Longitudinal and Multilevel Methods for Models with Discrete

Outcomes with Parametric and Non-Parametric Corrections for

Unobserved Heterogeneity

David K. Guilkey

Focus of this talk:

Binary dependent variablesUnordered categorical dependent variables

Models will be logit based – will not discuss probit, poisson or negative binomial models although STATA has methods for these estimators as well

Empirical example uses data from the Indonesian Family Life Survey:

Two outcomes:

Binary indicator for whether the respondent uses contraceptionUnordered categorical variable for method choice

Data Set Overview

Four waves of data: 1993, 1997, 2000, and 2007Individual level information on fertility, education, migrationCommunity and facility level data on health and family planning providers

Data from 321 enumeration areas – we will consider these communities

IFLS Longitudinal Sample SizeInitial Participation Cohort Survey Year

1993 1997 2000 2007 totalWave 1 Cohort 3520 2873 2684 1498 10575Wave 2 Cohort 2207 1742 1152 5101Wave 3 Cohort 1466 933 2399Wave 4 Cohort 2287 2287total observations 20362

IFLS Summary Statistics

mean

s.d.

Dependent Variables Contraceptive Use .588 .492

Method Choice no method .412 temporary modern .397 long Lasting modern .168 traditional .023 Independent Variables

highest ed grade school 0.669

0.470 highest ed high school 0.169

0.375

highest ed college 0.049

0.217 age 34.099

8.842

muslim 0.891

0.312 number of posyandus 7.507

6.251

Observations 20,000

Basic Model for Longitudinal Logit:

Where:

Yti: observed binary variable (respondent i from time period t)

Xti: time varying explanatory variables (age and education level)

Pti: time varying program variable (posyandus)

Zi: time invariant regressors (Muslim)

i=1,2,…N (individuals)

t=1,2,…Ti (observations per individual -- unbalanced panel)

( 1 | )ln

( 0 | )ti i

ti ti i i

ti i

P YX P Z

P Y

Assumptions:

for the parametric logit in STATA (xtlogit, melogit, and one variant of GLLAMM)and:

Note that observations for the same individual will be correlated because of the time invariant error – sometimes referred to as unobserved heterogeneity

Given the assumptions, estimation options are:

1. Simple logit yields consistent point estimates but incorrect SE’s2. Simple logit with cluster option corrects SE’s3. Parametric or semi-parametric maximum likelihood

~ (0,1)iN

( ) ( ) ( ) 0E X E P E Zti i ti i i i

The likelihood function for this model is derived as follows:

This is the probability that individual i at time t is using contraception conditional on time invariant heterogeneity.

For individual i, we observe Ti binary responses that we can write as:

Yi = (1,0,0,1) for a woman that is observed for 4 time periods andused contraception at times 1 and 4.

( 1| )1

ti ti i i

ti ti i i

X P Z

ti i X P Z

eP Y

e

Let Yi be the set of observed outcomes for individual i, then:

Joint probability must be approximated -- approximatingthe area under a curve. With the assumption of normalitythe approximation method is Gaussian Quadrature or Hermiteintegration

Points:

1. More accurate with more Hermite points – but execution time is longer.

2. You need more points as Ti gets larger.

1

1( ) ( 1| ) (1 ( 1| ) ( )

iti ti

TY Y

i ti i ti i i itP Y P Y P Y f d

Hermite integration replaces the integral with a sum:

where the weights (wm ’s) and the masspoints (μm’s) are known because of the assumption of normality

Alternative:

The discrete factor approximation searches over weights and mass points along with the other parameters of the model.

Must impose a normalization;

1. Weights sum to one2. Either set one mass point to zero (fortran program) or set mean of distribution to zero (GLLAMM)

1

1 1( ) ( 1| ) (1 ( 1| )

iti ti

TMY Y

i m ti m ti mm tP Y w P Y P Y

Simple Logit

Simple Logit with Corrected Se’s

logit cont_use posyandus age grade_school high_school college muslim Logistic regression Number of obs = 20000 LR chi2(6) = 494.70 Prob > chi2 = 0.0000 Log likelihood = -13304.212 Pseudo R2 = 0.0183 ------------------------------------------------------------------------------ cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- posyandus | .0246126 .00295 8.34 0.000 .0188306 .0303946 age | -.0267579 .0016964 -15.77 0.000 -.0300828 -.0234329 grade_school | .488657 .0469818 10.40 0.000 .3965744 .5807396 high_school | .3713791 .0569712 6.52 0.000 .2597175 .4830406 college | .3879987 .0789006 4.92 0.000 .2333564 .542641 muslim | -.0432766 .0468967 -0.92 0.356 -.1351924 .0486392 _cons | .7282074 .0909596 8.01 0.000 .5499298 .906485 ------------------------------------------------------------------------------

logit cont_use posyandus age grade_school high_school college muslim, cluster(ind_id) Logistic regression Number of obs = 20000 Wald chi2(6) = 346.25 Prob > chi2 = 0.0000 Log pseudolikelihood = -13304.212 Pseudo R2 = 0.0183 (Std. Err. adjusted for 9351 clusters in ind_id) ------------------------------------------------------------------------------ | Robust cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- posyandus | .0246126 .0035385 6.96 0.000 .0176772 .031548 age | -.0267579 .0019687 -13.59 0.000 -.0306165 -.0228993 grade_school | .488657 .0573944 8.51 0.000 .376166 .601148 high_school | .3713791 .0679893 5.46 0.000 .2381225 .5046357 college | .3879987 .0930589 4.17 0.000 .2056065 .5703909 muslim | -.0432766 .057396 -0.75 0.451 -.1557708 .0692176 _cons | .7282074 .1084511 6.71 0.000 .5156472 .9407677 ------------------------------------------------------------------------------

Parametric Maximum Likelihood

gllamm cont_use posyandus age grade_school high_school college muslim, i(ind_id) family(binomial) link(logit) nip(20) ip(g) trace dot log likelihood = -12661.672 ------------------------------------------------------------------------------ cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- posyandus | .0316745 .0049243 6.43 0.000 .022023 .0413259 age | -.0378263 .0026797 -14.12 0.000 -.0430785 -.0325741 grade_school | .651587 .0775775 8.40 0.000 .4995379 .8036361 high_school | .453883 .0939311 4.83 0.000 .2697814 .6379847 college | .4845458 .12741 3.80 0.000 .2348268 .7342648 muslim | .0000928 .0809837 0.00 0.999 -.1586322 .1588179 _cons | .9398335 .1461969 6.43 0.000 .6532929 1.226374 ------------------------------------------------------------------------------ Variances and covariances of random effects ------------------------------------------------------------------------------ ***level 2 (ind_id) var(1): 2.6610493 (.1476163) ------------------------------------------------------------------------------

Semi-Parametric Maximum Likelihood

gllamm cont_use posyandus age grade_school high_school college muslim, i(ind_id) family(binomial) link(logit) nip(3) ip(f) trace dot log likelihood = -12660.352 ------------------------------------------------------------------------------ cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- posyandus | .0311694 .0048755 6.39 0.000 .0216135 .0407253 age | -.0379544 .002707 -14.02 0.000 -.04326 -.0326488 grade_school | .6591399 .0788521 8.36 0.000 .5045927 .8136871 high_school | .4674408 .0945268 4.95 0.000 .2821716 .65271 college | .4973757 .1278639 3.89 0.000 .2467671 .7479843 muslim | .0008321 .0812183 0.01 0.992 -.1583529 .160017 _cons | 1.020194 .1998693 5.10 0.000 .6284575 1.411931 ------------------------------------------------------------------------------ Probabilities and locations of random effects ------------------------------------------------------------------------------ ***level 2 (ind_id) loc1: -2.0306, 2.9674, .16649 var(1): 2.780105 prob: 0.2982, 0.1744, 0.5274 ------------------------------------------------------------------------------

Multilevel Panel Models

Basic Form of the model:

where

j=1,2,…,J (communities)

i=1,2,…,Nj (individuals from community j)

t=1,2,…,Tij (observations for person i for community j)

1 2

( 1 | , )ln

( 0 | , )tij ij j

tij tij j ij j

tij ij j

P YX P Z

P Y

Xtij: individual level variables (some could be fixed through time)

Ptij: time varying program variable

Zj: time invariant community level variables

μij: time invariant individual level unobserved heterogeneity

λj: time invariant community level unobserved heterogeneity

This model allows observations on the same individual to be correlated and observations from the same community to be correlated.

Assumptions:

1. Simple logit yields consistent point estimates but incorrect SE’s2. Simple logit with cluster option corrects SE’s (at community level)3. Parametric or semi-parametric maximum likelihood

Maximum likelihood estimator is a straight forward extension of the longitudinal data model:

( ) ( ) ( ) 0tij j tij j ij j

E X E P E Z

( ) ( ) ( ) 0tij ij tij ij ij ij

E X E P E Z

1 2

1 2( 1| , )

1

tij tij j ij j

tij tij j ij j

X P Z

tij ij j X P Z

eP Y

e

You need the unconditional joint probability of the observedset of outcomes for the set of individuals in each community:

Conditional on the unobservables at the community level, the probability of the set of observed outcomes for person i from community j are:

The unconditional joint probability of the set of observed outcomes for all individuals in community j is then:

We then either use Hermite integration or the discrete factor method to approximate the integral.

1

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

ijtij tij

TY Y

ij j tij ij j tij ij j ij ijtP Y P Y P Y f d

1( ) ( | ) ( )

iN

ij tij j j jiP Y P Y f d

Simple logit

Simple Logit with Corrected SE’s

logit cont_use posyandus age grade_school high_school college muslim Logistic regression Number of obs = 20000 LR chi2(6) = 494.70 Prob > chi2 = 0.0000 Log likelihood = -13304.212 Pseudo R2 = 0.0183 ------------------------------------------------------------------------------ cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- posyandus | .0246126 .00295 8.34 0.000 .0188306 .0303946 age | -.0267579 .0016964 -15.77 0.000 -.0300828 -.0234329 grade_school | .488657 .0469818 10.40 0.000 .3965744 .5807396 high_school | .3713791 .0569712 6.52 0.000 .2597175 .4830406 college | .3879987 .0789006 4.92 0.000 .2333564 .542641 muslim | -.0432766 .0468967 -0.92 0.356 -.1351924 .0486392 _cons | .7282074 .0909596 8.01 0.000 .5499298 .906485 ------------------------------------------------------------------------------

logit cont_use posyandus age grade_school high_school college muslim, cluster(com_id) Logistic regression Number of obs = 20000 Wald chi2(6) = 263.28 Prob > chi2 = 0.0000 Log pseudolikelihood = -13304.212 Pseudo R2 = 0.0183 (Std. Err. adjusted for 313 clusters in com_id) ------------------------------------------------------------------------------ | Robust cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- posyandus | .0246126 .0052652 4.67 0.000 .014293 .0349322 age | -.0267579 .0022948 -11.66 0.000 -.0312555 -.0222603 grade_school | .488657 .0796778 6.13 0.000 .3324914 .6448226 high_school | .3713791 .0929568 4.00 0.000 .1891871 .553571 college | .3879987 .1057477 3.67 0.000 .180737 .5952603 muslim | -.0432766 .1257938 -0.34 0.731 -.2898279 .2032747 _cons | .7282074 .1919567 3.79 0.000 .3519792 1.104436 ------------------------------------------------------------------------------

Parametric Maximum Likelihood

gllamm cont_use posyandus age grade_school high_school college muslim, i(ind_id com_id) family(binomial) link(logit) nip(20) ip(g) trace dot number of level 1 units = 20000 number of level 2 units = 9394 number of level 3 units = 313 gllamm model log likelihood = -12548.522 ------------------------------------------------------------------------------ cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- posyandus | .0228368 .0069701 3.28 0.001 .0091757 .036498 age | -.037996 .0026758 -14.20 0.000 -.0432405 -.0327516 grade_school | .6367873 .0786581 8.10 0.000 .4826202 .7909543 high_school | .4122478 .0975244 4.23 0.000 .2211036 .6033921 college | .4165882 .1299495 3.21 0.001 .1618919 .6712844 muslim | .0376821 .1052797 0.36 0.720 -.1686623 .2440266 _cons | 1.00658 .1701569 5.92 0.000 .6730791 1.340082 ------------------------------------------------------------------------------ Variances and covariances of random effects ------------------------------------------------------------------------------ ***level 2 (ind_id) var(1): 2.2860509 (.13570515) ***level 3 (com_id) var(1): .34625941 (.04611334) ------------------------------------------------------------------------------

Non-parametric Maximum Likelihood

gllamm cont_use posyandus age grade_school high_school college muslim, i(ind_id com_id) family(binomial) link(logit) nip(3) ip(f) trace dot number of level 1 units = 20000 number of level 2 units = 9394 number of level 3 units = 313 gllamm model log likelihood = -12546.725 ------------------------------------------------------------------------------ cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- posyandus | .0208781 .0067149 3.11 0.002 .0077171 .0340391 age | -.037949 .0027013 -14.05 0.000 -.0432435 -.0326546 grade_school | .637163 .0795007 8.01 0.000 .4813445 .7929815 high_school | .4185102 .097197 4.31 0.000 .2280075 .6090129 college | .4177381 .1293205 3.23 0.001 .1642745 .6712016 muslim | -.0883427 .1015703 -0.87 0.384 -.2874169 .1107315 _cons | 1.164577 .1836346 6.34 0.000 .8046593 1.524494 ------------------------------------------------------------------------------ Probabilities and locations of random effects ------------------------------------------------------------------------------ ***level 2 (ind_id) loc1: -1.9001, 2.6348, .23361 var(1): 2.2386873 prob: 0.295, 0.1648, 0.5402 ***level 3 (com_id) loc1: -1.4082, .65457, -.1872 var(1): .33048135 prob: 0.0826, 0.3421, 0.5753 ------------------------------------------------------------------------------

Testing for Program Targeting

Programs may target high need areas or areas where they feelresidents would be receptive to family planning

For example: family planning programs may concentrate on highfertility areas

Result is that simple methods may understate or overstate program impact

Statistical Implication of program targeting:

( ) 0tij j

E P

Solutions:

Explicitly model program placement and estimate placement simultaneously with program impact equations (Angeles, Guilkey,and Mroz, 1998)

Treat as fixed effects and include dummies for communitiesor some other fixed effects method (Gertler and Molyneau, 1994)

Angeles, Guilkey, and Mroz show that the joint modeling approach yields smaller standard errors in Tanzania but the two methodsgave similar results

j

Example (fixed effects) plus Hausman Test for endogenous placement:

Efficient estimator under the null of no endogeneity (random effects):melogit cont_use posyandus age grade_school high_school college muslim ||prov_id: || ind_id:,intp(20) Mixed-effects logistic regression Number of obs = 20000 ----------------------------------------------------------- | No. of Observations per Group Group Variable | Groups Minimum Average Maximum ----------------+------------------------------------------ prov_id | 16 11 1250.0 3116 ind_id | 9507 1 2.1 4 ----------------------------------------------------------- Integration method: mvaghermite Integration points = 20 Wald chi2(6) = 468.70 Log likelihood = 5142.1765 Prob > chi2 = 0.0000 -------------------------------------------------------------------------------- cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------------+---------------------------------------------------------------- posyandus | .0260336 .0036554 7.12 0.000 .018869 .0331981 age | -.0279207 .0019312 -14.46 0.000 -.0317057 -.0241357 grade_school | .603515 .0539052 11.20 0.000 .4978628 .7091672 high_school | .4773575 .0663403 7.20 0.000 .3473329 .607382 college | .5571055 .0914372 6.09 0.000 .3778918 .7363192 muslim | .2747446 .0685264 4.01 0.000 .1404353 .4090539 _cons | .3397094 .1159908 2.93 0.003 .1123716 .5670472 ---------------+---------------------------------------------------------------- prov_id | var(_cons)| .264253 .1427983 .0916312 .7620726 ---------------+---------------------------------------------------------------- prov_id>ind_id | var(_cons)| 2.878393 .2659968 2.401537 3.449935 -------------------------------------------------------------------------------- LR test vs. logistic regression: chi2(2) = 36892.78 Prob > chi2 = 0.0000 Note: LR test is conservative and provided only for reference. . estimates store efficient

Consistent estimator under the alternate (fixed effects):

xi: melogit cont_use posyandus age grade_school high_school college muslim i.prov_id || ind_id:,intp(20) Integration method: mvaghermite Integration points = 20 Wald chi2(21) = 485.84 Log likelihood = -12574.068 Prob > chi2 = 0.0000 -------------------------------------------------------------------------------- cont_use | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------------+---------------------------------------------------------------- posyandus | .0279454 .0059279 4.71 0.000 .0163268 .0395639 age | -.0368115 .0026797 -13.74 0.000 -.0420635 -.0315595 grade_school | .6695194 .0781108 8.57 0.000 .516425 .8226137 high_school | .5039237 .0954551 5.28 0.000 .3168351 .6910123 college | .5033064 .1282156 3.93 0.000 .2520084 .7546044 muslim | .0992815 .1055572 0.94 0.347 -.1076069 .3061698 _Iprov_id_13 | .6485017 .1559989 4.16 0.000 .3427495 .9542539 . . _Iprov_id_76 | -.5329505 .8731102 -0.61 0.542 -2.244215 1.178314 _cons | .2371631 .1764231 1.34 0.179 -.1086197 .582946 ---------------+---------------------------------------------------------------- ind_id | var(_cons)| 2.564441 .1439339 2.297299 2.862648 -------------------------------------------------------------------------------- LR test vs. logistic regression: chibar2(01) = 1223.43 Prob>=chibar2 = 0.0000 estimates store consistent

Hausman test results:

hausman consistent efficient ---- Coefficients ---- | (b) (B) (b-B) sqrt(diag(V_b-V_B)) | consistent efficient Difference S.E. -------------+---------------------------------------------------------------- posyandus | .0279454 .0260336 .0019118 .0046667 age | -.0368115 -.0279207 -.0088908 .0018577 grade_school | .6695194 .603515 .0660044 .056529 high_school | .5039237 .4773575 .0265662 .0686342 college | .5033064 .5571055 -.0537991 .0898804 muslim | .0992815 .2747446 -.1754631 .0802898 ------------------------------------------------------------------------------ b = consistent under Ho and Ha; obtained from meglm B = inconsistent under Ha, efficient under Ho; obtained from meglm Test: Ho: difference in coefficients not systematic chi2(6) = (b-B)'[(V_b-V_B)^(-1)](b-B) = 31.45 Prob>chi2 = 0.0000

State Dependence and Unobserved Heterogeneity

Consider the simple model:

Note:

Implies:

Unless (no time invariant unobserved heterogeneity)

Now consider:

Now:

Very difficult to distinguish between the two models

Yti i ti

1, 1,Y it i t i

1,( ) 0ti t icorr Y Y

0i

1,t iY Yti ti

1,( ) 0ti t icorr Y Y

Same problem would exist if the unobserved heterogeneitywere at the community level

Solution is to estimate a comprehensive model:

Initial conditions problem:

Must either be able to set or jointly estimate the equation of interest with an equation of theform:

1, 1 2

( 1 | , )ln

( 0 | , )tij ij j

t ij tij tij j ij j

tij ij j

P YY X P Z

P Y

1 0ijY

0 0 0 0 01

1 1 2

1

( 1)ln

( 0)ij

ij tij j ij j

ij

P YX P Z

P Y

Often it is reasonable to set the initial value:

Observations start at the beginning of the woman’s child bearing years

In this example, it is not since women enter the year one data set atdifferent ages

Joint estimation is basically a simultaneous equations problem subjectto standard identification issues.

However, time varying exogenous variables provide identification (age and education in this case)

Example follows:

Estimation with no controls for unobserved heterogeneity and initial conditions:

DEPENDENT VARIABLE (LOGIT TYPE EQUATION): cont_use UNCONDITIONAL RESULTS LOG ODDS OF CATEGORY 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD one 1.68868 0.1512 11.168 0.193E+00 -0.205E+04 posyandus 0.01820 0.0044 4.162 0.141E+01 -0.154E+06 grade_school 0.30074 0.0664 4.532 0.136E+00 -0.140E+04 high_school 0.38269 0.0873 4.385 0.254E-01 -0.293E+03 college 0.65160 0.1258 5.178 0.753E-02 -0.834E+02 age -0.06683 0.0030 -22.524 0.747E+01 -0.302E+07 muslim -0.11749 0.0710 -1.655 0.172E+00 -0.183E+04 cont_use_lag 1.55126 0.0481 32.257 0.193E+00 -0.142E+04

Estimation with Controls:

RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER: 1 DEPENDENT VARIABLE (LOGIT TYPE EQUATION): cont_use UNCONDITIONAL RESULTS LOG ODDS OF CATEGORY 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD one 0.58041 0.4776 1.215 0.506E-02 -0.263E+03 posyandus 0.03158 0.0134 2.350 0.297E-01 -0.197E+05 grade_school 0.38268 0.1948 1.965 0.370E-02 -0.200E+03 high_school 0.29902 0.2368 1.263 0.730E-03 -0.385E+02 college 0.48432 0.3531 1.372 0.200E-03 -0.101E+02 age -0.03249 0.0089 -3.654 0.145E+00 -0.283E+06 muslim -0.31757 0.2206 -1.439 0.476E-02 -0.230E+03 OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl 1.75070 0.3575 4.897 -0.249E-03 -0.244E+02 OMEGAcl 0.12941 0.3337 0.388 0.336E-02 -0.107E+03 OMEGAi 0.0 -- NORMALIZED AT ZERO OMEGAi 8.78497 2.3058 3.810 0.536E-03 -0.941E-04

Estimation with Controls (continued) RESULTS FOR LOGIT-TYPE EQUATION -- NUMBER: 2 DEPENDENT VARIABLE (LOGIT TYPE EQUATION): cont_use UNCONDITIONAL RESULTS LOG ODDS OF CATEGORY 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD one 1.14288 0.2293 4.984 0.114E-02 -0.130E+04 posyandus 0.02006 0.0065 3.070 0.138E-01 -0.900E+05 grade_school 0.32652 0.0831 3.929 0.146E-02 -0.107E+04 high_school 0.38546 0.1017 3.790 0.161E-04 -0.259E+03 college 0.66607 0.1411 4.721 0.397E-03 -0.814E+02 age -0.07099 0.0033 -21.261 0.376E-01 -0.195E+07 muslim -0.02557 0.1082 -0.236 0.130E-02 -0.114E+04 cont_use_lag 1.37790 0.0633 21.774 -0.861E-03 -0.109E+04 OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl 1.00568 0.2002 5.024 -0.619E-04 -0.250E+03 OMEGAcl 0.58970 0.0829 7.114 0.140E-02 -0.572E+03 OMEGAi 0.0 -- NORMALIZED AT ZERO OMEGAi 0.51559 0.1148 4.492 -0.123E-02 -0.281E+03 HETEROGENEITY INFORMATION COMMUNITY SPECIFIC DISTRIBUTION POINT # PROBABILITY WEIGHT 1 0.31240645 2 0.17598629 3 0.51160726 INDIVIDUAL SPECIFIC DISTRIBUTION POINT # PROBABILITY WEIGHT 1 0.57174255 2 0.42825745

Basic Model Longitudinal Multinomial Logit with 3 Choices:

Individual i at time t time makes choice 3 (for example) if :

If we assume that the ε’s follow independent extreme value distributions and impose the restriction that:

1 1 1 1 1 1ti ti ti i i tiU X P Z

2 2 2 2 2 2ti ti ti i i tiU X P Z

3 3 3 3 3 3ti ti ti i i tiU X P Z

3 1 3 2( )

ti ti ti tiP U U and U U

So that the probabilities sum to one then:

for k=2,3.

The discrete factor model allows a more general pattern of correlation:

for m=1,2…,M and a common set of weights: allows for correlation in the μ’s

1 1 1 10

( | )ln

( 1 | )ti i

ti k k ti i k k i

ti i

P Y kX P Z

P Y

( | )ln

( 1 | )ti km

ti k k ti i k km

ti km

P Y kX P Z

P Y

mw

Unfortunately, GLLAMM estimates a needlessly restrictive version of the model:

Parametric:

If there are more than 3 choices, all ρ’s are restricted

Non-parametric:

for all m.

2 3

2 3m m

Extension to Multilevel Panel Model:

Parametric:

Semi-parametric:

1 2

( | , )ln

( 1 | , )tij ij j

tij k k tij j k k ij k j

tij ij j

P Y kX P Z

P Y

( | , )ln

( 1 | , )tij km kn

tij k k tij j k km kn

ti km kn

P Y kX P Z

P Y

The empirical example estimates a model with four choices:

1= Non use

2=Temporary Methods (pill, condom, injection)

3=Long Lasting Methods (IUD, sterilization)

4=Traditional Methods

We show the complete results for the most general model and then report partial results for other models:

DEPENDENT VARIABLE (LOGIT TYPE EQUATION): new_meth UNCONDITIONAL RESULTS LOG ODDS OF CATEGORY 2 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD Posyandus 0.01467 0.0102 1.437 -0.233E+00 -0.120E+06 age -0.06430 0.0034 -18.883 -0.109E+01 -0.246E+07 grade_school 0.62999 0.0992 6.348 -0.219E-01 -0.163E+04 high_school 0.37284 0.1190 3.132 -0.678E-02 -0.351E+03 college 0.21653 0.1497 1.446 -0.171E-02 -0.908E+02 muslim 0.75840 0.1621 4.678 -0.300E-01 -0.201E+04 constant 0.18753 0.3810 0.492 -0.336E-01 -0.227E+04 Community Heterogeneity OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl 0.27663 0.2840 0.974 -0.102E-01 -0.889E+03 OMEGAcl 1.07322 0.1974 5.437 -0.181E-01 -0.116E+04 Individual Heterogeneity OMEGAi 0.0 -- NORMALIZED AT ZERO OMEGAi 1.41424 0.1658 8.530 -0.154E-01 -0.627E+03 OMEGAi -1.28669 0.1762 -7.302 -0.182E-01 -0.816E+03 LOG ODDS OF CATEGORY 3 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD Posyandus 0.03780 0.0188 2.006 -0.972E-01 -0.699E+05 age 0.01783 0.0066 2.702 -0.451E+00 -0.100E+07 grade_school 0.35402 0.1371 2.582 -0.838E-02 -0.555E+03 high_school 0.21220 0.1968 1.078 -0.270E-02 -0.130E+03 college 0.48971 0.2856 1.715 -0.892E-03 -0.433E+02 muslim -0.92293 0.2182 -4.229 -0.108E-01 -0.648E+03 constant 1.59472 0.5112 3.120 -0.134E-01 -0.760E+03 Community Heterogeneity OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl -2.75334 0.4127 -6.671 0.524E-04 -0.139E+03 OMEGAcl -0.57495 0.4030 -1.427 -0.700E-02 -0.430E+03 Individual Heterogeneity OMEGAi 0.0 -- NORMALIZED AT ZERO OMEGAi -3.49818 0.3190 -10.965 -0.898E-03 -0.114E+03 OMEGAi -4.50909 0.1863 -24.207 -0.399E-02 -0.153E+03

LOG ODDS OF CATEGORY 4 RELATIVE TO CATEGORY 1 RHS. VAR. COEFFICIENT STD. ERR. T-SCORE FPD SPD Posyandus 0.05196 0.0139 3.734 0.431E-01 -0.845E+05 age 0.02778 0.0051 5.405 0.174E+00 -0.949E+06 grade_school 1.03143 0.2152 4.793 0.271E-02 -0.427E+03 high_school 1.55313 0.2487 6.246 0.140E-02 -0.117E+03 college 1.70120 0.2826 6.021 0.535E-03 -0.419E+02 muslim -0.72360 0.1498 -4.831 0.375E-02 -0.584E+03 constant -4.48901 0.4868 -9.221 0.490E-02 -0.711E+03 Community Heterogeneity OMEGAcl 0.0 -- NORMALIZED AT ZERO OMEGAcl -0.36298 0.3285 -1.105 0.136E-02 -0.167E+03 OMEGAcl -0.25288 0.3296 -0.767 0.231E-02 -0.235E+03 Individual Heterogeneity OMEGAi 0.0 -- NORMALIZED AT ZERO OMEGAi -1.15542 0.5734 -2.015 0.681E-03 -0.104E+02 OMEGAi 0.09344 0.2834 0.330 0.355E-02 -0.517E+03 HETEROGENEITY INFORMATION COMMUNITY SPECIFIC DISTRIBUTION POINT # PROBABILITY WEIGHT 1 0.25422817 2 0.25159735 3 0.49417448 INDIVIDUAL SPECIFIC DISTRIBUTION POINT # PROBABILITY WEIGHT 1 0.23493000 2 0.32619060 3 0.43887940

Comparison of Posyandu effects across estimation methods: Coefficient Std Error Z statistic

Multinomial Logit 2 versus 1 .01127 .0027824 4.05 3 versus 1 .0348281 .0031497 11.06 4 versus 1 .03814 .0062596 6.09

Multinomial Logit with community corrected SE’s 2 versus 1 .01127 .0042326 2.66 3 versus 1 .0348281 .0058663 5.94 4 versus 1 .03814 .0081891 4.66

Parametric Random effects Multilevel Multinomial Logit (GLLAMM restrictions)

2 versus 1 .0114064 .005755 1.98 3 versus 1 .0348275 .0059394 5.86 4 versus 1 .0384119 .0080835 4.75

Heterogeneity Community .35132246 .04663305 7.53 Individual 2.3408954 .13771143 17.00

Non-Parametric Random effects Multilevel Multinomial Logit (GLLAMM restrictions)

2 versus 1 .0116273 .0059026 1.97 3 versus 1 .035 .0060997 5.74 4 versus 1 .0385815 .0082056 4.70

Heterogeneity Community Mass points -1.3963 .6822 -.17666 Weights 0.083 0.3236 0.5934 Individual Mass points -1.9467 2.6613 .24876 Weights 0.2916 0.1622 0.5462

Non-Parametric Random effects Multilevel Multinomial Logit (Fortran) 2 versus 1 0.01467 0.0102 1.44 3 versus 1 0.03780 0.0188 2.01 4 versus 1 0.05196 0.0139 3.74