An estimator of 𝜎𝜎𝑖𝑖2 is given by 𝑒𝑒𝑖𝑖2 , the square of the OLS residual 𝑒𝑒𝑖𝑖 = 𝑦𝑦𝑖𝑖 − 𝑥𝑥𝑖𝑖′ 𝑏𝑏.
The square roots of the diagonal elements are the White standard errors.
Model for variance
𝜎𝜎𝑖𝑖2 = ℎ(𝑧𝑧𝑖𝑖′ 𝛾𝛾)
⋅ ℎ: a known function
⋅ 𝑧𝑧 = �1, 𝑧𝑧1 , … , 𝑧𝑧𝑝𝑝 �′, a vector of 𝑝𝑝 observed variables that influence the variances
⋅ 𝛾𝛾: a vector of 𝑝𝑝 unknown parameters
where 𝑣𝑣𝑖𝑖 > 0 is known and 𝜎𝜎 2 is an unknown scalar parameter.
Weighted least squares (WLS)
The transformed model satisfies assumptions 1 to 6.
Therefore, the best linear unbiased estimator (BLUE) of 𝛽𝛽 is obtained by applying least squares in the
transformed model.
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