Model Refutation Isn’t the End: Economists Push New Paths for Inference After Failed Specification Tests

When an econometric model fails a specification test, the immediate implication is clear: at least one identifying assumption is inconsistent with the observed data. What is not clear—and has long been a practical and philosophical problem in applied economics—is what a researcher should do next. A growing body of research argues that “model refutation” is not the end of inference, but a fork in the road: there may be multiple ways to relax assumptions to restore compatibility with the data, and those different “repairs” can point to different conclusions about the underlying structural parameters.

A new study introduces a framework aimed at making inference more honest and more stable in exactly this situation, proposing a method it calls Post-Relaxation Projection (PR) for partially identified models. The central claim is that once the original model is rejected, inference should not condition on a single preferred repair—because the choice of repair is itself uncertain and often under-justified. Instead, PR is designed to deliver uniform, post-relaxation inference that remains valid across a set of admissible repairs.

The core problem: many repairs, many answers

In standard practice, researchers who confront refutation typically “fix” the model by relaxing an assumption: perhaps allowing limited measurement error, weakening an exclusion restriction, adding some form of unobserved heterogeneity, or permitting small violations of monotonicity or rank invariance. Often, however, several relaxations are equally defensible given institutional knowledge and the evidence. The new study emphasizes that in this common situation, the true post-refutation model is not identified: the data indicate something must change, but do not uniquely determine which assumption is wrong or how much it should be relaxed.

This creates a subtle inferential hazard. Even if a researcher conducts exact inference under one repaired model, the resulting confidence set for a structural parameter can exclude values that would be supported under a different, equally plausible repair. In other words, the inference may be internally correct “conditional on the chosen repair,” yet misleading when viewed through the lens of the broader uncertainty introduced by refutation.

The paper’s contribution is to treat the repair as an additional unknown, and to design a procedure that carries coverage guarantees through that uncertainty.

Post-Relaxation Projection: one confidence event, many models

PR starts from a simple principle: after refutation, inference should account for the entire collection of relaxed models that meet a researcher-defined minimal standard of credibility. Rather than selecting one repair and proceeding as if it were correct, PR constructs a single simultaneous confidence event—an inferential statement designed to hold over a range of possible relaxations—and then projects that event onto the structural object of interest.

Technically, the setting is partial identification, where the data and maintained assumptions imply a set of feasible parameter values rather than a point. PR operates by (i) defining a family of “repairs” that restore data compatibility, (ii) building a confidence set that is valid uniformly over that family, and (iii) mapping that confidence information into a confidence region for the target parameter that reflects both sampling uncertainty and repair uncertainty.

The result is meant to be robust to the researcher’s inability—or unwillingness—to uniquely specify the correct post-refutation model. Instead of producing a narrow answer that depends heavily on a potentially arbitrary repair choice, PR yields inference that remains valid across all repairs in the admissible class.

A shift in the burden of argument

The study’s framing also shifts what researchers must defend. Under conventional workflows, the burden is to justify a particular repair and then report results under it. Under PR, the key empirical and substantive judgment becomes: what is the credible set of repairs? This is not a trivial change. It pushes applied work toward transparency about which assumption violations are considered plausible, and it discourages “repair shopping,” where inference is inadvertently driven by the modeling tweak that produces the cleanest answer.

Economists who study identification under weak assumptions have increasingly argued that such transparency is crucial, especially in policy settings where the consequences of overconfident inference can be large. The PR approach fits within that broader movement: accept that the model has been refuted, allow assumptions to relax, but ensure the resulting inference remains disciplined.

Implications for empirical practice

If adopted, PR could reshape how empirical papers present post-specification-test results. Instead of relegating failed tests to footnotes or shifting to an alternative specification without fully characterizing the implied uncertainty, researchers could report post-relaxation confidence regions that explicitly incorporate multiple plausible fixes. This is especially relevant in applications where identification hinges on contestable restrictions—instrument validity, selection assumptions, functional forms, or behavioral invariances—where small deviations can substantially alter conclusions.

At a broader level, the study treats model refutation not as an embarrassment to be patched over, but as information that should widen uncertainty in a principled way.

Why it matters

Modern empirical economics relies heavily on identifying assumptions that are inherently disputable. When data reject a model, continuing with inference after a single ad hoc fix can create a false sense of certainty. Methods like Post-Relaxation Projection aim to make post-refutation conclusions more credible by ensuring they remain valid across multiple plausible ways the model could be wrong.