Deriving Quasi-Demeaning For Random Effects

Why Random-Effects GLS Transforms the Data

In a panel model with a random unit effect, observations from the same unit share a component of their error. Consequently, their errors are correlated, and ordinary least squares on the untransformed observations does not use the error covariance structure in the way generalized least squares (GLS) does. In the balanced random-effects model, GLS can be implemented by a particularly simple transformation: subtract a fraction of each unit’s time average from each observation. This is called quasi-demeaning.

The fraction is neither arbitrary nor generally equal to one. It is determined by the relative sizes of the unit-effect and idiosyncratic error variances and by the number of observations per unit. Deriving it from the covariance matrix explains both why the transformation is GLS and why it only partially removes unit means.

Model and Covariance Structure

Consider a balanced panel with \(N\) units and \(T\) observations for each unit. For unit \(i\) at time \(t\), let

Here \(y_{it}\) is the outcome, \(x_{it}\) is a vector of regressors, \(\beta\) is the slope vector, \(a_i\) is a time-constant random unit effect, and \(u_{it}\) is the idiosyncratic error. Write \(X_i\) for the \(T\)-row regressor matrix and \(y_i\) for the \(T\)-element outcome vector of unit \(i\).

The covariance matrix has \(\sigma_u^2+\sigma_a^2\) on its diagonal and \(\sigma_a^2\) off the diagonal. For distinct times \(t\) and \(s\), the shared \(a_i\) gives \(\operatorname{Cov}(a_i+u_{it},a_i+u_{is}\mid X_i)=\sigma_a^2\). Thus the within-unit observations are not independent, even though the idiosyncratic errors are uncorrelated.

Deriving the Quasi-Demeaning Parameter

Define the projection onto the unit-mean direction by \(P=\mathbf{1}\mathbf{1}'/T\). It maps a vector to a vector whose entries are all that vector’s time average. The complementary projection \(I_T-P\) removes the unit mean. These two directions separate the covariance structure:

To check this decomposition, note that \(P^2=P\), \((I_T-P)^2=I_T-P\), and \(P(I_T-P)=0\). Also, \(\mathbf{1}\mathbf{1}'=TP\). Along directions with zero time mean, the covariance has eigenvalue \(\sigma_u^2\); along the unit-mean direction, it has eigenvalue \(\sigma_u^2+T\sigma_a^2\). GLS whitens these directions by dividing by the square roots of their respective eigenvalues.

Factor out \(1/\sigma_u\), and define \(\theta\) so the remaining matrix has the form \(I_T-\theta P\). Matching its coefficient on \(P\) gives

This parameter lies between zero and one under the stated positive-variance assumptions. More importantly for interpreting the operation, the time average of the transformed vector is

Thus quasi-demeaning removes the fraction \(\theta\) of the unit mean and retains the fraction \(1-\theta\). It does not generally remove the mean completely. The common scalar \(1/\sigma_u\) in the whitening matrix can be omitted when estimating by least squares: multiplying every transformed observation by the same nonzero scalar multiplies the least-squares objective by a constant and leaves its minimizer unchanged.

From the Transformation to GLS

Let \(y_i^*=y_i-\theta\bar y_i\mathbf{1}\) and transform each column of \(X_i\) in the same way to obtain \(X_i^* = X_i-\theta\mathbf{1}\bar x_i'\), where \(\bar x_i=T^{-1}X_i'\mathbf{1}\). Since the whitening matrix is proportional to \(I_T-\theta P\), GLS on the original panel has the same coefficient estimate as least squares using the transformed outcomes and regressors. In particular, the unit effect is transformed along with the observations:

The transformed error is not obtained by simply discarding \(a_i\). Rather, the mean-direction part of the composite error has been rescaled to have the appropriate GLS weight. If the model includes a common intercept, its regressor is a column of ones. That column transforms to \((1-\theta)\mathbf{1}\), not to zero except in the limiting case of complete demeaning.

Applying Quasi-Demeaning to Panel Data

Checking the GLS Objective

A direct numerical check makes the GLS connection concrete. For a residual vector \(r_i=y_i-X_i\beta\), GLS weights its squared deviations by \(\Omega_i^{-1}\), giving \(r_i'\Omega_i^{-1}r_i\). The quasi-demeaning matrix must reproduce this same quadratic objective, up to the common scale factor \(1/\sigma_u^2\).

What the Transformation Does to Unit Means

Quasi-demeaning lies between two familiar treatments of unit means. When \(\theta=0\), no part of a unit mean is removed. As \(\theta\) increases, a larger fraction is subtracted. Complete demeaning would subtract the entire mean; quasi-demeaning instead retains the fraction \(1-\theta\) dictated by the covariance weighting. This distinction matters when a regressor is constant over time: its transformed values are all \((1-\theta)x_i\), so it is not automatically eliminated by the transformation.

The covariance derivation also explains the role of panel length. The variance in the mean direction is \(\sigma_u^2+T\sigma_a^2\), whereas the variance in a zero-mean direction is \(\sigma_u^2\). GLS weights these two directions differently. The formula for \(\theta\) is precisely the adjustment that makes the transformed mean direction receive that relative weight. The transformation does not itself establish whether the random-effects assumptions are appropriate; it follows from the specified covariance structure.

Common Pitfall