teffects nnmatch - Stata

teffects nnmatch — Nearest-neighbor matching 9
̂ξ 2 τ =
and an ATET estimate of ξ 2 δ as
̂ξ 2 δ =
1
2 ∑ n
i w i
1
2 ∑ n
i t iw i
⎡ ∑ ⎤
wj {y n∑
i − y j (1 − t i ) − ̂τ 1 } 2
j∈Ω(i)
w i
⎢
⎥
⎣
⎦
i=1
n ∑
i=1
∑
wj
j∈Ω(i)
⎡ ∑
tj w j {y i − y j (1 − t i ) − ̂δ
⎤
1 } 2
j∈Ω(i)
t i w i
⎢
⎣
∑ ⎥
tj w j
⎦
j∈Ω(i)
If the conditional outcome variance is dependent on the covariates or treatment, we require an
estimate for ξi
2 at each observation. In this case, we require a second matching procedure, where we
match on observations within the same treatment group.
Define the within-treatment matching set
Ψ x h(i) = {j 1 , j 2 , . . . , j hi | t jk = t i , ‖x i − x jk ‖ S < ‖x i − x l ‖ S , t l = t i , l ≠ j k }
where h is the desired set size. As before, the number of elements in each set, h i = |Ψ x h
(i)|, may
vary depending on ties and the value of the caliper. You set h using the vce(robust, nn(#)) option.
As before, we will use the abbreviation Ψ(i) = Ψ x h
(i) where convenient.
We estimate ξ 2 i
by
∑
wj (y j − y Ψi ) 2
̂ξ 2 t i
(x i ) =
j∈Ψ(i)
∑
wj − 1
where y Ψi =
j∈Ψ(i)
∑
wj y j
j∈Ψ(i)
∑
wj − 1
j∈Ψ(i)
Bias-corrected matching estimator
When matching on more than one continuous covariate, the matching estimator described above
is biased, even in infinitely large samples; in other words, it is not √ n-consistent; see Abadie and
Imbens (2006, 2011). Following Abadie and Imbens (2011) and Abadie et al. (2004), teffects
nnmatch makes an adjustment based on the regression functions µ t (˜x i ) = E(y t | ˜x = ˜x i ), for
t = 0, 1 and the set of covariates ˜x i = (˜x i,1 , . . . , ˜x i,q ). The bias-correction covariates may be the
same as the NNM covariates x i . We denote the least-squares estimates as ̂µ t (˜x i ) = ̂ν t + ̂β′ t˜x i, where
we regress {y i | t i = t} onto {˜x i | t i = t} with weights w i K m (i), for t = 0, 1.
Given the estimated regression functions, the bias-corrected predictions for the potential outcomes
are computed as ⎧
y i
if t i = t
⎪⎨ ∑
ŷ ti = wj {y j + ̂µ t (˜x i ) − ̂µ t (˜x j )}
j∈Ω
⎪⎩
x m (i) ∑
wj − 1
otherwise
j∈Ω x m (i)
The biasadj(varlist) option specifies the bias-adjustment covariates ˜x i .