Panel Data Models: Fixed & Random Effects, Hausman Test
Panel Data Models: Key Concepts & Estimators
1. General Panel Data Model
The general panel data model is expressed as:
$y_{it} = \alpha_i + X_{it}’\beta + \epsilon_{it}$
$\alpha_i$: Represents the individual-specific, time-invariant effect.
Goal: Estimate $\beta$ despite unobserved heterogeneity.
2. Fixed Effects (FE) Estimator
Methods:
LSDV (Least Squares Dummy Variables): Not feasible for large $N$.
Within Estimator:
$y_{it} – \bar{y}_i = (X_{it} – \bar{X}_i)\beta + (\epsilon_{it} – \bar{\epsilon}_i)$
First-Difference (FD) Estimator:
$y_{i2} – y_{i1} = (X_{i2} – X_{i1})\beta + (\epsilon_{i2} – \epsilon_{i1})$
Note: Controls for correlation between regressors and $\alpha_i$.
Caution: Time-invariant regressors are dropped.
3. Random Effects (RE) Estimator
The random effects model is given by:
$y_{it} = \alpha + X_{it}’\beta + \nu_i + \epsilon_{it}$
Assumes: $\text{Cov}(X_{it}, \nu_i) = 0$
GLS transformation:
$y_{it} – \theta \bar{y}_i = (X_{it} – \theta \bar{X}_i)\beta + u_{it}$
with
$\theta = 1 – \frac{\sigma_\epsilon}{\sqrt{T\sigma^2_\nu + \sigma^2_\epsilon}}$
Key Panel Data Formulas
Panel Data Model
The fundamental panel data model is:
$y_{it} = \alpha_i + X_{it}’\beta + \epsilon_{it}$
Within Transformation
The within transformation for FE estimation is:
$y_{it} – \bar{y}_i = (X_{it} – \bar{X}_i)\beta + (\epsilon_{it} – \bar{\epsilon}_i)$
First Difference Transformation
The first difference transformation is:
$\Delta y_{it} = \Delta X_{it} \cdot \beta + \Delta \epsilon_{it}$
Random Effects Transformation
The transformed equation for RE estimation is:
$y_{it} – \theta \bar{y}_i = (X_{it} – \theta \bar{X}_i)\beta + u_{it}$
where
$\theta = 1 – \frac{\sigma_\epsilon}{\sqrt{T\sigma_\nu^2 + \sigma_\epsilon^2}}$
Hausman Test Statistic
The Hausman test statistic is used to choose between FE and RE models:
$W = (\hat{\beta}_{FE} – \hat{\beta}_{RE})’ [\text{Var}(\hat{\beta}_{FE}) – \text{Var}(\hat{\beta}_{RE})]^{-1} (\hat{\beta}_{FE} – \hat{\beta}_{RE})$