. Author manuscript; available in PMC: 2011 Jul 6.

Published in final edited form as: J Econ Lit. 2010 Jun 1;48(2):356–398. doi: 10.1257/jel.48.2.356

Table 3.

MTE Weights For Different Treatment Parameter and IVs. (F_p is the distribution of P. f_P is its density.)

Treatment Parameter Weights	$h^{A T E} (u_{D}) = 1$
	$h^{T T} (u_{D}) = [\int_{u_{D}}^{1} f_{P} (p) d p] \frac{1}{E (P)}$
	$h^{T T} (u_{D}) = [\int_{0}^{u_{D}} f_{P} (p) d p] \frac{1}{E (1 - P)}$
	$h^{P R T E} (u_{D}) = F_{P}^{b} (u_{D}) - F_{P}^{a} (u_{D})$

IV Weights	$h^{I V} (u_{D}) = [\int_{u_{D}}^{1} (p - E (P)) f_{P} (p) d p] \frac{1}{V a r (P)}$ for P(Z) as an instrument
	$h^{I V} (u_{D}) = \frac{\int_{u_{D}}^{1} \int_{- \infty}^{\infty} (j - E (J)) f_{J, P} (j, p) d j d p}{Cov (J, P)}$ for a general instrument J(Z)^*, a function of Z

f_J,P(j, p) is the joint density of J and P. For derivations of these weights, see Heckman and Vytlacil (1999, 2005, 2007b).