Many empirical questions concern target parameters selected through optimization. For example, researchers may be interested in the effectiveness of the best policy found in a randomized trial, or the best-performing investment strategy based on historical data. Such settings give rise to a winner’s curse, where conventional estimates are biased and conventional confidence intervals are unreliable. This paper develops optimal confidence intervals and median-unbiased estimators that are valid conditional on the target selected and so overcome this winner’s curse. If one requires validity only on average over targets that might have been selected, we develop hybrid procedures that combine conditional and projection confidence intervals to offer further performance gains relative to existing alternatives.

Inference on winners

This paper studies optimal decision rules, including estimators and tests, for weakly identified GMM models. We derive the limit experiment for weakly identified GMM, and propose a theoretically-motivated class of priors which give rise to quasi-Bayes decision rules as a limiting case. Together with results in the previous literature, this establishes desirable properties for the quasi-Bayes approach regardless of model identification status, and we recommend quasi-Bayes for settings where identification is a concern. We further propose weighted average power-optimal identification-robust frequentist tests and confidence sets, and prove a Bernstein-von Mises-type result for the quasi-Bayes posterior under weak identification.

Optimal Decision Rules for Weak GMM

Supply chain disturbances can lead to substantial increases in production costs. To mitigate these risks, firms may take steps to reduce their reliance on volatile suppliers. We construct a model of endogenous network formation to investigate how these decisions affect the structure of the production network and the level and volatility of macroeconomic aggregates. When uncertainty increases in the model, producers prefer to purchase from more stable suppliers, even though they might sell at higher prices. The resulting reorganization of the network leads to less macroeconomic volatility, but at the cost of a decline in aggregate output. The model also predicts that more productive and stable firms have higher Domar weights—a measure of their importance as suppliers—in the equilibrium network. We calibrate the model to U.S. data and find that the mechanism can account for a sizable decline in expected GDP during periods of high uncertainty like the Great Recession. JEL Classifications: E32, C67, D57, D80, D85 We thank Ivan Alfaro, Ryan Chahrour, Roger Farmer, Simon Gilchrist, Pablo Kurlat, David Zeke and seminar participants at Cornell University, New York University, the University of Southern California, the University of Pennsylvania, the University of Virginia, Yale University, the Toulouse School of Economics and the University of Hong Kong for useful comments and suggestions. This paper was previously circulated under the title “Endogenous Production Networks under Uncertainty”. We gratefully acknowledge financial support from the Cornell Center for Social Sciences.

Endogenous Production Networks under Supply Chain Uncertainty∗

Symmetry-based neural networks often constrain the architecture in order to achieve invariance or equivariance to a group of transformations. In this paper, we propose an alternative that avoids this architectural constraint by learning to produce canonical representations of the data. These canonicalization functions can readily be plugged into non-equivariant backbone architectures. We offer explicit ways to implement them for some groups of interest. We show that this approach enjoys universality while providing interpretable insights. Our main hypothesis, supported by our empirical results, is that learning a small neural network to perform canonicalization is better than using predefined heuristics. Our experiments show that learning the canonicalization function is competitive with existing techniques for learning equivariant functions across many tasks, including image classification, $N$-body dynamics prediction, point cloud classification and part segmentation, while being faster across the board.

Equivariance with Learned Canonicalization Functions

This paper studies optimal decision rules, including estimators and tests, for weakly identified GMM models. We derive the limit experiment for weakly identified GMM, and propose a theoretically‐motivated class of priors which give rise to quasi‐Bayes decision rules as a limiting case. Together with results in the previous literature, this establishes desirable properties for the quasi‐Bayes approach regardless of model identification status, and we recommend quasi‐Bayes for settings where identification is a concern. We further propose weighted average power‐optimal identification‐robust frequentist tests and confidence sets, and prove a Bernstein‐von Mises‐type result for the quasi‐Bayes posterior under weak identification. 

/pdf/optimal-decision-rules-for-weak-gmm-2agf3jph.pdf

This paper establishes the argmin of a random objective function to be unique almost surely. This paper first formulates a general result that proves almost sure uniqueness without convexity of the objective function. The general result is then applied to a variety of applications in statistics. Four applications are discussed, including uniqueness of M-estimators, both classical likelihood and penalized likelihood estimators, and two applications of the argmin theorem, threshold regression and weak identification.

Almost sure uniqueness of a global minimum without convexity

Almost Sure Uniqueness of a Global Minimum Without Convexity

When parameters are weakly identified, bounds on the parameters may provide a valuable source of information. Existing weak identification estimation and inference results are unable to combine weak identification with bounds. Within a class of minimum distance models, this paper proposes identification-robust inference that incorporates information from bounds when parameters are weakly identified. The inference is based on limit theory that combines weak identification theory (Andrews and Cheng (2012)) with parameter-on-the-boundary theory (Andrews (1999)) via a new argmax theorem. This paper characterizes weak identification in low-dimensional factor models (due to weak factors) and demonstrates the role of the bounds and identification-robust inference in two example factor models. This paper also demonstrates the identification-robust inference in an empirical application: estimating the effects of a randomized intervention on parental investments in children, where parental investments are modeled by a factor model.

Gregory Cox

Papers

Almost sure uniqueness of a global minimum without convexity

Almost Sure Uniqueness of a Global Minimum Without Convexity

Weak Identification with Bounds in a Class of Minimum Distance Models