16.3 Random effects
If \(\alpha _i\) is uncorrelated with the regressors, it can be treated as part of the disturbance: \(u_{it} = \alpha _i + \varepsilon _{it}\), with \(\alpha _i\) independent across units with variance \(\sigma _\alpha ^2\). Least squares is then consistent but inefficient, because the disturbances of the same unit are correlated: for unit \(i\), \[ \Var (u_i) = \Omega _T = \sigma _\varepsilon ^2I_T + \sigma _\alpha ^2\iota \iota ', \qquad \Corr (u_{it}, u_{is}) = \frac {\sigma _\alpha ^2}{\sigma _\alpha ^2 + \sigma _\varepsilon ^2} \quad (t \neq s), \] where \(\iota \) is a column of \(T\) ones. The common correlation is the intraclass correlation.
Theorem 16.3.1. Let \(\bar J = \iota \iota '/T\) and \[ \theta = 1 - \frac {\sigma _\varepsilon }{\sqrt {\sigma _\varepsilon ^2 + T\sigma _\alpha ^2}} . \] Then \(\Omega _T^{-1/2} = \sigma _\varepsilon ^{-1}(I_T - \theta \bar J)\), and the GLS (random effects) estimator is least squares on the quasi-demeaned data \[ y_{it} - \theta \bar y_i = (1 - \theta )\mu + (x_{it} - \theta \bar x_i)'\beta + (z_i - \theta z_i)'\gamma + \text {error} . \]
Proof. \(\bar J\) is symmetric and idempotent, and so is \(I - \bar J\), with \(\bar J(I - \bar J) = 0\). Then \(\Omega _T = \sigma _\varepsilon ^2(I - \bar J) + (\sigma _\varepsilon ^2 + T\sigma _\alpha ^2)\bar J\), a spectral decomposition with eigenvalues \(\sigma _\varepsilon ^2\) and \(\sigma _\varepsilon ^2 + T\sigma _\alpha ^2\). Hence \[ \Omega _T^{-1/2} = \frac {1}{\sigma _\varepsilon }(I - \bar J) + \frac {1}{\sqrt {\sigma _\varepsilon ^2 + T\sigma _\alpha ^2}}\bar J = \frac {1}{\sigma _\varepsilon }\bigl [I - (1 - \tfrac {\sigma _\varepsilon }{\sqrt {\sigma _\varepsilon ^2 + T\sigma _\alpha ^2}})\bar J\bigr ] = \frac {1}{\sigma _\varepsilon }(I - \theta \bar J), \] and \((I - \theta \bar J)y_i\) has elements \(y_{it} - \theta \bar y_i\). By Aitken’s Theorem 9.4.2, least squares on the transformed data is GLS; the factor \(1/\sigma _\varepsilon \) does not affect the estimates. □
The parameter \(\theta \) moves between the two extremes. If \(\sigma _\alpha ^2 = 0\), \(\theta = 0\) and random effects is pooled least squares. As \(T\sigma _\alpha ^2/\sigma _\varepsilon ^2 \to \infty \), \(\theta \to 1\) and random effects approaches fixed effects. Random effects keeps the time-invariant regressors and uses the between-unit variation as well as the within, so it is more efficient than fixed effects when its assumption holds. That assumption is the catch: if \(\alpha _i\) is correlated with the regressors, random effects is inconsistent, for the same reason as pooled least squares, although its bias is smaller since it gives less weight to the between variation.
In practice the variance components are estimated. The within residuals give \(\hat \sigma _\varepsilon ^2\), as in Theorem 16.2.1. The between regression of \(\bar y_i\) on \(\bar x_i\) and \(z_i\) has disturbance \(\alpha _i + \bar \varepsilon _i\), with variance \(\sigma _\alpha ^2 + \sigma _\varepsilon ^2/T\), so its residual variance minus \(\hat \sigma _\varepsilon ^2/T\) estimates \(\sigma _\alpha ^2\) (the method of Swamy and Arora, 1972). If this is negative, it is set to zero.
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