brycewang-stanford Panel Data Analysis Guide
Estimate and interpret fixed effects, random effects, and dynamic panel models using Stata, R, and Python for longitudinal/panel datasets.
What Is Panel Data?
Panel data (also called longitudinal or cross-sectional time-series data) tracks the same units (individuals, firms, countries) across multiple time periods. This structure enables:
- Controlling for unobserved heterogeneity (time-invariant omitted variables)
- Studying dynamic relationships (how X at time t affects Y at time t+1)
- Increased statistical power through more observations
Data Structure
| unit_id | year | gdp_growth | investment | trade_openness |
|---------|------|-----------|------------|----------------|
| USA | 2015 | 2.9 | 20.5 | 28.3 |
| USA | 2016 | 1.7 | 20.1 | 27.1 |
| USA | 2017 | 2.3 | 20.8 | 27.5 |
| CHN | 2015 | 6.9 | 43.3 | 39.9 |
| CHN | 2016 | 6.7 | 42.7 | 37.2 |
| CHN | 2017 | 6.9 | 43.1 | 38.1 |
Key notation:
- i = unit (cross-sectional dimension): i = 1, ..., N
- t = time period: t = 1, ..., T
- Y_it = dependent variable for unit i at time t
Model Specification
Pooled OLS
Y_it = alpha + beta * X_it + epsilon_it
Ignores panel structure; assumes no unit-specific effects. Rarely appropriate.
Fixed Effects (FE) Model
Y_it = alpha_i + beta * X_it + epsilon_it
Each unit has its own intercept (alpha_i) that captures all time-invariant unobserved heterogeneity. The "within" estimator removes alpha_i by demeaning.
Random Effects (RE) Model
Y_it = alpha + beta * X_it + u_i + epsilon_it
The unit-specific effect u_i is treated as random and uncorrelated with X_it.
Estimation in Stata
Setting Up Panel Data
* Declare panel structure
xtset country_id year
* Summarize within and between variation
xtsum gdp_growth investment trade_openness
Fixed Effects
* Fixed effects regression
xtreg gdp_growth investment trade_openness, fe
* Store results for Hausman test
estimates store FE
* Fixed effects with robust standard errors (clustered by unit)
xtreg gdp_growth investment trade_openness, fe vce(cluster country_id)
* Test joint significance of fixed effects
testparm i.country_id
Random Effects
* Random effects regression
xtreg gdp_growth investment trade_openness, re
* Store results for Hausman test
estimates store RE
Hausman Test (FE vs. RE)
* Hausman specification test
hausman FE RE
* If p < 0.05: reject RE, use FE
* If p > 0.05: RE is consistent and efficient, prefer RE
First Differences
* First-differenced regression (alternative to FE)
reg D.gdp_growth D.investment D.trade_openness, vce(cluster country_id)
Estimation in R (plm Package)
library(plm)
# Convert to panel data frame
pdata <- pdata.frame(mydata, index = c("country_id", "year"))
# Fixed effects
fe_model <- plm(gdp_growth ~ investment + trade_openness,
data = pdata, model = "within")
summary(fe_model)
# Random effects
re_model <- plm(gdp_growth ~ investment + trade_openness,
data = pdata, model = "random")
summary(re_model)
# Hausman test
phtest(fe_model, re_model)
# Clustered standard errors
library(lmtest)
library(sandwich)
coeftest(fe_model, vcov = vcovHC(fe_model, type = "HC1", cluster = "group"))
# Time fixed effects
fe_twoway <- plm(gdp_growth ~ investment + trade_openness + factor(year),
data = pdata, model = "within")
# Test for time fixed effects
pFtest(fe_twoway, fe_model)
Estimation in Python (linearmodels)
import pandas as pd
from linearmodels.panel import PanelOLS, RandomEffects, compare
# Set multi-index for panel structure
data = data.set_index(["country_id", "year"])
# Fixed effects
fe = PanelOLS.from_formula(
"gdp_growth ~ investment + trade_openness + EntityEffects",
data=data
)
fe_result = fe.fit(cov_type="clustered", cluster_entity=True)
print(fe_result.summary)
# Random effects
re = RandomEffects.from_formula(
"gdp_growth ~ investment + trade_openness + 1",
data=data
)
re_result = re.fit()
print(re_result.summary)
# Two-way fixed effects (entity + time)
twoway = PanelOLS.from_formula(
"gdp_growth ~ investment + trade_openness + EntityEffects + TimeEffects",
data=data
)
twoway_result = twoway.fit(cov_type="clustered", cluster_entity=True)
print(twoway_result.summary)
# Compare models
print(compare({"FE": fe_result, "RE": re_result, "Two-way FE": twoway_result}))
Diagnostic Tests
Testing for Panel Effects
| Test | Stata | R | Null Hypothesis |
|---|---|---|---|
| F-test for FE | Built into xtreg, fe |
pFtest() |
All alpha_i = 0 (pooled OLS is appropriate) |
| Breusch-Pagan LM | xttest0 |
plmtest() |
Var(u_i) = 0 (pooled OLS vs. RE) |
| Hausman | hausman FE RE |
phtest() |
RE is consistent (u_i uncorrelated with X) |
Testing for Serial Correlation
* Wooldridge test for serial correlation in panel data
xtserial gdp_growth investment trade_openness
* If p < 0.05: serial correlation present; use clustered SE or AR(1) correction
# Wooldridge test
pbgtest(fe_model) # Breusch-Godfrey test for serial correlation
Testing for Heteroskedasticity
* Modified Wald test for groupwise heteroskedasticity
xttest3
* If p < 0.05: heteroskedasticity present; use robust/clustered SE
Advanced Panel Models
Dynamic Panel (Arellano-Bond GMM)
When a lagged dependent variable is included as a regressor:
* Arellano-Bond one-step GMM
xtabond gdp_growth investment trade_openness, lags(1) vce(robust)
* System GMM (Blundell-Bond) - more efficient
xtdpdsys gdp_growth investment trade_openness, lags(1) vce(robust)
* Sargan/Hansen test for overidentifying restrictions
* AR(2) test for second-order serial correlation
Difference-in-Differences (DID)
* Basic DID with two-way fixed effects
xtreg outcome treated##post, fe vce(cluster unit_id)
* Event study specification
xtreg outcome i.relative_time##treated, fe vce(cluster unit_id)
Reporting Results
Table X: Panel Regression Results (Fixed Effects)
Dependent Variable: GDP Growth (%)
(1) (2) (3)
FE RE Two-way FE
Investment 0.125*** 0.118*** 0.131***
(0.032) (0.029) (0.035)
Trade Openness 0.045** 0.051** 0.038*
(0.018) (0.017) (0.020)
Entity FE Yes No Yes
Time FE No No Yes
Observations 850 850 850
R-squared (within) 0.234 0.228 0.267
Hausman test (p) -- 0.003 --
Notes: Robust standard errors clustered at the country level in
parentheses. * p<0.10, ** p<0.05, *** p<0.01.