GLS / FGLS — Generalized Least Squares

GLS (Generalized Least Squares) generalizes OLS to handle heteroskedasticity and/or autocorrelation through the error covariance matrix $\Omega$. When $\Omega$ is unknown and must be estimated from data, we use FGLS (Feasible GLS).

When to use

Use GLS/FGLS when errors have a complex variance/correlation structure (e.g., AR(1) autocorrelation, grouping). WLS is the special case of GLS when $\Omega$ is diagonal.

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Model specification

GLS estimates:

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

where $\Omega = \mathrm{Var}(\varepsilon \mid X)$ is the error covariance matrix. If $\Omega = \sigma^2 I$, GLS reduces to OLS.

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GLS vs FGLS

	GLS	FGLS
$\Omega$	Known	Estimated from data
Property	Efficient (if $\Omega$ correct)	Asymptotically efficient (large sample)
In practice	Rarely know $\Omega$	More common

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Running in EcoLab

1. Modeling module → Classical linear regression family → GLS or FGLS.
2. Select $Y$, the $X$ variables, and the covariance structure (e.g., AR(1), grouping).
3. Run and read the Estimation and Diagnostics tabs; export the replication code.

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Replication code

Stata

* ---- GLS / FGLS ----
* Load panel data (illustrative)
use "panel_data.dta", clear
xtset id time

* FGLS with heteroskedastic panels and AR(1) correlation
xtgls y x1 x2, panels(hetero) corr(ar1)

R

# ---- GLS / FGLS ----
library(nlme)

# Load data (illustrative)
df <- read.csv("panel_data.csv")

# GLS with AR(1) correlation and heteroskedastic variance
model_gls <- gls(y ~ x1 + x2,
                 correlation = corAR1(),
                 weights = varPower(),
                 data = df)
summary(model_gls)

Python

# ---- GLS ----
import numpy as np
import statsmodels.api as sm

# Load data (illustrative)
import pandas as pd
df = pd.read_csv("panel_data.csv")

X = sm.add_constant(df[["x1", "x2"]])
y = df["y"]

# GLS with a known covariance matrix Omega
# (In practice, estimate Omega from OLS residuals for FGLS)
Omega = np.eye(len(y))  # placeholder — replace with estimated Omega
model_gls = sm.GLS(y, X, sigma=Omega).fit()
print(model_gls.summary())

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Limitations

• FGLS can be biased in small samples if $\Omega$ is poorly estimated.
• If you only need robust inference, OLS + robust/clustered standard errors is often simpler and safer.

Video tutorial

Video Tutorial: Running GLS in EcoLab

See also

• OLS · WLS · Model catalog