A thread on BUE & BLUE🔥
Gauss-Markov condition:
1) y=Xβ+ε
2) E[ε|X]=0
3) Cov(ε|X)=σ^2Σ
Standard GM:
4) Σ=I
The GM thm shows that OLS/GLS is BL(inear)UE.
Hansen (’20) shows it holds for all unbiased est (inc. nonlinear) w/ an elegant proof (tilted density + Cramer-Rao)
1/n
Gauss-Markov condition:
1) y=Xβ+ε
2) E[ε|X]=0
3) Cov(ε|X)=σ^2Σ
Standard GM:
4) Σ=I
The GM thm shows that OLS/GLS is BL(inear)UE.
Hansen (’20) shows it holds for all unbiased est (inc. nonlinear) w/ an elegant proof (tilted density + Cramer-Rao)
1/n
https://twitter.com/CavaliereGiu/status/1492454733948362753
Call F_2 & F_2^0 the classes of dists satisfying 1-3 & 1-4.
Hansen proves if \hat{β} is unbiased under F_2 for all Σ, then GLS (OLS) is BUE under F_2 (F_2^0).
An intriguing Q. raised by @jmwooldridge & @CavaliereGiu is
Does there exist nonlinear unbiased est under F_2?
2/n
Hansen proves if \hat{β} is unbiased under F_2 for all Σ, then GLS (OLS) is BUE under F_2 (F_2^0).
An intriguing Q. raised by @jmwooldridge & @CavaliereGiu is
Does there exist nonlinear unbiased est under F_2?
2/n
Turns out no nonlinear est can be unbiased under F_2!
This can be proved using a deep result by Koopmann (’82) and restated in Gnot et al. (’92)
tandfonline.com/doi/abs/10.108…
Roughly speaking, an estimator that is unbiased under F_2 w/ a fixed Σ must be linear+quadratic.
3/n
This can be proved using a deep result by Koopmann (’82) and restated in Gnot et al. (’92)
tandfonline.com/doi/abs/10.108…
Roughly speaking, an estimator that is unbiased under F_2 w/ a fixed Σ must be linear+quadratic.
3/n
Specifically, an unbiased estimator under F_2 w/ a fixed Σ must be
Ay+(y’H_1y,…,y’H_py), tr(H_i Σ)=0, X’H_i X=0
Thus, if it is unbiased under F_2, tr(H_i Σ)=0 for any Σ. Since {Σ: Σ∈F_2} spans R^{n^2}, H_i must be 0.
Therefore, only linear est can be unbiased under F_2
4/n
Ay+(y’H_1y,…,y’H_py), tr(H_i Σ)=0, X’H_i X=0
Thus, if it is unbiased under F_2, tr(H_i Σ)=0 for any Σ. Since {Σ: Σ∈F_2} spans R^{n^2}, H_i must be 0.
Therefore, only linear est can be unbiased under F_2
4/n
This implies F_2 rules out all nonlinear unbiased est while Koopmann (’82) implies F_2^0 can’t.
A follow-up Q. is: what is the smallest class F that rules out all nonlinear unbiased est?
As above, F does if {Σ: Σ∈F} spans R^{n^2}. Two practically relevant special cases👇
5/n
A follow-up Q. is: what is the smallest class F that rules out all nonlinear unbiased est?
As above, F does if {Σ: Σ∈F} spans R^{n^2}. Two practically relevant special cases👇
5/n
Case 1 (slightly heteroscedastic class): F={(1)+(2)+(3) with Var(ε_i)∈[σ^2±0.01]}
Case 2 (slightly serially correlated): F={(1)+(2)+(3) with Var(ε_i)=σ^2 & Cov(ε_i, ε_j)∈[a,b]}
Then any estimator that is unbiased under F must be linear! And thus OLS/GLS are BUE!
6/n
Case 2 (slightly serially correlated): F={(1)+(2)+(3) with Var(ε_i)=σ^2 & Cov(ε_i, ε_j)∈[a,b]}
Then any estimator that is unbiased under F must be linear! And thus OLS/GLS are BUE!
6/n
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