Quasi-Demeaning in Random-Effects GLS

Why Random-Intercept GLS Uses Quasi-Demeaning

In panel data, observations from the same unit can share an unobserved component. A random intercept represents that shared component, so the resulting errors are correlated within units. Ordinary least squares on the untransformed observations does not account for this covariance pattern. Generalized least squares (GLS) does so by changing the scale of the common, between-unit direction relative to the within-unit directions. In this model, that operation has a particularly simple form: subtract a fraction of each unit’s time average from each observation.

The fraction is neither arbitrary nor generally equal to zero or one. It is determined by the variance of the random intercept, the variance of the idiosyncratic error, and the number of observations for that unit. Deriving it makes clear both why the transformation works and what its limiting cases mean.

Model and Assumptions

For unit \(i=1,\ldots,N\) and period \(t=1,\ldots,T_i\), consider the random-intercept model

Here \(y_{it}\) is the outcome, \(x_{it}\) is a vector of observed regressors, and \(\beta\) is the coefficient vector. The unit-specific random intercept \(a_i\) is shared across periods for unit \(i\); \(\epsilon_{it}\) is the period-specific error. The panel may be unbalanced, so the number of observed periods \(T_i\) can vary across units. Define unit averages by \(\bar y_i=T_i^{-1}\sum_{t=1}^{T_i}y_{it}\) and \(\bar x_i=T_i^{-1}\sum_{t=1}^{T_i}x_{it}\).

For the covariance structure and transformation below, assume \(E[a_i]=0\), \(\operatorname{Var}(a_i)=\sigma_a^2\), \(E[\epsilon_{it}]=0\), and \(\operatorname{Var}(\epsilon_{it})=\sigma_\epsilon^2\). Assume idiosyncratic errors are uncorrelated across periods and units, and are uncorrelated with the random intercepts. Random intercepts are uncorrelated across units. Thus, within a unit, the composite error \(u_{it}=a_i+\epsilon_{it}\) has covariance matrix

For unit \(i\)'s transformed errors to have zero conditional mean given \(X_i\), also require \(E[a_i\mid X_i]=0\) and \(E[\epsilon_{it}\mid X_i,a_i]=0\), where \(X_i\) collects the regressors for unit \(i\). These exogeneity conditions are distinct from the covariance assumptions: a correct covariance transformation does not by itself guarantee that the regressors are exogenous.

Deriving the Transformation Parameter

Let \(P_i=\mathbf{1}_{T_i}\mathbf{1}_{T_i}'/T_i\), the projection onto the direction that is constant across unit \(i\)’s observed periods. The covariance matrix has eigenvalue \(\sigma_\epsilon^2\) in every direction orthogonal to \(\mathbf{1}_{T_i}\), and eigenvalue \(\sigma_\epsilon^2+T_i\sigma_a^2\) in the constant direction. A transformation that leaves the within-unit directions unchanged and multiplies the constant direction by \(1-\theta_i\) is \(Q_i=I_{T_i}-\theta_iP_i\). Since \(P_i\) is a projection, \(Q_i\) acts as the identity on the within directions and as multiplication by \(1-\theta_i\) on the constant direction.

To make the transformed covariance equal to \(\sigma_\epsilon^2 I_{T_i}\), choose \(1-\theta_i\) so that the variance in the constant direction is reduced to \(\sigma_\epsilon^2\). The resulting equation and solution are

The positive square root is used so that \(1-\theta_i\) is positive. Equivalently, \(Q_i\operatorname{Var}(u_i)Q_i'=\sigma_\epsilon^2I_{T_i}\). Also, \(P_i y_i\) is the vector whose entries are all \(\bar y_i\), so \(Q_i y_i\) has entries \(y_{it}-\theta_i\bar y_i\). The identical operation on each regressor gives the stated quasi-demeaned regression. Because the transformed errors have the same scalar covariance within every unit, least squares on all transformed observations is GLS when the variance components are known.

Worked Transformations

What the Transformation Retains

The decomposition into a unit’s average and its deviations from that average clarifies the transformation’s meaning. Write an observation as \(y_{it}=(y_{it}-\bar y_i)+\bar y_i\). Then

The within-unit deviation is retained in full, while the unit mean is multiplied by \(1-\theta_i\). When \(\theta_i=0\), both within and between variation remain unchanged, as in pooled least squares. As \(\theta_i\) approaches one, the between-unit component is increasingly attenuated; at \(\theta_i=1\), only deviations from unit means remain. Quasi-demeaning is therefore not simply “subtracting the random intercept”: the intercept is unobserved, and the transformation instead rescales the common direction in the way required by the covariance matrix.

This distinction matters for interpretation and estimation. Intermediate quasi-demeaning uses both within- and between-unit variation, with their relative contribution shaped by the covariance parameters. It does not, by itself, resolve possible correlation between \(a_i\) and the regressors. If the random-intercept exogeneity condition fails, the transformation can still whiten the specified covariance matrix, but the resulting coefficient estimator need not have the desired conditional mean.

Common Mistakes

• Using a single parameter in an unbalanced panel. Since \(\theta_i\) depends on \(T_i\), units with different numbers of observed periods generally require different transformations.
• Transforming only the outcome. GLS requires applying the same operation to every regressor and to any constant column. Otherwise the transformed equation is not the original model’s GLS regression.
• Calling quasi-demeaning full demeaning. Unless \(\theta_i=1\), the transformed observation still contains \((1-\theta_i)\bar y_i\), so between-unit information has not been removed entirely.
• Rounding too early. A rounded \(\theta_i\) may not reproduce displayed products or transformed values. Keep full precision during multiplication and round the final results consistently.
• Confusing covariance correction with exogeneity. The transformation handles the stipulated within-unit covariance. It does not make a random intercept uncorrelated with regressors if that assumption is false.