Stata clogit command versus logit with manual fixed effects not (quite) reproducible: Coefficients double

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Thank you for taking a crack at my question! I am implementing a conditional logistic regression in Stata. I have choice data in long format where every choice consists of two available options and the decision-maker can only pick one. I have implemented it using the Stata clogit command, which in my understanding creates fixed effects for every choice in the data and partials them out before regressing the dependent variable on remaining explanatory variables in the logistic regression. To convince myself that clogit does what I think it does, I tried to reproduce the results I got using the logit command and adding the fixed effects manually. FWIW that is not straightforward with large data sets given Stata's limit on the number of explanatory variables, but the problem persists in the following smaller MWE:

* Retrieve MWE data set
webuse lowbirth2, clear

* Add arbitrary cluster variable, because in my real problem the data is clustered
gen cluster = ceil(_n/14)

clogit low lwt smoke ptd ht ui i.race, group(pairid) cluster(cluster)
    Conditional (fixed-effects) logistic regression

                                                Number of obs     =        112
                                                Wald chi2(7)      =     211.55
                                                Prob > chi2       =     0.0000
Log pseudolikelihood = -25.794271               Pseudo R2         =     0.3355

                                (Std. Err. adjusted for 8 clusters in cluster)
------------------------------------------------------------------------------
             |               Robust
         low |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         lwt |  -.0183757   .0111176    -1.65   0.098    -.0401657    .0034144
       smoke |   1.400656   .4670183     3.00   0.003     .4853172    2.315995
         ptd |   1.808009   .6162347     2.93   0.003      .600211    3.015807
          ht |   2.361152   .9149873     2.58   0.010     .5678096    4.154494
          ui |   1.401929   .5968851     2.35   0.019     .2320559    2.571802
             |
        race |
      black  |   .5713643   .5699717     1.00   0.316    -.5457596    1.688488
      other  |  -.0253148   .5197248    -0.05   0.961    -1.043957    .9933272
------------------------------------------------------------------------------

logit low lwt smoke ptd ht ui i.race i.pairid, cluster(cluster)
Logistic regression                             Number of obs     =        112
                                                Wald chi2(6)      =          .
                                                Prob > chi2       =          .
Log pseudolikelihood = -51.588542               Pseudo R2         =     0.3355

                                (Std. Err. adjusted for 8 clusters in cluster)
------------------------------------------------------------------------------
             |               Robust
         low |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         lwt |  -.0367513   .0222351    -1.65   0.098    -.0803314    .0068288
       smoke |   2.801312   .9340365     3.00   0.003     .9706343     4.63199
         ptd |   3.616018   1.232469     2.93   0.003     1.200422    6.031613
          ht |   4.722303   1.829975     2.58   0.010     1.135619    8.308988
          ui |   2.803858    1.19377     2.35   0.019     .4641118    5.143605
             |
        race |
      black  |   1.142729   1.139943     1.00   0.316    -1.091519    3.376977
      other  |  -.0506296    1.03945    -0.05   0.961    -2.087913    1.986654
             |
      pairid | ... omitted for brevity ...

Looking at these two outputs, we can see that the coefficients, standard errors, and Log Likelihood are not only different, but double exactly, as if the dependent variable had been scaled by a factor 2. I should add that when I don't cluster the standard errors, they don't exactly double anymore. So, clogit doesn't seem to just partial out fixed effects on the backend, but what does it do? Neither the documentation nor the clogit.ado file itself have resolved this for me.

stata regression
1 Answers

The difference in estimates that you're observing is the bias due to the incidental parameters problem from estimating logit in a finite panel.

See Greene (2004) for a discussion of this bias, and note that the bias is (1) away from zero, and (2) as high as 100% when T=2.

In your case T is not time, but the number of individuals in the pair (i.e. 2), thus you should expect the bias to be in the neighbourhood of 100%.

If you reread the pdf documentation for clogit, in particular the second paragraph of the Fixed-effects logit section under Remarks and examples you'll see how clogit avoids this problem.

Since the bias from using logit as you do is 100% relative to clogit the relationship you document between estimated coefficients is the expected behaviour (or really misbehaviour in the logit case) of the two estimators.

user_57918260
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