Transcript Single and Multiple Spell Discrete Time Hazards Models with

Single and Multiple Spell Discrete
Time Hazards Models with
Parametric and Non-Parametric
Corrections for Unobserved
Heterogeneity
David K. Guilkey
Demographic Applications:
Single Spell
1.
2.
3.
4.
Time until death
Time until retirement
Time until first marriage
Time until first birth
Multiple Spell
1. Time until birth of each child
2. Duration of each spell of employment
We will use time until first birth and the timing of subsequent
births as an example throughout the presentation.
The variable of interest is: P(t ≤ T < t+n | T > t)
This is the conditional probability that an individual experiences the
event between t and t+n given that she has not experienced the event
until that time.
Example:
The dependent variable is the timing of a first birth. Suppose the
discrete time interval is a year and we observe each woman from the
beginning of her child bearing years:
0…..1…..2…..3
Consider three cases:
Person 1: Has a birth in year 1 (time 0 may be age 12)
Person 2: Has a birth in year 2
Person 3: Still has not had a birth at the end of the observation period
Some important notes:
1. Since we are following the woman from the beginning of her child
bearing years, we have eliminated the possibility of left censoring (the
event occurs before the observation period).
2. Left censoring combined with unobserved heterogeneity introduces
bias into the estimation results. The correction requires the estimation
of an “initial conditions” equation similar to Heckman selection
equation which are well known to yield unstable parameter estimates.
3. The third person is right censored. However, right censoring is easily
handled as part of the estimation process.
4. As will be seen below, the dependent variable in a discrete time
hazard model is dichotomous. Can use probit, logit or complementary
log log (cloglog) models. Logit and cloglog are most often used. I use
logit since one of the software packages needs logit – results were
nearly the same for cloglog in models where software allowed for both
(STATA).
The model:
Person 1 (birth occurs in the first interval):
 P(Y  1) 
ln 
X 

 P(Y  0) 
11
11
11
Which leads to:
X111
e
P(Y  1) 
1 e
11
X111
1
Person 2 (No birth in the first year and a birth in the second year):
 P(Y  1) 
ln 
X 

 P(Y  0) 
12
12
1
12
 P(Y  1| Y  0) 
ln 
X 

 P(Y  0 | Y  0) 
22
12
22
22
2
12
Joint probability is:
X 22  2
e
P(Y  1, Y  0)  P(Y  1| Y  0) P(Y  0) 
1 e
22
12
22
12
12
X 22  2
1
1 e
X12 1
Person 3: (No births in the observation period)
P(Y  0, Y  0, Y  0) 
33
23
13
1
1 e
X 33  3
1
1 e
X 23  2
1
1 e
X13 1
Estimation:
Time 1: 3 observations
Time 2: 2 observations
Time 3: 1 observation
The three sets of coefficients could be estimated in three separate logits
for the set of individuals at risk. This is true since there is no unobserved
heterogeneity that links the three time periods together.
Duration dependence
This is a concept similar to state dependence in a standard panel
data model.
Duration dependence occurs when the value of the hazard at any
point in time depends on the amount of time that has already
elapsed.
Relates to the propensity of a state towards self-perpetuation
Examples:
Mortality – hazard increases with time regardless of the values of
the other covariates
Unemployment duration – hazard of finding employment may
decrease as the length of the unemployment spell increases
Modeling Duration Dependence
In our current model, duration dependence is captured by the intercept
terms in the equation since they are allowed to differ at each point in
time.
To see more clearly, assume that the effects of the covariates is the
same at each point in time (the β’s are the same in the previous
equations).
Now define T1ti=1 if if t=1 and 0 otherwise – with T2ti and T3ti defined
similarly
Then we can write (no constant in the model):
 P(Y  1| Y  0 
ln 
  X   T  T  T
 P(Y  0 | Y  0 
ti
t 1,i
ti
ti
1
1 ti
2
2 ti
3
3 ti
t 1,i
Which allows for a very flexible pattern of duration dependence – can
be non-linear for example
A less flexible pattern that requires the estimation of fewer parameters
is:
 P(Y  1| Y  0 
ln 
  X   t
P
(
Y

0
|
Y

0


t 1,i
ti
ti
t 1,i
ti
In our example, we will be examining the birth hazard starting all
women at age 10 and so age and duration dependence are not
separately identified.
A parametric model which allows for non-linear duration dependence
is:
 P(Y  1| Y  0 
ln 
  X   t  t
P
(
Y

0
|
Y

0


ti
t 1,i
ti
ti
t 1,i
1
2
2
Empirical Example
Data from Indonesia Family Life Survey.
We first examine timing of first birth – women followed from age 10
until first birth.
Data set up:
personid
1002
1002
1002
1002
1002
1002
1003
1003
1003
1003
1003
1003
1003
1003
1003
1003
1003
yrcal
63
64
65
66
67
68
67
68
69
70
71
72
73
74
75
76
77
conce
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
age
10
11
12
13
14
15
10
11
12
13
14
15
16
17
18
19
20
Simple Models (linear and non-linear duration dependence):
logit
conce urb age yrsinsch
Logistic regression
Log likelihood =
-14109.13
Number of obs
LR chi2(3)
Prob > chi2
Pseudo R2
=
=
=
=
50115
1750.39
0.0000
0.0584
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------urb | -.2801102
.0341458
-8.20
0.000
-.3470347
-.2131856
age |
.1079184
.0025615
42.13
0.000
.102898
.1129388
yrsinsch |
.0025478
.0041978
0.61
0.544
-.0056798
.0107754
_cons | -4.084346
.0519202
-78.67
0.000
-4.186107
-3.982584
------------------------------------------------------------------------------
Logistic regression
Log likelihood = -12934.914
Number of obs
LR chi2(4)
Prob > chi2
Pseudo R2
=
=
=
=
50115
4098.82
0.0000
0.1368
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------urb | -.2597377
.0347479
-7.47
0.000
-.3278422
-.1916331
age |
1.044837
.0264714
39.47
0.000
.9929543
1.09672
age_sq | -.0219638
.0006486
-33.87
0.000
-.023235
-.0206927
yrsinsch | -.0448548
.0041327
-10.85
0.000
-.0529547
-.0367549
_cons | -13.08654
.2603383
-50.27
0.000
-13.59679
-12.57628
------------------------------------------------------------------------------
Non-parametric Duration Dependence (using duration or age
dummies):
Logistic regression
Log likelihood = -12885.309
Number of obs
LR chi2(22)
Prob > chi2
Pseudo R2
=
=
=
=
50115
4198.03
0.0000
0.1401
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------urb | -.2558662
.0346786
-7.38
0.000
-.3238349
-.1878975
agedum1 | -4.966854
.587953
-8.45
0.000
-6.119221
-3.814487
agedum2 | -2.098861
.1787642
-11.74
0.000
-2.449232
-1.748489
agedum3 | -1.773665
.1644908
-10.78
0.000
-2.096061
-1.451269
agedum4 | -1.262905
.1467448
-8.61
0.000
-1.550519
-.9752901
agedum5 | -.6653681
.1331233
-5.00
0.000
-.926285
-.4044513
agedum6 | -.1173774
.1256073
-0.93
0.350
-.3635632
.1288084
agedum7 |
.235519
.1228915
1.92
0.055
-.0053439
.476382
agedum8 |
.6506007
.1207185
5.39
0.000
.4139968
.8872045
agedum9 |
.8665138
.1209088
7.17
0.000
.6295368
1.103491
agedum10 |
1.034363
.121797
8.49
0.000
.7956458
1.273081
agedum11 |
1.171486
.1234081
9.49
0.000
.9296107
1.413361
agedum12 |
1.182254
.1267102
9.33
0.000
.9339064
1.430601
agedum13 |
1.373427
.1289028
10.65
0.000
1.120783
1.626072
agedum14 |
1.381945
.1340668
10.31
0.000
1.119179
1.644711
agedum15 |
1.419845
.1399481
10.15
0.000
1.145552
1.694138
agedum16 |
1.18434
.1526529
7.76
0.000
.8851458
1.483534
agedum17 |
1.330018
.15859
8.39
0.000
1.019187
1.640848
agedum18 |
1.191952
.174294
6.84
0.000
.8503424
1.533562
agedum19 |
.855728
.2002908
4.27
0.000
.4631651
1.248291
agedum20 |
.6546571
.2260183
2.90
0.004
.2116694
1.097645
yrsinsch | -.0438339
.004139
-10.59
0.000
-.0519462
-.0357216
_cons | -2.157563
.1115451
-19.34
0.000
-2.376188
-1.938939
------------------------------------------------------------------------------
Duration Dependence and Unobserved Heterogeneity
Review of dynamic panel data model:
Yti  i   ti
where we have a time varying error and a persistent error (sometimes
referred to as time invariant unobserved heterogeneity)
Define:
2

Then:
cor (YtiYt 1,i )  

2
2
   
Alternative model (state dependence):
Yti  Yt 1,i   ti
where |α|<1
Now:
cor (YtiYt 1,i )  
It is very difficult to distinguish between the models – so we use the
hybrid model:
Yti  Yt 1,i  i   ti
A problem is that this model is more difficult to estimate – neither
ordinary least squares nor fixed effects methods yield consistent
estimators – use maximum likelihood (with initial conditions problem)
or instrumental variables.
Return to first example and unobserved heterogeneity (using person 2
as the example):
Person 2 (No birth in the first year and a birth in the second year):
 P(Y  1|  ) 
ln 
 X  

 P(Y  0 |  ) 
12
2
12
12
1
2
2
 P(Y  1| Y  0,  ) 
ln 
 X  

 P(Y  0 | Y  0,  ) 
22
12
2
22
12
2
22
2
2
We can no longer estimate parameters time period by time period –
due to selection on unobservables (just as in standard Heckman
selectivity model)
Joint probability is now:
X 22  2  2
e
P(Y  1, Y  0 |  ) 
1 e
22
12
2
X 22  2  2
1
1 e
X12 1   2
The unconditional joint probability is:

P(Y  1, Y  0)   P(Y  1, Y  0 |  ) f (  )d 
22
12

22
12
2
2
2
Most commonly used distributional assumption for the unobserved
heterogeneity is the normal distribution. The integral is approximated
using Hermite point and weights (simply looked up in a table for the
normal distribution):
P(Y  1, Y  0)   w P(Y  1,Y  0 | h )
K
22
12
k 1
k
22
12
k
K is the number of interpolation points – more accurate to add more
but slower (STATA default is 12 – frequently not enough for rare events)
Heckman-Singer approach: Do not assume a distribution – directly
estimate the points and weights as part of the maximum likelihood
estimation process – referred to as the discrete factor approximation.
Identification
“it is somewhat heroic to think that we can distinguish between
duration dependence and unobserved heterogeneity when we
only observe a single cycle for each agent” (Wooldridge – page
705)
Example:
Model with no censoring estimated by OLS.
Can identify both using functional form – but the model
parameter estimates are frequently unstable.
Examples
Assume normality:
. xtlogit conce urb age age_sq yrsinsch,i(personid) intp(20)
Random-effects logistic regression
Group variable: personid
Number of obs
Number of groups
=
=
50115
4659
Random effects u_i ~ Gaussian
Obs per group: min =
avg =
max =
1
10.8
40
Log likelihood
= -12767.616
Wald chi2(4)
Prob > chi2
=
=
252.90
0.0000
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------urb | -.8368256
.1328742
-6.30
0.000
-1.097254
-.576397
age |
3.508179
.2292176
15.31
0.000
3.05892
3.957437
age_sq | -.0544224
.0035137
-15.49
0.000
-.0613092
-.0475356
yrsinsch | -.4246292
.0365909
-11.60
0.000
-.4963459
-.3529124
_cons |
-44.2749
2.861672
-15.47
0.000
-49.88368
-38.66613
-------------+---------------------------------------------------------------/lnsig2u |
3.424721
.1417704
3.146856
3.702585
-------------+---------------------------------------------------------------sigma_u |
5.542027
.3928477
4.823153
6.368046
rho |
.9032504
.0123892
.8761003
.9249606
-----------------------------------------------------------------------------Likelihood-ratio test of rho=0: chibar2(01) =
334.60 Prob >= chibar2 = 0.000
Cannot directly compare the coefficients with and without
heterogeneity correction because of possible scale differences
for discrete dependent variable models. However, scale effects
can be removed if you compare ratios of coefficients:
Without unobserved heterogeneity:
Age
1.04

 26.00
Education 0.04
With Unobserved heterogeneity:
Age
3.51

 8.36
Education 0.42
Use Discrete Factor Method
. gllamm conce urb age age_sq yrsinsch,i(personid) family(binomial) link(logit) nip(3) ip(f) trace dot
number of level 1 units = 50115
number of level 2 units = 4659
Condition Number = 6445.9732
gllamm model
log likelihood = -12719.097
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------urb | -.4086555
.0536189
-7.62
0.000
-.5137466
-.3035643
age |
1.022321
.0402732
25.38
0.000
.9433868
1.101255
age_sq | -.0156502
.0009014
-17.36
0.000
-.0174168
-.0138836
yrsinsch | -.1584645
.0086805
-18.26
0.000
-.1754779
-.1414511
_cons | -14.04514
.4201875
-33.43
0.000
-14.86869
-13.22158
-----------------------------------------------------------------------------Probabilities and locations of random effects
-----------------------------------------------------------------------------***level 2 (personid)
loc1: -4.494, -1.0551, .89329
var(1): 2.0312478
prob: 0.0588, 0.296, 0.6453
Multiple Spell Discrete Time Hazards Models
Model with no unobserved heterogeneity:
Allow for M births:
 P(Y  1| Y
ln 
 P(Y  0 | Y
tim
tim
t 1,im
t 1,im
 0, births  m  1 
  X   t  t
 0, births  m  1
tim
m
1m m
2
2m m
With no heterogeneity, estimate M+1 single spell hazards models
(or fully interacted model).
Results for fully interacted model (m=0,1,2,3,4):
logit conce parity_0 urb_0 age_0 age_sq_0 yrsinsch_0 parity_1 urb_1 age_1 age_sq_1 yrsinsch_1 parity_2
urb_2 age_2 age_sq_2 yrsinsch_2 parity_3 urb_3 age_3 age_sq_3 yrsinsch_3
> parity_4 urb_4 age_4 age_sq_4 yrsinsch_4,nocons
Logistic regression
Log likelihood = -35873.982
Number of obs
Wald chi2(25)
Prob > chi2
=
=
=
101157
31743.59
0.0000
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------parity_0 |
-13.0865
.2603375
-50.27
0.000
-13.59676
-12.57625
urb_0 | -.2597377
.0347479
-7.47
0.000
-.3278422
-.1916331
age_0 |
1.044834
.0264713
39.47
0.000
.9929509
1.096716
age_sq_0 | -.0219637
.0006486
-33.87
0.000
-.0232349
-.0206926
yrsinsch_0 | -.0448548
.0041327
-10.85
0.000
-.0529547
-.0367549
parity_1 | -3.542908
.3506967
-10.10
0.000
-4.23026
-2.855555
urb_1 |
.1140113
.0407575
2.80
0.005
.0341281
.1938945
age_1 |
.2507085
.0302698
8.28
0.000
.1913807
.3100363
age_sq_1 | -.0062789
.0006319
-9.94
0.000
-.0075174
-.0050403
yrsinsch_1 | -.0006391
.0050614
-0.13
0.900
-.0105593
.0092811
parity_2 | -3.061192
.4668261
-6.56
0.000
-3.976155
-2.14623
urb_2 |
.0445203
.046588
0.96
0.339
-.0467904
.1358311
age_2 |
.2043045
.0367711
5.56
0.000
.1322346
.2763745
age_sq_2 | -.0053638
.0007064
-7.59
0.000
-.0067484
-.0039792
yrsinsch_2 | -.0152708
.0060049
-2.54
0.011
-.0270402
-.0035013
parity_3 | -3.860511
.677871
-5.70
0.000
-5.189114
-2.531909
urb_3 |
.0284173
.0568567
0.50
0.617
-.0830197
.1398543
age_3 |
.2695803
.0493892
5.46
0.000
.1727792
.3663814
age_sq_3 | -.0065998
.0008828
-7.48
0.000
-.00833
-.0048696
yrsinsch_3 |
-.029156
.0077006
-3.79
0.000
-.0442488
-.0140632
parity_4 | -3.382981
.9368919
-3.61
0.000
-5.219256
-1.546707
urb_4 |
.0550487
.0717479
0.77
0.443
-.0855747
.195672
age_4 |
.2516846
.0645228
3.90
0.000
.1252223
.3781469
age_sq_4 | -.0063117
.001094
-5.77
0.000
-.0084558
-.0041676
yrsinsch_4 | -.0403808
.0099802
-4.05
0.000
-.0599416
-.02082
------------------------------------------------------------------------------
Simple Model with coefficients restricted to be the same (using all
available births for all women):
. logit conce urb age age_sq yrsinsch parity
Logistic regression
Log likelihood = -41237.438
Number of obs
LR chi2(5)
Prob > chi2
Pseudo R2
=
=
=
=
113995
7363.66
0.0000
0.0820
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------urb | -.0696638
.0191482
-3.64
0.000
-.1071936
-.0321341
age |
.6257251
.0092857
67.39
0.000
.6075256
.6439246
age_sq | -.0127468
.0001916
-66.51
0.000
-.0131224
-.0123712
yrsinsch | -.0275519
.0024491
-11.25
0.000
-.0323521
-.0227518
parity |
.0479533
.0065391
7.33
0.000
.0351368
.0607697
_cons | -8.751946
.1082908
-80.82
0.000
-8.964192
-8.5397
------------------------------------------------------------------------------
Add unobserved heterogeneity to the model:
 P(Y  1| Y
ln 
 P(Y  0 | Y
t 1,im
tim
t 1,im
tim
 0, births  m  1 
  X   t  t  
 0, births  m  1
2
tim
m
1m m
2m m
mi
In order to use STATA, must assume a restrictive form of
unobserved heterogeneity for both parametric and nonparametric forms.
Parametric:
mi  i ~ N (0, 2 )
More flexible specification would be:
mi ~ N (0, )
where Σ is m x m
Estimate assuming normally distributed unobserved
heterogeneity (restrict coefficients across births):
Random-effects logistic regression
Group variable: personid
Number of obs
Number of groups
=
=
113995
4659
Random effects u_i ~ Gaussian
Obs per group: min =
avg =
max =
4
24.5
40
Log likelihood
= -41151.302
Wald chi2(5)
Prob > chi2
=
=
4672.43
0.0000
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------urb |
-.118293
.0259885
-4.55
0.000
-.1692295
-.0673565
age |
.6851735
.0107436
63.78
0.000
.6641164
.7062305
age_sq | -.0130654
.0001963
-66.57
0.000
-.0134501
-.0126807
yrsinsch | -.0393268
.0034555
-11.38
0.000
-.0460993
-.0325542
parity | -.1675735
.0182618
-9.18
0.000
-.2033661
-.131781
_cons | -9.588753
.1333153
-71.93
0.000
-9.850046
-9.32746
-------------+---------------------------------------------------------------/lnsig2u | -1.123201
.1024267
-1.323954
-.9224486
-------------+---------------------------------------------------------------sigma_u |
.5702955
.0292067
.5158305
.6305112
rho |
.0899661
.0083859
.074827
.1078112
-----------------------------------------------------------------------------Likelihood-ratio test of rho=0: chibar2(01) =
172.27 Prob >= chibar2 = 0.000
Use the discrete factor model (restrict coefficients across births):
gllamm model
log likelihood = -41134.644
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------urb | -.1047852
.0244908
-4.28
0.000
-.1527862
-.0567841
age |
.6769451
.0102551
66.01
0.000
.6568455
.6970448
age_sq | -.0130116
.0001951
-66.68
0.000
-.013394
-.0126291
yrsinsch | -.0362991
.003322
-10.93
0.000
-.0428102
-.0297881
parity | -.1423997
.0153752
-9.26
0.000
-.1725345
-.1122649
_cons | -9.488505
.1247421
-76.06
0.000
-9.732995
-9.244015
-----------------------------------------------------------------------------Probabilities and locations of random effects
-----------------------------------------------------------------------------***level 2 (personid)
loc1: -2.0664, 1.0212, -.07852
var(1): .32189016
prob: 0.0394, 0.1426, 0.818
------------------------------------------------------------------------------
Normally distributed unobserved heterogeneity (unrestricted
coefficients):
Random-effects logistic regression
Group variable: personid
Number of obs
Number of groups
=
=
101157
4659
Random effects u_i ~ Gaussian
Obs per group: min =
avg =
max =
4
21.7
40
Log likelihood
= -35847.471
Wald chi2(25)
Prob > chi2
=
=
20608.01
0.0000
-----------------------------------------------------------------------------conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------parity_0 | -13.46569
.27736
-48.55
0.000
-14.00931
-12.92207
urb_0 | -.3062007
.0396614
-7.72
0.000
-.3839355
-.2284658
age_0 |
1.063886
.0274376
38.77
0.000
1.01011
1.117663
age_sq_0 | -.0215194
.0006603
-32.59
0.000
-.0228135
-.0202252
yrsinsch_0 | -.0571776
.005131
-11.14
0.000
-.0672342
-.047121
parity_1 | -4.654384
.4273876
-10.89
0.000
-5.492048
-3.81672
urb_1 |
.0922073
.0453369
2.03
0.042
.0033486
.181066
age_1 |
.3206417
.0347263
9.23
0.000
.2525795
.3887039
age_sq_1 |
-.006944
.0006775
-10.25
0.000
-.0082718
-.0056162
yrsinsch_1 | -.0121149
.0059108
-2.05
0.040
-.0236998
-.00053
parity_2 | -4.545196
.5651961
-8.04
0.000
-5.65296
-3.437432
urb_2 |
.0211196
.0510565
0.41
0.679
-.0789492
.1211884
age_2 |
.2891532
.0419981
6.88
0.000
.2068384
.3714679
age_sq_2 | -.0063295
.0007663
-8.26
0.000
-.0078314
-.0048275
yrsinsch_2 | -.0268969
.0068016
-3.95
0.000
-.0402278
-.013566
parity_3 | -5.841153
.7929613
-7.37
0.000
-7.395329
-4.286977
urb_3 |
.0121073
.0617448
0.20
0.845
-.1089103
.1331249
age_3 |
.3764971
.0551655
6.82
0.000
.2683748
.4846194
age_sq_3 | -.0079421
.0009533
-8.33
0.000
-.0098106
-.0060737
yrsinsch_3 | -.0436156
.0085985
-5.07
0.000
-.0604683
-.0267629
parity_4 | -5.610291
1.062393
-5.28
0.000
-7.692542
-3.528039
urb_4 |
.0415766
.0772656
0.54
0.591
-.1098611
.1930143
age_4 |
.3614516
.0705488
5.12
0.000
.2231785
.4997246
age_sq_4 | -.0076617
.0011692
-6.55
0.000
-.0099532
-.0053702
yrsinsch_4 | -.0565882
.0110086
-5.14
0.000
-.0781647
-.0350117
-------------+---------------------------------------------------------------/lnsig2u | -1.395746
.1873089
-1.762865
-1.028628
-------------+---------------------------------------------------------------sigma_u |
.4976426
.0466064
.4141892
.5979107
rho |
.0700062
.0121948
.0495613
.0980152
-----------------------------------------------------------------------------Likelihood-ratio test of rho=0: chibar2(01) =
53.02 Prob >= chibar2 = 0.000
Discrete factor model with two points of support (unrestricted
coefficients):
conce |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------parity_0 | -12.53528
.2736712
-45.80
0.000
-13.07167
-11.99889
urb_0 | -.2984987
.039212
-7.61
0.000
-.3753528
-.2216446
age_0 |
.957689
.0284762
33.63
0.000
.9018766
1.013501
age_sq_0 | -.0183521
.00073
-25.14
0.000
-.0197829
-.0169214
yrsinsch_0 | -.0759569
.0051296
-14.81
0.000
-.0860107
-.065903
parity_1 | -3.094125
.3708715
-8.34
0.000
-3.82102
-2.367231
urb_1 |
.1217308
.0454716
2.68
0.007
.032608
.2108535
age_1 |
.1805441
.0330432
5.46
0.000
.1157807
.2453076
age_sq_1 | -.0039468
.0007335
-5.38
0.000
-.0053844
-.0025091
yrsinsch_1 | -.0223432
.0061971
-3.61
0.000
-.0344893
-.0101971
parity_2 | -3.071576
.4702316
-6.53
0.000
-3.993213
-2.149939
urb_2 |
.0426571
.0476055
0.90
0.370
-.0506479
.1359621
age_2 |
.187491
.0371968
5.04
0.000
.1145867
.2603953
age_sq_2 | -.0048974
.0007201
-6.80
0.000
-.0063089
-.003486
yrsinsch_2 | -.0197252
.0062636
-3.15
0.002
-.0320017
-.0074487
parity_3 | -4.005294
.6792858
-5.90
0.000
-5.33667
-2.673918
urb_3 |
.0280673
.0570765
0.49
0.623
-.0838007
.1399352
age_3 |
.2659887
.0495095
5.37
0.000
.1689519
.3630255
age_sq_3 | -.0065128
.0008855
-7.36
0.000
-.0082483
-.0047774
yrsinsch_3 |
-.030071
.0077417
-3.88
0.000
-.0452443
-.0148976
parity_4 | -3.568676
.9377294
-3.81
0.000
-5.406592
-1.73076
urb_4 |
.0551082
.0718064
0.77
0.443
-.0856299
.1958462
age_4 |
.2515324
.0645679
3.90
0.000
.1249815
.3780832
age_sq_4 | -.0063062
.0010947
-5.76
0.000
-.0084519
-.0041606
yrsinsch_4 | -.0405498
.0099895
-4.06
0.000
-.0601288
-.0209707
-----------------------------------------------------------------------------Probabilities and locations of random effects
***level 2 (personid)
loc1: -1.7338, .18739
var(1): .32490087
prob: 0.0975, 0.9025
Add non-parametric unobserved heterogeneity with three points
of support – unrestricted across equations using fortran:
RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER:
1
DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_0
UNCONDITIONAL RESULTS
LOG ODDS OF CATEGORY
RHS. VAR.
one
urb_0
age_0
age_sq_0
yrsinsch
OMEGAcl
OMEGAcl
OMEGAcl
COEFFICIENT
-18.56915
-0.39914
1.02592
-0.01569
-0.15850
0.0
5.37715
3.44257
2 RELATIVE TO CATEGORY 1
STD. ERR.
T-SCORE
FPD
0.4491
-41.344 -0.872E-01
0.0580
-6.882 -0.368E-01
0.0431
23.792 -0.151E+01
0.0009
-17.165 -0.262E+02
0.0093
-17.034 -0.549E+00
-- NORMALIZED AT ZERO
0.2247
23.935 -0.590E-01
0.1670
20.611 -0.341E-01
RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER:
SPD
-0.174E+04
-0.751E+03
-0.589E+06
-0.279E+09
-0.637E+05
-0.124E+04
-0.563E+03
2
DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_1
UNCONDITIONAL RESULTS
LOG ODDS OF CATEGORY
RHS. VAR.
one
urb_1
age_1
age_sq_1
yrsinsch
OMEGAcl
OMEGAcl
OMEGAcl
COEFFICIENT
-2.76996
0.12949
0.25196
-0.00664
0.00529
0.0
-0.73376
-0.51652
2 RELATIVE TO CATEGORY 1
STD. ERR.
T-SCORE
FPD
0.3670
-7.547
0.907E-02
0.0409
3.166
0.380E-02
0.0293
8.603
0.193E+00
0.0006
-10.842
0.416E+01
0.0055
0.965
0.527E-01
-- NORMALIZED AT ZERO
0.1664
-4.410
0.561E-02
0.1566
-3.298
0.402E-02
SPD
-0.268E+04
-0.110E+04
-0.142E+07
-0.873E+09
-0.101E+06
-0.164E+04
-0.391E+03
Continued:
RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER:
3
DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_2
UNCONDITIONAL RESULTS
LOG ODDS OF CATEGORY
RHS. VAR.
one
urb_2
age_2
age_sq_2
yrsinsch
OMEGAcl
OMEGAcl
OMEGAcl
COEFFICIENT
-1.52150
0.05753
0.20727
-0.00586
-0.00559
0.0
-1.49893
-1.04100
2 RELATIVE TO CATEGORY 1
STD. ERR.
T-SCORE
FPD
0.4546
-3.347
0.147E-01
0.0447
1.287
0.607E-02
0.0358
5.797
0.353E+00
0.0007
-8.455
0.863E+01
0.0059
-0.947
0.784E-01
-- NORMALIZED AT ZERO
0.1513
-9.907
0.102E-01
0.1482
-7.025
0.521E-02
RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER:
SPD
-0.184E+04
-0.730E+03
-0.119E+07
-0.870E+09
-0.654E+05
-0.119E+04
-0.258E+03
4
DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_3
UNCONDITIONAL RESULTS
LOG ODDS OF CATEGORY
RHS. VAR.
one
urb_3
age_3
age_sq_3
yrsinsch
OMEGAcl
OMEGAcl
OMEGAcl
COEFFICIENT
-2.38295
0.04127
0.25481
-0.00675
-0.01997
0.0
-1.23289
-0.48938
2 RELATIVE TO CATEGORY 1
STD. ERR.
T-SCORE
FPD
0.6737
-3.537
0.181E-01
0.0565
0.730
0.724E-02
0.0475
5.365
0.478E+00
0.0008
-7.949
0.129E+02
0.0077
-2.590
0.856E-01
-- NORMALIZED AT ZERO
0.1889
-6.525
0.124E-01
0.1995
-2.453
0.574E-02
SPD
-0.123E+04
-0.488E+03
-0.937E+06
-0.794E+09
-0.389E+05
-0.814E+03
-0.168E+03
Continued:
RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER:
5
DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_4
UNCONDITIONAL RESULTS
LOG ODDS OF CATEGORY
RHS. VAR.
one
urb_4
age_4
age_sq_4
yrsinsch
OMEGAcl
OMEGAcl
OMEGAcl
COEFFICIENT
-1.63142
0.07063
0.25541
-0.00675
-0.03041
0.0
-1.76729
-1.03957
2 RELATIVE TO CATEGORY 1
STD. ERR.
T-SCORE
FPD
0.9123
-1.788
0.116E-01
0.0722
0.978
0.450E-02
0.0613
4.168
0.321E+00
0.0010
-6.453
0.905E+01
0.0101
-2.998
0.496E-01
-- NORMALIZED AT ZERO
0.2657
-6.652
0.914E-02
0.2705
-3.843
0.277E-02
PROBABILITY WEIGHT RESULTS
POINT #
1
2
3
PROBABILITY WEIGHT
0.06040246
0.63953684
0.30006070
SPD
-0.802E+03
-0.305E+03
-0.687E+06
-0.649E+09
-0.232E+05
-0.504E+03
-0.120E+03
Can use likelihood ratio test to compare model without
heterogeneity to:
1. Discrete factor model with two points of support using STATA
where we have a restricted form of heterogeneity
2. Discrete factor model with three points of support and
unrestricted heterogeneity using fortran.
Tests sequentially reject the simpler models with p levels close to
zero.