In one sentence
The Aitken estimator is the generalized least squares (GLS) estimator, used when regression errors are heteroskedastic and/or correlated so that ordinary least squares is inefficient.
The setup
Consider the linear model:
\[
y = X\beta + \varepsilon, \qquad \mathbb{E}[\varepsilon\mid X]=0, \qquad \mathrm{Var}(\varepsilon\mid X)=\Omega
\]
OLS is unbiased under standard conditions, but when \(\Omega\) is not proportional to the identity matrix (the errors are “non-spherical”), OLS is not efficient.
The GLS (Aitken) estimator
If \(\Omega\) is known (or treated as known), GLS is:
\[
\hat{\beta}_{GLS} = (X’\Omega^{-1}X)^{-1}X’\Omega^{-1}y
\]
Intuition: GLS “whitens” the errors by reweighting observations and accounting for correlations, so more informative observations get more weight.
Relationship to OLS and WLS
If \(\Omega=\sigma^2 I\), then \(\Omega^{-1}\) is proportional to \(I\) and GLS reduces to OLS.
If \(\Omega\) is diagonal (heteroskedastic but uncorrelated errors), GLS reduces to weighted least squares (WLS).
If you can find a matrix \(P\) such that \(P’P = \Omega^{-1}\) (e.g., via a Cholesky factorization), GLS is equivalent to running OLS on the transformed model:
\[
Py = PX\beta + P\varepsilon
\]
In the transformed equation, the error term has spherical variance, so OLS becomes efficient.
Feasible GLS (what is done in practice)
Usually \(\Omega\) is unknown. Feasible GLS (FGLS) estimates \(\Omega\) (or its parameters) first and then plugs the estimate into the GLS formula. FGLS can improve efficiency, but inference needs care because \(\Omega\) is estimated.
In many applications, economists instead use OLS with robust/HAC standard errors if the goal is valid inference rather than maximum efficiency.
Why it matters
When errors are correlated over time (time series) or have changing variance (cross-sections), GLS-style methods can substantially change estimates and standard errors, affecting conclusions about policy effects and economic relationships.
- Ordinary Least Squares (OLS): A method for estimating unknown parameters in a linear regression model by minimizing the sum of squared differences between observed and predicted values.
- Heteroscedasticity: A condition in statistical models where the variance of errors is not constant.
- Generalized Least Squares (GLS): Estimation that accounts for heteroskedasticity and/or autocorrelation via a covariance matrix \(\Omega\).
- Feasible GLS (FGLS): GLS where \(\Omega\) is estimated from the data.
- HAC Standard Errors: Robust standard errors that remain valid under heteroskedasticity and autocorrelation (under conditions).
Quiz
### What is another name for the Aitken estimator?
- [x] Generalized Least Squares (GLS) Estimator
- [ ] Ordinary Least Squares (OLS) Estimator
- [ ] Maximum Likelihood Estimator
- [ ] Mean Squared Error Estimator
> **Explanation:** The Aitken estimator is commonly known as the Generalized Least Squares (GLS) estimator.
### Why is the Aitken estimator preferred over OLS in some situations?
- [x] It corrects for heteroscedasticity and autocorrelation.
- [ ] It is easier to compute.
- [ ] It always provides better predictions.
- [ ] It can only be used for linear models.
> **Explanation:** The Aitken estimator is preferred over OLS because it corrects for heteroscedasticity and autocorrelation.
### What does heteroscedasticity refer to?
- [ ] Residuals being perfectly correlated.
- [ ] Constant variance of residuals.
- [x] Variance of residuals changing across observations.
- [ ] Mean of residuals being zero.
> **Explanation:** Heteroscedasticity refers to the variance of residuals changing across observations.
### In which fields is the Aitken estimator commonly used?
- [ ] Only in medical research.
- [ ] Only in engineering.
- [x] Economics, finance, and social sciences.
- [ ] Only in pure mathematics.
> **Explanation:** The Aitken estimator is commonly used in economics, finance, and social sciences to address heteroscedasticity.
### What is autocorrelation?
- [x] Correlation of residuals with each other.
- [ ] No correlation at all.
- [ ] A perfect linear relationship between variables.
- [ ] Zero variance in residuals.
> **Explanation:** Autocorrelation refers to the correlation of residuals with each other in time series data.
### Who is the Aitken estimator named after?
- [ ] David Aitken
- [ ] Craig Aitken
- [x] Alexander Craig Aitken
- [ ] William Aitken
> **Explanation:** The Aitken estimator is named after Alexander Craig Aitken, a numerical analyst and statistician.
### Which method minimizes the sum of the squared differences between observed and predicted values?
- [x] Ordinary Least Squares (OLS)
- [ ] Generalized Least Squares (GLS)
- [ ] Maximum Likelihood
- [ ] Bayesian Estimation
> **Explanation:** The Ordinary Least Squares (OLS) method minimizes the sum of the squared differences between observed and predicted values.
### What does GLS stand for?
- [x] Generalized Least Squares
- [ ] General Linear Statistics
- [ ] Great Linear System
- [ ] Generated Least Squares
> **Explanation:** GLS stands for Generalized Least Squares.
### What kind of problems can the Aitken estimator address?
- [x] Heteroscedasticity and autocorrelation
- [ ] Only non-linear regression issues
- [ ] Perfect multicollinearity
- [ ] Homoscedasticity
> **Explanation:** The Aitken estimator addresses heteroscedasticity and autocorrelation problems.
### What does OLS assume about the variance of errors?
- [ ] The variance changes with observations.
- [x] The variance is constant across observations.
- [ ] There is no variance.
- [ ] The variance is correlated with residuals.
> **Explanation:** OLS assumes that the variance of errors is constant across observations (homoscedasticity).