# Panel data: fixed individual and random time effect

I am currently working with panel data. So far, I specified a fixed effects model with two fixed effects: firm fixed effects (I'm working with company data) and time fixed effects. After performing some Hausman tests, the results suggest that the firm effect is indeed fixed, but my time effect seems to be random.

To get to this conclusion, I tested my two-way fixed effects model against (1) the two-way random effects model, (2) a model with fixed time and random firm effects, and (3) a model with fixed firm and random time effects. All but the third model suggested the use of the two-way fixed effects model. So I concluded that I should use the model with the fixed firm and random time effects.

Now my questions:

1. Is it even possible to have one random and one fixed effect?

2. Do my tests make sense in this way?

3. How would you estimate such a model? I simply added a full set of firm/time dummies to my regression and then used the random effects estimators, but I really don't know if this makes sense.

4. Can you recommend any literature in which a model with one fixed and one random effect is explained?

Thank you very much!

• EViews User guide mentions this mixed model (should be accessible online). The text book by Baltagi mentions it as well. Sep 18, 2017 at 19:04
• Thank you! I think I just found the section in the book by Baltagi. You are speaking about "Econometric Analysis of Panel Data", especially the "Problems" section in Chapter 3, right? I will also check out the EViews user guide! Sep 19, 2017 at 8:57
• Yes, it is mentioned first in 3.7 "Computational Note". But take care, the term mixed effects is also often used in the ML context, afaik. Sep 19, 2017 at 11:48
• Thank you @Helix123, I did not see that he also mentioned it there! Good point, I was seriously confused when I first googled mixed models. :D Sep 19, 2017 at 19:17

1. Yes, this is a "mixed effects" model.

2. I am not a fan of diagnostic testing as a way of "discovering the question" your model is answering. A caveat to consider is the non-transitivity of tests. When you go about testing 3 (or more) models, you may end up with an ourobos where model A > model B, model B > model C, yet model C > model A. It makes no sense. The choice of whether to apply fixed or random effects is based on the scientific rationale. So I say your tests do not make sense.

3. Generally, if there is power to fit a fixed effects model, we prefer that approach because the model enables direct inference and prediction from estimated effects with fewer asymptotic results. A GEE blends many of these properties: easy inference, prediction, but accounting for correlated observations. I would fit a GEE knowing little else about your problem.

4. Virtually any data analysis text on the subject of longitudinalk data discusses this. "Longitudinal data analysis" by Diggle, Heagerty, Liang, Zeger is one such text I frequently access.

• Thank you very much for your answer @AdamO! I've read the name mixed effects model before. However, I was a little confused since the term "fixed effect" in these models sometimes seems to refer to an explanatory variable for which the slope coefficient needs to be estimated. I always thought of a fixed effect as an unobserved, cluster specific variable that is correlated with the explanatory variables. But maybe this is too imprecise? Sep 19, 2017 at 9:10
• Further, I feel similar to you. I am quite sure that I need to apply a fixed effects model. However, my supervisor probably wants to see at least one test. So if I really wanted to do so, would you recommend to run the Hausman test of two-ways fixed vs two-ways random? Next, I'd like to thank you for suggesting the GEE approach. I honestly didn't hear about it before and I am currently looking into it. As I understand so far, it is an estimation technique and a different approach as FE or RE estimation. And thank you for the book recommendation. Sep 19, 2017 at 9:15
• @HansLeifson a fixed effect is not a variable as the name would suggest :). A FE is any model coefficient that has no "randomness" (assume Frequentist inference here): it may be a site-based mean difference (or slope), but it also may be a mean difference for income, sex, or age-groups, or even between time periods. Sep 19, 2017 at 11:43
• @HansLeifson I really must push back on the concept of statistical testing: making inference only works when one has but one final question in mind. Conducting statistical tests because your advisor expects it makes me question the advisor and not the model. It sounds to me like you're doing a more exploratory data analysis. If that's the case, I see no problem in using statistics and data analysis to propose one or more models that have the same desired application but different approaches. To answer the question of 2 REs, 1 RE & 1 FE, vs 2 FEs consider finding which model is BIC optimal. Sep 19, 2017 at 11:46
• Again thank you very much for your answer @AdamO! I guess you are right and I will definitely talk to him about that. Also, I didn't think of the possibility to use such a criterion and it sounds like a good way around the testing procedure. Sep 19, 2017 at 18:55