Feasible Generalized Least Squares (FGLS)

 Feasible Generalized Least Squares (FGLS) is a two-step econometric method used to obtain efficient and unbiased estimates when the classical OLS (Ordinary Least Squares) assumptions are violated, especially when:

• Heteroskedasticity (non-constant variance of errors), or

• Autocorrelation (correlated error terms)
  is present in the regression model.

---

📌 Concept Overview:

In standard OLS, the Gauss-Markov theorem assures that OLS is BLUE (Best Linear Unbiased Estimator) only when the error terms are homoscedastic and uncorrelated. When this assumption fails:

• OLS is still unbiased, but no longer efficient.

• FGLS adjusts for the structure of the variance-covariance matrix of errors.

---

⚙️ How FGLS Works (Steps)

Step 1: Estimate OLS

Run OLS on your model:

$$y = X\beta + u$$

Obtain the residuals $\hat{u}$.

Step 2: Estimate the Variance-Covariance Matrix

Use $\hat{u}$ to estimate the structure of the error variance $\Omega$. For example:

• For heteroskedasticity, estimate a model for $\text{Var}(u_i)$.

• For autocorrelation, estimate parameters of the autocorrelation process (like AR(1)).

Step 3: Transform the Model

Apply a transformation to both $y$ and $X$ using the estimated $\hat{\Omega}^{-1/2}$ to get:

$$\hat{y}^* = \hat{X}^*\beta + \hat{u}^*$$

Step 4: Estimate via GLS

Run OLS on the transformed model:

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

---

✅ When to Use FGLS:

• Panel data models (cross-sectional time-series).

• Time-series regressions with autocorrelation.

• Cross-sectional models with known or suspected heteroskedasticity.

---

📊 Advantages:

• More efficient than OLS when non-spherical error terms exist.

• Can handle heteroskedasticity and autocorrelation simultaneously.

---

⚠️ Limitations:

• Requires correct specification of the error structure.

• If the estimate of $\Omega$ is poor, FGLS may perform worse than OLS.

• Not BLUE unless $\hat{\Omega} \to \Omega$ consistently.

---

📚 Example Use Case:

In panel data models (e.g., analyzing bank performance over 10 years across 20 banks), FGLS corrects for:

• Cross-sectional heteroskedasticity (different variance for each bank).

• Autocorrelation within panels (bank data over time).

---

🔍 Related Concepts:

Method	Handles	Efficiency	Notes
OLS	No hetero / auto	Low (if assumptions fail)	Simple and unbiased
GLS	Known error structure	High	Theoretical model
FGLS	Estimated error structure	Medium–High	Practical alternative