Joachim Freyberger, Julius Kappenberg
arXiv 26 Sep 2026 · Econometrics
arXiv:2609.32725 · PDF · Extracted main text
Economic theory frequently implies linear inequality restrictions on parameters or functions of interest. A common way to impose such restrictions is to project an unrestricted estimator onto the feasible set. Projection estimators arise naturally from constrained least squares, instrumental variables, generalized method of moments, maximum likelihood, and related extremum procedures. When the sampling covariance, loss function, and projection criterion induce different geometries, the choice of projection geometry can substantially affect risk. We study this choice in a fixed-dimensional local Gaussian experiment under quadratic loss. At exact-boundary configurations where only one maintained inequality binds, inverse-covariance projection is pointwise optimal. When at most two inequalities are locally relevant, it weakly improves on the unrestricted estimator throughout the corresponding local experiment and is minimax over exact-boundary configurations. For an arbitrary number of inequalities, we provide a sufficient condition for boundary minimaxity, but show by counterexample that inverse-covariance projection need not be boundary minimax once three inequalities can bind. Motivated by these results, we propose selecting the projection geometry to minimize worst-case boundary risk subject to a local no-harm condition relative to unrestricted estimation. We develop a feasible implementation and study its finite-sample performance in simulations and an application to gasoline demand.
appendix boundary found by appendix_command · 62% of the source is main text. Read the extracted text to check this.
The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Chetverikov, Denis and Wilhelm, Daniel (2017) Nonparametric Instrumental Variable Estimation Under Monotonicity | 1.000 | 5 | 4 | 100% |
| 2 | Rueda, Cristina and Salvador, Bonifacio (1995) Reduction of Risk Using Restricted Estimators | 0.737 | 3 | 3 | 67% |
| 3 | Donald W. K. Andrews (1999) Estimation When a Parameter is on a Boundary | 0.644 | 2 | 2 | 100% |
| 4 | V. Chernozhukov and I. Fernández-Val and A. Galichon (2009) Improving point and interval estimators of monotone functions by rearrangement | 0.644 | 2 | 2 | 100% |
| 5 | Geyer, Charles J (1994) On the Asymptotics of Constrained $M$-Estimation | 0.644 | 2 | 2 | 100% |
| 6 | Shinozaki, Nobuo and Chang, Yuan-Tsung (1999) A Comparison of Maximum Likelihood and Best Unbiased Estimators in the Estimation of Linear Combinations of Positive Normal Means | 0.644 | 2 | 2 | 100% |
| 7 | Theil, Henri (1971) Principles of Econometrics | 0.644 | 2 | 2 | 100% |
| 8 | Blundell, Richard and Horowitz, Joel L and Parey, Matthias (2012) Measuring the price responsiveness of gasoline demand: Economic shape restrictions and nonparametric demand estimation | 0.644 | 2 | 2 | 100% |
| 9 | Joel L. Horowitz and Sokbae Lee (2017) Nonparametric estimation and inference under shape restrictions | 0.511 | 2 | 1 | 100% |
| 10 | Amelunxen, Dennis and Lotz, Martin and McCoy, Michael B. and Tropp,… (2014) Living on the Edge: Phase Transitions in Convex Programs with Random Data | 0.405 | 1 | 1 | 100% |
Showing the top 10 of 49 scored citations.