teffects nnmatch - Stata

10 teffects nnmatch — Nearest-neighbor matching
Propensity-score matching estimator
The propensity-score matching (PSM) estimator uses a treatment model (TM), p(z i , t, γ), to model
the conditional probability that observation i receives treatment t given covariates z i . The literature
calls p(z i , t, γ) a propensity score, and PSM matches on the estimated propensity scores.
When matching on the estimated propensity score, the set of nearest-neighbor indices for observation
i, i = 1, . . . , n, is
Ω p m(i) = {j 1 , j 2 , . . . , j mi | t jk = 1 − t i , |̂p i (t) − ̂p jk (t)| < |̂p i (t) − ̂p l (t)|, t l = 1 − t i , l ≠ j k }
where ̂p i (t) = p(z i , t, ̂γ). As was the case with the NNM estimator, m i is the smallest number such
that the number of elements in each set, m i = |Ω p m(i)| = ∑ j∈Ω p m(i) w j, is at least m, the desired
number of matches, set by the nneighbors(#) option.
We define the within-treatment matching set analogously,
Ψ p h (i) = {j 1, j 2 , . . . , j hi | t jk = t i , |̂p i (t) − ̂p jk (t)| < |̂p i (t) − ̂p l (t)|, t l = t i , l ≠ j k }
where h is the desired number of within-treatment matches, and h i = |Ψ p h
(i)|, for i = 1, . . . , n, may
vary depending on ties and the value of the caliper. The sets Ψ p h
(i) are required to compute standard
errors for ̂τ 1 and ̂δ 1 .
Once a matching set is computed for each observation, the potential-outcome mean, ATE, and ATET
computations are identical to those of NNM. The ATE and ATET standard errors, however, must be
adjusted because the TM parameters were estimated; see Abadie and Imbens (2012).
PSM, ATE, and ATET variance adjustment
The variances for ̂τ 1 and ̂δ 1 must be adjusted because we use ̂γ instead of γ. The adjusted variances
for ̂τ 1 and ̂δ 1 have the following forms, respectively:
̂σ 2 τ,adj = ̂σ 2 τ + ĉ ′ τ ̂V γ ĉ τ
̂σ δ,adj 2 = ̂σ δ 2 − ĉ ′ ̂V δ γ ĉ δ + ̂∂δ 1 ̂V
̂∂δ1
∂γ ′ γ
∂γ
In both equations, the matrix ̂V γ is the TM coefficient variance–covariance matrix.
The adjustment term for ATE can be expressed as
ĉ τ =
1
∑ n
i=1 w i
n∑
i=1
( ĉov
w i f(z ′ (zi , ŷ i1 )
îγ)
+ ĉov (z )
i, ŷ i0 )
̂p i (1) ̂p i (0)
where
f(z ′ îγ) = d p(z i, 1, ̂γ)
d(z ′ îγ)