= Fixed Effects Model =

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

<<TableOfContents>>

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== Description ==

This model is used for panel analysis.

A good starting point for modeling with panel data is the [[Statistics/PooledOrdinaryLeastSquaresModel|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,,''.

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.



=== Least Squares Dummy Variable Estimator ===

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,, = β,,0,, + β,,1,,X,,it,, + β,,2,,Z,,it,, + ε,,it,,'', consider ''Y,,it,, = α,,i,, + β,,1,,X,,it,, + β,,2,,Z,,it,, + u,,it,,''.

The model is fit using [[Statistics/OrdinaryLeastSquares|OLS]] and the intercept is actually the first unit's intercept, ''α,,1,,''. 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



=== De-meaned Estimator ===

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

For a model specified as {{attachment:model1.svg}}, the within-unit average is:

{{attachment:model2.svg}}

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. This is also sometimes referred to as the '''within estimator'''.

This model is more commonly expressed as:

{{attachment:model3.svg}}

The overall averages are often re-added so as to give the model a relevant intercept, for interpretation.

{{attachment:model4.svg}}

Note that this is a simple evolution of the idea behind [[Statistics/FirstDifferencedEstimator|first-differenced estimators]].



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