Econometrics 2 / Econometrie 2 (Summary)

Econometrics 2 / Econometrie 2

Bachelor Econometrics & Operations Research



Heteroskedastic disturbances

𝑦𝑦 = 𝑋𝑋𝑋𝑋 + 𝜀𝜀

𝐸𝐸 [𝜀𝜀 ] = 0

𝜎𝜎12 0 ⋯ 0
𝐸𝐸 [𝜀𝜀𝜀𝜀 ′ ] = Ω = ⎛ 0 𝜎𝜎22 ⋯ 0⎞
⋮ ⋮ ⋱ ⋮
⎝0 0 ⋯ 𝜎𝜎𝑛𝑛2 ⎠

Violation of assumption 3 (homoskedasticity).


White standard errors

⋅ White standard errors are heteroskedasticity consistent.

𝑛𝑛

� (𝑏𝑏) =
var (𝑋𝑋 ′ 𝑋𝑋) −1
�� 𝜎𝜎𝑖𝑖2 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖′ � (𝑋𝑋′𝑋𝑋)−1
𝑖𝑖=1


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


Additive heteroskedasticity: ℎ(𝑧𝑧′𝛾𝛾) = 𝑧𝑧′𝛾𝛾
′
Multiplicative heteroskedasticity: ℎ(𝑧𝑧′𝛾𝛾) = 𝑒𝑒 𝑧𝑧 𝛾𝛾
⋅ Multiplicative heteroskedasticity model always gives positive variances.




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, Transformed linear model

′
𝑦𝑦𝑖𝑖∗ = 𝑥𝑥𝑖𝑖∗ 𝛽𝛽 + 𝜀𝜀𝑖𝑖∗

2
𝐸𝐸�𝜀𝜀𝑖𝑖∗ � = 𝜎𝜎 2

𝑦𝑦𝑖𝑖 1 𝜀𝜀𝑖𝑖
𝑦𝑦𝑖𝑖∗ = 𝑥𝑥𝑖𝑖∗ = 𝑥𝑥𝑖𝑖 𝜀𝜀𝑖𝑖∗ =
�𝑣𝑣𝑖𝑖 �𝑣𝑣𝑖𝑖 �𝑣𝑣𝑖𝑖


This follows because
𝜎𝜎𝑖𝑖2 = 𝜎𝜎 2 𝑣𝑣𝑖𝑖

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.

𝑛𝑛 −1 𝑛𝑛 𝑛𝑛 −1 𝑛𝑛
′ ′ 1 1
𝑏𝑏∗ = (𝑋𝑋∗ 𝑋𝑋∗ )−1 𝑋𝑋∗′ 𝑦𝑦∗ = �� 𝑥𝑥𝑖𝑖∗ 𝑥𝑥𝑖𝑖∗ � � 𝑥𝑥𝑖𝑖∗ 𝑦𝑦𝑖𝑖∗ = �� 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖′ � �� 𝑥𝑥 𝑦𝑦 �
𝑣𝑣𝑖𝑖 𝑣𝑣𝑖𝑖 𝑖𝑖 𝑖𝑖
𝑖𝑖=1 𝑖𝑖=1 𝑖𝑖=1 𝑖𝑖=1

𝑒𝑒∗ = 𝑦𝑦∗ − 𝑋𝑋∗ 𝑏𝑏∗


Weighted least squares criterion

Function to be minimalized:
𝑛𝑛 𝑛𝑛
′ 2 (𝑦𝑦𝑖𝑖 − 𝑥𝑥𝑖𝑖′ 𝛽𝛽)2
𝑆𝑆(𝛽𝛽) = ��𝑦𝑦𝑖𝑖∗ − 𝑥𝑥𝑖𝑖∗ 𝛽𝛽� =�
𝑣𝑣𝑖𝑖
𝑖𝑖=1 𝑖𝑖=1




Expectation and variance of weighted least squares estimators

𝐸𝐸(𝑏𝑏∗ ) = 𝛽𝛽
𝐸𝐸(𝑒𝑒∗ ) = 0
𝐸𝐸(𝑠𝑠∗2 ) = 𝜎𝜎 2


𝑛𝑛 −1
1
var(𝑏𝑏∗ ) = 𝜎𝜎 2 (𝑋𝑋∗′ 𝑋𝑋∗ )−1 = 𝜎𝜎 �� 𝑥𝑥𝑖𝑖 𝑥𝑥𝑖𝑖′ �
2
𝑣𝑣𝑖𝑖
𝑖𝑖=1




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