Manski (1997) and Manski and Pepper (2000) gave sharp bounds on causal effects under the assumptions of monotone treatment response (MTR) and monotone treatment selection (MTS). VanderWeele (2008) provided bounds for binary treatment under an assumption of monotone confounding (MC). We discuss the relation between MC and MTS and provide bounds under various combinations of these assumptions. We show that MC and MTS coincide for a binary treatment, but MC does not imply MTS for a treatment variable with more than two levels.
Keywords: bias; endogeneity; nonparametric bounds; partial identification.