追蹤資料分析 (Panel Data Analysis)

Overview

Panel data analysis exploits both cross-sectional and temporal variation to estimate causal effects while controlling for unobserved heterogeneity. Fixed effects eliminate time-invariant confounders through within-entity demeaning, while random effects assume unobserved heterogeneity is uncorrelated with regressors, yielding more efficient estimates when valid.

When to Use

• Data has repeated observations for the same entities (firms, individuals, countries) over time
• Unobserved time-invariant factors likely confound the relationship of interest
• Testing whether a policy or treatment effect varies across time periods
• Dynamic models where the lagged dependent variable is a regressor (use GMM)

When NOT to Use

• Pure cross-sectional data with no time dimension
• Interest is in estimating the effect of time-invariant variables (FE eliminates these)
• Panel is extremely short (T = 2) with many endogenous regressors
• Attrition is non-random and creates survivorship bias

Assumptions

IRON LAW: Fixed effects ONLY controls for TIME-INVARIANT unobservables —
time-varying confounders remain a threat. FE does not solve all
endogeneity problems.

Key assumptions:

1. Strict exogeneity for FE/RE: past, current, and future errors are uncorrelated with regressors
2. No serial correlation in idiosyncratic errors (or use cluster-robust SEs)
3. RE additionally assumes individual effects are uncorrelated with regressors
4. For dynamic GMM: instruments are valid and not too many (instrument proliferation)

Methodology

Step 1 — Explore Panel Structure

Report N (entities), T (time periods), balance status. Check within vs between variation for key variables. Visualize entity-level trends.

Step 2 — Estimate FE and RE Models

Run fixed effects (within estimator) and random effects (GLS). Include time fixed effects if common shocks exist. Use cluster-robust standard errors at the entity level.

Step 3 — Hausman Test for Model Selection

Test H₀: RE is consistent (individual effects uncorrelated with regressors). Rejection favors FE. See references/ for test statistic derivation.

Step 4 — Dynamic Extensions (if needed)

If lagged DV is included, use Arellano-Bond or System GMM. Report AR(1), AR(2) tests and Hansen/Sargan test for instrument validity. Monitor instrument count.

Output Format

## Panel Data Analysis: [Study Title]

### Panel Structure
| Dimension | Value |
|-----------|-------|
| Entities (N) | xxx |
| Time periods (T) | xxx |
| Balanced? | [Yes/No] |

### Estimation Results
| Variable | FE (β) | RE (β) | GMM (β) |
|----------|--------|--------|---------|
| [var] | x.xx (x.xx) | x.xx (x.xx) | x.xx (x.xx) |

### Model Selection
| Test | Statistic | p-value | Decision |
|------|-----------|---------|----------|
| Hausman | x.xx | x.xx | [FE/RE] |
| AR(2) | x.xx | x.xx | [pass/fail] |
| Hansen J | x.xx | x.xx | [pass/fail] |

### Key Findings
- [Interpretation]

### Limitations
- [Note any assumption violations]

Gotchas

• FE discards all between-entity variation; if most variation is between, FE estimates are imprecise
• Hausman test has low power in small samples — insignificance does not validate RE
• Dynamic panel GMM with too many instruments causes overfitting and weakens the Hansen test
• Nickell bias afflicts FE estimates with a lagged DV when T is small
• Two-way FE (entity + time) is often necessary but rarely the default in software
• Cluster-robust standard errors require a sufficient number of clusters (N ≥ 50 as guideline)

References

• Wooldridge, J. M. (2010). Econometric Analysis of Cross Section and Panel Data (2nd ed.). MIT Press.
• Arellano, M., & Bond, S. (1991). Some tests of specification for panel data. Review of Economic Studies, 58(2), 277-297.
• Baltagi, B. H. (2013). Econometric Analysis of Panel Data (5th ed.). Wiley.