Table 3:

Weights for Constructing MPRTE(x) from MTE(x, u_D)

Parameters of Interest	Weight
MPRTE(x): P* = P + α	hMPRTE(x, u) = fP(Z)|X=x(u)
MPRTE(x): P* = (1 + α)P	$h_{\text{MPRTE}}(x,u)=\frac{u{f}_{P(Z)\mid X=x}(u)}{\mathbb{E}(P(Z)\mid X=x)}$
$\text{MPRTE}(x):Z_k^{*}=Z_k+\alpha$	$h_{\text{MPRTE}}(x,u)=\frac{f_{P(\text{Z})\mid X=x}(u)f_V\left[F_V^{-1}(u)\right]}{\mathbb{E}\left[f_V\left({\gamma }^{\prime }Z\right)\right]}$

Source: Carneiro, Heckman and Vytlacil (2011)