MORISTAT: Chapter 3 — Panel Data Models

3.1 What Is Panel Data?

Definition: Panel data (also called longitudinal data) consists of observations on the same individuals (entities) over multiple time periods.

Panel Data Structure:

Entity i = 1, 2, ..., N (individuals, firms, countries) Time t = 1, 2, ..., T (years, quarters, months) Yᵢₜ = β₀ + β₁X₁ᵢₜ + β₂X₂ᵢₜ + ... + βₖXₖᵢₜ + μᵢ + εᵢₜ

Entity	Time	Y	X₁	X₂	...	Xₖ
1	1	y₁₁	x₁₁₁	x₂₁₁	...	xₖ₁₁
1	2	y₁₂	x₁₁₂	x₂₁₂	...	xₖ₁₂
1	T	y₁T	x₁₁T	x₂₁T	...	xₖ₁T
2	1	y₂₁	x₁₂₁	x₂₂₁	...	xₖ₂₁
2	2	y₂₂	x₁₂₂	x₂₂₂	...	xₖ₂₂
N	T	yNT	x₁NT	x₂NT	...	xₖNT

Types of Panel Data:

Type	Description	Example
Balanced Panel	All entities have the same number of time periods	N=100, T=10, total=1000 obs
Unbalanced Panel	Some entities have missing observations	Firms enter/exit the sample
Short Panel	N large, T small (N >> T)	Many firms, few years
Long Panel	N small, T large (N << T)	Few countries, many years

Why Use Panel Data?

Advantage	Description
Control for Unobserved Heterogeneity	Entity-specific effects are captured
More Variation	Both cross-section and time variation
Dynamic Effects	Study adjustment over time
Better Identification	Control for omitted variables bias
More Observations	N × T vs N (cross-section) or T (time)

3.2 The Panel Data Model

The General Panel Model:

Yᵢₜ = β₀ + β₁X₁ᵢₜ + β₂X₂ᵢₜ + ... + βₖXₖᵢₜ + μᵢ + εᵢₜ

Where:

Yᵢₜ = Dependent variable for entity i at time t
Xⱼᵢₜ = Independent variables for entity i at time t
βⱼ = Coefficients (the same for all entities)
μᵢ = Entity-specific unobserved heterogeneity (time-invariant)
εᵢₜ = Idiosyncratic error (varies across i and t)

The Error Components:

uᵢₜ = μᵢ + εᵢₜ

Component	Description	Properties
μᵢ	Entity-specific effect	Time-invariant, captures unobserved heterogeneity
εᵢₜ	Idiosyncratic error	Varies across both i and t, assumed i.i.d.

Key Assumptions:

E(εᵢₜ | X, μᵢ) = 0 — Idiosyncratic error is uncorrelated with regressors
E(εᵢₜ² | X, μᵢ) = σ² — Homoskedasticity
E(εᵢₜεᵢₛ | X, μᵢ) = 0 for t≠s — No serial correlation
E(εᵢₜεⱼₛ | X, μᵢ) = 0 for i≠j — No cross-sectional correlation

The Entity Effect μᵢ

The entity effect μᵢ captures all unobserved, time-invariant factors that affect Y:

Examples of μᵢ:

Firms: Management quality, corporate culture, location
Countries: Geography, institutions, culture, history
Individuals: Ability, motivation, family background

Key Question: Is μᵢ correlated with the regressors?

If yes → Fixed Effects
If no → Random Effects

3.3 Pooled OLS

What Is Pooled OLS?

Pooled OLS simply ignores the panel structure and estimates:

Yᵢₜ = β₀ + β₁X₁ᵢₜ + β₂X₂ᵢₜ + ... + βₖXₖᵢₜ + uᵢₜ

It pools all observations together (N × T observations).

The Problem with Pooled OLS

Pooled OLS is valid only if:

No entity-specific effects (μᵢ = 0)
Or μᵢ is uncorrelated with all regressors

If μᵢ ≠ 0 and correlated with X: Pooled OLS is biased and inconsistent.

When to Use Pooled OLS:

Situation	Recommendation
μᵢ = 0 (no entity effects)	Use Pooled OLS
μᵢ ≠ 0 but uncorrelated with X	Use Random Effects (more efficient)
μᵢ ≠ 0 and correlated with X	Use Fixed Effects

3.4 Fixed Effects (FE)

The Fixed Effects Model:

Yᵢₜ = β₁X₁ᵢₜ + β₂X₂ᵢₜ + ... + βₖXₖᵢₜ + μᵢ + εᵢₜ

Key Assumption: μᵢ is correlated with X (allows for arbitrary correlation).

Estimation Methods:

1. Within Transformation (Demeaning)

Subtract the entity-specific mean from each variable:

Yᵢₜ - Ȳᵢ = β₁(X₁ᵢₜ - X̄₁ᵢ) + β₂(X₂ᵢₜ - X̄₂ᵢ) + ... + βₖ(Xₖᵢₜ - X̄ₖᵢ) + (εᵢₜ - ε̄ᵢ)

Where:

Ȳᵢ = (1/T)ΣₜYᵢₜ (entity-specific mean)
X̄ⱼᵢ = (1/T)ΣₜXⱼᵢₜ (entity-specific mean)

This removes μᵢ because μᵢ - μᵢ = 0.

2. First-Difference Transformation

Subtract the previous period:

(Yᵢₜ - Yᵢ,ₜ₋₁) = β₁(X₁ᵢₜ - X₁ᵢ,ₜ₋₁) + β₂(X₂ᵢₜ - X₂ᵢ,ₜ₋₁) + ... + βₖ(Xₖᵢₜ - Xₖᵢ,ₜ₋₁) + (εᵢₜ - εᵢ,ₜ₋₁)

This also removes μᵢ because μᵢ - μᵢ = 0.

3. Least Squares Dummy Variable (LSDV)

Include entity-specific dummies:

Yᵢₜ = β₀ + β₁X₁ᵢₜ + β₂X₂ᵢₜ + ... + βₖXₖᵢₜ + δ₂D₂ + δ₃D₃ + ... + δₙDₙ + εᵢₜ

Where Dⱼ = 1 if entity = j, 0 otherwise.

Properties of FE:

Property	Description
Consistent	As N → ∞ or T → ∞
Unbiased	Only if T is fixed (within transformation)
Efficiency	OLS on transformed data is BLUE
Time-Invariant Variables	Cannot be estimated (they are differenced out)

Advantages of FE:

Controls for all time-invariant omitted variables
Does not require μᵢ to be uncorrelated with X
Robust to many forms of omitted variable bias

Disadvantages of FE:

Cannot estimate time-invariant variables (e.g., gender, race)
Can be inefficient if μᵢ is actually uncorrelated with X
Requires variation within entities over time

3.5 Random Effects (RE)

The Random Effects Model:

Yᵢₜ = β₀ + β₁X₁ᵢₜ + β₂X₂ᵢₜ + ... + βₖXₖᵢₜ + μᵢ + εᵢₜ

Key Assumption: μᵢ is uncorrelated with X (Cov(X, μ) = 0).

The Error Structure:

uᵢₜ = μᵢ + εᵢₜ

Variance Structure:

Var(uᵢₜ) = σ²_μ + σ²_ε
Cov(uᵢₜ, uᵢₛ) = σ²_μ (for t≠s) — correlation within entity
Cov(uᵢₜ, uⱼₛ) = 0 (for i≠j) — no correlation across entities

Estimation: GLS (Feasible GLS)

RE uses Generalized Least Squares (GLS):

β̂_RE = (X'Ω⁻¹X)⁻¹X'Ω⁻¹Y

Where Ω is the variance-covariance matrix of the errors.

In practice, we use Feasible GLS (FGLS) with estimated variance components:

Estimate σ²_ε from FE
Estimate σ²_μ from between variation
Compute θ = 1 - σ_ε / √(Tσ²_μ + σ²_ε)
Apply GLS transformation

The GLS Transformation:

Y*ᵢₜ = Yᵢₜ - θȲᵢ X*ⱼᵢₜ = Xⱼᵢₜ - θX̄ⱼᵢ

Where:

θ = 1 - σ_ε / √(Tσ²_μ + σ²_ε)

Special cases:

If σ²_μ = 0 → θ = 1 → RE = Pooled OLS
If σ²_ε = 0 → θ = 0 → RE = FE

Properties of RE:

Property	Description
Consistent	If Cov(X, μ) = 0
Efficient	Uses both within and between variation
Can Estimate	Time-invariant variables
Biased	If Cov(X, μ) ≠ 0

Advantages of RE:

More efficient than FE (uses more variation)
Can estimate time-invariant variables
Better for long panels (T large)

Disadvantages of RE:

Requires Cov(X, μ) = 0 (strong assumption)
Biased and inconsistent if assumption fails

3.6 Hausman Test

Choosing Between FE and RE:

The Hausman test determines whether the RE assumption (Cov(X, μ) = 0) is valid.

The Test:

Null Hypothesis: H₀: Cov(X, μ) = 0 (RE is consistent)

Alternative: H₁: Cov(X, μ) ≠ 0 (RE is inconsistent, FE is consistent)

The Test Statistic:

H = (β̂_FE - β̂_RE)'[Var(β̂_FE) - Var(β̂_RE)]⁻¹(β̂_FE - β̂_RE)

Under H₀, H ~ χ²(k), where k = number of time-varying regressors.

Interpretation:

Result	Decision	Meaning
p ≥ 0.10	Use RE	RE is consistent and efficient
p < 0.05	Use FE	RE is inconsistent
0.05 ≤ p < 0.10	Marginal	Use FE to be safe

Intuition

If μᵢ is uncorrelated with X → FE and RE are both consistent → RE is more efficient → Use RE
If μᵢ is correlated with X → FE is consistent, RE is not → Use FE

3.7 Within vs Between Variation

Understanding the Components:

Panel data contains two types of variation:

1. Within Variation

Variation of a variable around its entity-specific mean:

Xᵢₜ - X̄ᵢ

Captures: Changes within entities over time

Used by: Fixed Effects (FE)

2. Between Variation

Variation of entity-specific means:

X̄ᵢ - X̄

Captures: Differences across entities

Used by: Between Estimator

3. Total Variation

Xᵢₜ - X̄ = (Xᵢₜ - X̄ᵢ) + (X̄ᵢ - X̄)

Which Estimator Uses What?

Estimator	Variation Used	Interpretation
Pooled OLS	Total	Mix of within and between
FE	Within only	Changes within entities over time
RE	Weighted mix	Both within and between
Between	Between only	Differences across entities

3.8 Dynamic Panel Models

The Dynamic Panel Model:

Many economic processes are dynamic:

Yᵢₜ = αYᵢ,ₜ₋₁ + β₁X₁ᵢₜ + β₂X₂ᵢₜ + ... + βₖXₖᵢₜ + μᵢ + εᵢₜ

Where Yᵢ,ₜ₋₁ is the lagged dependent variable.

Why Dynamics Matter:

State dependence: Past outcomes affect current outcomes
Adjustment costs: Adjustment to equilibrium takes time
Habit formation: Current behavior depends on past behavior
Partial adjustment: Gradual adjustment to desired levels

The Problem with Dynamic Panels

Nickell Bias (1981): In short panels (T small), FE is biased and inconsistent.

Source of bias: The lagged dependent variable Yᵢ,ₜ₋₁ is correlated with μᵢ, and the within transformation induces correlation with εᵢ,ₜ₋₁.

Cov(Yᵢ,ₜ₋₁, μᵢ) ≠ 0 → Correlated with entity effect Cov(Yᵢ,ₜ₋₁, εᵢₜ) ≠ 0 → After within transformation

Arellano-Bond GMM

The Arellano-Bond Difference GMM estimator addresses the Nickell bias.

Difference GMM:

First-difference the model to remove μᵢ:

ΔYᵢₜ = αΔYᵢ,ₜ₋₁ + β₁ΔX₁ᵢₜ + β₂ΔX₂ᵢₜ + ... + βₖΔXₖᵢₜ + Δεᵢₜ

Instruments:

Use lagged values of Y and X as instruments:

For ΔYᵢₜ = Yᵢₜ - Yᵢ,ₜ₋₁: Instruments: Yᵢ,ₜ₋₂, Yᵢ,ₜ₋₃, ..., Xᵢ,ₜ₋₁, Xᵢ,ₜ₋₂, ...

Validity: Lagged values are correlated with Yᵢ,ₜ₋₁ but uncorrelated with Δεᵢₜ.

System GMM

Blundell-Bond System GMM improves efficiency by combining:

Difference equations (like Arellano-Bond)
Level equations (with lagged differences as instruments)

Better for: Persistent series (Y is close to a random walk)

GMM Diagnostics:

Test	Purpose	Rule
AR(1)	Test for first-order autocorrelation	Expected (p < 0.05)
AR(2)	Test for second-order autocorrelation	Should not be significant (p > 0.05)
Sargan/Hansen J	Overidentification test	p > 0.10 (instruments valid)

3.9 Applications and Examples

Example 1: Production Function Estimation

Research Question: What is the elasticity of output with respect to inputs?

Dataset: Firm-level panel data (N=5000 firms, T=10 years)

Model:

Log(Output)ᵢₜ = β₁Log(Capital)ᵢₜ + β₂Log(Labor)ᵢₜ + β₃Log(Materials)ᵢₜ + μᵢ + εᵢₜ

Endogeneity Concern: Input choices are correlated with productivity (μᵢ)

Solution: FE or IV with lagged inputs

Variable	OLS	FE	System GMM
Capital	0.12**	0.08**	0.09**
Labor	0.45**	0.38**	0.40**
Materials	0.35**	0.32**	0.33**

Example 2: Economic Growth

Research Question: What determines economic growth?

Dataset: Country-year panel (N=100 countries, T=30 years)

Model:

Growthᵢₜ = β₁InitialGDPᵢₜ + β₂Investmentᵢₜ + β₃Educationᵢₜ + β₄Tradeᵢₜ + μᵢ + εᵢₜ

Endogeneity Concern: Growth affects investment (simultaneity)

Solution: FE + GMM

Variable	FE	System GMM
Initial GDP	-0.02**	-0.01**
Investment	0.15**	0.12**
Education	0.08**	0.06**
Trade	0.05**	0.04**

Example 3: Healthcare Utilization

Research Question: How does insurance affect healthcare utilization?

Dataset: Individual-year panel (N=10,000 individuals, T=5 years)

Model:

DoctorVisitsᵢₜ = β₁Insuredᵢₜ + β₂Ageᵢₜ + β₃Incomeᵢₜ + μᵢ + εᵢₜ

Endogeneity Concern: Health status affects both insurance and utilization

Solution: FE (controls for individual health status)

Variable	Pooled OLS	FE
Insured	2.45**	1.12**
Age	0.03**	0.02**
Income	0.01**	0.00
N	50,000	50,000

Interpretation: The OLS estimate is biased upward because healthier people are more likely to be insured.

3.10 Implementation in Practice

Checklist for Panel Data Analysis:

Understand panel structure (balanced/unbalanced, N vs T)
Transform variables appropriately (logs, ratios)
Test for entity-specific effects (F-test for μᵢ)
Run Pooled OLS as baseline
Run FE and RE
Conduct Hausman test
If FE chosen, consider time effects (year dummies)
Check for serial correlation and heteroskedasticity
For dynamic models, use Arellano-Bond GMM
Report results with appropriate standard errors

Common Mistakes to Avoid:

Mistake	Why It's Wrong	What To Do Instead
Using Pooled OLS with entity effects	Biased and inconsistent	Use FE or RE
Using FE for time-invariant variables	Dropped from estimation	Use RE or between estimator
Ignoring serial correlation	Inefficient SEs	Use clustered SEs
Overinterpreting FE estimates	FE uses only within variation	Interpret as changes over time
Using GMM with too many instruments	Overfitting	Limit instruments

3.11 Decision Tree for Panel Data

Do you have panel data? └─── No → Use cross-section or time series methods └─── Yes → Proceed Are entity effects μᵢ present? └─── No (F-test not significant) → Use Pooled OLS └─── Yes → Proceed Is μᵢ correlated with X? └─── No (Hausman test p ≥ 0.10) → Use Random Effects (RE) └─── Yes (Hausman test p < 0.05) → Use Fixed Effects (FE) Is the model dynamic? (has lagged Y) └─── No → Use FE or RE └─── Yes → Use Arellano-Bond GMM Is your panel short (T small)? └─── No (T large) → FE is OK └─── Yes (T small) → Use System GMM

3.12 Quick Reference Cards

Panel Models Summary:

Model	Key Feature	Assumption	Estimation
Pooled OLS	Ignore panel structure	μᵢ = 0	OLS
Fixed Effects (FE)	Entity-specific intercepts	Cov(X, μ) ≠ 0	Within
Random Effects (RE)	Random entity effects	Cov(X, μ) = 0	GLS
Between	Between variation only	Cov(X, μ) ≠ 0	OLS on means

Hausman Test Summary:

H = (β̂_FE - β̂_RE)'[Var(β̂_FE) - Var(β̂_RE)]⁻¹(β̂_FE - β̂_RE) If p ≥ 0.10 → Use RE (consistent and efficient) If p < 0.05 → Use FE (RE is inconsistent)

FE vs RE Comparison:

Aspect	FE	RE
Assumption	Cov(X, μ) ≠ 0	Cov(X, μ) = 0
Time-invariant variables	Cannot estimate	Can estimate
Efficiency	Less efficient	More efficient
Consistency	Always consistent	Consistent only if assumption holds

3.13 Chapter Summary

Key Takeaways:

Panel data combines cross-section and time series — More variation and better identification
Unobserved heterogeneity is the key issue — μᵢ captures time-invariant entity effects
FE controls for any correlation between μᵢ and X — Uses within variation
RE is more efficient but requires Cov(X, μ) = 0 — Uses both within and between variation
Hausman test chooses between FE and RE — If p < 0.05, use FE
Dynamic panels require GMM — Arellano-Bond addresses Nickell bias
Always check for serial correlation — Use clustered standard errors
Report both FE and RE — Compare to assess robustness

Checklist Before Proceeding:

Understand the panel structure of your data
Know the difference between FE and RE
Can implement Hausman test
Understand the within transformation
Know when to use dynamic panel methods
Ready for Advanced Topics (Chapter 4)

3.14 What's Next?

📘 Chapter 4: Advanced Topics in Econometrics

In the next chapter, we will cover:

Limited Dependent Variable Models — Logit, Probit, Tobit
Binary Choice Models — Interpretation and marginal effects
Censored and Truncated Models — Tobit regression
Count Data Models — Poisson and Negative Binomial
Duration Models — Survival analysis
Mixture Models — Finite mixtures and latent class analysis
Spatial Econometrics — Spatial autocorrelation and dependence
Machine Learning in Econometrics — Ridge, Lasso, and beyond

📘 MORISTAT: Advanced Econometrics

Chapter 3: Panel Data Models