[Jeff Wooldridge]

There is a natural confusion between estimation methods and models. I think one can best understand the traditional RE, FE, and Mundlak approaches as different estimation strategies. The model is best called something like an "unobserved effects model."

As Eric observed, the Mundlak (also called correlated random effects) approach produces the FE estimates on the time-varying X -- and it does not matter what other time-constant controls, the sector dummies in Guest's case -- are included. As a bonus, the CRE approach allows one to estimate coefficients on the time-constant (sector) dummies. However, one should not read too much into these as they are unlikely to be "causal" in any sense.

As a way to test RE versus FE, it is very important to include the sector dummies in the Mundlak regression and then test the time averages. The time-constant controls (sector dummies) can and should be included in the RE estimation and they should also be included in the CRE estimation. It is easy to think of cases where including the sector dummies could render RE sufficient. Guest included them and also made the standard errors fully robust, so the outcome of the test can be trusted. You are rejecting RE in favor of FE because those are the only coefficients we can compare. That one can estimate coefficients on the sector dummies is, relatively speaking, not especially important. They need to be included. And you can report them if you must. But the point of FE is that you don't have to include any time-constant variables. Having firm effects when you use FE is much more general than sector effects, and that is why they drop out. So you are doing the Mundlak version of FE. I'm not sure we needed a new phrase, "hybrid model." I just find that it confuses people.

If you want a fairly simple source for all of this, I discuss in in Chapter 14 of my introductory econometrics text, "Introductory Econometrics: A Modern Approach," 7e, 2019. Some of the discussion is in earlier editions, too.

[Jeff Wooldridge]

Guest: On your last point, you ARE rejecting RE. The conclusion is you have to estimate the coefficients on X1, X2, and so on by FE. Whether you use usual FE or the Mundlak approach is irrelevant: the estimates and standard errors (made robust) are the same. The only issue is whether you want to report the coefficients on the sector dummies. If that's important to you, then report them. But the other estimates are the FE estimates.

[Carlo Lazzaro]

Thanks Jeff for your detailed explanation.
I really hoped your chiming in this interesting thread.

[Eric de Souza]

Thanks for providing all the details, Jeff. I ought to have done so but was trying to finish something else when I saw this thread yesterday.
And, indeed, your 7th edition was very helpful for my students in this regard.
On Edit: you say that "You are rejecting RE in favor of FE because those are the only coefficients we can compare." This point about "only being able to compare the coefficients on the time varying variable" I missed.

[Guest]

Jeff Wooldridge Thank you for your clarifying response, Mr Wooldridge. I really appreciate it. Many thanks to you too Carlo Lazzaro and Eric de Souza. I learned a lot from your feedbacks too.