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.

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.

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 .

Strong assumptions about the (co-)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_itε_is]

    • Cov(ε_it, ε_is) = E[(u_it + α_i)(u_is + α_i)] = E[u_itu_is + u_itα_i + u_isα_i + α_i²]

• The components of errors are independent.

  • The above simplifies to σ_α².

  • For the same reasons, the variance of errors simplifies to σ_α² + σ_u².

• 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:

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

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 de-meaned within estimator. The diagonal members can be summed and averaged to arrive at σ_ε²:

There are a few different estimators for σ_α², but the simplest intuition is summing and averaging the off-diagonal members.

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:

And the random effects model can be formulated as:

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 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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