MTE with Misspecification^††thanks: We thank Vitor Possebom, Yixiao Sun, Kaspar Wuthrich, and seminar participants at UC San Diego for helpful comments. All remaining errors are ours.

Marginal treatment effects (MTEs) have unified the identification theory of several policy parameters. While the MTE framework is essentially non-parametric,¹¹1Linearity is sometimes assumed to facilitate estimation. See, e.g., Appendix B in Heckman, Urzua, and Vytlacil (2006) it is required that the recipient’s participation into treatment follows a (generalized) Roy model. This is often referred to as additive separability: an “additive” comparison of costs and benefits determines selection. On the other hand, identification of the MTE is achieved via the local instrumental variable (LIV) approach (Heckman and Vytlacil (2001, 2005)). An excellent survey is provided by Mogstad and Torgovitsky (2018). An early effort to analyze MTE under misspecification can be found in the appendix of the seminal paper by Heckman and Vytlacil (2001). They consider a case where the additive separability in the selection equation does not hold. The most serious consequence is that the LIV approach does not identify the MTE curve.

In this paper we analyze a different type of misspecification. We model a situation in which, under additive separability, a proportion of the population does not take into account the instrumental variable when deciding whether to take up treatment or not. We refer to them as non-responders. To analyze the resulting bias, we define a pseudo-MTE curve which results from the LIV approach. Under no misspecification, the pseudo-MTE curve would coincide with the MTE curve. The resulting bias can be interpreted as a location-scale change of the MTE curve, parameterized by the proportion of non-responders and their propensity score.

We have two main results. The first one shows that the ability to recover the conditional average treatment effect (CATE) for the subpopulation of responders depends on the proportion of non-responders only through the support of the responders’ propensity score. Indeed, when the support of the propensity score is the unit interval, it is possible to identify the CATE without having to recover the true MTE curve in the first place. In a nutshell, ignoring misspecification and integrating under the pseudo-MTE curve over the support of observed propensity score yields the correct CATE for the subpopulation of responders.

While the previous identification result for the CATE is independent of the proportion of non-responders, this is not true of the MTE curve and other parameters derived from it such as LATE and MPRTE. However, in our second result, we show how to recover the MTE curve for responders by undoing the location-scale change induced by the presence of non-responders. The correction is based on an estimate of the support of the propensity score and requires only observable data. It gives an estimator of the policy parameter of interest that is simple to implement. Cases where the propensity score is fully supported are relevant in practice. For a recent example, see the survey approach of Briggs, Caplin, Leth-Petersen, Tonetti, and Violante (2020) the probability of having a child is supported on the full unit interval.

Recently, Acerenza, Ban, and Kedagni (2021) and Possebom (2021) focus on the effect of measurement error in treatment status on the MTE curve. We complement such results by noting that a simple change to our setup can cover the case of misclassification. In a setting where treatment status is misclassified, the observed outcome is generated with the true treatment status. In our setting of misclassification, the observed outcome can be regarded as a mixture of responders and non-responders. The proportion of non-responders is analogous to the proportion of misreporters. Indeed, our results also hold if instead of having a fraction of non-responders, we have a fraction of misreporters.

Another consequence of the presence of non-responders in the sample is that the effect of the instrumental variable on the propensity score is attenuated. Motivated by this, we model a situation where the proportion of non-responders approaches 1, analogous to the setting of weak instruments of Staiger and Stock (1997). Thus, we can derive weak-instrument-like asymptotic distributions for the parameters derived from the MTE curve.

The rest of the paper is organized as follows: section 2 introduces the model; section 3 contains the main identification results; section 4 provides bounds for the case where the propensity score is not fully supported in the unit interval; section 5 traces the connection to the weak IV literature; and section 6 concludes. While this paper only deals with identification, we expect to extend our results to cover estimation and inference.

In this section we introduce our model for misspecification in the MTE framework (Bjorklund and Moffitt (1987), Heckman and Vytlacil (2001, 2005)). We analyze the consequences of misspecification from the identification point of view.

We now introduce our two main results. We show that, for any subpopulation $X=x$ where the instrument is strong enough to induce a propensity score supported on the full unit interval $[0,1]$, the associated $CATE(x)$ can be identified for responders. This is true even if the $MTE^{*}(u,x,\delta_{x})$ curve is biased for $MTE(u,x)$. We note that the identified $CATE(x)$ parameters corresponds to the subpopulation of responders.

Assumption 3 says that the incentive in the instrument $Z$ is strong enough to induce any individual in the $X=x$ subpopulation into or out of treatment. Perhaps surprisingly, the $\text{CATE}(x)$, can be recovered only by resorting to the full support assumption. That is, to correctly compute the $\text{CATE}(x)$ we do not need to recover the true MTE curve for responders.

Unfortunately, the automatic “de-biasing" in Theorem 1 does not hold for the other policy parameters that can be obtained via the MTE curve. On the other hand, we show that the full support assumption can be used to identify $\delta_{x}$ which allows an explicit “de-biasing" procedure. Given that $\mathcal{P}^{*}_{x}:=[\underline{p_{x}^{*}},\overline{p_{x}^{*}}]=[\delta_{x}\tilde{P}(x),(1-\delta_{x})+\delta_{x}\tilde{P}(x)]$ we can actually identify both $\delta_{x}$ and $\tilde{P}(x)$. It follows then from Lemma 2 that we can recover the $\text{MTE}(u,x)$ curve.

The intuition for this result is simple. Because the original propensity score $P(Z,x)$, for any fixed $x$, is supported on the unit interval, the observed support $\mathcal{P}^{*}_{x}=[\underline{p_{x}^{*}},\overline{p_{x}^{*}}]$ will contain enough information to identify $\delta_{x}$. This is summarized Figure 1.

Having identified $\delta_{x}$, then we use Equation (8) to identify the MTE curve.

This corollary provides the correct “de-biasing” to be performed on the observed MTE curve to match the true MTE curve. However, it is possible to recover parameters that are based on the MTE curve without having to recover the MTE curve in the first place. We provide two examples.

In the previous examples, proceeding as if there were no misspecification, yields biased parameters. Thus, the automatic “de-biasing” in CATE is the exception rather than the rule.

Instead of assuming full support, now we allow for limited support of the propensity score $P(x,Z)$, but we still require that it is an interval.

In general, it is not possible to provide an upper bound for $\delta_{x}$. This is similar to the case of misclassification. Following that literature (see Assumption 4 in Acerenza, Ban, and Kedagni (2021), and references therein), we assume it is known that for some $\overline{\delta}_{x}$: $\delta_{x}\leq\overline{\delta}_{x}<1$. Thus, we can write $1-(\overline{p_{x}^{*}}-\underline{p_{x}^{*}})\leq\delta_{x}\leq\overline{\delta}_{x}.$ The correction factor in Examples 2 and 3 is $(1-\delta_{x})$. Now, it bounded by $1-\overline{\delta}_{x}\leq 1-\delta_{x}\leq\overline{p_{x}^{*}}-\underline{p_{x}^{*}}$. Thus, we can bound both LATE and MPRTE using this:

	$\displaystyle(1-\overline{\delta})\text{LATE}^{*}(x,P^{*}(x,z),P^{*}(x,z^{\prime}))$	$\displaystyle\leq\text{LATE}(x,P(x,z),P(x,z^{\prime}))$
		$\displaystyle\leq(\overline{p_{x}^{*}}-\underline{p_{x}^{*}})\text{LATE}^{*}(x,P^{*}(x,z),P^{*}(x,z^{\prime})),$

and

$\displaystyle(1-\overline{\delta})\text{MPRTE}^{*}(x)\leq\text{MPRTE}(x)\leq(\overline{p_{x}^{*}}-\underline{p_{x}^{*}})\text{MPRTE}^{*}(x).$

Naturally, if $\overline{\delta}_{x}$ is not known, we can only provide upper bounds.

Again, we stress that is not necessary to bound the MTE curve in the first place. Such a bound can be complicated to obtain since, by Lemma 2, $\delta_{x}$ enters in three different ways in the observed MTE curve.

We can frame our model as the triangular scheme of Staiger and Stock (1997) and consider a sequence $\left\{\delta_{x,n}\right\}_{n=1}^{\infty}$ such that $\lim_{n\to\infty}\delta_{x,n}=1$ at a certain rate as $n\to\infty$. Thus, as $n\to\infty$, the instrument becomes irrelevant in the model. A possible indicator of the presence of a large value of $\delta_{x,n}$ can be the average derivative of the observed propensity score. This equals an attenuated version of the average derivative of the true propensity score. For a given value of $\delta_{x,n}$, by equation (4), we have

$\displaystyle E\left[\frac{\partial P^{*}(x,Z)}{\partial z}\right]=(1-\delta_{x,n})E\left[\frac{\partial P(x,Z)}{\partial z}\right]$

Thus a “small” value can be an indication that $\delta_{x,n}$ is close to 1. This is similar to a first stage regression in the linear model. We take the derivative with respect to $z$ to get rid of the propensity score that does not respond to $Z$. We average, because this likely to be a non-linear expression. Thus, $(1-\delta_{x,n})$ can be thought of as the counterpart of $C/\sqrt{T}$ in the notation of Staiger and Stock (1997). Indeed, define

$\displaystyle Cov_{x}(Z,D^{*}):=E[ZD^{*}|X=x]-E[Z|X=x]E[D^{*}|X=x].$

We have

	$\displaystyle E[ZD^{*}|X=x]$	$\displaystyle=E[ZSD|X=x]+E[Z(1-S)\tilde{D}|X=x]$
		$\displaystyle=E[ZSD|X=x]+E[Z|X=x]E[(1-S)\tilde{D}|X=x]$

and

$\displaystyle E[D^{*}|X=x]=E[SD|X=x]+E[(1-S)\tilde{D}|X=x]$

Thus,

	$\displaystyle Cov_{x}(Z,D^{*})$	$\displaystyle=E[ZSD|X=x]-E[Z|X=x]E[SD|X=x]$
		$\displaystyle+E[Z|X=x]E[(1-S)\tilde{D}|X=x]-E[Z|X=x]E[(1-S)\tilde{D}|X=x]$
		$\displaystyle=Cov_{x}(Z,SD)$

which is the covariance between the instrument and treatment status for the responders with $X=x$. To see the role of the rate at which $\delta_{x,n}$ converges to 1, suppose for a second that we know the functional form of $P^{*}(x,Z)$, and we estimate the average derivative using a sample mean:

$\displaystyle\hat{E}\left[\frac{\partial P^{*}(x,Z)}{\partial z}\right]=\frac{1}{n}\sum_{i=1}^{n}\frac{\partial P^{*}(x,Z_{i})}{\partial z}=(1-\delta_{x,n})\frac{1}{n}\sum_{i=1}^{n}\frac{\partial P(x,Z_{i})}{\partial z}$

Then

$\displaystyle\hat{E}\left[\frac{\partial P^{*}(x,Z)}{\partial z}\right]-E\left[\frac{\partial P^{*}(x,Z)}{\partial z}\right]=(1-\delta_{x,n})\left(\frac{1}{n}\sum_{i=1}^{n}\frac{\partial P(x,Z_{i})}{\partial z}-E\left[\frac{\partial P(x,Z)}{\partial z}\right]\right)$