Correlated Random Effects (CRE) / Mundlak approach

[Wulan Lika]

Dear all,

I am working with panel data and I would like to apply the Correlated Random Effects (Mundlak) approach to estimate time-invariant variables as independent variables.

Here is my specification:
xtset id year
xtreg gr x1 x2 x3 x4 d1, cre vce(cluster id)

I also performed the Mundlak test:
estat mundlak

The result rejects the null hypothesis (Prob > chi2 = 0.0000), indicating correlation between regressors and individual effects.

My question is:
1. Have I correctly implemented the Mundlak (CRE) approach in this case, particularly to handle the issue of time-invariant variables?
2. Does this result imply that the CRE (Mundlak) model is the appropriate model to use, instead of the standard Random Effects model?
3. How should I correctly interpret the coefficients from this CRE model?

Thank you very much for your guidance.

[Erik Ruzek]

Wulan,

It is hard to answer your questions because we do not know which of the presented variables are time-varying and which are time-invariant. Let's assume that the x variables time-varying and d variables are time invariant. To your questions...

1. The correlated random effects model estimates a model in which the cluster mean for each of the time-varying predictors is modeled along with the uncentered time-varying variables. xtreg, cre does this for you automatically.

2. A significant test indicates that the time-varying and time-invariant effects of the group of time-varying variables are different from each other. That is, you cannot assume that the within- and between-effects are the same. Accordingly, you need to use either the CRE model or a fixed effects model. Using a fixed effects model would force you to abandon your time invariant predictor as it is perfectly correlated with the indicators for cluster. You could only estimate a cluster indicator by time invariant predictor interaction to look at the extent to which the time varying effect differs by cluster.

3. The coefficients of the time-varying variables are interpreted in the same way as they would be in fixed effects. They tell you the mean difference in the outcome for a 1-unit increase in the predictor within clusters. The coefficients for the cluster means of the time-varying variables tell you the difference in the between and within effect of the regressor. To get the pure between (time-invariant) cluster effect, you would need to add these two coefficients together.

[Wulan Lika]

Dear Erik Ruzek,

Thank you very much for your clear and helpful explanation.

Yes, in my case, the x variables are time-varying, while d1 is a time-invariant variables. Based on your explanation, I understand that the CRE automatically includes the cluster means of the time-varying variables, allowing me to separate the within and between effects.

Regarding the Mundlak test result (Prob > chi2 = 0.0000), I understand that this indicates a significant difference between the within and between effects. Therefore, the standard Random Effects assumption is violated, and using either the CRE or Fixed Effects model would be more appropriate. However, since I am interested in estimating the effect of a time-invariant variable (d1), the CRE model is preferable in this case.

For the interpretation, I understand that:

• The coefficients of the time-varying variables represent the within effects, similar to Fixed Effects.
• The coefficients of the cluster means capture the difference between the within and between effects.
  
  Regression results

  gr	Coef.		St.Err.	t-value		p-value	[95% Conf		Interval]		Sig
  xit_vars
  x1	-.149		.031	-4.82		0	-.21		-.089		***
  x2	.003		.001	2.48		.013	.001		.006		**
  x3	-.001		.001	-0.85		.395	-.003		.001
  x4	.003		.003	1.09		.275	-.003		.01
  d1	.026		.016	1.65		.1	-.005		.057		*
  constant	.802		.198	4.05		0	.414		1.19		***
  xt_means
  x1	.129		.036	3.57		0	.058		.201		***
  x2	.001		.008	0.11		.916	-.014		.016
  x3	-.001		.002	-0.57		.571	-.005		.003
  x4	-.018		.01	-1.76		.079	-.037		.002		*
  d1	0		.	.		.	.		.
  Mean dependent var		0.328			SD dependent var			0.032
  Overall r-squared		0.455			Number of obs			108
  Chi-square		151.682			Prob > chi2			.
  R-squared within		0.399			R-squared between			0.605
  *** p<.01, ** p<.05, * p<.1

Is it correct that the main interpretation should be based on the coefficients of the time-varying variables (xit_vars) regarding the Mundlak test result, as they represent the within effects?


However, I would like to clarify one more point regarding the model selection procedure.

Is it appropriate to first perform a Chow test to choose between pooled OLS and fixed effects, and then proceed with the CRE estimation followed by the Mundlak test to determine the appropriate model?

Or should the CRE (Mundlak) approach be considered as an alternative framework to the traditional Chow–Hausman testing sequence?

Thank you for your guidance.

Best regards,
Wulan

[Carlo Lazzaro]

Wulan:
you cannot perform -hausman- with non-default standard errors, but you need to the community-contributed module -xtoverid-.

[Stephen Jenkins]

Wulan Lika Erik Ruzek Carlo Lazzaro Regarding -xtreg, cre- models, you might be interested in an article that's just appeared online, open access, Time Invariant Variables in the Mundlak and Hausman–Taylor Panel Data Models by Badi H. Baltagi, Long Liu, OBES: https://doi.org/10.1111/obes.70056

[Carlo Lazzaro]

Thanks, Stephen!

[Jeff Wooldridge]

Assuming that the reported chi-square statistic is the test of joint significant of the time averages, it is fully robust because vce(cluster id) has been used. That's the robust, regression-based version of the Hausman test. It is not numerically identical but the same idea as xtoverid. But I'm not entirely sure that it's only testing the time averages, rather than all variables. Oddly, it doesn't report a degrees-of-freedom.

[Wulan Lika]

Thank you for your guidance Carlo Lazzaro Stephen Jenkins Jeff Wooldridge

I would also like to ask, if my model does not show any signs of heteroskedasticity or autocorrelation, would it still be recommended to use robust (e.g., clustered) standard errors -xtreg, cre vce(cluster id)-, or would conventional standard errors be sufficient?

Thank you again for your guidance!

Best wishes,

Wulan

[Carlo Lazzaro]

Wulan:
did you check for cross-sectional dependence?

[Wulan Lika]

Carlo Lazzaro
I have already check for cross-sectional dependence, and the results show that most variables are significant, indicating the presence of cross-sectional dependence. Does this mean that I should use robust standard errors in correlated random effects (CRE) model with robust standard errors?

[Carlo Lazzaro]

Wulan:
see Robust Standard Errors for Panel Regressions with Cross-Sectional Dependence