Testing and relaxing the exclusion restriction in the control function approach
Introduction
The control function approach is a popular way to use instrumental variables (IV) in nonlinear models with endogeneity. An important reason is that, as Imbens and Newey (2009) demonstrated in a seminal paper, it applies to a large class of models, including nonseparable, nonparametric ones with possibly multi-dimensional unobservables. Like most common IV methods, the control function approach crucially relies on the exclusion restriction that the IV is not a part of the structural outcome equation. Exclusion restrictions, however, are frequently controversial. The aim of this paper is to show that, in certain circumstances, the conditions usually imposed in the control function approach make it possible to test this exclusion restriction as well as to relax it.
Specifically, suppose that the endogenous variable is continuous and an additional condition on the instrument , which we term “local irrelevance condition”, holds. In this context, we derive a testable implication for the validity of the exclusion restriction. Roughly speaking, the local irrelevance condition, which is itself testable, requires that for a subset of individuals, possibly of measure zero, a change in does not affect their . This condition does not contradict the usual relevance conditions on the IV if, basically, has an heterogeneous effect on .1 We develop a bootstrap test for our testable implication, and show its size control and consistency.
Second, we show that average causal effects of and on may be identified without exclusion restrictions. We do so in what are arguably the two most common specifications of the general nonseparable model analyzed in Imbens and Newey (2009): random coefficients models and single index models. Remarkably, in both cases, point identification of some causal effects of interest already holds with a discrete instrument. A richer support of then leads to identification of additional causal effects.
Though the arguments differ in important aspects from one model to another, the general identifying strategy is the same in both cases. In a first step, we exploit the local irrelevance condition to identify the direct effect of on . Intuitively, if the instrument is locally irrelevant, there is a subpopulation for which a change in will affect only directly, and not indirectly through the effect of on . In a second step, we consider other subpopulations for which has an effect on . For such subpopulations, the change in is due to the direct effect of but also to its indirect effect, i.e., through . Because we have already identified the direct effect of in step one, we can recover the indirect effect and thus, at the end, the causal effect of on . While the arguments used to establish identification in these models share this general strategy, there are also profound differences between models, entailing separate formal identification results across the models.
The control function approach has a long tradition in econometrics since, at least, the work of Heckman (1979) (see Wooldridge, 2015 for a recent survey). Historically, this approach has mainly relied on two sets of restrictions: (1) functional-form restrictions and (2) exclusion restrictions. As Imbens and Newey (2009) made clear, functional-form restrictions are actually superfluous for the control function approach to work. Exclusion restrictions, on the other hand, still remain essential in their framework. The same is true for the rest of the control function literature, with the exception of Klein and Vella (2010), which we discuss below. The models of Chesher, 2003, Vytlacil and Yildiz, 2007, Hoderlein and Sasaki, 2013, D’Haultfoeuille and Février, 2015 and Torgovitsky (2015), to cite but a few, all feature exclusion restrictions, and are concerned with other parts of the model specification.
However, the exclusion restriction is controversial. In their critique of natural experiments, Rosenzweig and Wolpin (2000) discuss several important examples where the instrument, while exogenous in our sense, may have a direct effect on the outcome variable. Also, van den Berg (2007) considers randomized controlled trials for which some time elapses between the moment the agents realizes she may be treated and the moment when the treatment takes place. In such common situations, the agent has an incentive to learn the value of the instrument. Then, van den Berg shows that the exclusion restriction can be violated if the interaction between the effort of the agent and the treatment affects the outcome variable. In the binary treatment case, Jones (2015) presents several examples of economic models where the exclusion restriction is likely to be violated, especially among inframarginal agents such as always- and never-takers. In the very typical analyses of returns to educations where college subsidy is used as an instrument, for example, the college subsidy can generate an income effect for always takers and thus may affect labor market outcomes (Jones, 2015 Section 3.1).2
Thus, our paper contributes to the literature by first showing that under some natural and testable restrictions on the distribution of , the exclusion restriction can be tested within the control function approach. We are not the first, however, to investigate testability without excluding instruments in models with endogenous variables. In the context of linear IV models, Bound and Jaeger (2000) and Altonji et al. (2005) suggest a test of the exclusion restriction based on a similar idea as ours. Specifically, if the first-stage effect is zero for subgroups defined by covariates, the effect of in the reduced form equation should be zero as well. The zero first-stage effect is close in spirit to our local irrelevance condition, but in our context subgroups are constructed with and only. Also, our test works beyond linear models — actually, it does not rely on any functional form restrictions on the structural equation, though it does impose conditions (monotonicity) on the first stage.
The related test of Kitagawa (2015) does not impose any functional forms either, and may also be seen as a test of the exclusion restriction if we maintain exogeneity and a monotonicity condition. The two procedures have also important differences. First and foremost, the test of Kitagawa (2015) applies to the framework of Imbens and Angrist (1994) with binary treatment and binary instrument, whereas we consider the case of continuous treatment, with either discrete or continuous instrument. Second, the monotonicity restrictions are different in the two settings. Caetano et al. (2016) also develop a nonparametric test of the exclusion restriction,3 but under a different condition on . Specifically, they assume that admits a mass point at the boundary of its support, whereas we leverage on the aforementioned local irrelevance condition.
Turning to identification, several papers have shown how to recover causal effects without exclusion restrictions in linear models. In particular, van Kippersluis and Rietveld (2018) show that if we assume homogeneous effects across subgroups and we have a zero first stage effect, we can identify both the effects of and . Their idea is related to our main identification idea for the random coefficients models. The main difference is that we allow for heterogeneous treatment effects in this model.4 Still in linear models, Rigobon (2003) and Lewbel (2012) show that second-order moment conditions have enough identifying power in systems of simultaneous equations, provided the model displays some heteroskedasticity. Klein and Vella (2010), who rely on a control function specification, exploit heteroskedasticity as well. However, their approach crucially hinges on the linearity of the structural and first-stage equations, whereas we establish identification in possibly nonlinear or nonparametric models. Our paper is also related to recent papers showing identification in nonseparable models when instruments has limited support. Newey and Stouli ( 2018, 2020) show that under restrictions on the structural functions, including random coefficients models similar to that considered below, identification can be achieved with a discrete instrument (see also Masten and Torgovitsky, 2016 for a similar result). While we consider less general effects than them, our approach does not require any exclusion restriction. Finally, our paper is related to Feng (2020) in that both exploit variations in covariates to identify causal effects. But they are quite different otherwise, as Feng (2020) considers a discrete and still relies on exclusion restrictions.
Section snippets
The general model
We introduce in this section the general class of triangular models that we discuss throughout this paper. The class of models is formally defined through the following system of equations: where, for simplicity, we assume that both and are scalar variables. We could allow both equations to depend in addition on a random vector of exogenous regressors denoted by . In line with the treatment effect literature, we omit this dependence, as the analysis can be done
Testable implications
In many applications, a candidate variable that may satisfy the exclusion restriction is available, but it is uncertain whether it satisfies it or not. In this section, we establish that the exclusion restriction is actually testable under the local irrelevance condition below.
Assumption 4 Locally Irrelevant Instrument There exists such that and .
This assumption may hold even if is binary. Also, since it only involves observed variables, the condition is testable.
Identification without exclusion restriction
Theorem 1 suggests that, under Assumptions 1–4, the exclusion restriction may not be necessary for the identification of causal effects. We provide results in this direction in this section, under other restrictions on . Such results may be useful in particular if we reject the previous test, but still Assumptions 1–3 appear credible.7
Conclusion
In this paper, we first show that under the control function approach, we can test the exclusion restriction in nonseparable triangular models as soon as a local irrelevance condition on the instrument holds, and we devise and analyze such a test. We also show that causal effects can be identified without exclusion restrictions in the linear random coefficients models or under index restrictions. Identification of some effects can be achieved even if the instrument is binary, but other effects
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