# 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.