Simple Inference on a Simplex-Valued Weight

TL;DR

This paper introduces a straightforward, tuning-free method for constructing confidence sets for simplex-valued weights in statistical models, accommodating both point and set identification, with proven asymptotic validity.

Contribution

It proposes an adaptive, projection-based confidence set construction method for simplex weights that is simple, tuning-free, and valid under various identification scenarios.

Findings

Method is asymptotically uniformly valid.

No tuning parameters or simulations needed for critical values.

Applicable to empirical examples with simplex weights.

Abstract

In many applications, the parameter of interest involves a simplex-valued weight which is identified as a solution to an optimization problem. Examples include synthetic control methods with group-level weights and various methods of model averaging and forecast combinations. The simplex constraint on the weight poses a challenge in statistical inference due to the constraint potentially binding. In this paper, we propose a simple method of constructing a confidence set for the weight using an adaptive test based on the projection on a polyhedral cone and prove that the method is asymptotically uniformly valid. The procedure does not require tuning parameters or simulations to compute critical values. The confidence set accommodates both the cases of point-identification or set-identification of the weight. We illustrate the method with an empirical example.