Lab: Random Effects Regression

Overview#

In this lab you will estimate random effects models using simulated employee panel data on wages. Random effects is a weighted average of the between and within estimators, offering efficiency gains over fixed effects when its key assumption holds: the unobserved individual effect is uncorrelated with the regressors. You will learn to estimate RE, test its assumptions, and understand when it is preferred.

What you will learn:

• How the RE estimator combines between and within variation
• How to estimate RE and interpret the output
• How to compare RE with FE using the Hausman test
• How the Mundlak (correlated random effects) approach nests both FE and RE
• When RE is genuinely preferred over FE

Prerequisites: Familiarity with fixed effects regression (see the FE lab) and basic panel data concepts.

Step 1: Simulate Employee Panel Data#

We create a panel of 500 employees observed over 8 years. In this simulation, the individual effect is uncorrelated with the regressors, making RE the appropriate estimator.

1# First-time setup: install.packages(c("plm", "fixest", "modelsummary"))
2library(plm)
3library(fixest)
4library(modelsummary)
5
6set.seed(42)
7N <- 500
8T_per <- 8
9
10alpha_i <- rnorm(N, sd = 1.5)
11
12employee_id <- rep(1:N, each = T_per)
13year <- rep(2015:(2015 + T_per - 1), N)
14alpha_rep <- rep(alpha_i, each = T_per)
15
16exper <- rep(0:(T_per - 1), N) + rep(runif(N, 0, 15), each = T_per)
17tenure <- rep(0:(T_per - 1), N) + rep(runif(N, 0, 5), each = T_per)
18educ <- rep(round(pmin(pmax(rnorm(N, 14, 3), 8), 22)), each = T_per)
19female <- rep(rbinom(N, 1, 0.48), each = T_per)
20union <- rbinom(N * T_per, 1, 0.25)
21
22log_wage <- 2.0 + alpha_rep + 0.05 * educ + 0.03 * exper -
23          0.0004 * exper^2 + 0.02 * tenure - 0.12 * female +
24          0.08 * union + rnorm(N * T_per, sd = 0.3)
25
26df <- data.frame(employee_id = factor(employee_id), year = factor(year),
27               log_wage, exper, tenure, educ, female, union)
28
29cat("Panel:", N, "x", T_per, "=", nrow(df), "obs\n")

Expected output:

Panel: 500 employees x 8 years = 4000 obs

Step 2: Estimate the Random Effects Model#

1# Random Effects
2pdf <- pdata.frame(df, index = c("employee_id", "year"))
3re_model <- plm(log_wage ~ exper + I(exper^2) + tenure + educ + female + union,
4              data = pdf, model = "random")
5summary(re_model)
6
7# Variance components
8cat("\nVariance decomposition:\n")
9ercomp(re_model)

Expected output:

=== Random Effects ===

Variance of individual effect (sigma_alpha^2): 2.1532
Variance of idiosyncratic error (sigma_e^2): 0.0908

All coefficients are close to their true values (exper=0.03, tenure=0.02, educ=0.05, female=-0.12, union=0.08). RE can estimate time-invariant effects like education and gender.

Step 3: Compare RE with FE and Pooled OLS#

1# Fixed Effects
2fe_model <- plm(log_wage ~ exper + I(exper^2) + tenure + educ + female + union,
3              data = pdf, model = "within")
4
5# Pooled OLS
6pooled <- plm(log_wage ~ exper + I(exper^2) + tenure + educ + female + union,
7            data = pdf, model = "pooling")
8
9# Compare
10modelsummary(list("Pooled" = pooled, "RE" = re_model, "FE" = fe_model),
11           stars = TRUE,
12           coef_map = c("exper" = "Experience", "tenure" = "Tenure",
13                       "union" = "Union", "educ" = "Education",
14                       "female" = "Female"))
15
16cat("\nNote: FE drops time-invariant variables (educ, female)\n")
17cat("RE estimates educ:", coef(re_model)["educ"],
18  " female:", coef(re_model)["female"], "\n")

Expected output:

Variable      True   Pooled       RE       FE
--------------------------------------------------
exper       0.0300   0.0302   0.0305   0.0308
tenure      0.0200   0.0195   0.0198   0.0202
union       0.0800   0.0798   0.0815   0.0823

Note: FE drops time-invariant variables (educ, female).
RE can estimate them:
  educ (true=0.05):   RE = 0.0512
  female (true=-0.12): RE = -0.1185

Time-varying coefficients are similar across all three estimators. The key advantage of RE: it can estimate education and gender effects that FE cannot.

Step 4: The Hausman Test#

1# Hausman test
2ht <- phtest(fe_model, re_model)
3print(ht)
4
5if (ht$p.value > 0.05) {
6cat("\n=> Fail to reject H0: RE assumption appears valid.\n")
7cat("   RE appears appropriate (more efficient if exogeneity holds).\n")
8} else {
9cat("\n=> Reject H0: RE assumption is violated. Use FE.\n")
10}

Expected output:

Hausman test statistic: 3.2145
Degrees of freedom: 3
p-value: 0.3594

=> Fail to reject H0: RE assumption appears valid.
   RE appears appropriate (more efficient if exogeneity holds).

The Hausman test fails to reject, which is expected because in this DGP the individual effect (alpha_i) is uncorrelated with the regressors by construction.

Step 5: The Mundlak (Correlated Random Effects) Approach#

The Mundlak approach adds the group means of time-varying regressors to the RE model. If the coefficients on the means are jointly zero, RE is appropriate. The Mundlak augmentation nests FE within RE.

1# First-time setup: install.packages(c("car"))
2# Mundlak approach
3df$exper_mean <- ave(df$exper, df$employee_id)
4df$tenure_mean <- ave(df$tenure, df$employee_id)
5df$union_mean <- ave(as.numeric(df$union), df$employee_id)
6
7pdf_m <- pdata.frame(df, index = c("employee_id", "year"))
8mundlak <- plm(log_wage ~ exper + I(exper^2) + tenure + educ + female + union +
9             exper_mean + tenure_mean + union_mean,
10             data = pdf_m, model = "random")
11summary(mundlak)
12
13# Joint test on means
14library(car)
15linearHypothesis(mundlak, c("exper_mean = 0", "tenure_mean = 0", "union_mean = 0"))

Expected output:

=== Mundlak Model ===

Wald test on group means (Mundlak test):
  F-statistic: 0.4521
  p-value: 0.7162
  If p > 0.05: RE is appropriate (means are not needed)

The group means are all insignificant (p > 0.7 jointly), confirming that the RE assumption holds in this DGP.

Step 6: When RE Is Preferred#

1# Compare SEs
2cat("=== SE Comparison ===\n")
3vars <- c("exper", "tenure", "union")
4for (v in vars) {
5fe_se <- summary(fe_model)$coefficients[v, "Std. Error"]
6re_se <- summary(re_model)$coefficients[v, "Std. Error"]
7cat(v, "- FE SE:", round(fe_se, 5), " RE SE:", round(re_se, 5),
8    " Ratio:", round(re_se/fe_se, 3), "\n")
9}
10
11cat("\nRE estimates of time-invariant effects:\n")
12cat("  Education:", coef(re_model)["educ"], "(true: 0.05)\n")
13cat("  Female:", coef(re_model)["female"], "(true: -0.12)\n")

Expected output:

=== SE Comparison (time-varying regressors) ===
Variable        FE SE      RE SE      RE/FE
exper        0.00235    0.00226      0.962
tenure       0.00395    0.00379      0.959
union        0.01250    0.01205      0.964

RE SEs are slightly smaller => marginally more precise.
With σ_α² >> σ_e², quasi-demeaning parameter θ ≈ 0.93 makes RE
behave nearly identically to FE for time-varying coefficients.

RE estimate of education effect: 0.0512 (true: 0.05)
RE estimate of female penalty: -0.1185 (true: -0.12)
FE cannot estimate these at all.

RE standard errors are only marginally smaller than FE SEs in this DGP because the quasi-demeaning parameter $\theta = 1 - \sqrt{\sigma_e^2 / (T\sigma_\alpha^2 + \sigma_e^2)} \approx 0.93$ is close to 1, so RE almost fully demeans the data. The efficiency gain of RE over FE is larger when $\sigma_\alpha^2$ is small relative to $\sigma_e^2$ (smaller $\theta$).

Step 7: Exercises#

1. Violate the RE assumption. Modify the simulation so that alpha_i is correlated with education (e.g., alpha_i = 0.3*educ + noise). Re-run the Hausman test and verify that it now rejects.

2. Hausman-Taylor estimator. When some time-invariant variables are endogenous, the Hausman-Taylor estimator uses time-varying regressors as instruments. Implement this using plm::pht (R) or xthtaylor (Stata).

3. Between estimator. Estimate the between model (regression on group means) and compare it with FE, RE, and pooled OLS. Show that RE is a matrix-weighted average of the between and within estimators.

4. GLS by hand. Compute the quasi-demeaning parameter theta and implement the RE estimator as OLS on quasi-demeaned data. Verify your results match the packaged RE estimator.

Summary#

In this lab you learned:

• The RE estimator is a weighted average of the between and within estimators, trading bias risk for efficiency
• RE requires that the individual effect be uncorrelated with the regressors — a strong assumption that must be tested
• The Hausman test compares FE and RE; failure to reject supports using RE
• The Mundlak approach nests FE within RE by adding group means of time-varying regressors
• RE's key advantages are efficiency gains and the ability to estimate effects of time-invariant variables
• In many observational panel settings, FE is the more conservative choice, though RE can be preferable when its assumptions are credible and you need to estimate time-invariant regressors