Set Identification of Causal Effects When Point Identification Fails in the Common-Confounding Instrumental-Variables Framework of Tien (2023)

In the instrumental-variables (IV) framework of Tien (2023), instruments are assumed exogenous only conditional on unobserved common confounders, which are measured by proxy variables that may themselves directly affect both treatment and outcome. The paper proves point identification of average causal effects under either a linearly separable outcome or strict first-stage monotonicity with common support, and states that when these conditions fail the approach can still identify informative bounds on causal effects — but explicitly leaves this set-identification exercise to future work. This statement formulates that open question: characterize the partially identified set, and derive informative (ideally sharp) bounds, for the average causal effect $\theta_0=\int Y(a)\pi(a)\,d\mu_A(a)$ from the observable joint distribution of $(Y,A,Z,W)$ under the framework's maintained assumptions, or prove that only trivial bounds are available. The problem is open.

Background

Instrumental-variables (IV) methods identify causal effects in the presence of unobserved confounding by exploiting a variable $Z$ — an instrument, such as proximity to a college or pre-college test scores in the returns-to-education literature — that is relevant for the treatment $A$ and exogenous, meaning statistically independent of the unobserved determinants of the outcome $Y$ and with no direct effect on $Y$ (the exclusion restriction). Exogeneity is the assumption practitioners find hardest to defend: in the returns-to-education problem, pre-college test scores are plausibly excluded with respect to selection into work but are endogenous to unobserved ability.

Two existing literatures relax parts of this. Proximal causal inference (Miao, Geng, and Tchetgen Tchetgen 2018) assumes the analyst observes proxy variables for the unobserved confounders and identifies causal effects through bridge functions — functions of the proxies and treatment that solve integral equations of the observed distribution and substitute for the unobserved confounders — requiring a completeness condition — a rank-type condition stating that the proxies are sufficiently informative about the confounders (formally, $E[g(U)\mid W]=0$ only when $g(U)=0$); however, in that framework the proxies have no direct effect on treatment or outcome. A separate literature bounds causal effects when the exclusion restriction itself is violated through a direct instrument effect (Conley, Hansen, and Rossi 2012).

Tien (2023) — building on the same author's "Instrumented Common Confounding" (2022) — combines these ideas in a new model class, best understood as IV with mismeasured confounders. There are instruments $Z$ whose exogeneity holds only conditional on unobserved common confounders $U$ (formally, $Y(a)\perp Z \mid U$), and proxies $W$ that are relevant for $U$ (complete) but, unlike in proximal causal inference, are allowed to be economically meaningful variables with their own direct effects on both $A$ and $Y$ — for instance risky-behavior indicators proxying ability. Identification rests on an index-sufficiency assumption: there exists a control function $\tau$ with $T=\tau(Z)$ such that $U\perp Z \mid T$, where $\tau$ can be learned from the observable dependence between $Z$ and $W$, together with completeness of $Z$ for $A$ given $T$ (after netting out the $T$-variation of $Z$, enough instrument variation remains to move $A$). A control function is a variable that absorbs the endogenous part of the variation in a regressor so that, conditional on it, the remaining variation is exogenous.

Under these assumptions the paper establishes two point-identification results for average causal quantities $\theta_0=\int Y(a)\pi(a)\,d\mu_A(a)$, where $\pi$ is a contrast function (for example $\pi$ concentrated on two treatment values gives an average treatment effect). Theorem 1 handles outcomes linearly separable in treatment and disturbance. Theorems 2–3 handle nonseparable outcomes via strict monotonicity of the first stage $A=h(Z,\eta)$ in a scalar disturbance $\eta$, in the spirit of Imbens and Newey (2009), using the control functions $(V_T,T)$ with $V_T=F_{A\mid Z;T}(A,Z)$ plus a common-support condition (the joint support of the control functions must equal their support conditional on any treatment value carrying nonzero contrast weight); the identified quantities integrate out the unobservables. For the linear case the paper derives doubly robust (valid if either the outcome regression or the nuisance function is correctly specified) and Neyman-orthogonal (insensitive to small estimation errors in the nuisances) moment conditions permitting $\sqrt{n}$ estimation despite nuisances defined by Fredholm integral equations of the first kind (ill-posed inverse problems), where nuisances are functions solving integral equations of the observed joint distribution and are identified only up to a non-unique class.

Partial (set) identification asks what can be learned when point identification fails: one characterizes the set of parameter values compatible with everything observable — here the joint law of $(Y,A,Z,W)$ — under the maintained assumptions, and derives bounds on the target. The paper's conclusion asserts that "when point identification is impossible, this approach can still identify informative bounds on causal effects" but leaves this exercise entirely to future work. Related relaxation questions have adjacent answers elsewhere: D’Haultfœuille, Hoderlein, and Sasaki (2024) test and relax the exclusion restriction within the control-function approach, and Ghassami, Zhang, Shpitser, and Tchetgen Tchetgen (2023) derive partial-identification bounds in the standard proximal framework without completeness.

Problem Statement

In the common-confounding IV model of Tien (2023) — with treatment $A$, outcome $Y$, instruments $Z$ satisfying conditional exogeneity $Y(a)\perp Z\mid U$ with respect to unobserved common confounders $U$, proxies $W$ with $W\perp Z\mid U$ that are complete for $U$ (that is, $E[g(U)\mid W]=0$ only when $g(U)=0$), an index-sufficiency control function $T=\tau(Z)$ with $U\perp Z\mid T$, and completeness of $Z$ for $A$ given $T$ (after netting out the $T$-variation of $Z$, enough instrument variation remains to move $A$; formally, $E[g(A,T)\mid Z]=0$ only when $g(A,T)=0$) — point identification of the average causal quantity $\theta_0=\int_{\mathcal A} Y(a)\pi(a)\,d\mu_A(a)$ is established only under linear separability of the outcome in treatment and disturbance, or under strict monotonicity of the first stage (treatment equation) $A=h(Z,\eta)$ in a scalar disturbance $\eta$ together with a common-support condition (the support of the control functions equals their support conditional on any treatment value with nonzero contrast weight). The open problem is the set-identification exercise the source asserts but does not perform: when neither of these point-identification routes is available, characterize the identified set of values of $\theta_0$ compatible with the observable joint distribution of $(Y,A,Z,W)$ under the maintained assumptions, and derive valid, informative (ideally sharp) bounds on $\theta_0$ as functionals of that observable distribution. The maintained assumptions are those listed above; validity requires the bounds to hold for every data-generating process in the resulting model class; sharpness of the bounds is required; and informativeness requires them to be strictly tighter than the trivial range implied by the support of $Y$ alone, under explicit conditions. Demonstrating instead that only trivial bounds are available without additional assumptions also resolves the question, by correcting the source's claim that the approach 'can still identify informative bounds'.

References

1. Tien, Christian (2023). "Relaxing Instrument Exogeneity with Common Confounders." arXiv:2301.02052, DOI 10.48550/arXiv.2301.02052, https://arxiv.org/abs/2301.02052
2. Tien, Christian (2022). "Instrumented Common Confounding." arXiv:2206.12919, DOI 10.48550/arXiv.2206.12919, https://arxiv.org/abs/2206.12919
3. Ghassami, AmirEmad; Zhang, Yunshu; Shpitser, Ilya; Tchetgen Tchetgen, Eric (2023). "Partial Identification of Causal Effects Using Proxy Variables." arXiv:2304.04374, DOI 10.48550/arXiv.2304.04374, https://arxiv.org/abs/2304.04374
4. Sverdrup, Erik; Cui, Yifan (2023). "Proximal Causal Learning of Conditional Average Treatment Effects." arXiv:2301.10913, DOI 10.48550/arXiv.2301.10913, https://arxiv.org/abs/2301.10913
5. Zhang, Zhiheng (2022). "Partial Identification with Proxy of Latent Confoundings via Sum-of-ratios Fractional Programming." arXiv:2210.09885, DOI 10.48550/arXiv.2210.09885, https://arxiv.org/abs/2210.09885
6. Miao, Wang; Geng, Zhi; Tchetgen Tchetgen, Eric J. (2018). "Identifying causal effects with proxy variables of an unmeasured confounder." Biometrika 105(4):987–993, DOI 10.1093/biomet/asy038, https://doi.org/10.1093/biomet/asy038
7. Imbens, Guido W.; Newey, Whitney K. (2009). "Identification and estimation of triangular simultaneous equations models without additivity." Econometrica 77(5):1481–1512, DOI 10.3982/ecta7108, https://doi.org/10.3982/ecta7108
8. D’Haultfœuille, Xavier; Hoderlein, Stefan; Sasaki, Yuya (2024). "Testing and relaxing the exclusion restriction in the control function approach." Journal of Econometrics 240(2):105075, DOI 10.1016/j.jeconom.2020.09.012, https://doi.org/10.1016/j.jeconom.2020.09.012
9. Conley, Timothy G.; Hansen, Christian B.; Rossi, Peter E. (2012). "Plausibly Exogenous." Review of Economics and Statistics 94(1), DOI 10.1162/rest_a_00139, https://doi.org/10.1162/rest_a_00139