Formula for weighted simple linear regression

This wiki page Simple linear regression has formulas to calculate α and β. Could anyone tell me how to derive the formulas in weighted case?

Answer

Think of ordinary least squares (OLS) as a “black box” to minimize

ni=1(yi(α1+βxi))2

for a data table whose ith row is the tuple (1,xi,yi).

When there are weights, necessarily positive, we can write them as w2i. By definition, weighted least squares minimizes

ni=1w2i(yi(α1+βxi))2

=ni=1(wiyi(αwi+βwixi))2.

But that’s exactly what the OLS black box is minimizing when given the data table consisting of the “weighted” tuples (wi,wixi,wiyi). So, applying the OLS formulas to these weighted tuples gives the formulas you seek.

Attribution
Source : Link , Question Author : Wei Shi , Answer Author : whuber

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