st: unobserved hetereogeneity and duration: interpreting pgmhaz8 andxtclog

[Gijs Dekkers <gd@plan.be>]

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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