Which statistical violations survive a model’s room to adjust?

Economic models do not always identify a single parameter value. Sometimes they identify a set of possibilities consistent with restrictions on population averages, such as equalities or inequalities. A specification test then asks whether that set is empty. A related test asks whether a proposed value for one component of the parameter is compatible with the model.

In the preprint Inference under partial identification with minimax test statistics, Isaac Loh of the University of North Carolina Wilmington studies a broad family of such tests. The contribution is mathematical distribution theory, supported by simulations—not an empirical finding about a particular economy.

The violations that cannot be adjusted away

The tests start by measuring how much estimated restrictions are violated at each candidate parameter value. They then minimize that measure over the parameter space. In a minimax representation, the inner maximization scores violations and the outer minimization chooses the parameter most consistent with the data.

Understanding the statistic’s sampling distribution requires accounting for both operations. A restriction that binds at a population-compatible parameter need not contribute fully to the final statistic: moving the parameter slightly may absorb its sampling noise.

Loh describes the surviving contributions through a mathematical object called a contact set. It retains directions that both attain the population violation score and cannot be reduced to first order by a permitted local parameter movement. For convex losses, these directions combine supporting slopes of the loss with a condition describing which changes the parameter can produce.

A familiar example is the generalized method of moments overidentification test. Under regular point identification, a full-rank derivative and efficient weighting, the paper’s geometry recovers the classical chi-squared limit for the squared statistic, with degrees of freedom equal to the number of moments minus the parameter dimension. Allowing parameters to adjust is what removes those degrees of freedom.

The general formula is an asymptotic upper bound, not automatically the exact distribution. Exactness follows under additional conditions: convexity of the indexed population criterion in the parameter, a sufficiently strong polynomial lower bound on how violation grows away from the identified set, or an appropriate continuity condition on the contact sets. The framework also accommodates function-valued moments in Banach spaces.

Re-solving rather than estimating the geometry

Although the theory uses derivatives and contact sets, the bootstrap need not estimate them. Instead, it adds a scaled random perturbation to the estimated criterion or moments and solves the minimum-violation problem again.

This re-optimization bootstrap is an existing procedure. Loh connects it to Hong and Li’s numerical delta method and, for finitely many studentized moments with a sum-of-squares criterion, to Bugni, Canay and Shi’s penalize-resampling statistic. The preprint extends the justification for these methods rather than introducing re-optimization itself.

Under the stated conditions, the bootstrap consistently estimates the exact limiting distribution. Its scale parameter must diverge more slowly than the statistic’s normalization rate. For the common square-root-of-sample-size normalization, that leaves an intermediate range between a constant and the square root of sample size.

A separate result concerns uniform validity: controlling erroneous rejection across a whole class of null configurations, including configurations that change with sample size. Under a suitable uniform lower-growth condition and the other assumptions, the bootstrap supplies uniformly valid critical values. The general guarantee adds an arbitrarily small positive slack to the bootstrap quantile; continuity conditions can permit dropping it.

What the simulations establish

The main simulations use a nonlinear, two-parameter Gaussian model with six moments, combining equalities and inequalities. Four null designs vary the geometry of the compatible parameter set. Each uses 2,500 replications at sample sizes of 500, 1,000, 5,000 and 10,000, with 599 bootstrap draws per replication.

At the reported intermediate tuning choice, re-optimization rejects true nulls between 2.5% and 5.6% of the time across those designs, against a nominal 5% level. The existing minimum-resampling test, Test MR, ranges from 2.9% to 5.1%. These results support comparable performance in this design, not a universal ranking. Supplementary simulations also show that tuning performance depends on the chosen loss.

The paper’s boundary result is equally important. When uniform separation fails—allowing nearly zero violations far from the identified set—it constructs a Gaussian family where pointwise exactness and uniform validity cannot generally coexist. Accurate calibration at each fixed configuration therefore does not, by itself, establish reliable inference across nearby configurations.