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  • Manual -xtprobit, fe- and -xtoprobit, fe-?

    Dear listers,

    after estimating fixed effect regressions for an outcome with a limited number of values with -xtreg- and -xtologit-, I'd ideally like to explore how robust results are to Probit instead of Logit or the Linear Probability Model.
    I've noted that Stata's -xtoprobit- or -xtprobit- do not allow for the fixed effect option, and I'm not sure whether moving to random effects instead would make much sense or how I would sensibly interpret such results then.

    Technically I can circumvent that constraint by just manually adding the respective set of fixed effects to the list of controls, as the number of fixed effects would still allow me to compute it given my computer and enough hours of patience (I take it that there is no faster option like -areg- for Probit either?).

    However, I'm wondering whether doing so makes sense if the number of fixed effects is not too large, or whether results would be biased so that I should abstain from using (O)Probit whenever I also wish to use fixed effects?

    Thanks so much,
    PM




  • #2
    As you have noted, there is no fixed effects probit (nor xtprobit nor xtoprobit) estimator in Stata. As far as I know, they don't exist in any statistical package.

    Running these models using indicator variables for your panels to the list of predictor variables is not equivalent to a fixed-effects probit (etc.) estimator. That trick is only valid for linear regression. And a random effects model is estimating completely different things from a fixed-effects model, so using that as a robustness check would be completely misleading.

    I think the only thing you can do is compare -probit- and -logit-. I can tell you in advance that it is very rare for these models to give markedly different results in the sense of the z-statistics. The coefficients themselves will never be equal (unless they are zero) because the residual error in a probit model has variance 1, whereas the residual error in a logit model is pi2/3. What you can expect to find is that the coefficients will differ by a ratio of approximately pi/sqrt(3) (logit:probit). If you do find a marked deviation from this, it suggests that you are working with very "pathological" data that is calculating large numbers of error terms in the far tails of the corresponding distributions. If that's the case, the conclusion would probably be that neither model is appropriate for your data. Usually, in the real world, that implies, in turn, that there is something really wrong with your data, though it is possible that the data are right and some drastically different model is required.

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    • #3
      Clyde means that there is no conditional fixed effects probit estimator. But there is an unconditional fixed effects probit estimator (i.e., probit with dummies). The incidental parameter bias is declining with T. So if you have large T and substantial computing resources, you may be able to estimate a fixed effects model with negligible incidental parameter bias. Bill Greene did some Monte Carlo analysis of the fixed effects probit estimator some years back and found that with small T, using the random effects estimator is worse than either ignoring heterogeneity or using the unconditional fixed effects estimator. You can download the paper for free here.

      Reference:

      Greene, W.H., 2004, The behaviour of the maximum likelihood estimator of limited dependent variable models in the presence of fixed effects, Econometrics Journal 7, 98–119.

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      • #4
        In addition to the incidental parameters problem if T is not large is the issue of computing proper standard errors, especially for the average partial effects.

        You can always try a correlated random effects probit. In the linear case, CRE reproduces the FE estimates. It often works well in the context of probit, too.

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        • #5
          Dear Clyde, Andrew and Jeff,

          thank you all three so much for these very helpful responses!

          Then I conclude that it clearly does not make sense to provide any sort of probit estimations to test the robustness of my LPM and logit estimations, and rather I shall rely on the latter two only. It is good to be sure about this and know better now how to explain it.

          Thanks so much and best regards, PM

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          • #6
            Peter: I didn’t quite say that. In fact, if you track down a copy of my MIT Press book you’ll see that I have a table that reports the LPM, fixed effects logit, and two versions of CRE probit: pooled and joint MLE. The pooled MLE and the LPM give remarkably similar results. I think the pooled MLE probit provides a good robustness check.

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            • #7
              You talk about different estimators, but you do not say what your data looks like. And you do say something which is irrelevant " if the number of fixed effects is not too large".

              It does not matter whether the number of fixed effects is large or not, the key is whether you have enough data points per fixed effect. In other words if you create manually a dummy for each fixed effect in the cross sectional dimension, the key is how many T data points you have per fixed effect.

              At the extreme if your T=2, you do have the accidental parameters problem, and manually including dummies for the fixed effects in your probit does not make sense.

              (What I am saying is pretty much what Andrew is saying, he just phrased is as disappearing bias as T grows. I more think of this as "manually including dummies makes sense if you think your T is large". )

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