Random Effects

Motivating Example: Cross-Country Growth Regressions

You want to understand what drives economic growth across countries. Your panel has 100 countries observed over 30 years, and you are interested in the effects of institutions, trade openness, and human capital on GDP growth.

The challenge is that some of your most important variables — geography, colonial history, legal origin — are time-invariant. A fixed effects model would absorb all of these into the country fixed effects, making them impossible to estimate. But these time-invariant variables are precisely the ones you care about.

If you are willing to assume that the unobserved country-level factors (captured by the country effect $\alpha_i$) are uncorrelated with your regressors, the random effects model lets you estimate the effects of both time-varying and time-invariant variables. This assumption is strong, but in some settings it is defensible — and the payoff is substantial.

A. Overview: What Random Effects Does

The Model

The panel data model is the same as in fixed effects:

where $Z_i$ are time-invariant covariates (like geography) and $\alpha_i$ is the unobserved unit effect. The key difference is in the assumption about $\alpha_i$.

• Fixed effects treats $\alpha_i$ as a parameter to be estimated (or differenced away) and allows it to be arbitrarily correlated with $X_{it}$.
• Random effects treats $\alpha_i$ as a random variable drawn from a distribution, and assumes it is uncorrelated with $X_{it}$ and $Z_i$.

Why Does the RE Assumption Matter?

If $\alpha_i$ is uncorrelated with the regressors, RE produces more efficient estimates (smaller standard errors) than FE. Intuitively, RE uses both within-unit and between-unit variation, while FE discards the between-unit information entirely.

RE also allows you to estimate the coefficients on time-invariant variables ($\gamma$), which FE cannot.

The RE Estimator as a Weighted Average

The RE estimator is a matrix-weighted average of the within (FE) estimator and the between estimator. In the balanced-panel, single-regressor case, this simplifies to a scalar weight:

where $\omega$ depends on the ratio of the idiosyncratic variance $\sigma^2_\varepsilon$ to the unit-effect variance $\sigma^2_\alpha$ and the number of time periods $T$. When $\sigma^2_\alpha$ is large (lots of persistent between-unit heterogeneity), RE puts more weight on the within estimator and approaches FE. In the general multivariate or unbalanced-panel case, $\omega$ becomes a matrix weight and the expression above no longer applies as a simple scalar formula.

Common Confusions

B. Identification: The RE Assumption

The Core Assumption

The unobserved unit effect is uncorrelated with all regressors in all time periods. This condition is the same as saying there is no selection bias from unobserved time-invariant factors.

When Is the RE Assumption Plausible?

RE is most defensible when:

1. Treatment is randomly assigned — In an RCT with panel data, the unit effect $\alpha_i$ is, by design, uncorrelated with treatment. RE is efficient here.
2. The between and within estimates agree — If the Hausman test does not reject, and the coefficients are substantively similar, RE may be appropriate.
3. You are studying time-invariant variables — If the research question demands estimating time-invariant effects, RE (or Mundlak) is your only option.
4. Prediction is the goal — If you want to predict outcomes (not estimate causal effects), RE can be more useful than FE.

When Is It NOT Plausible?

RE is rarely credible when:

1. Selection into treatment is likely — If units self-select into treatment based on unobserved characteristics.
2. Cross-country regressions with endogenous institutions — Country-level unobservables (culture, geography) likely affect both institutions and growth.
3. The Hausman test strongly rejects — A large test statistic means FE and RE estimates differ substantially, and FE is the safer choice.

C. Visual Intuition

Recall the FE picture: each unit has its own intercept, and FE fits separate within-unit regression lines. RE does something subtler. It partially pools the unit-specific intercepts toward the overall mean. Units with more observations (or more precise data) keep more of their own intercept; units with less data are pulled more toward the grand mean.

Think of RE as a compromise: it does not fully trust the between-unit comparisons (like pooled OLS), but it does not fully discard them either (like FE). It uses the data to find a well-suited blend.

This partial pooling is exactly what hierarchical/multilevel models do. RE and hierarchical linear models (HLM) are the same thing, just with different jargon. Econometricians say "random effects." Psychometricians and education researchers say "multilevel models." The math is identical.

D. Mathematical Derivation

E. Implementation

1library(plm)
2
3# Set up panel data
4pdata <- pdata.frame(df, index = c("country_id", "year"))
5
6# Random effects
7re_fit <- plm(growth ~ trade_openness + human_capital + institutions,
8            data = pdata, model = "random")
9summary(re_fit)
10
11# Fixed effects
12fe_fit <- plm(growth ~ trade_openness + human_capital,
13            data = pdata, model = "within")
14
15# Hausman test
16phtest(fe_fit, re_fit)
17
18# Mundlak approach
19library(fixest)
20df$mean_trade <- ave(df$trade_openness, df$country_id)
21df$mean_hcap <- ave(df$human_capital, df$country_id)
22mundlak <- feols(growth ~ trade_openness + human_capital + institutions +
23               mean_trade + mean_hcap | 0,
24               data = df, vcov = ~country_id)
25summary(mundlak)

F. Diagnostics

The Hausman Test in Practice

Run the Hausman test and report the statistic and p-value. But interpret it carefully:

• p < 0.05: Reject RE. Use FE. The unobserved effect likely correlates with regressors.
• p > 0.05: Cannot reject RE. But this does not prove RE is correct — you may simply lack power.

The Mundlak Test (A Better Alternative)

The Mundlak (1978) approach adds group means of time-varying regressors to the RE model. If the coefficients on the group means are jointly significant, the RE assumption is violated. This approach is asymptotically equivalent to the Hausman test under homoskedasticity, but is more intuitive and easier to implement with robust or clustered SEs (under which the classical Hausman test is invalid).

Breusch-Pagan LM Test

Tests whether the variance of the unit effect ($\sigma^2_\alpha$) is zero. If it is, there are no unit effects and pooled OLS is fine. If $\sigma^2_\alpha > 0$, either FE or RE is needed.

Interpreting Results

• RE coefficients on time-varying variables reflect both within-unit and between-unit variation. They are efficient but potentially biased if the RE assumption fails.
• RE coefficients on time-invariant variables (like geography or gender) are identified entirely from between-unit variation. They are only valid if the RE assumption holds.
• If FE and RE estimates are substantively similar (not just statistically indistinguishable), this agreement is reassuring. Report both.
• The Mundlak approach gives you the strengths of both approaches: FE-consistent estimates of time-varying effects plus estimates of time-invariant effects, all in one model.

G. What Can Go Wrong

H. Practice

I. Swap-In: When to Use Something Else

• Fixed effects: When the RE assumption (unit effects uncorrelated with regressors) is implausible — FE allows arbitrary correlation between unit effects and covariates at the cost of discarding between-unit variation.
• Correlated Random Effects (Mundlak): When you want to test the RE assumption while still estimating time-invariant effects — add group means of time-varying regressors to the RE model.
• First differencing: Equivalent to FE with two periods. With more periods, FE is generally more efficient unless errors follow a random walk.
• Arellano-Bond GMM: For dynamic panels with a lagged dependent variable, where both FE and RE are biased.

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