fixedeffects

Fixed effects is a modeling approach used with panel (longitudinal) data to account for unobserved heterogeneity that is constant over time within an entity. It allows the intercept to vary by entity, capturing time-invariant characteristics that may be correlated with explanatory variables.

The standard fixed-effects model can be written as y_it = alpha_i + X_it beta + u_it, where i indexes entities (for example individuals or firms) and t indexes time. Alpha_i are entity-specific intercepts. Because alpha_i may be correlated with X_it, pooled OLS can be biased. The within or fixed-effects estimator removes alpha_i by demeaning the data (or by including entity dummies), focusing on within-entity variation. It yields consistent estimates under appropriate exogeneity assumptions and sufficient time periods.

Interpretation and comparison: Fixed effects absorb all time-invariant differences across entities, so the estimated coefficients reflect within-entity effects. Random effects can be used if the entity effects are uncorrelated with regressors, but fixed effects is robust to such correlation. The Hausman test is commonly used to decide between fixed and random effects. Extensions include two-way fixed effects (entity and time) to control for common shocks, and clustered standard errors at the entity level to address serial correlation and heteroskedasticity. The Mundlak device links fixed and random effects by modeling alpha_i as a function of the entity means of regressors.

Limitations and extensions: time-invariant regressors cannot be estimated separately in a pure fixed-effects model; dynamic models with lagged dependent variables can introduce bias unless specialized estimators (for example, Arellano–Bondd) are used. In nonlinear models, the incidental parameters problem can affect consistency. Fixed effects remain a central tool for causal inference with panel data when unobserved, time-invariant factors are potentially correlated with the regressors.