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| 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 === |
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=== De-meaned Estimator === |
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| 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. | 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: |
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| The overall averages are often re-added so as to give the model a relevant intercept, for interpretation. {{attachment:model4.svg}} |
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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. |
Fixed Effects Model
A fixed effects model utilizes repeated observations (i.e., panel data) to remove between-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 εit into time-variant and time-invariant components: uit and αi.
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(uit, xis) = 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(uit, xit) = 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 Yit = β0 + β1Xit + β2Zit + εit, consider Yit = αi + β1Xit + β2Zit + uit.
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
De-meaned Estimator
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 αi term. This is also sometimes referred to as the within estimator.
This model is more commonly expressed as:
The overall averages are often re-added so as to give the model a relevant intercept, for interpretation.
Note that this is a simple evolution of the idea behind first-differenced estimators.