The Aitken estimator is another name for the generalized least squares estimator. It is used in linear regression when the error terms are not spherical, meaning they are heteroskedastic, correlated, or both.
The Basic Problem
Start with the linear model:
[ y = X\beta + \varepsilon, \qquad E[\varepsilon|X] = 0, \qquad Var(\varepsilon|X) = \Omega ]
If \Omega is proportional to the identity matrix, ordinary least squares is efficient. If not, OLS may remain unbiased under standard conditions, but it no longer uses the data in the most efficient way.
The Aitken Or GLS Formula
When \Omega is known, the Aitken estimator is:
[ \hat{\beta}_{GLS} = (X’\Omega^{-1}X)^{-1}X’\Omega^{-1}y ]
This reweights the regression to account for unequal variance and correlation across observations.
Why It Matters
The point of GLS is that some observations contain more usable information than others once error structure is taken into account. A well-specified GLS estimator can reduce estimator variance relative to OLS.
If \Omega is diagonal, GLS becomes weighted least squares. If \Omega must be estimated rather than known, the method becomes feasible GLS.