Weighted least squares (WLS) is least squares in which each observation has its own weight. It is the usual tool when errors are independent (or treated as independent) but do not share one common variance: some y values are measured more precisely than others. High-precision observations get a higher weight; noisy observations get a lower weight.
Weighted least squares (WLS) is least squares in which each observation has its own weight. It is the usual tool when errors are independent (or treated as independent) but do not share one common variance: some y values are measured more precisely than others. High-precision observations get a higher weight; noisy observations get a lower weight.
Annual regional demand might be more precisely recorded in later years than in early years with incomplete meters. WLS can down- weight the early, noisier years instead of treating every year as equally precise.
WLS minimizes Σ wi (yi − ŷi)2
If all weights equal 1, WLS reduces to OLS. Weights are often taken as inversely proportional to an estimated variance. WLS does not automatically fix every heteroscedasticity problem: the weights must be reasonable, and leftover autocorrelation is a different issue (then think GLS or a time- series error model, not only WLS).
| OLS | WLS | GLS | |
|---|---|---|---|
| Main error concern | Equal variance; independence | Unequal variances (weights) | General covariance, including correlation |
| Criterion | Σ ei2 | Σ wi ei2 | Uses the full error covariance idea |
| Relation | Equal-weight special case | A special case of GLS when errors are uncorrelated but variances differ | Broader; WLS is not the same as modelling lag-1 residual correlation |