I am not sure if I asked the question correctly. I am analysing longitudinal data. If I include time-invariant independent variable in the fixed-effect model, it will be just ignored because there is no variation within individuals.

The Hausman test is not defined in this case as there is nothing to test (FE estimation identifies only coefficients of time varying variables).

If the regressor is time invariant there is no point in running hausman test as it checks hetrogeniety across individuals and time....

Agreed with the previous comment. If you have a theory that tells you you should have the time invariant regressors or if the coef of these variables are of interest to you then use random effects and if you re worried about the correlation between the random intercept and your regressors then add a Mundlak correction. If you re interested in the dynamics you ll have to also correct for initial conditions. Check Wooldridge. It also work if your dependent variable is dichotomic

To add, that the Hausman test was not intended to be a means to decide between RE and FE modelling procedures. Instead it is a method to understand how much within and between variation there is in the data. The RE vs FE choice should be guided by the theory behind your question and the information that you wish to uncover. (some useful information here https://www.researchgate.net/publication/233756428_Explaining_Fixed_Effects_Random_Effects_modelling_of_Time-Series_Cross-Sectional_and_Panel_Data?ev=prf_pub)

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...

If I interpret your question correctly, in your RE specification you have both time-varying and time-constant variables. Provided that your RE and FE models share the same set of time-varying variables, you can just carry out the Hasuman on the coefficients of the time-varying variables as they are identified in both specifications. Your case is quite usual, also because the RE model, unlike the FE, identifies the constant. Finally, note that having less regressors does not make your FE model less general since the fixed effects absorb the effects of *every* time-constant variable, observed or not.

Most of my independent variables are time-variant. When I run the FE and RE models only with these time-variant variables I get quite similar results and the Hausman tests also suggest that both models are consistent. The problem comes when I add this important time-invariant control variable in the RE model because in this case the coefficient of my key explanatory variable decreases significantly and the Hausman test shows that the difference between FE and RE is statistically significant. Does it mean that I should use the FE? In any case this additional control variable doesn't effect the consistency of the FE model.

I have always thought that the Hausman test serves as a guidance as to which model to choose from - FE or RE. Am I really wrong about this? Is there other opinions?

Thanks, Giovanni. You got my question right and this is answer I wanted to read.

There is nothing wrong in carrying out the Hausman test when the RE effect specification contains time-constant variables. After all, this is one of the main reason to focus on the RE. But if the test shows signficant differences between the two models, then the FE should be preferred.

Thank you Plamen with your question. Me too, I learned that you can run a Hausman test in order to decide between fixed or random effects. The null hypothesis is that the preferred model is random effects vs. the alternative the fixed effects (see Green, 2008, chapter 9). In other words, you test whether the unique

errors are correlated with the regressors.

Giovanni, Tharcisse - I disagree. If you were interested in time-constant variables, you would be fool to choose a method (FE) that cannot answer your research question. As my paper that David linked above shows (https://www.researchgate.net/publication/233756428), there is an easy correction for the omitted variable bias in a RE model, meaning RE will always give *exactly* the same answer as FE, making the Hausman Test as test of FE vs RE redundant. It is a common misconception that the Hausman Test is a test of FE vs RE, but it is a misconception nonetheless.

The above paper is currently in review, but If you want something a bit more published that shows this, this is Mundlak in his original (1978) paper:

“when the model is properly specified, the GLSE [that is RE] is identical to the "within" [that is, the FE] estimator. Thus there is only one estimator. The whole literature which has been based on an imaginary difference between the two estimators ... is based on an incorrect specification which ignores the correlation between the effects and the explanatory variables."

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...

Andrew, if you are interested in the RE model, because of one important time-constant variable, and the hausman test rejects, this is no good news for your RE. Mundlak makes it clear that RE can be obtained as a restricted GLS and that FE is an unrestricted GLS (including group means). Indeed, you can use Mundlak as a test between RE and FE and with homoskedastic errors this is just Hausman test. Then, of course one can purge the RE from the bias, but this is another matter (I did not know your paper and I will have a look at it, thanks.).

You can use the Hausman-Taylor estimator, which can accommodate random effects panel models when there is endogeneity of the regressors like in your case. Basically, the Hausman test leads you to FE, but FE will not generate coefficients for time-invariant regressors, then the Hausman-Taylor will be appropriate. Stata can do the job.

Andrew, I don't remember saying that I am interested in estimating the time-constant variables. I am just wondering why this extra control variable changes the coefficient of my key explanatory variable (which is time-variant) in the RE model. In your citation by Mundlak it is mentioned "if the model is properly specified..." - I guess that means that you need to have all confounders available in the data set, which is rarely the case. After all, the great advantage of the FE is that it controls for unobservable heterogeneity or for anything that is time-constant.

Omar, xttylor could be a good alternative, but in my case I have dichotomous outcome variable which complicated the model too much. But thanks for your suggestion.

Thank you all for the nice discussion!

Plamen - 'properly specified' means including the group (individual) mean in your regression as an additional regressor. So you don't need all confounders. The group mean controls everything that you need to allowing you to get unbiased estimators (that are the same as FE) whilst keeping all the other advantages of RE. It is explained more fully in my paper.

For you, where the interest is in the time-varying variable only, you could use either model (the results will be the same), but the RE model will allow you to answer some additional questions regarding how individual context matters. This could be valuable, and given you get the same answers either way, why not do so?

It is interesting that the addition of time invariant variables biases your results. It suggests something like: your observed t-invariant variables are, say, positively correlated with the higher level residual, but your unobserved t-invaraint variables are negatively correlated. When neither are in the model they cancel each other out in their effect of the estimates, producing an apparently unbiased result, but when you control for only some of the important variables, the bias from the unobserved variables comes through. Either way, including group means will sort you out and give unbiased estimates.

****

Giovanni - you are right that Mundlak's formulation can be used as a form of Hausman Test, but note that the formulation is estimated with a RE estimator, and itself gives you *the same* parameter estimates as FE. Why would you run this model and then throw it away in favour of FE, when you could just use the Mundlak estimates and keep the time-invariant estimates which might be valuable?

Dear Plamen, if I understand it correctly, the Hausman test indicates that your RE estimates for the time-varying regressors deviate from the FE estimates - supposedly because there is both correlation between your time-invariant regressor and the time-constant error component on the one hand and your time-invariant regressor and the means/initial values of your other regressors... you could try to include more time-constant regressors in order to reduce the non-explained part of the model, or employ a factor model to get rid of the time-invariant error component (see Bai's 2013 ECTA article) without omitting all time constant information.

Best

Norbert

If the true model if RE, then any time invariant regressors that you include in the RE model will be subsumed in the intercepts of the FE model. So, there should be no significant difference in the coefficients on the time-varying regressors between the two models (you just lose efficiency by estimating all those intercepts). If there is a significant difference between RE and FE it implies the true model is not RE.

By the way, a little noted fact is that both RE and FE require strict exogeneity of the regessors for consistency (And this is a very strong assumption that is not required by OLS). If RE and FE differ it may just be because failure of strict exogeneity biases the two estimators in different ways. My 1992 JBES paper with Runkle talks about this.

Andrew, the Mundlak specification is no magic trick solving the endogeneity problem of the time-constant variables (although it is a convenient device to solve heteroskedasticity problems in the Hausman test). Mundlak models correlated individual heterogeneity by including the group means of the explanatory variables. But the group mean of a time constant is exactly that time constant and so your coefficient estimates on time constant variables, in Mundlak, capture both the structural impact of interest and the impact of the correlated latent heterogeneity. In other words, the estimated coefficients on the time-constant variables in Mundlak are biased (unless, of course, you assume zero correlation with the latent individual components). In fact, you can easily verify that the estimated coefficients on the time constant variables, in Mundlak, are exactly the same coefficients that you would obtain by running a between regression, not exactly a robust estimator.

The Hausman-Taylor estimator, suggested by Omar can be a solution, but it still requires strong identifying assumptions and its validity is usually tested against the FE estimator through the Hausman test, again limited to the common coefficients.

Giovanni - This is very true, but hardly a reason to choose FE as suggested by the Hausman Test, given that that cannot estimate time constant variables at all. That a model isn't perfect doesn't mean that it shouldn't be used, just that researchers need to, as with any model, think carefully about possible sources of bias (omitted variables, etc).

You might be interested in this paper which applies the principles of the Hausman Taylor estimator to the Mundlak specification: http://www.touteconomie.org/afse/index.php/congres/2013/paper/viewFile/228/331

Andrew, the Mundlak specification is very useful for a number of applications. My point is only that, by itself -that is is by adding the group means and then implementing FGLS estimation-it does not solve the endogeneity problem that would arise from the correlation of the time-invariant regressors and the latent heterogeneity. For that one needs other estimators.

Best

Giovanni

"That a model isn't perfect doesn't mean that it shouldn't be used, just that researchers need to, as with any model, think carefully about possible sources of bias (omitted variables, etc)."

This is a good point !!!