= Random Effects Model =

A '''random effects model''' utilizes repeated observations (i.e., panel data) to decompose and correct for within-group and between-group 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.

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,,''.

Also, consider ''N'' to the total number of observations. If using a balanced panel, i.e. all individuals ''i'' have ''T'' observations, this is simply ''nT''. More generally though, the calculation is {{attachment:n.svg}}.

Strong assumptions about the variance structure are made.

 * Errors are distributed about 0, i.e. ''E[ε,,it,,] = 0''.
   * Therefore the covariance of errors between two measurements of the same individual is:
     * ''Cov(ε,,it,,, ε,,is,,) = E[(ε,,it,, - 0)(ε,,is,, - 0)] = E[ε,,it,,ε,,is]''
     * ''Cov(ε,,it,,, ε,,is,,) = E[(u,,it,, + α,,i,,)(u,,is,, + α,,i,,)] = E[u,,it,,u,,is,, + u,,it,,α,,i,, + u,,is,,α,,i,, + α,,i,,^2^]''
 * The components of errors are independent.
   * The above simplifies to ''σ,,α,,^2^''.
   * For the same reasons, the '''variance''' of errors (i.e., the covariance between a measurement and itself) simplifies to ''σ,,α,,^2^ + σ,,u,,^2^''.
 * There is zero covariance between the errors and any predictor.

The first two lead to a ''T,,i,,'' by ''T,,i,,'' covariance matrix for any individual ''i'':

{{attachment:sigma.svg}}

Furthermore, the covariance matrix for all individuals and all measurements can be fully expressed in a ''N'' by ''N'' covariance matrix like:

{{attachment:omega.svg}}

Note that all off-diagonal covariances are zero unless individuals ''i'' and ''j'' are the same.

The final assumption is important because the total errors, '''''ε''''' composed of ''ε,,it,,'', can then be calculated using a [[Statistics/FixedEffectsModel#De-meaned_Estimator|de-meaned within estimator]]. The diagonal members can be summed and averaged to arrive at ''σ,,ε,,^2^'':

{{attachment:within1.svg}}

There are a few different estimators for ''σ,,α,,^2^'', but the simplest intuition is summing and averaging the off-diagonal members.

[[Statistics/GeneralizedLeastSquares|Feasible GLS]] is used to fit the random effects model. This can be interpreted as transforming the space by weights, '''''θ''''' composed of ''θ,,i,,'', that mix observations with individual-level averages.

The weights are specified as:

{{attachment:theta.svg}}

And the random effects model can be formulated as:

{{attachment:re.svg}}

As ''θ,,i,,'' approaches 1, this model converges to the fixed effects model. As ''θ,,i,,'' approaches 0, this model converges to a pooled OLS model.

Because of this nesting and the fact that the fixed effects model is less efficient but must be consistent, a [[Statistics/HausmanTest|Hausman test]] should be performed with the null hypothesis that the random effects model is consistent. If rejected, the fixed effects model should be used instead.



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