Lecture 3

Panel Data Models

Byeong-Hak Choe
bchoe@geneseo.edu

SUNY Geneseo

January 28, 2026

👥📈 Panel Data Models

👥📈 Panel Data Models: POLS → FE → TWFE → RE

Learning goals

  • Understand what panel data is (and why we love it)
  • Compare Pooled OLS, Fixed Effects, Two-Way FE, and Random Effects
  • Learn what variation each model uses and what it controls for
  • Understand country-clustered standard errors (and why we need them)

What is Panel Data?

Panel data tracks the same units over multiple time periods:

  • Unit (\(i\)): country (USA, Korea, India, …)
  • Time (\(t\)): year (1990–2022, …)

Example (CO\(_2\) panel): - Outcome: CO\(_2\) emissions per capita - Unit ID: iso_code - Time ID: year

Tip

Panel data gives us two kinds of variation: - Between variation: differences across countries - Within variation: changes within a country over time

Why Use Panel Models (Not Only Pooled OLS)?

Pooled OLS can be misleading because it ignores unobserved country differences that don’t change much over time, such as:

  • 🌍 geography / climate / natural resources
  • 🏭 industrial structure and history
  • 🏛️ institutions and regulation
  • ⚡ energy infrastructure and technology path
  • 🧑‍🤝‍🧑 culture and long-run lifestyle patterns

These factors can affect both

  • emissions (or other outcomes) and
  • GDP per capita (or other regressors)

➡️ That creates omitted variable bias when we use Pooled OLS.

🧠 Unobserved Heterogeneity (The Core Issue)

Suppose the true relationship includes an unobserved country component:

\[ y_{it} = \beta x_{it} + \underbrace{\alpha_i}_{\text{unobserved country trait}} + \epsilon_{it} \]

If we ignore \(\alpha_i\) and estimate Pooled OLS:

\[ y_{it} = \beta x_{it} + u_{it} \quad \text{where} \quad u_{it} = \alpha_i + \epsilon_{it} \]

⚠️ If \(Cov(\alpha_i, x_{it}) \neq 0\), then:

  • \(x_{it}\) is correlated with the error term
  • Pooled OLS is biased

Fixed Effects is designed to solve exactly this problem.

1️⃣ Pooled OLS (POLS)

Pooled OLS: Baseline Model

Model \[ y_{it} = \beta_0 + \beta_1 x_{it} + \epsilon_{it} \]

Meaning

  • We pretend the dataset is one big cross-section
  • We ignore the panel structure (country identity)

Uses all variation

  • between-country variation
  • within-country variation

What POLS is Really Doing

When you estimate POLS, the slope \(\beta_1\) reflects a combination of:

  • Cross-country differences (rich vs. poor countries)
  • Within-country changes (one country over time)

So interpretation becomes:

“On average, countries with higher \(x\) have higher (or lower) \(y\), and/or when \(x\) rises over time, \(y\) moves…”

⚠️ That can be fine only if unobserved country traits don’t confound the relationship.

⚠️ The Big Risk in POLS: Omitted Variable Bias

If richer countries also have different institutions, energy systems, or historical industrialization paths, then:

  • \(x_{it}\) (GDP per capita) is correlated with \(\alpha_i\) (unobserved traits)
  • Pooled OLS mixes together the effect of \(x_{it}\) and the influence of \(\alpha_i\), so the estimated \(\beta\) can be misleading.

Panel methods help because they explicitly control for \(\alpha_i\).

2️⃣ Fixed Effects (FE)

Country Fixed Effects (FE)

Model \[ y_{it} = \beta x_{it} + \alpha_i + \epsilon_{it} \]

  • \(\alpha_i\) = country fixed effect
  • captures all time-invariant country characteristics
    • geography, long-run institutions, baseline culture, etc.

FE answers a within-country question:

“If a country’s \(x\) changes over time, how does \(y\) change within that same country?”

🔎 What FE Controls For (Automatically)

Because each country gets its own intercept \(\alpha_i\), FE controls for:

  • any factor that is constant within a country
  • even if you do not observe or measure it

Examples (time-invariant or very slow-moving):

  • distance to equator
  • landlocked status
  • baseline political system
  • deep energy infrastructure

📌 FE is powerful because it controls for a lot without measuring those things.

FE Uses Only Within-Country Variation

FE does not use differences between countries:

  • It does not compare USA vs India levels
  • It compares USA over time to itself

So FE is best when:

  • the key confounding problem is time-invariant
  • we care about within-country causal interpretation

⚠️ FE cannot estimate effects of variables that do not change over time, e.g.:

  • landlocked, continent, distance_to_equator

3️⃣ Two-Way Fixed Effects (TWFE)

Two-Way Fixed Effects (Country + Year FE)

Model \[ y_{it} = \beta x_{it} + \alpha_i + \gamma_t + \epsilon_{it} \]

  • \(\alpha_i\): country FE (time-invariant traits)
  • \(\gamma_t\): year FE (global shocks / common trends)

TWFE answers:

“Within a country, how does \(y\) change with \(x\), after removing global year shocks?”

🌍 Why Add Year Fixed Effects?

Year effects capture shocks that hit many countries at once:

  • 🛢️ oil price spikes
  • 📉 global recessions (e.g., 2008)
  • 🦠 pandemics
  • 💡 global technology trends
  • 🌐 global climate agreements

If we don’t include \(\gamma_t\), the regression may mistakenly attribute these common shocks to \(x_{it}\).

Year FE helps remove this confounding.

4️⃣ Random Effects (RE)

Random Effects (RE)

Model \[ y_{it} = \beta x_{it} + \alpha_i + \epsilon_{it} \]

Same structure as FE… but different assumption:

  • RE treats \(\alpha_i\) as random
  • and assumes it is uncorrelated with regressors:

\[ Cov(\alpha_i, x_{it}) = 0 \]

If that assumption holds:

  • RE is more efficient than FE (smaller standard errors)
  • RE uses both within and between variation

⚠️ When RE Fails (Bias Risk)

If \(Cov(\alpha_i, x_{it}) \neq 0\), then RE is biased.

In many economic settings, this correlation is very plausible:

  • richer countries may also have better institutions (inside \(\alpha_i\))
  • energy system differences affect both GDP and emissions

So FE is often the safer default in applied work.

🧠 Standard Errors in Panel Data

🧠 Why We Cluster Standard Errors (Country-Clustered SE)

The problem: errors are correlated within countries

In panel data, observations inside a country are rarely independent:

  • shocks persist over time
  • policies change gradually
  • measurement errors repeat
  • emissions and GDP move smoothly

That means residuals may satisfy serial correlation:

\[ Cov(\epsilon_{it}, \epsilon_{is}) \neq 0 \quad \text{for } t \neq s \]

If we ignore this, standard errors are often too small → false “significance”

Cluster-Robust Standard Errors (by Country)

Country-clustered SEs allow:

  • arbitrary correlation over time within a country
  • heteroskedasticity across countries

Interpretation:

“We allow any kind of serial correlation within each country.”

In R (plm), this is typically implemented using robust VCOV options, e.g. vcovHC() with clustering by group.

  • vcovHC: variance–covariance, heteroskedasticity-consistent

Model Summary (What Each One Controls For)

Model Controls for country traits? Controls for year shocks? Uses between-country variation?
Pooled OLS ❌ No ❌ No Yes
Country FE Yes (time-invariant) ❌ No ❌ No
Two-way FE Yes (time-invariant) Yes ❌ No
Random Effects Yes (modeled as random intercept) Optional Yes

Final Takeaway (Panel Models)

We use panel data models to estimate relationships within units over time, while controlling for:

  • hidden country characteristics (\(\alpha_i\))
  • global year shocks (\(\gamma_t\))

And we use clustered SEs because panel residuals are often correlated within each unit.

🌿 Environmental Kuznets Curve (EKC): Growth vs. Pollution

🌿 Environmental Kuznets Curve (EKC): Growth vs. Pollution

Note

Now we add the key EKC idea: a nonlinear relationship between income and pollution.

What is the EKC?

The Environmental Kuznets Curve (EKC) hypothesis suggests:

As a country becomes richer, pollution first increases,
but after some income level, pollution decreases.

It is often described as an inverted-U relationship between: - pollution (e.g., CO\(_2\) per capita) - income (GDP per capita)

📉📈 Draw the EKC Curve (Inverted-U)

EKC Panel Regression (Country-Year Data)

A common EKC specification:

\[ \log(CO2pc_{it}) = \beta_1 \log(GDPpc_{it}) + \beta_2 \left[\log(GDPpc_{it})\right]^2 + \alpha_i + \gamma_t + \epsilon_{it} \]

  • \(i\): country
  • \(t\): year
  • \(\alpha_i\): country fixed effects
  • \(\gamma_t\): year fixed effects

How to Interpret EKC Coefficients

  • If \(\beta_1 > 0\) and \(\beta_2 < 0\)
    emissions rise at first, then eventually fall
    → EKC pattern (inverted-U)

  • If \(\beta_2 = 0\)
    → relationship is linear

  • If \(\beta_2 > 0\)
    → U-shape (emissions accelerate with income)

EKC Turning Point (Income Level)

The turning point occurs where the slope becomes zero:

\[ \frac{\partial \log(CO2pc)}{\partial \log(GDPpc)} = \beta_1 + 2\beta_2 \log(GDPpc) = 0 \]

So the turning point in log income is:

\[ \log(GDPpc^*) = -\frac{\beta_1}{2\beta_2} \]

Convert back to the income level:

\[ GDPpc^* = \exp\left(-\frac{\beta_1}{2\beta_2}\right) \]

At \(GDPpc^*\), CO\(_2\) per capita stops rising and begins falling (if EKC holds).

Intuition: Why Might EKC Happen?

As income rises, countries often experience:

  • Scale effect 📈: more production → more emissions (early stage)
  • Composition effect 🏗️➡️🧑‍💼: shift from industry → services (middle stage)
  • Technique effect 🧪⚡: cleaner technology + regulation (later stage)

EKC is the balance of:

  • growth pressures that increase emissions
  • development forces that can reduce emissions

Important Caveat for CO\(_2\)

EKC may be weaker for CO\(_2\) than for local air pollutants because:

  • CO\(_2\) is deeply tied to energy systems
  • benefits are global, costs are often local
  • emissions may shift across borders via trade (carbon leakage)

So EKC is a hypothesis to test, not a guaranteed pattern.

Final Takeaway (EKC)

EKC tests whether pollution follows an inverted-U with income.

Panel FE/TWFE help make that test more credible by controlling for:

  • country traits (\(\alpha_i\))
  • year shocks (\(\gamma_t\))