Estimating economic bounds when sharper models become statistically fragile

Economic models do not always identify a single treatment effect, cost or other quantity. Instead, data and maintained assumptions may identify a range of possibilities. Its sharp bounds are the tightest endpoints those restrictions support. In “The Linear Programming Approach to Partially Identified Models,” UCLA economist Andrei Voronin proposes a method for estimating such endpoints. The work is an arXiv preprint, presenting theoretical results and simulation evidence rather than new empirical estimates.

Why tighter restrictions can cause trouble

The paper studies bounds calculated through linear programming: minimizing a linear objective subject to linear restrictions. Those restrictions describe combinations of unknown quantities compatible with the model. Their coefficients are often estimated from data.

Simply replacing population coefficients with sample estimates can fail. Sampling noise may make the estimated restrictions mutually incompatible, leaving no feasible solution. Even when a solution exists, its value need not converge to the true bound. This becomes particularly troublesome when restrictions nearly or exactly pin down some combination of parameters. A model that narrows the possibilities can therefore make conventional bound estimation less reliable.

Another approach expands the estimated feasible set to accommodate sampling error. That can restore consistency, but may push an estimated lower bound unnecessarily downward. Such conservativeness matters when researchers compare an economic benefit with an implementation cost.

Penalize first, then remove the penalty

Voronin’s estimator first permits violations of the estimated restrictions, charging a penalty for them. The penalty increases with sample size, but more slowly than its square root, balancing enforcement of the model against sensitivity to sampling noise.

The second step reports the original economic objective without the penalty charge. If several solutions minimize the penalized objective, it selects the largest original objective among them—the least conservative lower-bound estimate within that group. Computing the estimator requires two linear programs.

The reasoning exploits restrictions that bind at a selected solution. The paper shows that, with probability approaching one at a fixed underlying distribution, these restrictions select equalities that identify the population bound. Removing the penalty from the reported value removes the slower-converging penalty contribution.

Under the stated assumptions, the estimation error is of order one divided by the square root of sample size. The main formulation requires a nonempty population feasible set contained in a known compact region, coefficient estimates with the same statistical rate, and either a polyhedral region or population optima away from its boundary. An extension addresses an unknown bounding region while retaining boundedness as a regularity condition.

A guarantee with an important boundary

The paper distinguishes pointwise performance, which holds for each fixed data-generating distribution, from uniform performance across a class of distributions. It proves that no estimator can be uniformly consistent over the unrestricted class considered: nearly indistinguishable distributions can have substantially different bounds.

The proposed estimator nevertheless retains a one-sided safeguard. Under uniform coefficient-estimation and suitable penalty-rate conditions, overstatement of a sharp lower bound vanishes uniformly. Understatement can persist along difficult sequences.

The paper establishes full uniform consistency under an additional “delta-condition.” This is a sufficient condition, not a demonstrated necessary condition for every estimator or every restricted distribution class. It keeps every full-rank square system drawn from restrictions binding at a population vertex sufficiently well-conditioned. This rules out a form of weak identification analogous to weak instruments, while permitting point identification, redundant binding restrictions and multiple optimal solutions.

For uncertainty assessment, one procedure selects restrictions using half the sample and evaluates an orthogonal estimating function using the other half. Its exact asymptotic coverage is pointwise and requires additional assumptions, including asymptotic normality and nondegenerate variation. A full-sample alternative screens additional candidate systems and combines their lower confidence bounds. Its uniform coverage theorem assumes the delta-condition, uniform Gaussian approximation and covariance-estimation conditions, among other restrictions; coverage can be conservative.

What the simulations establish—and do not

The simulations are calibrated to Thomas Holmes’s 2011 study of Walmart store locations and distribution-related savings. Voronin treats 12 inequalities estimated from replication data as population restrictions, then generates Gaussian perturbations around them. Modified designs progressively flatten the feasible set or collapse it to a point. These are controlled experiments, not new estimates of Walmart’s actual costs.

At the row-collapse limit with 3,176 observations, the minimum-violation benchmark’s root-mean-squared error was 2.21 times that of the proposed estimator. Its upward bias in the lower bound was about 20% of the original identified interval’s length. Set expansion instead exhibited substantial downward bias.

Finite-sample confidence results were mixed: sample-splitting coverage fell to 91% in some designs targeting 95%, while full-sample coverage reached 100% near point identification. Computation was often much faster than test inversion, but full-sample search can grow exponentially around flat optimal faces. The calibration also omits first-stage demand-estimation uncertainty because the necessary data were unavailable. The evidence supports a promising conditional method, not a universal cure for unstable economic bounds.