Panel Data Guide

| 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           |

Y_it = alpha + beta * X_it + epsilon_it

Y_it = alpha_i + beta * X_it + epsilon_it

Y_it = alpha + beta * X_it + u_i + epsilon_it

* Declare panel structure
xtset country_id year

* Summarize within and between variation
xtsum gdp_growth investment trade_openness

* Check for gaps in panel
gen gap = year - l.year if l.year != .
tab gap  // Should be all 1's for balanced annual panels

* Create balanced subsample
by country_id: gen T_i = _N
keep if T_i == max_T  // Keep only units observed in all periods

* Attrition analysis
gen in_panel = 1
tsfill, full
replace in_panel = 0 if missing(in_panel)

* 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

* Entity and time fixed effects (fast, memory-efficient)
reghdfe gdp_growth investment trade_openness, ///
    absorb(country_id year) cluster(country_id)

* Two-way clustering (entity and year)
reghdfe gdp_growth investment trade_openness, ///
    absorb(country_id year) cluster(country_id year)

* Random effects regression
xtreg gdp_growth investment trade_openness, re

* Store results for Hausman test
estimates store 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

* Mundlak (1978): add group means to RE model (robust to heteroskedasticity)
foreach var of varlist investment trade_openness {
    bysort country_id: egen m_`var' = mean(`var')
}
xtreg gdp_growth investment trade_openness ///
    m_investment m_trade_openness, re cluster(country_id)
test m_investment m_trade_openness
* Rejection => FE preferred; failure to reject => RE acceptable

* First-differenced regression (alternative to FE)
reg D.gdp_growth D.investment D.trade_openness, vce(cluster country_id)

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)

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}))