Generalised Least Squares

Generalised Least Squares (GLS)

The Generalised Least Squares (GLS) estimator

Introduction

We make the same assumptions as OLS.

\(\mathbf {y}=\mathbf {X}\theta+\boldsymbol {\epsilon }\)

We assume:

  • \(E[\epsilon |\boldsymbol X]=0\)

  • \(Cov [\epsilon |\boldsymbol X]=\boldsymbol \Omega \)

The GLS estimator

GLS estimator is:

\(\hat \theta_{GLS} = argmin_b (y-Xb)^T\Omega^{-1}(y-Xb)\)

\(\hat \theta_{GLS}=(X^T\Omega ^{-1}X)^{-1}X^T\Omega^{-1}y\)

This is the vector that minimises the Mahalanobis distance.

This is equivalent to doing OLS on a linearly transformed version of the data.

Identifying \(\Omega \)

If \(\Omega \) is known, we can proceed. Generally, however, \(\Omega \) is not known, and so the GLS estimate in infeasible.

Feasible Generalised Least Squares (FGLS)

The Feasible Generalised Least Squares (FGLS) estimator

Introduction

We do OLS to get a consistent estimate of \(\Omega \), \(\hat \Omega \).

We then plug this into the GLS estimator.

Heteroskedasticity

Weighted least squares