Projected Instruments Offer a Conditional Route to Estimating Causal Effects

An auxiliary variable can help researchers distinguish cause from correlation—but only if its relationship with treatment and outcome meets demanding conditions. A preprint by Shuyuan Chen and Yifan Cui of Zhejiang University asks whether useful causal information can survive when those conventional conditions fail. Their projected instrumental variable framework offers an answer under alternative, explicitly stated assumptions.

Why imperfect instruments matter

Researchers estimating the effect of education, medical treatment or another intervention often face unmeasured confounding: factors that influence both who receives treatment and what outcome follows. Instrumental variables offer one way to address this problem using auxiliary variables associated with treatment.

Conventionally, an instrument must predict treatment, be independent of unmeasured confounders and affect the outcome only through treatment. These requirements can be difficult to defend. A candidate instrument might, for example, influence the outcome directly.

In their preprint, Shuyuan Chen and Yifan Cui propose projected instrumental variables, or PIVs. They show that average treatment effects can be identified under their conditions even when none of the candidate instruments meets classical validity requirements. This is an alternative set of assumptions, not a way to dispense with assumptions altogether.

Separating background structure from useful variation

The central idea is a mathematical projection: finding the best approximation to a function within a specified class of functions. The authors choose a space that represents allowable relationships between the candidate instruments, observed covariates and the mean outcome without treatment.

They then remove the component of a weighting function that lies within that space. The remaining component is orthogonal to the specified background structure—meaning its expected product with functions in that space is zero.

A toy example makes this concrete. Suppose an instrument has a direct linear relationship with the outcome, violating the exclusion restriction. Removing constant and linear components from a function of that instrument can eliminate their contribution to the identifying calculation. Treatment-related variation remaining outside those components may still identify the treatment effect.

That strategy requires sufficient remaining variation. In the single-instrument linear toy example, a binary instrument leaves no residual information after projection. At least three distinct instrument values are necessary for identification to be possible, although that alone is not sufficient.

What the framework still requires

The PIV conditions impose two substantive restrictions. First, the conditional mean of the untreated potential outcome must belong to the chosen function space. Second, treatment multiplied by the residualized weighting function must be conditionally uncorrelated with individual treatment effects, given observed covariates.

The framework also requires projected relevance: the instrument’s relationship with treatment must contain enough variation outside the chosen space. The authors formalize weighting conditions that keep the identifying denominator away from zero and control residual variation.

Choosing the space therefore creates a trade-off. A larger space accommodates more background relationships but can remove useful treatment information, increasing instability. A smaller space preserves more information but risks leaving background relationships that invalidate identification.

The authors also express identification through treatment and outcome bridge functions—functions solving moment equations—and connect this perspective to Riesz representation and inverse problems.

Learning effects and quantifying uncertainty

The preprint develops two debiased estimators: one with a prespecified weighting function and another that learns the conditional treatment probability from data. Cross-fitting trains supporting prediction models on separate observations from those used to evaluate the estimating equations.

The authors establish consistency and asymptotic normality under stated boundedness, relevance and estimation-error conditions. Convergence faster than the sample size raised to minus one-quarter for the supporting functions is a sufficient route to the required error rates.

Their variance-optimality result also has a boundary: under a generalized homoskedasticity condition, the learned treatment-probability weight minimizes asymptotic variance within the compared class of fixed-weight functionals. The paper explicitly distinguishes this from establishing the efficiency bound in the more restrictive causal model.

Evidence and practical boundaries

The authors report six simulation settings, each with 5,000 observations and 500 replications. Two PIV specifications produced small absolute biases and confidence-interval coverage between 93.0% and 96.4%. A more restrictive specification produced substantial bias, illustrating the consequences of overstating instrument validity. The simulation implementation omitted the inner cross-fitting step described in the nuisance-estimation procedure.

An application uses 5,661 observations from the 1984 and 1985 Australian Longitudinal Survey waves to study education beyond 12 years and hourly wages. Two PIV specifications yielded estimated increases of 2.75 and 2.84 wage units, with standard errors of 1.51 and 1.29. A third yielded 6.27; the authors interpret the discrepancy as suggesting bias from invalid instruments, rather than treating it as stronger evidence of a larger effect.

The framework also recovers an instrumented difference-in-differences functional as a special case. Its contribution is a shared mathematical language for several identification strategies, while retaining the need to justify background structure, treatment-effect restrictions and usable instrument strength.