Binary Models with Endogenous Explanatory Variables 1 ... - Cemfi

B Consistent standard errors for two-step estimators
Consider an estimator b θ that maximizes an objective function that depends on estimated parameters:
b θ =argmax
θ
NX
`i (θ, bγ) .
i=1
The estimated parameters themselves are obtained by solving
bγ =argmax
γ
NX
ζi (γ) .
i=1
Denote true values as (θ0, γ 0), and consider the following notation for score and Hessian terms:
` θ i (θ, γ) = ∂`i (θ, γ)
, ζ
∂θ
γ
i (γ) =∂ζ i (γ)
∂γ
Hθθ = − plim
N→∞
1
N
NX
i=1
∂ 2 `i (θ0, γ 0)
∂θ∂θ 0 , Hθγ = − plim
N→∞
Moreover, assume that the sample scores are asymptotically normal:
NX
Ã
1 `θ i (θ0, γ
√
0)
N ζ γ
i (γ ! " Ã
d Υθθ
→ N 0,
0)
Υγθ
Υθγ
Υγγ
!#
.
i=1
1
N
NX
i=1
∂ 2 `i (θ0, γ 0)
∂θ∂γ 0 , Hγγ = − plim
N→∞
Under standard regularity conditions, an expansion of the first-order conditions for b θ gives
0= 1
√ N
NX
i=1
` θ i
³ ´
bθ, bγ = 1
√
N
A similar expansion of the first-order conditions for bγ gives
0= 1
√ N
or equivalently
NX
i=1
ζ γ 1
i (bγ) = √
N
√ N (bγ − γ0) =H −1
γγ
1
√
N
Substituting (4) into (3) we get
and
√ ³ ´
Hθθ N bθ − θ0 = 1
√
N
√ ³ ´
N bθ − θ0
= H −1
θθ
NX
i=1
NX
i=1
NX
i=1
NX
i=1
1
N
NX
i=1
∂ 2 ζ i (γ 0)
∂γ∂γ 0 .
` θ √ ³ ´ √
i (θ0, γ0) − Hθθ N bθ − θ0 − Hθγ N (bγ − γ0)+op (1) . (3)
ζ γ
i (γ √
0) − Hγγ N (bγ − γ0)+op (1)
ζ γ
i (γ 0)+op (1) . (4)
` θ i (θ0, γ 0) − HθγH −1
γγ
¡
I,−HθγH −1¢
1
γγ √N
i=1
1
√
N
NX
i=1
ζ γ
i (γ 0)+op (1)
NX
Ã
`θ i (θ0, γ0) ζ γ
i (γ !
+ op (1) .
0)
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