Generalized Least Squares (GLS) Estimator

This post deals with the generalized least squares (GLS) estimator. When deriving the Black-Litterman (BL) model, the Theil mixed estimator is used, which is a kind of GLS.

Generalized Least Squares (GLS)

Black-Litterman (BL) model is based on Theil mixed estimator which is a kind of the GLS estimator.

Since BL model combines two regressions (market + views) which have different variance terms respectively, the conbined BL regression model has two different variance terms naturally. Unlike OLS with a constant (homogenous) variance term, when variance terms are not constant (heterogeneous), it is natural to apply the GLS estimator to get estimated parameters.

The starting point is a linear model.

Linear model

OLS estimator

As we are familar, OLS estimator has the following form with a constant variance term across all observations.

Hance, OLS estimator is of the following formula.

GLS estimator

Our focus in on the GLS estimator for the linear regression model with non-constant variance terms (Σ = σ²Ω) across all observations, which means that residuals are heteroscedastic and/or serially dependent.

GLS estimator is of the following formula.

or

Derivation of GLS estimator

When Ω is symmetric, using eigen decomposition, Ω can be expressed as follows,

Here, A and Λ are eigenvector and eigenvalue matrix respectively.

Now Ω can be transformed into the identity matrix (I_n) by multiplying P⁻¹ in both sides

Multiplying P⁻¹ on both sides of y = βX + u  results in

Therefore, the linear regression model above is rewritten as

Least squares as the minimization problem is implemented as follows.

GLS estimator is

GLS estimator can be also expressed with the OLS estimator

Concluding Remarks

This post derived the GLS estimator which will be used when deriving the Black-Litterman model.

Originally posted on SH Fintech Modeling blog.