One of the problems with the word endogeniety is that it can be used for a set of very different issues/problems.

In a repeated measures analysis with occasions (level 1, repeated measures) nested with 'individuals' at level 2, fixed effects have have historically been the gold standard as this gets rid of any effect (including unmeasured effects) of the individual. This therefore gets rid of any omitted variable bias which is one form of endogeneity. But, because you have in effect dummied out all the information about individuals, you cannot include what are often very interesting questions about time invariant variables like the gender of the individual. Social science is full of such questions.

Traditionally, random effects has been seen as the evil twin. You can include time- invariant variables and have interactions between time-varying and time invariant variables so that you can model such questions as whether the gender gap in salaries is increasing over time. But, you are open to the charge of omitted variable bias affecting the time-varying effects.

But we (see below) have argued that it is possible to have your cake and eat it by using a particular specification of the random effects model called the within between (REWB) model. This gives exactly the same estimates of the within ( time- varying) effect as a fixed effects analysis but additionally allows you to estimate the cross-sectional effect, include time-invariant variables, (and explicitly model different forms of heterogeneity).

Importantly this requires no new estimation procedures and you can use standard random effects model software. But what is required is a re-specification of the models. The level 1 variable is de-meaned by the individual average, and this group mean value is additionally included as a level 2 variable. It is really easy to do in practice. [Another variant is the Mundlak model in which the group level variable is included by the level 1 variable is not de-meaned].

From this perspective, the oft-used Hausman test does not distinguish between random effects but rather that the cross-sectional between effect is different from the time-varying effect ; which is often interesting and worth knowing.

For some this is apostasy as we are abandoning the gold standard of fixed effects. For differing reactions see

https://en.wikipedia.org/wiki/Kelvyn_Jones#Academic_controversies

section beginning He (with colleagues) has challenged the 'gold standard' that fixed effects should be the standard approach to the analysis of Panel data.

The three papers that set this all out are

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

Article Fixed and Random effects models: making an informed choice

Article Understanding and misunderstanding Group mean centering: a c...

The latter summarizes the different models in Table 1 (see attachment).

In a more practical sense, these may also be useful

Book Developing multilevel models for analysing contextuality, he...

Book Developing multilevel models for analysing contextuality, he...

see chapters on the analysis of repeated measures .

This site

https://strengejacke.github.io/mixed-models-snippets/random-effects-within-between-effects-model.html#the-simple-random-effect-within-between-model-rewb-and-mundlak-model

sets out the argument and included snippets of code in R. Thus, the following code snippets show how to translate the Equations from (Bell, Fairbrother, and Jones 2018) into R-code, using lmer() from the lme4-package. That is it implements the equations of Table 1 (see attachment).

To be clear, REWB is not a magic panacea and we need to question its assumptions and try and build better models, on which see

https://www.researchgate.net/post/When_should_I_check_lmer_assumptions

and

https://www.researchgate.net/post/assumption_test_in_conducting_multilevel_modeling

Finally on assuming that the random effects are uncorrelated with the regressors, see this recent paper

Article On Ignoring the Random Effects Assumption in Multilevel Mode...

• which brings us back to using random effects and including group means............

One of the problems with the word endogeniety is that it can be used for a set of very different issues/problems.

In a repeated measures analysis with occasions (level 1, repeated measures) nested with 'individuals' at level 2, fixed effects have have historically been the gold standard as this gets rid of any effect (including unmeasured effects) of the individual. This therefore gets rid of any omitted variable bias which is one form of endogeneity. But, because you have in effect dummied out all the information about individuals, you cannot include what are often very interesting questions about time invariant variables like the gender of the individual. Social science is full of such questions.

Traditionally, random effects has been seen as the evil twin. You can include time- invariant variables and have interactions between time-varying and time invariant variables so that you can model such questions as whether the gender gap in salaries is increasing over time. But, you are open to the charge of omitted variable bias affecting the time-varying effects.

But we (see below) have argued that it is possible to have your cake and eat it by using a particular specification of the random effects model called the within between (REWB) model. This gives exactly the same estimates of the within ( time- varying) effect as a fixed effects analysis but additionally allows you to estimate the cross-sectional effect, include time-invariant variables, (and explicitly model different forms of heterogeneity).

Importantly this requires no new estimation procedures and you can use standard random effects model software. But what is required is a re-specification of the models. The level 1 variable is de-meaned by the individual average, and this group mean value is additionally included as a level 2 variable. It is really easy to do in practice. [Another variant is the Mundlak model in which the group level variable is included by the level 1 variable is not de-meaned].

From this perspective, the oft-used Hausman test does not distinguish between random effects but rather that the cross-sectional between effect is different from the time-varying effect ; which is often interesting and worth knowing.

For some this is apostasy as we are abandoning the gold standard of fixed effects. For differing reactions see

https://en.wikipedia.org/wiki/Kelvyn_Jones#Academic_controversies

section beginning He (with colleagues) has challenged the 'gold standard' that fixed effects should be the standard approach to the analysis of Panel data.

The three papers that set this all out are

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

Article Fixed and Random effects models: making an informed choice

Article Understanding and misunderstanding Group mean centering: a c...

The latter summarizes the different models in Table 1 (see attachment).

In a more practical sense, these may also be useful

Book Developing multilevel models for analysing contextuality, he...

Book Developing multilevel models for analysing contextuality, he...

see chapters on the analysis of repeated measures .

This site

https://strengejacke.github.io/mixed-models-snippets/random-effects-within-between-effects-model.html#the-simple-random-effect-within-between-model-rewb-and-mundlak-model

sets out the argument and included snippets of code in R. Thus, the following code snippets show how to translate the Equations from (Bell, Fairbrother, and Jones 2018) into R-code, using lmer() from the lme4-package. That is it implements the equations of Table 1 (see attachment).

To be clear, REWB is not a magic panacea and we need to question its assumptions and try and build better models, on which see

https://www.researchgate.net/post/When_should_I_check_lmer_assumptions

and

https://www.researchgate.net/post/assumption_test_in_conducting_multilevel_modeling

Finally on assuming that the random effects are uncorrelated with the regressors, see this recent paper

Article On Ignoring the Random Effects Assumption in Multilevel Mode...

• which brings us back to using random effects and including group means............

Kelvyn Jones

Thank you Professor

Thanks a lot

professor!