# Generalized least squares

In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model. The GLS is applied when the variances of the observations are unequal (heteroscedasticity), or when there is a certain degree of correlation between the observations. In these cases ordinary least squares can be statistically inefficient, or even give misleading inferences.

## Method outline

In a typical linear regression model we observe data $\{y_i,x_{ij}\}_{i=1..n,j=1..p}$ on n statistical units. The response values are placed in a vector Y = (y1, ..., yn)′, and the predictor values are placed in the design matrix X = [[xij]], where xij is the value of the jth predictor variable for the ith unit. The model assumes that the conditional mean of Y given X is a linear function of X, whereas the conditional variance of the error term given X is a known matrix Ω. This is usually written as

$Y = X\beta + \varepsilon, \qquad \mathrm{E}[\varepsilon|X]=0,\ \operatorname{Var}[\varepsilon|X]=\Omega.$

Here β is a vector of unknown “regression coefficients” that must be estimated from the data.

Suppose b is a candidate estimate for β. Then the residual vector for b will be Y − Xb. Generalized least squares method estimates β by minimizing the squared Mahalanobis length of this residual vector:

$\hat\beta = \underset{b}{\rm arg\,min}\,(Y-Xb)'\,\Omega^{-1}(Y-Xb),$

Since the objective is a quadratic form in b, the estimator has an explicit formula:

$\hat\beta = (X'\Omega^{-1}X)^{-1} X'\Omega^{-1}Y.$

### Properties

The GLS estimator is unbiased, consistent, efficient, and asymptotically normal:

$\sqrt{n}(\hat\beta - \beta)\ \xrightarrow{d}\ \mathcal{N}\!\left(0,\,(X'\,\Omega^{-1}X)^{-1}\right).$

GLS is equivalent to applying ordinary least squares to a linearly transformed version of the data. To see this, factor Ω = BB′, for instance using the Cholesky decomposition. Then if we multiply both sides of the equation Y = + ε by B−1, we get an equivalent linear model Y* = X*β + ε*, where Y* = B−1Y, X* = B−1X, and ε* = B−1ε. In this model Var[ε*] = B−1Ω(B−1)′ = I. Thus we can efficiently estimate β by applying OLS to the transformed data, which requires minimizing

$(Y^*-X^*b)'(Y^*-X^*b) = (Y-Xb)'\,\Omega^{-1}(Y-Xb).$

This has the effect of standardizing the scale of the errors and “de-correlating” them. Since OLS is applied to data with homoscedastic errors, the Gauss–Markov theorem applies, and therefore the GLS estimate is the best linear unbiased estimator for β.

## Weighted least squares[1]

A special case of GLS called weighted least squares (WLS) occurs when all the off-diagonal entries of Ω are 0. This situation arises when the variances of the observed values are unequal (i.e. heteroscedasticity is present), but where no correlations exist among the observed variances. The weight for unit i is proportional to the reciprocal of the variance of the response for unit i.

## Feasible generalized least squares

In practice, the method cannot be applied since the covariance of the errors is generally unknown. One strategy for building an implementable version of GLS is the Feasible Generalized Least Squares (FGLS) estimator, where we proceed in two stages: (1) in the first step the model is estimated by OLS or another consistent (but inefficient) estimator, and the residuals are used to build a consistent estimator of the errors covariance matrix (to do so, we often need to examine the model adding additional constraints, for example if the errors follow a time series process, we generally need some theoretical assumptions on this process to ensure that a consistent estimator is available); and (2) using the consistent estimator of the covariance matrix of the errors, we implement GLS ideas. In general this estimator has different properties than GLS. For large samples (i.e., asymptotically) all properties are (under appropriate conditions) common with respect to GLS, but for finite samples the properties of FGLS estimators are unknown: they vary dramatically with each particular model, and as a general rule their exact distributions cannot be derived analytically. For finite samples FGLS may be even less efficient than OLS in some cases. Thus, while GLS can be made feasible, it is not always wise to apply this method when the sample is small. A method sometimes used to improve the accurancy of the estimators in finite samples is to iterate, i.e. taking the residuals from FGLS to update the errors covariance estimator, and then updating the FGLS estimation, applying the same idea iteratively until the estimators vary less than some tolerance. But this method actually does not necessarily improve the efficiency of the estimator very much if the original sample was small. A reasonable option when samples are not too large is to apply OLS, but throwing away the classical variance estimator

$\sigma^2*(X'X)^{-1}$

(which is inconsistent in this framework) and using a HAC (Heteroskedasticity and Autocorrelation Consistent) estimator. For example, in autocorrelation context we can use the Bartlett estimator (often known as Newey-West estimator since these authors popularized the use of this estimator among econometricians in their 1987 Econometrica article), and in heteroskedastic context we can use the Eicker–White estimator ( Eicker–White ). This approach is much safer, and it is the appropriate path to take unless the sample is large, and "large" is sometimes a slippery issue (e.g. if the errors distribution is asymmetric the required sample would be much larger).

The ordinary least squares (OLS) estimator is calculated as usual by

$\widehat \beta_{OLS} = (X' X)^{-1} X' y$

and estimates of the residuals $\widehat{u}_j= (Y-Xb)_j$ are constructed.

For simplicity consider the model for heteroskedastic errors. Assume that the variance-covariance matrix $\Omega$ of the error vector is diagonal, or equivalently that errors from distinct observations are uncorrelated. Then each diagonal entry may be estimated by the fitted residuals $\widehat{u}_j$ so $\widehat{\Omega}_{OLS}$ may be constructed by

$\widehat{\Omega}_{OLS} = \operatorname{diag}(\widehat{\sigma}^2_1, \widehat{\sigma}^2_2, \dots , \widehat{\sigma}^2_n).$

It is important to notice that the squared residuals cannot be used in the previous expression, we need an estimator of the errors variances. To do so we can use a parametric heteroskedasticity model, or a nonparametric estimator. Once this step is fulfilled, we can proceed:

Estimate $\beta_{FGLS1}$ using $\widehat{\Omega}_{OLS}$ using weighted least squares

$\widehat \beta_{FGLS1} = (X'\widehat{\Omega}^{-1}_{OLS} X)^{-1} X' \widehat{\Omega}^{-1}_{OLS} y$

The procedure can be iterated, the first iteration is given by

$\widehat{u}_{FGLS1} = Y - X \widehat \beta_{FGLS1}$
$\widehat{\Omega}_{FGLS1} = \operatorname{diag}(\widehat{\sigma}^2_{FGLS1,1}, \widehat{\sigma}^2_{FGLS1,2}, \dots ,\widehat{\sigma}^2_{FGLS1,n})$
$\widehat \beta_{FGLS2} = (X'\widehat{\Omega}^{-1}_{FGLS1} X)^{-1} X' \widehat{\Omega}^{-1}_{FGLS1} y$

This estimation of $\widehat{\Omega}$ can be iterated to convergence.

Under regularity conditions any of the FGLS estimator (or that of any of its iterations, if we iterate a finite number of times) is asymptotically distributed as

$\sqrt{n}(\hat\beta_{FGLS} - \beta)\ \xrightarrow{d}\ \mathcal{N}\!\left(0,\,V\right).$

where n is the sample size and

$V = \text{p-lim}(X'\Omega^{-1}X/T)$

here p-lim means limit in probability