# 微观经济计量学 Microeconometrics ECON60052T

which under our assumptions implies
$$h(x, z)=F_{v}^{-1}(P(x, z)) .$$
Following the analysis in Matzkin (1992), we can recover both $h(x, z)$ and $F_{v}()$ nonparametrically up to normalization.

Next, take the conditional (on $X, Z$ ) expectation of the outcome for the treated group
$$E(Y \mid X=x, Z=z, D=1)=g_{1}(x)+E\left(\varepsilon_{1} \mid X=x, Z=z, D=1\right)$$
We can write the last term as
$$E\left(\varepsilon_{1} \mid X=x, Z=z, D=1\right)=E\left(\varepsilon_{1} \mid v<h(x, z)\right)=E\left(\varepsilon_{1} \mid v<F_{v}^{-1}(P(x, z))\right)$$
That is, we can write it as a function of the known $h(x, z)$ or, equivalently, as a function of the probability of selection $P(x, z)$,
$$E(Y \mid X=x, Z=z, D=1)=g_{1}(x)+K_{1}(P(x, z))$$

## ECON60052T COURSE NOTES ：

$$R(\theta, \delta)=E_{\theta}[L(\theta, \delta(Z, U))]=\int_{0}^{1} \int L(\theta, \delta(z, u)) d P_{\theta}(z) d u .$$
A rule $\delta$ is admissible if there exists no other rule $\delta^{\prime}$ with
$$R\left(\theta, \delta^{\prime}\right) \leq R(\theta, \delta), \quad \forall \theta \in \Theta,$$
and
$$R\left(\theta, \delta^{\prime}\right)<(\theta, \delta) \text { for some } \theta \text {. }$$