xtnbreg, re - loglikelihood discontinuous region encountered (not so for conditional or unconditional fe)

[Chiara Pollio]

Dear Statalists,
Sorry in advance for the long message, there are few Stata output that I need to include for clarity.
I am dealing with a panel dataset in which I want to estimate the territories' patenting activity on the basis of some performance indicators. I have a balanced panel dataset of 32 towns for 14 years. Giving the features of my dependent variable, I am using a negative binomial model, and I want to exploit the longitudinal feature of the data.

I have successfully ran the model with some selected independent variables as a poisson (non panel), negative binomial (non panel), unconditional possion fixed effect (glm command), unconditional negative binomial fixed effect (glm command) , conditional poisson fixed effect (xtpoisson, fe), conditional negative binomial fixed effect (xtnbreg, fe) models, random poisson model (xtpoisson, re). This is why I found it strange when I could not get any results from the negative binomial random effect (xtnbreg, re) model.
I post here the - xtnbreg, fe -, the - xtpoisson, re - and the - xtnbreg, re - codes, together with some of the output:

*Negative binomial, conditional fixed effect

xtnbreg SIPO_Pat ICT nospec cap_on_emp Nr_individual l1.Exports Nr_foreign l1.Grad_sec_sch pre* post*, fe
note: post_2005 omitted because of collinearity

Iteration 0:   log likelihood = -8759.6522  (not concave)
...
Iteration 24:  log likelihood = -2481.9586  (not concave)
Iteration 25:  log likelihood = -2356.2314  
Iteration 26:  log likelihood = -2219.1313  
Iteration 27:  log likelihood = -2199.4056  
Iteration 28:  log likelihood = -2198.8179  
Iteration 29:  log likelihood = -2198.8158  
Iteration 30:  log likelihood = -2198.8158  

Conditional FE negative binomial regression     Number of obs      =       416
Group variable: TOWN_ID                         Number of groups   =        32

                                                Obs per group: min =        13
                                                               avg =      13.0
                                                               max =        13

                                                Wald chi2(10)      =   1405.40
Log likelihood  = -2198.8158                    Prob > chi2        =    0.0000

--------------------------------------------------------------------------------------------------
                   SIPO_Pat_Appl |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------------------------+----------------------------------------------------------------
                             ICT |  -.1426851    .082568    -1.73   0.084    -.3045154    .0191452
                          nospec |  -.2190276   .0679435    -3.22   0.001    -.3521944   -.0858607
             cap_on_employee_ads |   .0012927   .0013914     0.93   0.353    -.0014343    .0040197
Nr_individual_comm_hous_enterpri |   1.60e-06   1.60e-06     0.99   0.320    -1.55e-06    4.74e-06
                                 |
                    Exports_10th |
                             L1. |   4.31e-07   2.13e-07     2.03   0.043     1.41e-08    8.48e-07
                                 |
         Nr_foreign_funded_firms |   .0002219   .0001048     2.12   0.034     .0000164    .0004274
                                 |
                Grad_sec_sch_pop |
                             L1. |   7.776467   2.578928     3.02   0.003      2.72186    12.83107
                                 |
                        pre_2005 |  -.8859314    .081811   -10.83   0.000    -1.046278   -.7255848
                       post_2005 |          0  (omitted)
                       post_2008 |   .6558707   .0543943    12.06   0.000     .5492597    .7624817
                       post_2011 |   .2776123    .049691     5.59   0.000     .1802197    .3750049
                           _cons |   1.541551   .1224057    12.59   0.000      1.30164    1.781461
--------------------------------------------------------------------------------------------------




*Poisson, random effect
xtpoisson SIPO_Pat ICT nospec cap_on_emp Nr_individual l1.Exports Nr_foreign l1.Grad_sec_sch pre* post*, re

Fitting Poisson model:

Iteration 0:   log likelihood =  -31973.28  
Iteration 1:   log likelihood = -30094.781  
Iteration 2:   log likelihood = -30089.312  
Iteration 3:   log likelihood = -30089.312  

Fitting full model:

Iteration 0:   log likelihood = -17908.496  
Iteration 1:   log likelihood = -11730.035  
Iteration 2:   log likelihood = -11570.704  
Iteration 3:   log likelihood = -11545.794  
Iteration 4:   log likelihood = -11543.718  
Iteration 5:   log likelihood = -11543.687  
Iteration 6:   log likelihood = -11543.687  

Random-effects Poisson regression               Number of obs      =       416
Group variable: TOWN_ID                         Number of groups   =        32

Random effects u_i ~ Gamma                      Obs per group: min =        13
                                                               avg =      13.0
                                                               max =        13

                                                Wald chi2(10)      =  55875.28
Log likelihood  = -11543.687                    Prob > chi2        =    0.0000

--------------------------------------------------------------------------------------------------
                   SIPO_Pat_Appl |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------------------------+----------------------------------------------------------------
                             ICT |  -.1028622   .0140955    -7.30   0.000    -.1304888   -.0752356
                          nospec |  -.1438835    .012527   -11.49   0.000     -.168436    -.119331
             cap_on_employee_ads |   .0039368   .0002509    15.69   0.000      .003445    .0044287
Nr_individual_comm_hous_enterpri |   1.81e-06   2.57e-07     7.04   0.000     1.31e-06    2.32e-06
                                 |
                    Exports_10th |
                             L1. |   3.20e-07   3.04e-08    10.53   0.000     2.61e-07    3.80e-07
                                 |
         Nr_foreign_funded_firms |   .0002136   .0000139    15.32   0.000     .0001863    .0002409
                                 |
                Grad_sec_sch_pop |
                             L1. |   11.57073   .4436957    26.08   0.000      10.7011    12.44036
                                 |
                        pre_2005 |  -1.116745   .0140921   -79.25   0.000    -1.144365   -1.089125
                       post_2008 |    .739617   .0080947    91.37   0.000     .7237516    .7554824
                       post_2011 |   .1076197   .0078306    13.74   0.000     .0922719    .1229674
                           _cons |   5.178147   .1205244    42.96   0.000     4.941924    5.414371
---------------------------------+----------------------------------------------------------------
                        /lnalpha |  -.7870015   .2338881                     -1.245414   -.3285894
---------------------------------+----------------------------------------------------------------
                           alpha |   .4552077   .1064676                      .2878218    .7199386
--------------------------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0: chibar2(01) =  3.7e+04 Prob>=chibar2 = 0.000

. 
*Negative Binomial random effect
 xtnbreg SIPO_Pat ICT nospec cap_on_emp Nr_individual l1.Exports Nr_foreign l1.Grad_sec_sch pre* post*, re

Fitting negative binomial (constant dispersion) model:

Iteration 0:   log likelihood =  -31973.28  
Iteration 1:   log likelihood = -30094.781  
Iteration 2:   log likelihood = -30089.312  
Iteration 3:   log likelihood = -30089.312  

Iteration 0:   log likelihood = -50019.689  
Iteration 1:   log likelihood = -6160.8259  
Iteration 2:   log likelihood = -4620.1901  (backed up)
Iteration 3:   log likelihood = -2957.8093  
Iteration 4:   log likelihood = -2928.1561  
Iteration 5:   log likelihood = -2926.5559  
Iteration 6:   log likelihood = -2926.5532  
Iteration 7:   log likelihood = -2926.5532  

Iteration 0:   log likelihood = -2926.5532  (not concave)
Iteration 1:   log likelihood = -2825.9998  
Iteration 2:   log likelihood = -2699.4153  
Iteration 3:   log likelihood = -2674.8744  
Iteration 4:   log likelihood = -2672.2632  
Iteration 5:   log likelihood = -2672.2598  
Iteration 6:   log likelihood = -2672.2598  

Fitting full model:

Iteration 0:   log likelihood = -8326.6712  (not concave)
Iteration 1:   log likelihood = -5825.2809  (not concave)
Iteration 2:   log likelihood = -5782.6175  
Iteration 3:   log likelihood = -4181.4293  (not concave)
Iteration 4:   log likelihood = -3677.2887  
Iteration 5:   log likelihood = -3110.3723  (not concave)
Iteration 6:   log likelihood = -3103.9361  (not concave)
Iteration 7:   log likelihood =  -3096.597  (not concave)
Iteration 8:   log likelihood = -3096.0307  (not concave)
Iteration 9:   log likelihood = -3095.7554  (not concave)
... 

At first, the loglikelhood function yelded a non concave function. I waited up to the iteration 1380 but it did not seem to get to a solution. Therefore I tried to use various solutions that I found in the forum:
- using the - difficult - option
- changing the algorithm technique to the dfp and bfgs ones (I did not go through the bhhh as it reported it does not work with d2 evaluators).

The only "improvement" that I could get is that the problem is no longer the not concave function but the message "loglikelihood discontinuous region encountered":

xtnbreg SIPO_Pat ICT nospec cap_on_emp Nr_individual l1.Exports Nr_foreign l1.Grad_sec_sch pre* post*, re difficult technique(nr dfp bfgs)
note: post_2005 omitted because of collinearity

Fitting negative binomial (constant dispersion) model:

(setting technique to nr)
Iteration 0:   log likelihood =  -31973.28  
Iteration 1:   log likelihood = -30094.781  
Iteration 2:   log likelihood = -30089.312  
Iteration 3:   log likelihood = -30089.312  

(setting technique to nr)
Iteration 0:   log likelihood = -50019.689  
Iteration 1:   log likelihood = -6160.8259  
Iteration 2:   log likelihood = -4620.1901  (backed up)
Iteration 3:   log likelihood = -2957.8093  
Iteration 4:   log likelihood = -2928.1561  
(switching technique to dfp)
Iteration 5:   log likelihood = -2926.5559  
Iteration 6:   log likelihood = -2926.5532  
Iteration 7:   log likelihood = -2926.5532  

(setting technique to nr)
Iteration 0:   log likelihood = -2926.5532  (not concave)
Iteration 1:   log likelihood = -2825.9998  
Iteration 2:   log likelihood = -2699.4153  
Iteration 3:   log likelihood = -2674.8744  
Iteration 4:   log likelihood = -2672.2632  
(switching technique to dfp)
DFP stepping has contracted, resetting DFP Hessian
Iteration 5:   log likelihood = -2672.2598  
DFP stepping has contracted, resetting DFP Hessian
Iteration 6:   log likelihood = -2672.2598  (backed up)
DFP stepping has contracted, resetting DFP Hessian
Iteration 7:   log likelihood = -2672.2598  (backed up)
DFP stepping has contracted, resetting DFP Hessian
Iteration 8:   log likelihood = -2672.2598  (backed up)
DFP stepping has contracted, resetting DFP Hessian
Iteration 9:   log likelihood = -2672.2598  (backed up)
(switching technique to bfgs)
BFGS stepping has contracted, resetting BFGS Hessian
Iteration 10:  log likelihood = -2672.2598  (backed up)
BFGS stepping has contracted, resetting BFGS Hessian
Iteration 11:  log likelihood = -2672.2598  (backed up)
BFGS stepping has contracted, resetting BFGS Hessian
Iteration 12:  log likelihood = -2672.2598  (backed up)
BFGS stepping has contracted, resetting BFGS Hessian
Iteration 13:  log likelihood = -2672.2598  (backed up)
BFGS stepping has contracted, resetting BFGS Hessian
Iteration 14:  log likelihood = -2672.2598  (backed up)
(switching technique to nr)
Iteration 15:  log likelihood = -2672.2598  (backed up)

Fitting full model:

(setting technique to nr)
Iteration 0:   log likelihood = -8326.6715  (not concave)
Iteration 1:   log likelihood = -5825.2839  (not concave)
Iteration 2:   log likelihood =  -5782.621  
Iteration 3:   log likelihood = -4181.1842  (not concave)
Iteration 4:   log likelihood = -3677.1854  
(switching technique to dfp)
DFP stepping has contracted, resetting DFP Hessian
Iteration 5:   log likelihood = -3110.5124  
cannot compute an improvement -- discontinuous region encountered
r(430); 

I can only run the model if I omit some specific variables:

 xtnbreg SIPO_Pat ICT nospec cap_on_emp Nr_foreign l1.Grad_sec_sch pre* post*, re difficult technique(nr dfp bfgs)
note: post_2005 omitted because of collinearity

Fitting negative binomial (constant dispersion) model:

(setting technique to nr)
Iteration 0:   log likelihood = -38755.247  
Iteration 1:   log likelihood = -37842.441  
Iteration 2:   log likelihood =  -37840.06  
Iteration 3:   log likelihood =  -37840.06  

(setting technique to nr)
Iteration 0:   log likelihood = -50019.689  
Iteration 1:   log likelihood = -6160.8259  
Iteration 2:   log likelihood = -4620.1901  (backed up)
Iteration 3:   log likelihood = -2957.8093  
Iteration 4:   log likelihood = -2928.1561  
(switching technique to dfp)
Iteration 5:   log likelihood = -2926.5559  
Iteration 6:   log likelihood = -2926.5532  
Iteration 7:   log likelihood = -2926.5532  

(setting technique to nr)
Iteration 0:   log likelihood = -2926.5532  
Iteration 1:   log likelihood = -2786.4174  (backed up)
Iteration 2:   log likelihood = -2743.1568  
Iteration 3:   log likelihood = -2723.0039  
Iteration 4:   log likelihood = -2722.8331  
(switching technique to dfp)
Iteration 5:   log likelihood =  -2722.833  

Fitting full model:

(setting technique to nr)
Iteration 0:   log likelihood = -9353.5423  (not concave)
Iteration 1:   log likelihood = -2769.5786  (not concave)
Iteration 2:   log likelihood = -2637.0027  (not concave)
Iteration 3:   log likelihood = -2519.5827  
Iteration 4:   log likelihood = -2512.9767  
(switching technique to dfp)
Iteration 5:   log likelihood = -2503.1936  
Iteration 6:   log likelihood = -2502.3942  
Iteration 7:   log likelihood = -2502.1686  
Iteration 8:   log likelihood = -2502.1571  
Iteration 9:   log likelihood = -2502.1567  
(switching technique to bfgs)
Iteration 10:  log likelihood = -2502.1567  

Random-effects negative binomial regression     Number of obs      =       416
Group variable: TOWN_ID                         Number of groups   =        32

Random effects u_i ~ Beta                       Obs per group: min =        13
                                                               avg =      13.0
                                                               max =        13

                                                Wald chi2(8)       =   1378.00
Log likelihood  = -2502.1567                    Prob > chi2        =    0.0000

-----------------------------------------------------------------------------------------
          SIPO_Pat_Appl |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------------+----------------------------------------------------------------
                    ICT |  -.1073468   .0817717    -1.31   0.189    -.2676165    .0529228
                 nospec |  -.2141712   .0666093    -3.22   0.001    -.3447231   -.0836194
    cap_on_employee_ads |   .0008254   .0014677     0.56   0.574    -.0020512    .0037021
Nr_foreign_funded_firms |   .0003357   .0001039     3.23   0.001      .000132    .0005394
                        |
       Grad_sec_sch_pop |
                    L1. |   8.398023   2.525613     3.33   0.001     3.447912    13.34813
                        |
               pre_2005 |  -.9107259   .0804405   -11.32   0.000    -1.068386   -.7530655
              post_2005 |          0  (omitted)
              post_2008 |   .6540546   .0534153    12.24   0.000     .5493626    .7587466
              post_2011 |   .3172153   .0452615     7.01   0.000     .2285043    .4059262
                  _cons |   1.553998   .1206744    12.88   0.000     1.317481    1.790516
------------------------+----------------------------------------------------------------
                  /ln_r |   .8607036   .2442626                      .3819576     1.33945
                  /ln_s |   4.170516   .2849279                      3.612068    4.728964
------------------------+----------------------------------------------------------------
                      r |   2.364824   .5776382                       1.46515    3.816942
                      s |   64.74885   18.44875                      37.04256    113.1783
-----------------------------------------------------------------------------------------
Likelihood-ratio test vs. pooled: chibar2(01) =   441.35 Prob>=chibar2 = 0.000 

There seem to be an issue with two variables (named Nr_individual and Nr_foreign). What I do not understand is why the code is able to run with fixed effects and not with the random effect model. Why is it so? should I suspect that something is not right in the fixed effect model either? Is there any other way to overcome the problem apart from those I already tried? Shall I try some transformation with the variables that are causing trouble?

Thanks to anyone that will help!

[Joao Santos Silva]

Dear Chiara,

I am sorry to answer your question with another question, but why are you estimating all these models? Most of the models you are estimating are quite fragile in the sense that they depend on unrealistic assumptions. Unless you are interested in computing probabilities of events, I suggest you just use Poisson regression with fixed effects.

Best regards,

Joao

[Chiara Pollio]

Dear Joao,
thank you for your reply.
i am running all these models for two reasons: first, the outcome variable is very overdispersed, nor the xtpoisson fe seems to deal properly with it (while the negative binomial seems toto work properly). Secondly,it seems that the outcome variable shows more within than between variability, so i supect that a fixed effect would not fit thethe data properly. Do you think xtpoisson fe is preferable anyway? Also it would be great if you had some specific references on this, i am using mainly the contributions of Hilbe.

thank you,
c

[Joao Santos Silva]

Dear Chiara,

The over-dispersion is a serious problem only if you want to compute probabilities of events. In most cases practitioners only want to estimate the conditional mean and in that case it is better to have a reliable estimator of the mean than an unreliable estimator that purportedly accounts for over-dispersion. The FE Poisson regression is remarkably robust, as shown by

Wooldridge, J. M. (1999): Distribution-Free Estimation of Some Nonlinear Panel Data Models. Journal of Econometrics (90), pp. 77-97

and, for me, that is the perfect workhorse for these problems. The NB estimator and any RE estimator will depended on very strong distributional assumptions about the nature of the over-dispersion and therefore are unreliable.

Best wishes,

Joao