teffects nnmatch - Stata

teffects nnmatch — Nearest-neighbor matching 11
′
is the derivative of p(z i , 1, ̂γ) with respect to z îγ, and
⎧
∑
wj (z j − z Ψi )(y j − y Ψi )
⎪⎨
ĉov (z i , ŷ ti ) =
⎪⎩
is a p × 1 vector with
j∈Ψ h (i)
∑
wj − 1
j∈Ψ h (i)
∑
wj (z j − z Ωi )(y j − y Ωi )
j∈Ω h (i)
∑
wj − 1
j∈Ω h (i)
if t i = t
otherwise
z Ψi =
∑
wj z j
j∈Ψ h (i)
∑
wj
z Ωi =
∑
wj z j
j∈Ω h (i)
∑
wj
and y Ωi =
∑
wj y j
j∈Ω h (i)
∑
wj
j∈Ψ h (i)
j∈Ω h (i)
j∈Ω h (i)
Here we have used the notation Ψ h (i) = Ψ p h (i) and Ω h(i) = Ω p h
(i) to stress that the within-treatment
and opposite-treatment clusters used in computing ̂σ
τ,adj 2 and ̂δ τ,adj 2 are based on h instead of the
cluster Ω p m(i) based on m used to compute ̂τ 1 and ̂δ 1 , although you may desire to have h = m.
The adjustment term c δ for the ATET estimate has two components, c δ = c δ,1 + c δ,2 , defined as
c δ,1 =
c δ,2 =
1
∑ n
i=1 t iw i
1
∑ n
i=1 t iw i
n ∑
i=1
∑
n
i=1
w i z i f(z ′ îγ)
(ỹ 1i − ỹ 0i − ̂δ
)
1
{
w i f(z ′ îγ) ĉov (z i , ŷ 1i ) + ̂p }
i(1)
̂p i (0)ĉov (z i, ŷ 0i )
where
⎧ ∑
wj y j
j∈Ψ h (−i)
∑
wj
⎪⎨ j∈Ψ h (−i)
ỹ ti =
∑
wj y j
if t = t i
⎪⎩
j∈Ω
∑ h
wj
j∈Ω h
otherwise
and the within-treatment matching sets Ψ h (−i) = Ψ p h (−i) are similar to Ψp h
(i) but exclude observation
i:
Ψ p h (−i) = {j 1, j 2 , . . . , j hi | j k ≠ i, t jk = t i , |̂p i − ̂p jk | < |̂p i − ̂p l |, t l = t i , l ∉ {i, j k }}
Finally, we cover the computation of ∂γ ̂∂δ1 in the third term on the right-hand side of ̂σ 2 ′
δ,adj . Here
we require yet another clustering, but we match on the opposite treatment by using the covariates
z i = (z i,1 , . . . , z i,p ) ′ . We will denote these cluster sets as Ω z m(i), for i = 1, . . . , n.
The estimator of the p × 1 vector (∂δ 1 )/(∂γ ′ ) is computed as
̂∂δ 1 1
∂γ ′ = ∑ n
i t iw i
n ∑
i=1
z i f(z ′̂γ)
{(2t i − 1)(y i − y Ω
z
m
i) − ̂δ
}
1