What Can Regression Reveal When Key Variables Are Only Known Within Ranges?

Replacing an uncertain measurement with a single number can make a regression look more decisive than the data justify. This preprint asks what researchers can conclude while preserving that uncertainty—and shows how using all the model’s restrictions together can keep robust conclusions informative.

Keeping uncertainty in the analysis

Economic datasets often contain brackets rather than exact measurements. Surveys report ranges, administrative records cap values, and participants leave studies before outcomes are recorded. Replacing a bracket with its midpoint or discarding incomplete records makes regression easier, but those choices can change what the analysis supports.

In their November 2026 preprint, Sharp Identification for Regressions with Interval-Observed and Missing Covariates, Martin Fankhauser and Gil Peled develop methods for asking what incomplete observations identify without assumptions about why they are incomplete. Their framework covers conditional moment models, including familiar linear, probit and logit regressions.

The central requirement is containment: an unobserved value must lie within its recorded interval. An exactly measured value has an interval of zero width; a missing value can be represented by its full possible range. Missingness and interval widths may depend on the underlying values themselves. The approach does not require missingness at random.

Why “sharp” does not mean precise

Instead of necessarily producing one regression coefficient, the method targets an identified set: all parameter values compatible with the observed data and the maintained regression model.

Calling this set sharp means that it includes every compatible parameter value and excludes every incompatible one. It does not mean the set is narrow, nor does it remove the need for model assumptions.

The crucial distinction is joint compatibility. One candidate distribution for the unknown values must satisfy every model restriction simultaneously. Checking restrictions separately can admit a parameter because each check can be passed using a different completion of the data—even when no single completion passes them all.

The authors translate this joint requirement into a geometric problem. They collect the possible model moments associated with each observation’s interval into a random set, then use support functions—descriptions of how far a set extends in different directions—to characterize compatibility using observed data alone.

Under their stated assumptions, they establish consistency for an estimator of the identified set and asymptotically uniform coverage for confidence regions for individual compatible parameter values. That coverage result does not guarantee simultaneous coverage of the entire identified set.

Making an infinite problem computable

The theoretical characterization involves infinitely many restrictions. Practical implementations retain finite collections, producing outer approximations that can contain additional parameter values beyond the sharp set.

For scalar-moment models with a monotone index, including the regression examples above, the authors derive an exact linear-programming representation of the finite estimation criterion. This does not directly solve the variance-adjusted test statistic used for inference.

They propose two implementations. An adaptive version selects informative directions on one half of the sample and tests on the other, reverses the roles, and intersects the confidence regions. A fixed version uses prespecified directions and the full sample.

Their simulations illustrate both robustness and its cost. When observation depends on the outcome, conventional methods can become more concentrated around incorrect values as samples grow. In one such design with 5,000 observations, both proposed implementations achieved 95.6% coverage for nominal 90% regions, compared with 2.7% for midpoint-imputed ordinary least squares and zero for complete-case analysis. Under random censoring, however, valid complete-case analysis was generally more powerful in their point-identified design.

A graduation association becomes inconclusive

The application revisits research relating children’s discount rates to on-time high-school graduation. The rates were elicited through choices between earlier and later payments. Because the offered amounts changed in discrete steps, even consistent answers reveal an interval rather than an exact discount rate.

For 547 participants, conventional midpoint-based probit analysis without additional controls produced an estimated marginal effect of −0.086, with a 95% confidence interval of [−0.152, −0.020]. The adaptive interval-aware procedure instead produced [−0.226, 0.020], which includes zero. This leaves the association inconclusive under that analysis; it does not establish that the association is absent.

The reported approximately 27% widening compares interval-aware inference with midpoint treatment under the same inference procedure—not with the narrower conventional probit interval. Changing the inference procedure itself also widens the reported uncertainty.

Allowing inconsistent switching responses and missing graduation outcomes introduces further ambiguity. Nevertheless, the authors’ implementation remains substantially tighter than their interval-data extension of an earlier non-sharp procedure.

What the results still depend on

Robustness to the observation process is not robustness to every modeling choice. The theory retains containment, bounded-domain and regularity assumptions. Larger samples cannot necessarily eliminate ambiguity inherent in interval measurement.

The application’s additional-controls analysis also holds estimated control coefficients and propensity weights fixed, omitting first-stage uncertainty. The authors explicitly state that their formal coverage guarantee does not extend to those results. Their contribution is a benchmark for what incomplete data and a maintained model support, rather than a promise of assumption-free precision.

Source note: The arXiv abstract page’s citation metadata gives the title as “Sharp Identification for Regression with Interval-Observed and Missing Covariates” and the author as “Peled, Gil Jan.” The full paper uses “Sharp Identification for Regressions with Interval-Observed and Missing Covariates” and “Gil Peled.” The sources do not explain the difference.