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Pooled OLS:
\[ y_{it} = \alpha + x_{it}'\beta + \varepsilon_{it} \]
Assumes a common intercept across entities and periods; ignores unobserved state heterogeneity.
One-way Fixed Effects:
\[ y_{it} = \alpha_i + x_{it}'\beta + \varepsilon_{it} \]
Uses within-state variation and controls for time-invariant state traits via entity intercepts.
Two-way Fixed Effects:
\[ y_{it} = \alpha_i + \lambda_t + x_{it}'\beta + \varepsilon_{it} \]
Controls both entity heterogeneity and common time shocks; regressors that vary only over time can be absorbed by \(\lambda_t\).
Random Effects:
\[ y_{it} = \alpha + u_i + x_{it}'\beta + \varepsilon_{it}, \quad \mathrm{Cov}(u_i, x_{it}) = 0 \]
Can be more efficient than FE if the orthogonality assumption holds.
A fixed effects estimator that removes unobserved, time-invariant individual effects by differencing adjacent time periods.
It is an alternative to the within (demeaning) estimator.
Original panel model:
\[ y_{it} = \alpha + \beta x_{it} + u_i + \varepsilon_{it} \]
where:
Take first difference:
\[ \Delta y_{it} = \beta \Delta x_{it} + \Delta \varepsilon_{it} \]
The fixed effect \(u_i\) is eliminated.
Test for fixed effects using F-test:
\[ H_0:u_i = 0 \]
Reject → fixed effects required
Hausman test:
\[ H_0: \text{Random Effects consistent} \]
Reject → use Fixed Effects / First Differences
Compare within vs first difference estimates:
All time-invariant variables drop out of the model.