Why do Poisson and Negative Binomial Regressions yield the same result?

[Mehrzad Baktash]

I am using the count data models as I have a count dependent variable (0, 1, 2, 3, 4).

There is no overdispersion present and no assumption of the Poisson model is violated. Pearson goodness-of-fit and Deviane goodness-of-fit both suggest the usage of Poisson regression instead of NB regression.

However, I am still interested to compare the results of both regressions and see if there are some changes. Surprisingly, the results are identical.

My question would be: Why do the Poisson and Negative Binomial Regressions both give the same results? Is it because of no violations of Poisson model assumptions? What should be done in this case?

Poisson results:

dnewoccposition3	.0989237	.028555	3.46	0.001	.042957	.1548904
dnewoccposition4	.1154875	.0467075	2.47	0.013	.0239424	.2070326
dnewoccposition5	.1718835	.0493519	3.48	0.000	.0751555	.2686115

NB results:

dnewoccposition3	.0989237	.028555	3.46	0.001	.042957	.1548904
dnewoccposition4	.1154875	.0467075	2.47	0.013	.0239424	.2070326
dnewoccposition5	.1718835	.0493519	3.48	0.000	.0751555	.2686115

Many thanks in advance!

Best,
Mehrzad

[Carlo Lazzaro]

Mehrzad:
yes, it is because there's no violation of the Poisson-related requirements (basically, equidispersion).
I can replicate the issue in the following toy-example:

. use http://www.stata-press.com/data/r16/airline.dta

. poisson injuries i.airline

Iteration 0:   log likelihood = -16.084082 
Iteration 1:   log likelihood = -15.980494 
Iteration 2:   log likelihood = -15.980112 
Iteration 3:   log likelihood = -15.980112 

Poisson regression                              Number of obs     =          9
                                                LR chi2(8)        =      31.86
                                                Prob > chi2       =     0.0001
Log likelihood = -15.980112                     Pseudo R2         =     0.4992

------------------------------------------------------------------------------
    injuries |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     airline |
          2  |  -.4519851   .4834938    -0.93   0.350    -1.399616    .4956453
          3  |  -.4519851   .4834938    -0.93   0.350    -1.399616    .4956453
          4  |   .5465437   .3788676     1.44   0.149    -.1960232    1.289111
          5  |  -.2006707   .4494666    -0.45   0.655    -1.081609    .6802676
          6  |  -1.011601   .5838742    -1.73   0.083    -2.155973    .1327715
          7  |  -1.299283   .6513389    -1.99   0.046    -2.575884   -.0226821
          8  |  -2.397895   1.044466    -2.30   0.022    -4.445011   -.3507797
          9  |  -1.299283   .6513389    -1.99   0.046    -2.575884   -.0226821
             |
       _cons |   2.397895   .3015113     7.95   0.000     1.806944    2.988847
------------------------------------------------------------------------------

. nbreg injuries i.airline

Fitting Poisson model:

Iteration 0:   log likelihood = -16.084082 
Iteration 1:   log likelihood = -15.980494 
Iteration 2:   log likelihood = -15.980112 
Iteration 3:   log likelihood = -15.980112 

Fitting constant-only model:

Iteration 0:   log likelihood = -27.260001 
Iteration 1:   log likelihood = -26.029258 
Iteration 2:   log likelihood =  -26.02905 
Iteration 3:   log likelihood =  -26.02905 

Fitting full model:

Iteration 0:   log likelihood = -22.032816 
Iteration 1:   log likelihood = -18.466959 
Iteration 2:   log likelihood = -17.617989 
Iteration 3:   log likelihood = -16.121217 
Iteration 4:   log likelihood = -15.980749 
Iteration 5:   log likelihood =  -15.98014 
Iteration 6:   log likelihood = -15.980111 
Iteration 7:   log likelihood = -15.980111  (not concave)
Iteration 8:   log likelihood =  -15.98011 

Negative binomial regression                    Number of obs     =          9
                                                LR chi2(8)        =      20.10
Dispersion     = mean                           Prob > chi2       =     0.0100
Log likelihood =  -15.98011                     Pseudo R2         =     0.3861

------------------------------------------------------------------------------
    injuries |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     airline |
          2  |  -.4519853   .4834938    -0.93   0.350    -1.399616    .4956452
          3  |  -.4519853   .4834938    -0.93   0.350    -1.399616    .4956452
          4  |   .5464595   .3788735     1.44   0.149    -.1961189    1.289038
          5  |  -.2006707   .4494666    -0.45   0.655    -1.081609    .6802676
          6  |  -1.011601   .5838742    -1.73   0.083    -2.155973    .1327715
          7  |  -1.299283    .651339    -1.99   0.046    -2.575884   -.0226821
          8  |  -2.397895   1.044466    -2.30   0.022    -4.445011   -.3507797
          9  |  -1.299283    .651339    -1.99   0.046    -2.575884   -.0226821
             |
       _cons |   2.397895   .3015114     7.95   0.000     1.806944    2.988847
-------------+----------------------------------------------------------------
    /lnalpha |  -19.25422   1404.153                     -2771.344    2732.835
-------------+----------------------------------------------------------------
       alpha |   4.35e-09   6.10e-06                             0           .
------------------------------------------------------------------------------
LR test of alpha=0: chibar2(01) = 4.7e-06              Prob >= chibar2 = 0.499

.

[Jeff Wooldridge]

To add to Carlo's helpful reply, let me emphasize that Poisson regression produces consistent estimators of the mean parameters with any kind of variance/mean relationship. Use -glm- with the fam(poisson) option and you'll be able to estimate the coefficient in the variance/mean relationship. I bet you'll find underdispersion. In this case, the negbin, fortunately, heads to the Poisson estimates, but this is not guaranteed. In more complicated settings, such as with panel data, I've seen xtnbreg hang up and fail to converge because the data are underdispersed. I've never seen this with the more robust xtpoisson estimator.

You should rarely use nbreg. In fact, I can't think of a case where I'd use it over Poisson regression unless I'm really interested in computing probabilities. For estimating mean effects, Poisson is much preferred. And always use a robust variance matrix. In cases of underdispersion, it actually reduces the standard errors!

Below is an example to a fertility data set (FERTIL2 with my introductory econometrics book). The estimate of the sigma parameter, which is based on the Pearson residuals, is 0.7859815. The nbreg estimates would not converge due do the underdispersion. You can see the robust standard errors are smaller than the usual MLE standard errors.

. glm children age c.age#c.age educ electric urban, fam(poisson) vce(robust)

Iteration 0:   log pseudolikelihood = -6700.6931  
Iteration 1:   log pseudolikelihood = -6586.2189  
Iteration 2:   log pseudolikelihood = -6585.9102  
Iteration 3:   log pseudolikelihood = -6585.9102  

Generalized linear models                         Number of obs   =      4,358
Optimization     : ML                             Residual df     =      4,352
                                                  Scale parameter =          1
Deviance         =  4086.463495                   (1/df) Deviance =   .9389852
Pearson          =  3420.591324                   (1/df) Pearson  =   .7859815

Variance function: V(u) = u                       [Poisson]
Link function    : g(u) = ln(u)                   [Log]

                                                  AIC             =     3.0252
Log pseudolikelihood = -6585.910155               BIC             =  -32382.29

------------------------------------------------------------------------------
             |               Robust
    children |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .3665522   .0092319    39.70   0.000     .3484579    .3846465
             |
 c.age#c.age |  -.0044698   .0001429   -31.28   0.000    -.0047499   -.0041897
             |
        educ |  -.0259419   .0025496   -10.18   0.000     -.030939   -.0209449
    electric |  -.1512171   .0332707    -4.55   0.000    -.2164264   -.0860077
       urban |  -.0761091   .0206833    -3.68   0.000    -.1166476   -.0355707
       _cons |  -5.710309   .1461743   -39.07   0.000    -5.996805   -5.423812
------------------------------------------------------------------------------

. glm children age c.age#c.age educ electric urban, fam(poisson)

Iteration 0:   log likelihood = -6700.6931  
Iteration 1:   log likelihood = -6586.2189  
Iteration 2:   log likelihood = -6585.9102  
Iteration 3:   log likelihood = -6585.9102  

Generalized linear models                         Number of obs   =      4,358
Optimization     : ML                             Residual df     =      4,352
                                                  Scale parameter =          1
Deviance         =  4086.463495                   (1/df) Deviance =   .9389852
Pearson          =  3420.591324                   (1/df) Pearson  =   .7859815

Variance function: V(u) = u                       [Poisson]
Link function    : g(u) = ln(u)                   [Log]

                                                  AIC             =     3.0252
Log likelihood   = -6585.910155                   BIC             =  -32382.29

------------------------------------------------------------------------------
             |                 OIM
    children |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         age |   .3665522   .0096188    38.11   0.000     .3476996    .3854047
             |
 c.age#c.age |  -.0044698   .0001422   -31.44   0.000    -.0047484   -.0041911
             |
        educ |  -.0259419   .0028118    -9.23   0.000    -.0314529    -.020431
    electric |  -.1512171   .0347648    -4.35   0.000    -.2193548   -.0830794
       urban |  -.0761091   .0215737    -3.53   0.000    -.1183927   -.0338255
       _cons |  -5.710309   .1596396   -35.77   0.000    -6.023197   -5.397421
------------------------------------------------------------------------------

[Mehrzad Baktash]

Dear Carlo and Jeff,

many thanks for the helpful and prompt replies. It helped me understand the issue much better.

Best regards,
Mehrzad

[Maxence Morlet]

Dear Professor Jeff Wooldridge,

Regarding your comment: "For estimating mean effects, Poisson is much preferred. And always use a robust variance matrix. In cases of underdispersion, it actually reduces the standard errors!" please could you explain mathematically how this is the case? That would be immensely helpful in order to better understand how vce(robust) affects standard error estimates in the case of a Poisson distribution.

Many thanks in advance!