provide a natural extension to imposing constraints on arbitrary PSEs, since these can be written in the form of ${\mathbb{E}}_x[m(X; \alpha)]$ for some $m(\cdot)$.

If outcomes are missing at random, the NDE is identified by ${\mathbb{E}}_x[ m(X;  \alpha)]$, where [ $$\begin{aligned}
        &m(X;  \alpha) = \sum_{M} \Big\{{\mathbb{E}}[Y | A = 1, M, X, R = 1;  \alpha_y] - \nonumber \\
        &\qquad {\mathbb{E}}[Y | A = 0, M, X, R = 1;  \alpha_y]\Big\} p(M | A = 0, X; \alpha_m).
    \end{aligned}$$ ]{}The resulting functional is then used in the profile empirical likelihood in (\[eq:hybrid\_fair\]).

Unlike the standard unconstrained prediction setting where it is common to use nonparametric methods to estimate an arbitrary regression function, our task requires a combination of prediction and estimation of the relevant causal parameter (constraint). Estimating the causal parameter requires estimating certain nuisance components (like $E[Y|Z]$ in eq. \[eq:est\_NDE\]) which we choose to do parametrically in part because we desire certain frequentist properties, namely fast rates of convergence. More fundamentally, the empirical likelihood optimization problem in (\[eq:hybrid\_fair\]) finds optimal parameter values $\alpha$, where $\alpha$ appears also in the constraint $\sum_i p_i m(X_i;p_i,\alpha)=0$. That is, the structure of the empirical likelihood optimization problem requires that $Y$ and $M$ models 