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  • Mixed Model - Random Effect nested in one of the Fixed Effects

    Dear Statalisters,

    I have a very short question and would be thankful if you could give a quick thought.

    I am a financial economist and currently study the performance of financial investors. For this purpose, I want to fit a mixed model of the following sort:

    Code:
    mixed investment_performance covariates i.investor_id || investor_id_investmet_year:
    investor_id_investmet_year is an interaction between the investor and the investment year (each investor makes several investments a year). I am not so familiar with mixed models but believe to understand the basic assumptions.
    In essence, I want to know whether it may cause any problem if the random effect (investor_id_investmet_year) is nested in one of the fixed effects (investor_id) of the model?
    I don’t want to use a model with nested random effects (i.e. … || investor_id: || investor_id_investmet_year as the investor-specific effects are correlated with the other covariates, thereby violating the exogeneity assumption.

    Happy to hear your thoughts on this.

    Best regards
    Peter

  • #2
    Most of the variance of the random effects (if not all) is absorbed by the investor fixed effects (included as dummies). How different are these two models?

    Code:
    mixed investment_performance covariates i.investor_id
    and

    Code:
    mixed investment_performance covariates i.investor_id || investor_id_investmet_year:
    The LR test at foot of the second model above should tell you whether they are equivalent.

    Comment


    • #3
      Thanks Andrew for your reply, I really appreciate it.

      The two models you proposed are in fact not significantly different unless I omit the other covariates (I consider that a result already).
      I understand that the investor fixed effects (included as dummies) absorb most of the variance and hence can interpret the results. I was just wondering whether I miss something and the whole specification is potentially invalid. But if I understand you correctly, you would not see a technical issue here?

      Thanks a lot.

      Best regards
      Peter

      Comment


      • #4
        If the variance of the random effect is zero, it means that you estimated a fixed effects model. In that case, you cannot say that you estimated a mixed model (with both fixed and random effects). What I will say is that you cannot have both fixed effects and random effects for the same entity. The relationship here is not perfect as one is an interaction of the other, but the results tell you that what you are estimating when including both is just the fixed effects model if in fact the variance of the random effect is zero.

        Comment


        • #5
          Understood. But for the model without covariates:
          Code:
           mixed investment_performance i.investor_id || investor_id_investmet_year:
          I actually do find a significant LR-Test. Would you then agree that this is a properly specified mixed model? Or do I miss something here? Sorry for being so persistent, but it really helps to hear your thoughts.

          Best regards
          Peter

          Comment


          • #6
            If you look into the theory, the motivation for fixed effects models lies in the ability to control for the possibly correlated, time-invariant heterogeneity without observing it. The rationale is that the coefficients on the time-varying variables are inconsistent if the unobserved characteristics are correlated with these variables. In the absence of such regressors, then the model makes no sense. For all intents and purposes, coefficients on the individual dummies are nuisance parameters.

            Comment


            • #7
              I understand where you're coming from.
              However, I am interested in the heterogeneity of the investors themselves (i.e. the individual fixed effects). Maybe I should have clarified that from the beginning. Accordingly, I would argue that a fixed effects model without covariates does make sense in this context (it just provides an investor-specific constant, essentially the difference in mean relative to the base category). Notwithstanding, I - of course - add the covariates in additional regressions, consistent with your reasoning. I was just merely wondering whether the interaction of year&investor as an additional random effect messes up the model in a way that I am unaware. From your comments, I would figure that this is not the case. Instead adding the interaction appears to constitute a very peculiar way of modeling the error structure (and to test whether the errors are correlated in the investor-year cluster when including the fixed effects). Would you agree?

              Thanks a lot for your reply - really helps to think this through!

              Comment


              • #8
                It's not a problem provided that the random effects assumption holds. In economics, we let the sample data make that call. That means that you have to use a Hausman test to compare the coefficients on the time-varying variables in the mixed model against those from the following model:

                Code:
                regress investment_performance indepvars i.investor_id i.investor_id_investmet_year
                If you cannot reject the null hypothesis of the Hausman, go right ahead.

                Comment


                • #9
                  Thanks!

                  Comment


                  • #10
                    Maybe a quick follow-up question, Andrew.
                    Code:
                     
                     mixed investment_performance covariates i.investor_id || investor_id_investmet_year:
                    Would I potentially run into the incidental parameters problem in the model above? I guess that is a rather general question: Can I actually estimate fixed effects (i.investor_id for instance) consistently in a mixed model, given that I use ML for the estimation? Do you happen to know that?

                    Cheers, Peter

                    Comment


                    • #11
                      The incidental parameters problem is only relevant in nonlinear limited dependent variable models. In a linear model, there is no issue with including fixed effects as indicators (except for the computational burden).
                      Last edited by Andrew Musau; 05 Nov 2021, 07:15.

                      Comment

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