Fixed Effects Model

A fixed effects model utilizes repeated observations (i.e., panel data) to remove within-group unobserved heterogeneity.

Description

This model is used for panel analysis.

A good starting point for modeling with panel data is the pooled OLS model. This model builds upon weaknesses of that methodology.

There are two, essentially-equivalent formulations of a fixed effects model. It is helpful to establish a decomposition for the unit error term ε_it into time-variant and time-invariant components: u_it and α_i.

The first formulation is to introduce dummy variables for each unit. This is also called a least squares dummy variable (LSDV) model.

The intuition here is that the intercept term is made to vary across units. Rather than specifying the model as Y_it = β₀ + β₁X_it + β₂Z_it + ε_it, consider Y_it = α_i + β₁X_it + β₂Z_it + u_it.

The model is fit using OLS and the intercept is actually the first unit's intercept, α₁. All subsequent units' intercepts are that term plus the estimated coefficient for their corresponding dummy variable. All time-invariant unit effects will be captured by these coefficients. This effectively removes

The second formulation is to normalize the measurements to unit means.

For a model specified as , the within-unit average is . In normalizing the data by subtracting the within-unit average, all terms that do not vary within-unit are removed. This importantly includes the α_i term.

Note that this is a simple evolution of the idea behind first-differenced estimators.

In either formulation, the important detail is that all time-invariant heterogeneity is removed. This importantly means that no other time-invariant predictors can be used in the model, as they would be colinear with the dummy variables.

The fixed effects estimators are BLUE if Cov(u_it, x_is) = 0 for all predictors x and for all units i and for all time periods t and s. But a weak exogeneity assumption is more often used, such that Cov(u_it, x_it) = 0, under which the estimators may be biased but are still consistent.

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