panel-data

Panel Data Models Skill

This skill covers panel data econometrics: pooled OLS, fixed effects (FE), random effects (RE), and two-way FE models. It guides model selection, assumption testing, and interpretation for longitudinal/panel datasets.

Key Terminology

• Panel dataset: observations on N units (individuals, firms, countries) over T time periods
• Balanced panel: every unit observed in every period
• Unbalanced panel: some unit-period observations missing
• Unobserved heterogeneity (αᵢ): time-invariant unit-specific factors (e.g., firm culture, individual ability)

Model Selection Framework

Start
 ├─ Is unobserved heterogeneity correlated with regressors?
 │    ├─ YES → Fixed Effects (FE)
 │    └─ NO  → Random Effects (RE) — test with Hausman
 │
 ├─ Are time effects important?
 │    ├─ YES → Two-Way FE (entity + time dummies)
 │    └─ NO  → One-Way FE
 │
 └─ Need to estimate effect of time-invariant variables?
      ├─ YES → Random Effects or Mundlak/Correlated RE
      └─ NO  → Fixed Effects preferred

Hausman Test Decision Rule

• H₀: RE is consistent (αᵢ uncorrelated with X)
• H₁: FE is consistent but RE is not (αᵢ correlated with X)
• p < 0.05: Use Fixed Effects
• p ≥ 0.05: Random Effects is efficient

Mundlak / Correlated Random Effects (CRE)

Use when you need RE to estimate time-invariant variable effects, but want to relax the strict exogeneity assumption of RE. CRE includes group means of time-varying regressors in the RE equation, making it equivalent to FE for those variables while still estimating time-invariant effects.

# R — Mundlak CRE approach
library(plm); library(dplyr)

# Compute entity means of time-varying regressors
panel_df_cre <- df %>%
  group_by(entity_id) %>%
  mutate(x1_mean = mean(x1),
         x2_mean = mean(x2)) %>%
  ungroup()

panel_cre <- pdata.frame(panel_df_cre, index = c("entity_id", "time_var"))

# RE model augmented with group means (Mundlak approach):
cre_model <- plm(y ~ x1 + x2 + time_invariant_var + x1_mean + x2_mean,
                 data   = panel_cre,
                 model  = "random")
summary(cre_model)
# Coefficients on x1_mean, x2_mean test the correlation between αᵢ and X
# (equivalent to Hausman test; joint significance = prefer FE)
# Coefficient on time_invariant_var is identified via between variation

* Stata — Mundlak CRE
* Step 1: compute entity means
bysort entity_id: egen x1_mean = mean(x1)
bysort entity_id: egen x2_mean = mean(x2)

* Step 2: RE model with group means added
xtreg y x1 x2 time_invariant_var x1_mean x2_mean, re

* Test joint significance of group means (Mundlak test):
testparm x1_mean x2_mean
* p < 0.05 → group means matter → prefer FE for time-varying regressors

Quick Code Templates

Fixed Effects

# Python (linearmodels)
from linearmodels.panel import PanelOLS
import pandas as pd

# Set multi-index: entity and time
df = df.set_index(['entity_id', 'time_var'])

model = PanelOLS(df['y'], df[['x1', 'x2']], entity_effects=True,
                 time_effects=True)  # Two-way FE
result = model.fit(cov_type='clustered', cluster_entity=True)
print(result.summary)

# R (plm)
library(plm)
panel_df <- pdata.frame(df, index = c("entity_id", "time_var"))

# One-way FE
fe_model <- plm(y ~ x1 + x2, data = panel_df, model = "within")

# Two-way FE
twfe_model <- plm(y ~ x1 + x2, data = panel_df, model = "within",
                  effect = "twoways")

# Clustered SE
library(lmtest); library(sandwich)
coeftest(fe_model, vcov = vcovHC(fe_model, cluster = "group"))

* Stata — Two-way FE with clustered SE
xtset entity_id time_var
xtreg y x1 x2 i.time_var, fe cluster(entity_id)

Random Effects

from linearmodels.panel import RandomEffects

re_model = RandomEffects(df['y'], df[['x1', 'x2']])
re_result = re_model.fit()
print(re_result.summary)

re_model <- plm(y ~ x1 + x2, data = panel_df, model = "random")
summary(re_model)

xtreg y x1 x2, re

Hausman Test

from linearmodels.panel import compare
# Compare FE vs RE
print(compare({'FE': fe_result, 'RE': re_result}))
# Or use statsmodels hausman

phtest(fe_model, re_model)
# p < 0.05 → prefer Fixed Effects

hausman fe_estimates re_estimates

Dynamic Panels and Arellano-Bond GMM

Use when the model includes a lagged dependent variable (Yᵢₜ₋₁) in short-T panels. The within (FE) estimator is biased in this case (Nickell 1981 bias). Arellano-Bond uses lagged levels as instruments for the differenced equation.

When to use: Short T (T < 10), panel includes lagged DV, suspicion of endogenous regressors.

# R — Arellano-Bond GMM (plm package)
library(plm)

# Difference GMM (Arellano-Bond 1991)
ab <- pgmm(
  y ~ lag(y, 1) + x1 + x2 | lag(y, 2:4),   # instruments: lags 2-4 of y
  data  = panel_df,
  effect = "individual",
  model  = "twosteps"   # two-step is asymptotically efficient
)
summary(ab, robust = TRUE)

# Key diagnostics:
# AR(1): should be significant (differencing induces MA(1))
# AR(2): should be insignificant (no serial correlation in levels)
# Hansen J test: p > 0.05 → instruments are valid

# Python — Arellano-Bond GMM (linearmodels)
# Note: linearmodels does not directly implement Arellano-Bond;
# use the dedicated BetterArellano approach or wrap via R
# Alternatively, use system GMM via a custom estimator:
# pip install pydynpd
import pydynpd

# pydynpd syntax
command_str = "y L1.y x1 x2 | gmm(y, 2 4) iv(x1 x2)"
results = pydynpd.regression.abond(command_str, df, ["entity_id", "time_var"])
print(results.summary)

* Stata — Arellano-Bond with xtabond2 (preferred)
ssc install xtabond2

xtset entity_id time_var

* Difference GMM:
xtabond2 y L.y x1 x2, gmm(L.y, lag(2 4)) iv(x1 x2) ///
    twostep robust noleveleq

* System GMM (adds level equation with lagged differences as instruments):
xtabond2 y L.y x1 x2, gmm(L.y, lag(2 4)) iv(x1 x2) twostep robust

* Diagnostics reported automatically:
* - AR(1), AR(2) tests
* - Hansen J test of overidentifying restrictions

Interpretation rules:

• AR(1) significant, AR(2) insignificant → no second-order serial correlation in levels ✓
• Hansen J p > 0.05 → instruments jointly valid ✓
• Too many instruments (> N) weakens the Hansen test — restrict lag range

Standard Errors for Panel Data

Situation	Recommended SE
Serial correlation within entities	Cluster by entity
Cross-sectional dependence	Driscoll-Kraay SE
Both serial + cross-sectional	Two-way clustering
Heteroskedasticity only	HC robust SE

Driscoll-Kraay and Two-Way Clustering Code

Driscoll-Kraay SE: Robust to cross-sectional dependence and serial correlation. Preferred for macro panels (small N, large T).

# R — Driscoll-Kraay SE (sandwich package)
library(plm); library(sandwich); library(lmtest)

fe_model <- plm(y ~ x1 + x2, data = panel_df, model = "within")

# Driscoll-Kraay SE (robust to cross-sectional and serial dependence):
coeftest(fe_model, vcov = vcovSCC(fe_model, type = "HC1", maxlag = 4))
# maxlag: number of lags for serial correlation (typically T^0.25)

* Stata — Driscoll-Kraay SE
xtscc y x1 x2, fe lag(4)
* lag(4) = bandwidth parameter; use T^0.25 as a rule of thumb

Two-way clustering: Clusters at both entity and time level. Use when treatment varies at both levels.

# R — Two-way clustering (sandwich)
library(sandwich); library(lmtest)

# Manually compute two-way clustered SE:
# V_twoway = V_entity + V_time - V_entity×time
vcov_entity <- vcovCL(fe_model, cluster = ~entity_id)
vcov_time   <- vcovCL(fe_model, cluster = ~time_var)
vcov_both   <- vcovCL(fe_model, cluster = ~entity_id + time_var)
coeftest(fe_model, vcov = vcov_both)

* Stata — Two-way clustering
xtreg y x1 x2 i.time_var, fe vce(cluster entity_id)  // cluster by entity only
* For two-way clustering (entity AND time):
reghdfe y x1 x2, absorb(entity_id time_var) vce(cluster entity_id time_var)

Interpreting Fixed Effects Results

• FE coefficients identify within-unit variation only
• Cannot estimate effect of time-invariant variables (absorbed by unit FEs)
• Two-way FE removes both unit trends and aggregate time trends
• Always report whether entity FE, time FE, or both are included

Reporting Standards

• State panel dimensions: N = [units], T = [periods], total obs
• Report whether SE are clustered (at entity level is standard)
• Specify which effects are included (entity, time, or both)
• Report F-test for joint significance of fixed effects
• Include Hausman test result when choosing FE over RE

For first-difference (FD) estimators and panel models with limited dependent variables (conditional logit, Poisson FE, Cox stratified hazard), see references/panel-ldv-advanced.md.

Common Pitfalls

• Using RE when FE is appropriate: If Hausman test rejects, RE is inconsistent — always test
• Clustering at the wrong level: Cluster SE at the level of treatment variation, not the individual level
• Nickell bias: Including lagged DV in short-T panels with FE is biased — use Arellano-Bond GMM
• Ignoring cross-sectional dependence: In macro panels (small N, large T), standard FE SE are invalid — use Driscoll-Kraay
• Interpreting FE coefficients as between-unit effects: FE estimates are purely within-unit; they cannot speak to cross-unit differences