The fixed-effect estimator runs OLS on demeaned data:
\[ (y_{it} - \bar y_i) = (X_{it} - \bar X_i) \beta + (\epsilon_{it} - \bar \epsilon_i) \tag{9.2}\] Both \(\alpha\) and \(v_i\) drop out after demeaning by unit.
9.3 Random effect
Random effects treat \(v_i + \epsilon_{it}\) as a composite error with two variance components: \(\hat \sigma^2_e\) (idiosyncratic) and \(\hat \sigma^2_u\) (individual). The GLS transformation is
\(z^*_{it} = z_{it} - \hat \theta_i \bar z_i\)
with \(\hat \theta_i = 1 - \sqrt{\hat \sigma^2_e / (T_i \hat \sigma^2_u + \hat \sigma^2_e)}\), where \(T_i\) is the number of observations for unit \(i\). Given estimated variance components, we run OLS on the transformed variables and iterate.
9.4 Correlated random effect
The correlated random-effect (CRE) model runs a random-effects regression of \(y_{it}\) on \(X_{it}\), \(\bar X_i\), time-invariant covariates \(w_i\), and a constant.
RE requires \(v_i\) and \(X_{it}\) to be uncorrelated — an assumption most economists reject. FE is consistent regardless, but it cannot estimate time-invariant covariates (race, gender, etc.). CRE has both advantages: it includes time-invariant variables and remains consistent. Mundlak and Wooldridge show that the CRE coefficients on \(X_{it}\) are identical to the FE estimates. Testing whether the coefficients on \(\bar X_i\) are jointly zero gives the Mundlak test for choosing between RE and CRE.
9.5 Example
Stata 18 added a cre option to xtreg. The following uses nlswork, 28,101 person-years from the National Longitudinal Survey of Young Women with complete data on the model’s variables. We regress log wage on tenure and age, which vary within a person, plus race, which does not — the covariate a fixed-effect model has to drop.
Code
webuse nlsworkxtreg ln_wage tenure age i.race, cre vce(cluster idcode)
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)
note: 2.race omitted from xt_means because of collinearity.
note: 3.race omitted from xt_means because of collinearity.
Correlated random-effects regression Number of obs = 28,101
Group variable: idcode Number of groups = 4,699
R-squared: Obs per group:
Within = 0.1296 min = 1
Between = 0.2346 avg = 6.0
Overall = 0.1890 max = 15
Wald chi2(4) = 1685.18
corr(xit_vars*b, xt_means*γ) = 0.5474 Prob > chi2 = 0.0000
(Std. err. adjusted for 4,699 clusters in idcode)
------------------------------------------------------------------------------
| Robust
ln_wage | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
xit_vars |
tenure | .0211313 .0012113 17.44 0.000 .0187572 .0235055
age | .0121949 .0007414 16.45 0.000 .0107417 .013648
|
race |
Black | -.1312068 .0117856 -11.13 0.000 -.1543061 -.1081075
Other | .1059379 .0593177 1.79 0.074 -.0103225 .2221984
|
_cons | 1.2159 .0306965 39.61 0.000 1.155736 1.276064
-------------+----------------------------------------------------------------
xt_means |
tenure | .0376991 .002281 16.53 0.000 .0332283 .0421698
age | -.0011984 .0013313 -0.90 0.368 -.0038077 .0014109
|
race |
Black | 0 (omitted)
Other | 0 (omitted)
-------------+----------------------------------------------------------------
sigma_u | .33334407
sigma_e | .29808194
rho | .55567161 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Mundlak test (xt_means = 0): chi2(2) = 331.5144 Prob > chi2 = 0.0000
CRE gives tenure = .0211313 and age = .0121949, and estimates race as well: Black women earn 13.1% less (\(-.1312068\), SE .0118), while the “Other” category is marginal (\(.1059\), \(p = 0.074\)). The coefficients on the group means are .0376991 for mean tenure and \(-.0011984\) for mean age.
Stata prints the Mundlak test at the foot of the table: \(\chi^2(2) = 331.51\), \(p < 0.0001\). The group means are jointly far from zero, so plain random effects is rejected — \(v_i\) and the regressors are correlated, as the reported \(\text{corr} = 0.5474\) between the two blocks says directly.
To compare with a fixed effect model:
Code
webuse nlsworkxtreg ln_wage tenure age i.race, fevce(cluster idcode)
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)
note: 2.race omitted because of collinearity.
note: 3.race omitted because of collinearity.
Fixed-effects (within) regression Number of obs = 28,101
Group variable: idcode Number of groups = 4,699
R-squared: Obs per group:
Within = 0.1296 min = 1
Between = 0.1916 avg = 6.0
Overall = 0.1456 max = 15
F(2, 4698) = 766.79
corr(u_i, Xb) = 0.1302 Prob > F = 0.0000
(Std. err. adjusted for 4,699 clusters in idcode)
------------------------------------------------------------------------------
| Robust
ln_wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
tenure | .0211313 .0012112 17.45 0.000 .0187568 .0235059
age | .0121949 .0007414 16.45 0.000 .0107414 .0136483
|
race |
Black | 0 (omitted)
Other | 0 (omitted)
|
_cons | 1.256467 .0194187 64.70 0.000 1.218397 1.294537
-------------+----------------------------------------------------------------
sigma_u | .39034493
sigma_e | .29808194
rho | .63165531 (fraction of variance due to u_i)
------------------------------------------------------------------------------
The coefficients on tenure and age are identical: .0211313 and .0121949 in both, with standard errors agreeing to the seventh decimal (.0012112 against .0012113). Race is reported as 0 (omitted) here, being collinear with the unit effects — the one thing FE cannot do and CRE can. This is the Mundlak result in practice: CRE gives you the fixed-effect estimates and the time-invariant coefficients from a single random-effects fit.
We can replicate this manually by including the group means in a RE model:
Code
webuse nlswork* Restrict the mean calculation to the estimation sample (listwise-deleted* on the model's variables), matching what xtreg's native cre option does* internally. Both tenure and age have missingvaluesin nlswork, so* computing age_mean/tenure_mean overallrowsfor an idcode (rather than* only rows with complete dataon ln_wage, tenure, age, and race) would* make the manual means diverge from cre's.egen age_mean = mean(age) if !missing(ln_wage, tenure, age, race), by(idcode)egen tenure_mean = mean(tenure) if !missing(ln_wage, tenure, age, race), by(idcode)xtreg ln_wage tenure tenure_mean age age_mean i.race, vce(cluster idcode)
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)
(433 missing values generated)
(433 missing values generated)
Random-effects GLS regression Number of obs = 28,101
Group variable: idcode Number of groups = 4,699
R-squared: Obs per group:
Within = 0.1296 min = 1
Between = 0.2346 avg = 6.0
Overall = 0.1890 max = 15
Wald chi2(6) = 2688.54
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
(Std. err. adjusted for 4,699 clusters in idcode)
------------------------------------------------------------------------------
| Robust
ln_wage | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
tenure | .0211313 .0012113 17.44 0.000 .0187572 .0235055
tenure_mean | .0376991 .002281 16.53 0.000 .0332283 .0421698
age | .0121949 .0007414 16.45 0.000 .0107417 .013648
age_mean | -.0011984 .0013313 -0.90 0.368 -.0038077 .0014109
|
race |
Black | -.1312068 .0117856 -11.13 0.000 -.1543061 -.1081075
Other | .1059379 .0593177 1.79 0.074 -.0103225 .2221984
|
_cons | 1.2159 .0306965 39.61 0.000 1.155736 1.276064
-------------+----------------------------------------------------------------
sigma_u | .33334407
sigma_e | .29808194
rho | .55567161 (fraction of variance due to u_i)
------------------------------------------------------------------------------
This is what Stata’s cre option does internally. Every number matches the first table: tenure .0211313, age .0121949, Black \(-.1312068\), Other .1059379, constant 1.2159, and the same variance components (\(\sigma_u\) .3333, \(\sigma_e\) .2981, \(\rho\) .5557). The cre option is a convenience, not a different estimator.
The one detail worth care is in the code above: the group means must be computed on the estimation sample. Both tenure and age have missing values in nlswork, so averaging over all of a woman’s rows rather than only her complete ones would give different means and break the equivalence.
Source Code
---title: "Correlated Random Effect"date: "2025-04-15"---## Panel dataThe standard panel-data setup is:$$y_{it} = \alpha + X_{it} \beta + v_i + \epsilon_{it} $$ {#eq-correlated-random-effect-1}## Fixed effectThe fixed-effect estimator runs OLS on demeaned data:$$ (y_{it} - \bar y_i) = (X_{it} - \bar X_i) \beta + (\epsilon_{it} - \bar \epsilon_i)$$ {#eq-correlated-random-effect-2}Both $\alpha$ and $v_i$ drop out after demeaning by unit.## Random effectRandom effects treat $v_i + \epsilon_{it}$ as a composite error with two variance components: $\hat \sigma^2_e$ (idiosyncratic) and $\hat \sigma^2_u$ (individual). The GLS transformation is$z^*_{it} = z_{it} - \hat \theta_i \bar z_i$with $\hat \theta_i = 1 - \sqrt{\hat \sigma^2_e / (T_i \hat \sigma^2_u + \hat \sigma^2_e)}$, where $T_i$ is the number of observations for unit $i$. Given estimated variance components, we run OLS on the transformed variables and iterate.## Correlated random effectThe correlated random-effect (CRE) model runs a random-effects regression of $y_{it}$ on $X_{it}$, $\bar X_i$, time-invariant covariates $w_i$, and a constant.RE requires $v_i$ and $X_{it}$ to be uncorrelated — an assumption most economists reject. FE is consistent regardless, but it cannot estimate time-invariant covariates (race, gender, etc.). CRE has both advantages: it includes time-invariant variables and remains consistent. Mundlak and Wooldridge show that the CRE coefficients on $X_{it}$ are identical to the FE estimates. Testing whether the coefficients on $\bar X_i$ are jointly zero gives the Mundlak test for choosing between RE and CRE.## ExampleStata 18 added a `cre` option to `xtreg`. The following uses `nlswork`, 28,101 person-years from the National Longitudinal Survey of Young Women with complete data on the model's variables. We regress log wage on `tenure` and `age`, which vary within a person, plus `race`, which does not — the covariate a fixed-effect model has to drop.```{r}#| label: setup#| include: falseknitr::opts_chunk$set(echo =TRUE)``````{r}#| include: falselibrary(Statamarkdown)stataexe <-find_stata()#stataexe <- "/usr/local/bin/stata"knitr::opts_chunk$set(engine.path=list(stata=stataexe))``````{stata}*| label: stata1*| echo: true*| collectcode: truewebuse nlsworkxtreg ln_wage tenure age i.race, cre vce(cluster idcode)```CRE gives `tenure` = .0211313 and `age` = .0121949, and estimates race as well:Black women earn 13.1% less ($-.1312068$, SE .0118), while the "Other" categoryis marginal ($.1059$, $p = 0.074$). The coefficients on the group means are.0376991 for mean tenure and $-.0011984$ for mean age.Stata prints the Mundlak test at the foot of the table: $\chi^2(2) = 331.51$,$p < 0.0001$. The group means are jointly far from zero, so plain randomeffects is rejected — $v_i$ and the regressors are correlated, as the reported$\text{corr} = 0.5474$ between the two blocks says directly.To compare with a fixed effect model:```{stata}*| label: stata2*| echo: true*| collectcode: truewebuse nlsworkxtreg ln_wage tenure age i.race, fe vce(cluster idcode)```The coefficients on `tenure` and `age` are identical: .0211313 and .0121949 inboth, with standard errors agreeing to the seventh decimal (.0012112 against.0012113). Race is reported as `0 (omitted)` here, being collinear with the uniteffects — the one thing FE cannot do and CRE can. This is the Mundlak result inpractice: CRE gives you the fixed-effect estimates *and* the time-invariantcoefficients from a single random-effects fit.We can replicate this manually by including the group means in a RE model:```{stata}*| label: stata3*| echo: true*| collectcode: truewebuse nlswork* Restrict the mean calculation to the estimation sample (listwise-deleted* on the model's variables), matching what xtreg's native cre option does* internally. Both tenure and age have missing values in nlswork, so* computing age_mean/tenure_mean over all rows for an idcode (rather than* only rows with complete data on ln_wage, tenure, age, and race) would* make the manual means diverge from cre's.egen age_mean = mean(age) if !missing(ln_wage, tenure, age, race), by(idcode)egen tenure_mean = mean(tenure) if !missing(ln_wage, tenure, age, race), by(idcode)xtreg ln_wage tenure tenure_mean age age_mean i.race, vce(cluster idcode)```This is what Stata's `cre` option does internally. Every number matches thefirst table: `tenure` .0211313, `age` .0121949, Black $-.1312068$, Other.1059379, constant 1.2159, and the same variance components ($\sigma_u$ .3333,$\sigma_e$ .2981, $\rho$ .5557). The `cre` option is a convenience, not adifferent estimator.The one detail worth care is in the code above: the group means must be computedon the estimation sample. Both `tenure` and `age` have missing values in`nlswork`, so averaging over all of a woman's rows rather than only her completeones would give different means and break the equivalence.