# st: unobserved hetereogeneity and duration: interpreting pgmhaz8 andxtclog

 From Gijs Dekkers To statalist@hsphsun2.harvard.edu Subject st: unobserved hetereogeneity and duration: interpreting pgmhaz8 andxtclog Date Fri, 23 Sep 2005 11:55:51 +0200

Dear fellow Stata-users,

I am estimating a discrete duration model, explaining the probability that a cohabiting (unmarried) individual (cohab=1) separates
i.e. no longer consensual union and not married after a certain time (the variable 'duration'). The dataset is the European Comunity Household Panel ECHP.

The variables are
pid: unique person identifier
duration: time (years)
cosep: 0 if the individual lives in consensual union, 1=if (s)he does not live in consensual union (and is not married)

The data is of the following form:
+----------------------------+
| pid duration cosep |
|----------------------------|
1. | 1028101 1 0 |
2. | 1028101 2 0 |
3. | 1028105 1 0 |
4. | 1028105 2 0 |
5. | 2053101 1 0 |
|----------------------------|
6. | 2053102 1 0 |
7. | 3023101 1 0 |
8. | 3023101 2 0 |
9. | 3023101 3 1 |
etc...

A first analysis (somewhat dissapointingly) showed that the only significant explanatory variables are a function of duration. In fact, the best model explains 'cosep' using 'duration' and its quadrature 'duration2'

. cloglog cosep duration duration2

Iteration 0: log likelihood = -377.28398 Iteration 1: log likelihood = -377.24866 Iteration 2: log likelihood = -377.24865
Complementary log-log regression Number of obs = 2410
Zero outcomes = 2319
Nonzero outcomes = 91

LR chi2(2) = 20.35
Log likelihood = -377.24865 Prob > chi2 = 0.0000

------------------------------------------------------------------------------
cosep | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
duration | .8310887 .2263603 3.67 0.000 .3874307 1.274747
duration2 | -.0825692 .0272601 -3.03 0.002 -.135998 -.0291405
_cons | -4.782091 .4091147 -11.69 0.000 -5.583941 -3.980241
------------------------------------------------------------------------------

However, I want to test for various parametric forms of frailty, using Jenkins' Lesson 7 on 'unobserved heterogeneity' (http://www.iser.essex.ac.uk/teaching/degree/stephenj/ec968/#_Toc520705914).
First, he suggests to test for heterogeneity assuming a normally distributed frailty term (page 14).
. xtclog cosep duration duration2, nolog i(pid)

Random-effects complementary log-log model Number of obs = 2410
Group variable (i): pid Number of groups = 739

Random effects u_i ~ Gaussian Obs per group: min = 1
avg = 3.3
max = 8

Wald chi2(2) = 18.20
Log likelihood = -377.24865 Prob > chi2 = 0.0001

------------------------------------------------------------------------------
cosep | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
duration | .8310886 .2263602 3.67 0.000 .3874307 1.274747
duration2 | -.0825692 .0272601 -3.03 0.002 -.135998 -.0291404
_cons | -4.782091 .4091147 -11.69 0.000 -5.583941 -3.980241
-------------+----------------------------------------------------------------
/lnsig2u | -14 . . .
-------------+----------------------------------------------------------------
sigma_u | .0009119 . . .
rho | 5.06e-07 . . .
------------------------------------------------------------------------------
Likelihood-ratio test of rho=0: chibar2(01) = 0.00 Prob >= chibar2 = 1.000

Now this already looks pretty strange to me, or is it my suspicious mind? Can I safely coclude that the hypothesis of normally distributed unobserved heterogeneity shoud (very much) be rejected?

Secondly, I used pgmhaz8 to test for gamma-distributed unobserved heterogeneity. I found the pgmhaz8-manual at http://ideas.repec.org/c/boc/bocode/s438501.html
If I understand this manual correctly (but I am not quite sure), the model should be

. pgmhaz8 duration2, id(pid) dead(cosep) seq(duration)

(anyway, the model pgmhaz8 duration duration2 etc. does not converge)

The results are:

PGM hazard model without gamma frailty

Generalized linear models No. of obs = 2410
Optimization : ML Residual df = 2408
Scale parameter = 1
Deviance = 769.387838 (1/df) Deviance = .3195132
Pearson = 2400.608032 (1/df) Pearson = .9969302

Variance function: V(u) = u*(1-u) [Bernoulli]
Link function : g(u) = ln(-ln(1-u)) [Complementary log-log]

AIC = .3209078
Log likelihood = -384.693919 BIC = -17982.63

------------------------------------------------------------------------------
| OIM
cosep | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
duration2 | .0135626 .0055006 2.47 0.014 .0027817 .0243435
_cons | -3.456552 .1401122 -24.67 0.000 -3.731167 -3.181937
------------------------------------------------------------------------------

Iteration 0: log likelihood = -385.00279 Iteration 1: log likelihood = -384.79069 Iteration 2: log likelihood = -384.73062 Iteration 3: log likelihood = -384.70334 Iteration 4: log likelihood = -384.69612 Iteration 5: log likelihood = -384.6944 Iteration 6: log likelihood = -384.69403 Iteration 7: log likelihood = -384.69394 Iteration 8: log likelihood = -384.69392 Iteration 9: log likelihood = -384.69392 Iteration 10: log likelihood = -384.69392
PGM hazard model with gamma frailty Number of obs = 2410
LR chi2() = .
Log likelihood = -384.69392 Prob > chi2 = .

------------------------------------------------------------------------------
cosep | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
hazard |
duration2 | .0135593 .0055347 2.45 0.014 .0027114 .0244072
_cons | -3.456778 .141079 -24.50 0.000 -3.733287 -3.180268
-------------+----------------------------------------------------------------
ln_varg |
_cons | -13.77345 952.569 -0.01 0.988 -1880.774 1853.228
-------------+----------------------------------------------------------------
Gamma var. | 1.04e-06 .0009935 0.00 0.999 0 .
------------------------------------------------------------------------------
LR test of Gamma var. = 0: chibar2(01) = -8.9e-06 Prob.>=chibar2 = .5

And here it is again: analogous to the results from the xtclog, the hypothesis of gamma-distributed unobserved heterogeneity should be rejected. However, again like the xtclog results, the above results of pgmhaz8 suspiciously look like some sort of corner solution, or an artefact.

And this (finally!) brings me to my question: can I trust these results and safely conclude that the hypotheses of unobserved hetereogeneity (both normally and gamma-distributed) should be rejected? Or is there something else going on? If so, any suggestions?

Any help would be appreciated!

Gijs

--
dr. Gijs Dekkers
Federal Planning Bureau
Kunstlaan 47-49
1000 Brussels, Belgium
++32/(0)2/5077413
fax 7373 gd@plan.be, gijs.dekkers@soc.kuleuven.be

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