Generalized least squares (GLS) estimates regression coefficients using a stated structure for the errors — typically correlation over time, or unequal variances, or both. OLS treats every residual as if it had the same variance and no correlation with other residuals. When that simple error story is wrong, OLS point estimates may still be usable, but they can be inefficient, and ordinary SEs can be invalid.
Generalized least squares (GLS) estimates regression coefficients using a stated structure for the errors — typically correlation over time, or unequal variances, or both. OLS treats every residual as if it had the same variance and no correlation with other residuals. When that simple error story is wrong, OLS point estimates may still be usable, but they can be inefficient, and ordinary SEs can be invalid.
Monthly sales on price may leave residuals that stay high for several months in a row (positive autocorrelation). GLS (or a close practical relative) uses that covariance pattern when estimating the price slope, instead of pretending the leftovers are independent.
| OLS | GLS | |
|---|---|---|
| Error story | Equal variance; independent errors | Uses a covariance structure (correlation and/or unequal variance) |
| How coefficients are estimated | Minimize Σ ei2 with equal weights | Minimize a weighted residual criterion that respects the covariance |
| When it is the natural tool | Simple independent-error setting | Autocorrelated or otherwise structured errors, if the structure is reasonable |
Conceptually, GLS “re-weights and re-combines” the observations using the error covariance so that leftover dependence is not ignored. The covariance structure must be specified or estimated (for example a simple AR(1) leftover). A badly chosen structure can be worse than OLS. This unit does not require matrix derivations of GLS.