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. Time until death2. Time until retirement3. Time until first marriage4. Time until first birth

Multiple Spell

1. Time until birth of each child2. 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 2Person 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):

Which leads to:

11

11 1

11

( 1)ln

( 0)

P YX

P Y

11 1

11 111( 1)

1

X

X

eP Y

e

Person 2 (No birth in the first year and a birth in the second year):

Joint probability is:

12

12 1

12

( 1)ln

( 0)

P YX

P Y

22 12

22 2

22 12

( 1| 0)ln

( 0 | 0)

P Y YX

P Y Y

22 2

22 2 12 122 12 22 12 12

1( 1, 0) ( 1| 0) ( 0)

1 1

X

X X

eP Y Y P Y Y P Y

e e

Person 3: (No births in the observation period)

Estimation:

Time 1: 3 observationsTime 2: 2 observationsTime 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.

33 23 2 13 133 23 13 3

1 1 1( 0, 0, 0)

1 1 1X X XP Y Y Y

e e e

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 covariatesUnemployment 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):

Which allows for a very flexible pattern of duration dependence – can be non-linear for example

1,

1 1 2 2 3 3

1,

( 1| 0ln

( 0 | 0ti t i

ti ti ti ti

ti t i

P Y YX T T T

P Y Y

A less flexible pattern that requires the estimation of fewer parameters is:

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:

1,

1,

( 1| 0ln

( 0 | 0ti t i

ti

ti t i

P Y YX t

P Y Y

21,

1 2

1,

( 1| 0ln

( 0 | 0ti t i

ti

ti t i

P Y YX t t

P Y Y

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 yrcal conce age

1002 63 0 101002 64 0 111002 65 0 121002 66 0 131002 67 0 141002 68 1 151003 67 0 101003 68 0 111003 69 0 121003 70 0 131003 71 0 141003 72 0 151003 73 0 161003 74 0 171003 75 0 181003 76 0 191003 77 1 20

Simple Models (linear and non-linear duration dependence):logit conce urb age yrsinsch Logistic regression Number of obs = 50115 LR chi2(3) = 1750.39 Prob > chi2 = 0.0000 Log likelihood = -14109.13 Pseudo R2 = 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 Number of obs = 50115 LR chi2(4) = 4098.82 Prob > chi2 = 0.0000 Log likelihood = -12934.914 Pseudo R2 = 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 Number of obs = 50115 LR chi2(22) = 4198.03 Prob > chi2 = 0.0000 Log likelihood = -12885.309 Pseudo R2 = 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:

where we have a time varying error and a persistent error (sometimes referred to as time invariant unobserved heterogeneity)

Define:

Then:

Yti i ti

2

2 2

1,( )ti t icor Y Y

Alternative model (state dependence):

where |α|<1

Now:

It is very difficult to distinguish between the models – so we use the hybrid model:

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.

1,ti t i tiY Y

1,( )ti t icor Y Y

1,ti t i i tiY Y

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):

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:

12 2

12 1 2

12 2

( 1| )ln

( 0 | )

P YX

P Y

22 12 2

22 2 2

22 12 2

( 1| 0, )ln

( 0 | 0, )

P Y YX

P Y Y

22 2 2

22 2 2 12 1 222 12 2

1( 1, 0 | )

1 1

X

X X

eP Y Y

e e

The unconditional joint probability is:

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):

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.

22 12 22 12 2 2 2( 1, 0) ( 1, 0 | ) ( )P Y Y P Y Y f d

22 12 22 121

( 1, 0) ( 1, 0 | )K

k kk

P Y Y w P Y Y h

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 Number of obs = 50115 Group variable: personid Number of groups = 4659 Random effects u_i ~ Gaussian Obs per group: min = 1 avg = 10.8 max = 40 Wald chi2(4) = 252.90 Log likelihood = -12767.616 Prob > chi2 = 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:

With Unobserved heterogeneity:

1.0426.00

0.04

Age

Education

3.518.36

0.42

Age

Education

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:

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):

21,

1 2

1,

( 1| 0, 1ln

( 0 | 0, 1tim t im

tim m m m m m

tim t im

P Y Y births mX t t

P Y Y births m

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 Number of obs = 101157 Wald chi2(25) = 31743.59 Log likelihood = -35873.982 Prob > chi2 = 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 Number of obs = 113995 LR chi2(5) = 7363.66 Prob > chi2 = 0.0000 Log likelihood = -41237.438 Pseudo R2 = 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:

In order to use STATA, must assume a restrictive form of unobserved heterogeneity for both parametric and non-parametric forms.

Parametric:

More flexible specification would be:

where Σ is m x m

21,

1 2

1,

( 1| 0, 1ln

( 0 | 0, 1tim t im

tim m m m m m mi

tim t im

P Y Y births mX t t

P Y Y births m

2~ (0, )mi i N

~ (0, )mi N

Estimate assuming normally distributed unobserved heterogeneity (restrict coefficients across births):

Random-effects logistic regression Number of obs = 113995 Group variable: personid Number of groups = 4659 Random effects u_i ~ Gaussian Obs per group: min = 4 avg = 24.5 max = 40 Wald chi2(5) = 4672.43 Log likelihood = -41151.302 Prob > chi2 = 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 Number of obs = 101157 Group variable: personid Number of groups = 4659 Random effects u_i ~ Gaussian Obs per group: min = 4 avg = 21.7 max = 40 Wald chi2(25) = 20608.01 Log likelihood = -35847.471 Prob > chi2 = 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 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD one -18.56915 0.4491 -41.344 -0.872E-01 -0.174E+04 urb_0 -0.39914 0.0580 -6.882 -0.368E-01 -0.751E+03 age_0 1.02592 0.0431 23.792 -0.151E+01 -0.589E+06 age_sq_0 -0.01569 0.0009 -17.165 -0.262E+02 -0.279E+09 yrsinsch -0.15850 0.0093 -17.034 -0.549E+00 -0.637E+05 OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl 5.37715 0.2247 23.935 -0.590E-01 -0.124E+04 OMEGAcl 3.44257 0.1670 20.611 -0.341E-01 -0.563E+03 RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER: 2 DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_1 UNCONDITIONAL RESULTS LOG ODDS OF CATEGORY 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD one -2.76996 0.3670 -7.547 0.907E-02 -0.268E+04 urb_1 0.12949 0.0409 3.166 0.380E-02 -0.110E+04 age_1 0.25196 0.0293 8.603 0.193E+00 -0.142E+07 age_sq_1 -0.00664 0.0006 -10.842 0.416E+01 -0.873E+09 yrsinsch 0.00529 0.0055 0.965 0.527E-01 -0.101E+06 OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl -0.73376 0.1664 -4.410 0.561E-02 -0.164E+04 OMEGAcl -0.51652 0.1566 -3.298 0.402E-02 -0.391E+03

Continued:RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER: 3 DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_2 UNCONDITIONAL RESULTS LOG ODDS OF CATEGORY 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD one -1.52150 0.4546 -3.347 0.147E-01 -0.184E+04 urb_2 0.05753 0.0447 1.287 0.607E-02 -0.730E+03 age_2 0.20727 0.0358 5.797 0.353E+00 -0.119E+07 age_sq_2 -0.00586 0.0007 -8.455 0.863E+01 -0.870E+09 yrsinsch -0.00559 0.0059 -0.947 0.784E-01 -0.654E+05 OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl -1.49893 0.1513 -9.907 0.102E-01 -0.119E+04 OMEGAcl -1.04100 0.1482 -7.025 0.521E-02 -0.258E+03 RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER: 4 DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_3 UNCONDITIONAL RESULTS LOG ODDS OF CATEGORY 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD one -2.38295 0.6737 -3.537 0.181E-01 -0.123E+04 urb_3 0.04127 0.0565 0.730 0.724E-02 -0.488E+03 age_3 0.25481 0.0475 5.365 0.478E+00 -0.937E+06 age_sq_3 -0.00675 0.0008 -7.949 0.129E+02 -0.794E+09 yrsinsch -0.01997 0.0077 -2.590 0.856E-01 -0.389E+05 OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl -1.23289 0.1889 -6.525 0.124E-01 -0.814E+03 OMEGAcl -0.48938 0.1995 -2.453 0.574E-02 -0.168E+03

Continued:

RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER: 5 DEPENDENT VARIABLE (LOGIT TYPE EQUATION): conce_4 UNCONDITIONAL RESULTS LOG ODDS OF CATEGORY 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD one -1.63142 0.9123 -1.788 0.116E-01 -0.802E+03 urb_4 0.07063 0.0722 0.978 0.450E-02 -0.305E+03 age_4 0.25541 0.0613 4.168 0.321E+00 -0.687E+06 age_sq_4 -0.00675 0.0010 -6.453 0.905E+01 -0.649E+09 yrsinsch -0.03041 0.0101 -2.998 0.496E-01 -0.232E+05 OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl -1.76729 0.2657 -6.652 0.914E-02 -0.504E+03 OMEGAcl -1.03957 0.2705 -3.843 0.277E-02 -0.120E+03 PROBABILITY WEIGHT RESULTS POINT # PROBABILITY WEIGHT 1 0.06040246 2 0.63953684 3 0.30006070

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.