Motivation
traditional different evaluation parameters are an average over parts of the distribution of impacts:
- The ATE averages over the entire distribution
- The ATT averages over the distribution of impacts for those who are somehow allocated into treatment
- LATE averages over the distribution of impacts for those who switch into treatment as a result of a reform or more precisely, as a result of a change of the value of some instrument a§ecting decisions to participate.
shortcomings:
- represent an aggregation over di§erent margins
- not comparable and they are di¢ cult to interpret from the perspective of general
Heckman and Vytlacil (2005) defined the MARGINAL TREATMENT EFFECT MTE unifying those treatment parameters
MTE is the effect of a treatment on the marginal individual entering treatment, focusing on the specified individual who is indifferent between entering the policy(treatment) or not. (in average measure)
Definition
Consider a discrete treatment T. The rule allocating to treatment may be written as:
For a particular value of ${Z_i}^{‘}\gamma$ the marginal individual is the one with
Now consider the e§ect of treatment for the ith individual (of course the ith individual can not have both $Y_i^1$ and $Y_i^0$ the discussion see the next subsection)
Then the marginal treatment effect can be defined by :
MTE has a very important property that all treatment parameters, such as ATE, ATT, AUT, can be written as weighted averages of MTE
Heckman defination
It is easy to understand if we can start from basic notation. For policy analysis, The generalized Roy Model is a basic choice-theoretic framework. Take the attending college for example.
- Let $Y_1$ be the potential return (log wage) if the individual were in the treatment group( were to attend college)
- Let $Y_0$ be the potential return (log wage) if the individual were in the control group( were not to attend college)
We have potential outcomes as :
where $\mu_{1}(X)=E(Y_{1}|X=x)$ and $\mu_{0}(X)=E(Y_{0}|X=x)$. The policy effect (return to schooling ) is $Y_{1} -Y_{0} = \beta = \mu_{1}(X)+U_{1} - \mu_{0}(X)+U_{0}$. Then average treatment effect conditional on $X=x$ is
and average effect of treatment (those who choose to attend college conditional on $X=x$) is
Next, we utlize the latent variable discrete choice model to represents the individual’s decision (such as attending school)
Causal Inference Problem
Suppose a policy $D$ (which is dichotomous in simplicity) affect the population group $U$.
- $D=1$ if a memebr is treated
- $D=0$ if a member is not treated
for ith member in U, we denote $Y_{i}^{1}$ as ith member potential outcome if treated ($d_i=1$). similiarly, $Y_{i}^{0}$ as ith member potential outcome if untreated ($d_i=0$)
$\beta_{i}$ represent hypothetical treatment effect for ith member. But the problem is that for a given individual i, we can only observe either $Y_{i}^{1}$ ($d_i=1$) or $Y_{i}^{0}$ ($d_i=0$), but not both.
如果个体i参加了项目($d_i=1$)，观察到$Y_{i}^{1}$，但看不到这个人的$Y_{i}^{0}$。除非可以把这个人送回“过去”，改写历史不让他参加项目($d_i=0$)，记录其$Y_{i}^{0}$。简单来说个体只能处于一种状态
regarding the fundamental problem, how can we estimate treatment effect? Holland describes two possible solutions: the “scientific solution” and the “statistical solution.”
- “scientific solution” : capitalizes on homogeneity in assuming that all members in U are the same, in either the treated state or the control state. So to estimate the effect we can just use two members (one in treatment group and the other one in untreatment group). this solution has little practical values
“statistical solution” : compute quantities of interest that reveal treatment effects only at the group level, which induce the ATE ATT ATUT method.
$ATE=E(Y_{i}^{1}-Y_{i}^{0})$
- $ATT=E(Y_{i}^{1}-Y_{i}^{0} | D=1)$
- $ATUT=E(Y_{i}^{1}-Y_{i}^{0} | D=0)$
Building MTE with more detailed justification
Here I borrow the case from Costas Meghir notes to illusatrate how to build MTE