood
-4283.166
|
| Chi squared
109.6320
|
| Degrees of freedom
6
|
| Prob[ChiSqd > value] =
.0000000
|
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Constant
-.01060549
.04902957
-.216
.8287
IMUM
.43885789
.14633344
2.999
.0027
.25275054
FDIUM
2.59443123
.39703852
6.534
.0000
.04580618
SP
.43672968
.11922200
3.663
.0002
.07428482
RAWMTL
.000000
.06217590
.000 1.0000
.08661417
INVGOOD
.000000
.03590410
.000 1.0000
.50236220
FOOD
.000000
.07923549
.000 1.0000
.04724409
Likelihood Ratio Test
LR = 2[LogL(unrestricted) - Logl(restricted)]
PROBIT ; Lhs = IP ; Rhs = X1,Sectors $
CALC ; LOGLU = Logl $
PROBIT ; Lhs = IP ; Rhs = X1 $
CALC ; LOGLR = Logl $
CALC ; List ; LRStat = 2*(LOGLU – LOGLR) $
Result is 164.878.
Using the Binary Choice Simulator
Fit the model with MODEL ; Lhs = … ; Rhs = …
Simulate the model with
BINARY CHOICE ; <same LHS and RHS > ; Start = B (coefficients)
; Model = the kind of model (Probit or Logit)
; Scenario: variable <operation> = value / (may repeat)
; Plot: Variable ( range of variation is optional)
; Limit = P* (is optional, 0.5 is the default) $
E.g.:
Probit ; Lhs = IP ; Rhs = One,LogSales,Imum,FDIum $
BinaryChoice ; Lhs = IP ; Rhs = One,LogSales,IMUM,FDIUM
; Model = Probit ; Start = B
; Scenario: LogSales * = 1.1 ; Plot: LogSales $
Estimated Model for Innovation
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Index function for probability
Constant
-1.89382186
.20520881
-9.229
.0000
LOGSALES
.16345837
.01766902
9.251
.0000
10.5400961
IMUM
.99773826
.14091020
7.081
.0000
.25275054
FDIUM
3.66322280
.37793285
9.693
.0000
.04580618
+---------------------------------------------------------+
|Predictions for Binary Choice Model. Predicted value is |
|1 when probability is greater than .500000, 0 otherwise.|
|------+---------------------------------+----------------+
|Actual|
Predicted Value
|
|
|Value |
0
1
| Total Actual
|
+------+----------------+----------------+----------------+
| 0
|
531 ( 8.4%)|
2033 ( 32.0%)|
2564 ( 40.4%)|
| 1
|
454 ( 7.1%)|
3332 ( 52.5%)|
3786 ( 59.6%)|
+------+----------------+----------------+----------------+
|Total |
985 ( 15.5%)|
5365 ( 84.5%)|
6350 (100.0%)|
+------+----------------+----------------+----------------+
Effect of logSales on Probability
Model Simulation:
logSales Increases by 10% for all Firms in the Sample
+-------------------------------------------------------------+
|Scenario 1. Effect on aggregate proportions. Probit
Model |
|Threshold T* for computing Fit = 1[Prob > T*] is .50000
|
|Variable changing = LOGSALES, Operation = *, value =
1.100 |
+-------------------------------------------------------------+
|Outcome
Base case
Under Scenario
Change
|
|
0
985 = 15.51%
300 =
4.72%
-685
|
|
1
5365 = 84.49%
6050 =
95.28%
685
|
| Total
6350 = 100.00%
6350 = 100.00%
0
|
+-------------------------------------------------------------+
Lab Session 3
Bivariate Extensions of the Probit Model
Bivariate Probit Model
Two equation model
General usage of



LHS = the set of dependent variables
RH1 = one set of independent variables
RH2 = a second set of variables
Economical use of namelists is useful here
Namelist ; x1=one,age,female,educ,married,working $
Namelist ; x2=one,age,female,hhninc,hhkids $
BivariateProbit ;lhs=doctor,hospital
;rh1=x1
;rh2=x2;marginal effects $
Heteroscedasticity in the Bivariate Probit Model
General form of heteroscedasticity in
LIMDEP/NLOGIT: Exponential
σi = σ exp(γ’zi) so that σi > 0


γ = 0 returns the homoscedastic case σi = σ
Easy to specify
Namelist ; x1=one,age,female,educ,married,working ; z1 = … $
Namelist ; x2=one,age,female,hhninc,hhkids
; z2 = … $
BivariateProbit ;lhs=doctor,hospital
;rh1=x1 ; hf1 = z1
;rh2=x2 ; hf2 = z2$
Heteroscedasticity in Marginal Effects
Univariate case:
 βx i 
E[y | x i , zi ] =  

 exp( γzi ) 
 βx i 
E[y | x i , zi ]
 
β

x i
exp
(
γ
z
)

i 
 βx i   βx i 
E[y | x i , zi ]
  
 γ



zi
exp
(
γ
z
)
exp
(
γ
z
)

i 
i 
If the variables are the same in x and z,
these terms are added.
Sign and magnitude are ambiguous
Vastly more complicated for the bivariate
probit case. NLOGIT handles it internally.
Marginal Effects: Heteroscedasticity
+------------------------------------------------------+
|
Partial Effects for Ey1|y2=1
|
+----------+---------------------+---------------------+
|
| Regression Function | Heteroscedasticity |
|
+---------------------+---------------------+
|
|
Direct | Indirect |
Direct | Indirect |
| Variable | Efct x1 | Efct x2 | Efct h1 | Efct h2 |
+----------+----------+----------+----------+----------+
|
AGE |
.00190 | -.00012 |
.00000 |
.00000 |
|
FEMALE |
.10215 |
.20688 | -.05880 | -.30944 |
|
EDUC | -.00247 |
.00000 |
.00000 |
.00000 |
| MARRIED |
.00103 |
.00000 |
.00064 |
.00476 |
| WORKING | -.02139 |
.00000 |
.00000 |
.00000 |
|
HHNINC |
.00000 |
.00154 |
.00000 |
.00000 |
|
HHKIDS |
.00000 |
.00005 |
.00000 |
.00000 |
+----------+----------+----------+----------+----------+
Marginal Effects: Total Effects
+-------------------------------------------+
| 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|
+---------+--------------+----------------+--------+---------+----------+
Constant
.000000
......(Fixed Parameter).......
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
Imposing Fixed Value and Equality Constraints
Used throughout LIMDEP in all models,
model parameters appear as a long list:
β1 β2 β3 β4 α1 α2 α3 α4 σ and so on.
M parameters in total.
Use ; RST = list of symbols for the model
parameters, in the right order
This may be used for nonlinear models. Not in
REGRESS. Use ;CLS:… for linear models
Use the same name for equal parameters
Use specific numbers to fix the values
BivariateProbit ;
;
;
;
lhs=doctor,hospital
rh1=one,age,female,educ,married,working
rh2=one,age,female,hhninc,hhkids
rst = beta1,beta2,beta3,be,bm,bw,
beta1,beta2,beta3,bi,bk, 0.4 $
--------+------------------------------------------------------------Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------|Index
equation for DOCTOR
Constant|
-1.69181***
.08938
-18.928
.0000
AGE|
.01244***
.00167
7.440
.0000
44.3352
FEMALE|
.38543***
.03157
12.209
.0000
.42277
EDUC|
.08144***
.00457
17.834
.0000
10.9409
MARRIED|
.42021***
.03987
10.541
.0000
.84539
WORKING|
.03310
.03910
.847
.3972
.73941
|Index
equation for HOSPITAL
Constant|
-1.69181***
.08938
-18.928
.0000
AGE|
.01244***
.00167
7.440
.0000
44.3352
FEMALE|
.38543***
.03157
12.209
.0000
.42277
HHNINC|
-.98617***
.08917
-11.060
.0000
.34930
HHKIDS|
-.09406**
.04600
-2.045
.0409
.45482
|Disturbance correlation
RHO(1,2)|
.40000
......(Fixed Parameter)......
--------+-------------------------------------------------------------
Miscellaneous Topics
Two Step Estimation
 Robust (Sandwich) Covariance
matrix
 Matrix Algebra – Testing for
Normality

Two Step Estimation
Murphy and Topel
This can usually easily be programmed using the models,
CREATE, CALC and MATRIX. Several leading cases are built in.
Two Step Estimation: Automated
Application: Recursive Probit
Hospital = bh’xh + c*Doctor + eh
Doctor = bd’xd
+ ed
Sample ; All $
Namelist ; xD=one,age,female,educ,married,working
; xH=one,age,female,hhninc,hhkids $
Reject ; _Groupti < 7 $
Probit
; lhs=hospital;rhs=xh,doctor$
Probit
; lhs=doctor;rhs=xd;prob=pd;hold$
Probit
; lhs=hospital;rhs=xh,pd;2step=pd$
Robust Covariance Matrix
Standard Covariance Matrix Estimator (General)
1
 n  2 logL 
ˆ = 
V

i=1 ˆ
ˆ






ML
ML 

'Robust' (Sandwich) Estimator
1
 n  2 logL   n   logL    logL    n  2 log L 
ˆ = 
V
 
    i=1
   i=1 

i=1 ˆ
ˆ
ˆ
ˆ
ˆ
ˆ


ML ML  












ML  
ML  
ML
ML 

TO WHAT SPECIFICATION 'ERRORS' IS THIS
ESTIMATOR ROBUST? IN THE PROBIT CASES
THE ESTIMATOR IS INCONSISTENT, SO NOT TO
(1) HETEROSCEDASTICITY
(2) OMITTED VARIABLES
(3) WRONG DISTRIBUTIONAL ASSUMPTION
POSSIBLY TO CROSS OBSERVATION CORRELATION.
1
Robust Covariance Matrix
; ROBUST
Using the health care data:
+---------------------------------------------+
| Binomial Probit Model
|
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
+---------+--------------+----------------+--------+---------+
|Index function for probability
Constant|
-.17336***
.05874
-2.951
AGE|
.01393***
.00074
18.920
FEMALE|
.32097***
.01718
18.682
EDUC|
-.01602***
.00344
-4.650
MARRIED|
-.00153
.01869
-.082
WORKING|
-.09257***
.01893
-4.889
Robust VC=<H>G<H> used for estimates.
Constant|
-.17336***
.05881
-2.948
AGE|
.01393***
.00073
19.024
FEMALE|
.32097***
.01701
18.869
EDUC|
-.01602***
.00345
-4.648
MARRIED|
-.00153
.01874
-.082
WORKING|
-.09257***
.01885
-4.911
.0032
.0000
.0000
.0000
.9347
.0000
43.5257
.47877
11.3206
.75862
.67705
.0032
.0000
.0000
.0000
.9348
.0000
43.5257
.47877
11.3206
.75862
.67705
Cluster Correction
PROBIT ; Lhs = doctor
; Rhs = one,age,female,educ,married,working
; Cluster = ID $
Normal exit:
4 iterations. Status=0. F=
17448.10
+---------------------------------------------------------------------+
| Covariance matrix for the model is adjusted for data clustering.
|
| Sample of 27326 observations contained
7293 clusters defined by |
| variable ID
which identifies by a value a cluster ID.
|
+---------------------------------------------------------------------+
Variable| Coefficient
Standard Error b/St.Er. P[|Z|>z]
Mean of X
--------+------------------------------------------------------------|Index function for probability
Constant|
-.17336**
.08118
-2.135
.0327
AGE|
.01393***
.00102
13.691
.0000
43.5257
FEMALE|
.32097***
.02378
13.497
.0000
.47877
EDUC|
-.01602***
.00492
-3.259
.0011
11.3206
MARRIED|
-.00153
.02553
-.060
.9521
.75862
WORKING|
-.09257***
.02423
-3.820
.0001
.67705
--------+-------------------------------------------------------------
Using Matrix Algebra
Namelists with the current sample serve 2 major functions:
(1) Define lists of variables for model estimation
(2) Define the columns of matrices built from the data.
NAMELIST ; X = a list ; Z = a list … $
Set the sample any way you like. Observations are now
the rows of all matrices. When the sample changes,
the matrices change.
Lists may be anything, may contain ONE, may overlap
(some or all variables) and may contain the same
variable(s) more than once
Matrix Functions
Matrix Product: MATRIX ; XZ = X’Z $
Moments and Inverse MATRIX ; XPX = X’X
; InvXPX = <X’X> $
Moments with individual specific weights in variable w.
Σi wi xixi’ = X’[w]X.
[Σi wi xixi’ ]-1 = <X’[w]X>
Unweighted Sum of Rows in a Matrix
Σi xi = 1’X
Column of Sample Means
(1/n) Σi xi = 1/n * X’1 or MEAN(X)
(Matrix function. There are over 100 others.)
Weighted Sum of rows in matrix
Σi wi xi = 1’[w]X
Normality Test for Probit
Testing for normality in the probit model:
x i  RHS variables. y i = LHS variable
Probit Model
Prob[y i  1 | x i ]  (βx i ), Normal CDF. (βx i )  density
zi  [x i , zi3 , zi4 ],
zi3  -(1/3)[(βx i )2  1], zi4  (1/4){(βx i )[3  (βx i )2 ]}
ei  y i  (x i ), di 
(βx i )
(βx i )[1  (βx i )]
Lagrange Multiplier Statistic. ^ = compute at MLE of β


ˆ iˆ
LM=  i=1 (e
di ) ˆ
zi '
n
 i=1 ˆdi 2 ˆzi
n
 
1
n
i=1
ˆi ˆ
(e
di ) ˆ
zi

Thanks to Joachim Wilde, Univ. Halle, Germany for suggesting this.
Normality Test for Probit
NAMELIST
CREATE
PROBIT
CREATE
CREATE
NAMELIST
CREATE
MATRIX
; XI = One,... $
; yi = the dependent variable $
; Lhs = yi ; Rhs = Xi ; Prob = Pfi $
; bxi = b'Xi ; fi = N01(bxi) $
; zi3 = -1/2*(bxi^2 - 1) ; zi4 = 1/4*(bxi*(bxi^2+3)) $
; Zi = Xi,zi3,zi4 $
; di = fi/sqr(pfi*(1-pfi)) ; ei = yi - pfi
; eidi = ei*di ; di2 = di*di $
; List ; LM = 1'[eidi]Zi * <ZI'[di2]Zi> * Zi'[eidi]1 $
Multivariate Probit
MPROBIT ; LHS = y1,y2,…,yM
; Eq1 = RHS for equation 1
; Eq2 = RHS for equation 2
…
; EqM = RHS for equation M $
Parameters are the slope vectors followed by the
lower triangle of the correlation matrix
Constrained Panel Probit
Sample ; 1 - 1270 $
MPROBIT ; LHS = IP84, IP85, IP86 ; MarginalEffects
; Eq1 = One,Fdium84,SP84
; Eq2 = One,Fdium85,SP85
; Eq3 = One,Fdium86,SP86
; Rst = b1,b2,b3,b1,b2,b3,b1,b2,b3,r45, r46, r56
; Maxit = 3 ; Pts = 15 $ (Reduces time to compute)
Estimated Multivariate Probit
+---------------------------------------------+
| Multivariate Probit Model: 3 equations.
|
| Number of observations
1270
|
| Log likelihood function
-2423.732
|
| Number of parameters
6
|
| Replications for simulated probs. = 15
|
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Index function for IP84
Constant
.13489406
.03467525
3.890
.0001
FDIUM84
.33571101
.47118274
.712
.4762
.05055702
SP84
.65662961
.13801209
4.758
.0000
.11012047
Index function for IP85
Constant
.13489406
.03467525
3.890
.0001
FDIUM85
.33571101
.47118274
.712
.4762
.05051809
SP85
.65662961
.13801209
4.758
.0000
.11014611
Index function for IP86
Constant
.13489406
.03467525
3.890
.0001
FDIUM86
.33571101
.47118274
.712
.4762
.05049439
SP86
.65662961
.13801209
4.758
.0000
.11016926
Correlation coefficients
R(01,02)
.46759312
.03716428
12.582
.0000
R(01,03)
.37251014
.03946383
9.439
.0000
R(02,03)
.46215054
.03721312
12.419
.0000
Endogenous Variable in Probit Model
PROBIT ; Lhs = y1, y2
; Rh1 = rhs for the probit model,y2
; Rh2 = exogenous variables for y2 $
SAMPLE
CREATE
PROBIT
;
;
;
;
;
All $
GoodHlth = Hsat > 5 $
Lhs = GoodHlth,Hhninc
Rh1 = One,Female,Hhninc
Rh2 = One,Age,Educ $
Lab Session 4
Panel Data
Binary Choice Models with
Panel Data
Telling NLOGIT You are Fitting a Panel Data Model
Balanced Panel
Model ; … ; PDS = number of periods $
REGRESS ; Lhs = Milk ; Rhs = One,Labor ; Pds = 6 ; Panel $
(Note ;Panel is needed only for REGRESS)
Unbalanced Panel
Model ; … ; PDS = group size variable $
REGRESS ; Lhs = Milk ; Rhs = One,Labor ; Pds = FarmPrds
; Panel $
FarmPrds gives the number of periods, in every period.
(More later about unbalanced panels)
Group Size Variables for
Unbalanced Panels
Farm
Milk
Cows
FarmPrds
1
23.3
10.7
3
1
23.3
10.6
3
1
25
9.4
3
2
19.6
11
2
2
22.2
11
2
3
24.7
11
4
3
25.4
12
4
3
25.3
13.5
4
3
26.1
14.5
4
4
55.4
22
2
4
63.5
22
2
Application to Spanish
Dairy Farms Dairy.lpj
N = 247 farms, T = 6 years (1993-1998)
Input
Units
Mean
Std. Dev.
Minimum
92,539
14,110
Milk
Milk production (liters)
131,108
Cows
# of milking cows
2.12
11.27
4.5
82.3
Labor
# man-equivalent units
1.67
0.55
1.0
4.0
Land
Hectares of land
devoted to pasture and
crops.
12.99
6.17
2.0
45.1
Feed
Total amount of
feedstuffs fed to dairy
cows (tons)
57,941
47,981
3,924.14
Maximum
727,281
376,732
Global Setting for Panels
SETPANEL ; Group = the name of the ID variable
; PDS = the name of the groupsize variable to create $
Subsequent model commands state ;PANEL
with no other specifications requred to set the panel.
Some other specifications usually required for the
specific model – e.g., fixed vs. random effects.
Dialog Boxes for Model Commands
Selecting PANEL from the Options Tab
Load the Probit Data Set
Data for this session are
PANELPROBIT.LPJ
Various Fixed and Random Effects
Models
Random Parameters
Latent Class
Fixed Effects Models
? Fixed Effects Probit.
? Looks like an incidental parameters problem.
Sample ; All $
Namelist ; X = IMUM,FDIUM,SP,LogSales $
Probit ; Lhs = IP ; Rhs = X ; FEM ; Marginal ; Pds=5 $
Probit ; Lhs = IP ; Rhs = X,one ; Marginal $
Logit Fixed Effects Models
Conditional and Unconditional FE
? Logit, conditional vs. unconditional
Logit ; Lhs = IP ; Rhs = X ; Pds = 5 $ (Conditional)
Logit ; Lhs = IP ; Rhs = X ; Pds = 5 ; Fixed $
Hausman Test for Fixed Effects
? Logit: Hausman test for fixed effects
?
Logit ; Lhs = IP ; Rhs = X ; Pds = 5 $
Matrix ; Bf = B ; Vf = Varb $
Logit ; Lhs = IP ; Rhs = X,One $
Calc ; K = Col(X) $
Matrix ; Bp = b(1:K) ; Vp = Varb(1:K,1:K) $
Matrix ; Db = Bf - Bp ; DV = Vf - Vp
; List ; Hausman = Db'<DV>Db $
Calc ; List ; Ctb(.95,k) $
Random Effects and Random Constant
? Random effects
? Quadrature Based (Butler and Moffitt) Estimator
Probit ; Lhs = IP ; Rhs = X,One ; Random ; Pds = 5 $
Calc ; List ; RhoQ = rho $
? Simulation Based Estimator
Probit ; Lhs = IP ; Rhs = X,one ; RPM ; Pds = 5
; Fcn = One(N) ; Halton ; Pts = 25 $
Calc ; List ; RhoRP = b(6)^2/(1+b(6)^2) ; RhoQ $
Unbalanced Panel Data Set
Load healthcare.lpj
Create group size variable
Examine Distribution of Group Sizes
Sample ; all$
Setpanel ; Group = id ; Pds = ti $
Histogram ; rhs = ti $
Group Sizes
A Fixed Effects Probit Model
Probit ;lhs=doctor ; rhs=age,hhninc,educ,married
; fem ; panel ; Parameters $
+---------------------------------------------+
| Probit
Regression Start Values for DOCTOR |
| Maximum Likelihood Estimates
|
These are the pooled data
| Dependent variable
DOCTOR
|
| Weighting variable
None
|
estimates used to obtain
| Number of observations
27326
|
starting values for the
| Iterations completed
10
|
| Log likelihood function
-17700.96
|
iterations to get the full
| Number of parameters
5
|
fixed effects model.
| Akaike IC=35411.927 Bayes IC=35453.005
|
| Finite sample corrected AIC =35411.929
|
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
AGE
.01538640
.00071823
21.423
.0000
43.5256898
HHNINC
-.09775927
.04626475
-2.113
.0346
.35208362
EDUC
-.02811308
.00350079
-8.031
.0000
11.3206310
MARRIED
-.00930667
.01887548
-.493
.6220
.75861817
Constant
.02642358
.05397131
.490
.6244
Fixed Effects Model
Nonlinear Estimation of Model Parameters
Method=Newton; Maximum iterations=100
Convergence criteria: max|dB|
.1000D-08, dF/F= .1000D-08, g<H>g= .1000D-08
Normal exit from iterations. Exit status=0.
+---------------------------------------------+
| FIXED EFFECTS Probit Model
|
| Maximum Likelihood Estimates
|
| Dependent variable
DOCTOR
|
| Number of observations
27326
|
| Iterations completed
11
|
| Log likelihood function
-9454.061
|
| Number of parameters
4928
|
| Akaike IC=28764.123 Bayes IC=69250.570
|
| Finite sample corrected AIC =30933.173
|
| Unbalanced panel has
7293 individuals.
|
| Bypassed 2369 groups with inestimable a(i). |
| PROBIT (normal) probability model
|
+---------------------------------------------+
+---------+--------------+----------------+--------+---------+----------+
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|
+---------+--------------+----------------+--------+---------+----------+
Index function for probability
AGE
.06334017
.00425865
14.873
.0000
42.8271810
HHNINC
-.02495794
.10712886
-.233
.8158
.35402169
EDUC
-.07547019
.04062770
-1.858
.0632
11.3602526
MARRIED
-.04864731
.06193652
-.785
.4322
.76348771
Computed Fixed Effects Parameters
Lab Session 5
Modeling Heterogeneity with Random
Parameters and Latent Classes
Random Parameters Model
? Random parameters specification
?
Logit ; Lhs = IP ; Rhs = One,IMUM,FDIUM,SP,LogSales
; Pds = 5 ; RPM ; Halton ; Pts = 25 ; Cor
; Fcn = One(n),IMUM(n),FDIUM(n) ; Marginal
; Parameters $
Sample ; 1 - 1270 $
Create ; bimum = 0 $
Matrix ; bi = beta_i(1:1270,2:2) $
Create ; bimum = bi $
Kernel ; Rhs = bimum $
4 .2 0
3 .3 6
De n s ity
2 .5 2
1 .6 8
.8 4
.0 0
1 .1 2 5 0
1 .2 0 0 0
1 .2 7 5 0
1 .3 5 0 0
1 .4 2 5 0
BIM UM
Ke rn e l d e n s i ty e s ti m a te fo r
BIM UM
1 .5 0 0 0
1 .5 7 5 0
Random Parameters with
Industry Heterogeneity
? Random parameters with industry heterogeneity
? Examine effect of industry heterogeneity.
Sample ; All $
Logit ; Lhs = IP ; Rhs = One,IMUM,FDIUM,SP,LogSales
; Pds = 5 ; RPM = InvGood,RawMtl
; Halton ; Pts = 15 ; Cor
; Fcn = One(n),IMUM(n),FDIUM(n) ; Marginal
; Parameters $
Create; Bimum = beta_i(firm,2) $
Regress ; Lhs = Bimum ; Rhs = one,InvGood,RawMtl $
Latent Class Models
? Latent class models
Sample ; All $
Logit ; Lhs = IP ; Rhs = X ; LCM ; Pds=5 ; Pts = 3 $
Logit ; Lhs = IP ; Rhs = X ; LCM=Invgood,Rawmtl
; Pds=5 ; Pts = 3 $
Logit ; Lhs = IP ; Rhs = X ; LCM ; Pds=5 ; Pts = 4 $
Logit ; Lhs = IP ; Rhs = X ; LCM ; Pds=5 ; Pts = 5 $