Optimal Decision Rules for Weak GMM

Abstract

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.

Macro‐Finance Decoupling: Robust Evaluations of Macro Asset Pricing Models

“…The optimal test in the presence of weakly identified baseline moments is beyond the scope of the paper. However, the literature has provided several encouraging power results for various conditional tests against the null hypothesis (e.g., Andrews, Moreira, and Stock (2006), Andrews and Mikusheva (2016a, forthcoming)) and we expect the conditional specification test to inherit these good properties. One may also apply the generic methods of Elliott, Müller, and Watson (2015) to evaluate the efficiency of an ad hoc test with correct size.…”

GMM is Inadmissible Under Weak Identification

“…In this limit experiment, as in the finite-sample problem, the goal is to choose an estimator δ, which now maps realizations of g(•) to estimates δ(g, Σ) ∈ A, in a way which yields a low risk E m [L(δ(g, Σ), θ * )], where E m [•] denotes the expectation taken under (2). Andrews and Mikusheva (2022) shows that the risk in the limit experiment lower-bounds the (appropriately scaled) asymptotic risk in the original problem.…”

“…Specifically, the moment condition E [φ(X, θ)] = 0 should be uniquely solved at θ * , and the sample moment function g n (θ) should be wellseparated from zero, asymptotically, outside infinitesimal neighborhoods of θ * . These point-and strong-identification assumptions are a poor fit for many economic applications, so in Andrews and Mikusheva (2022) we derived an alternative asymptotic efficiency theory for moment condition models with weak and partial identification. There, we showed that under mild conditions the problem of inference on θ * under weak identification reduces, asymptotically, to observing a single realization of a Gaussian process…”

“…can be unreliable, and weak-identification approximations, which model informativeness of the data as limited even in large samples, often provide a better description of finitesample behavior (Staiger and Stock 1997, D. Andrews and Cheng 2012, Andrews and Mikusheva 2022. Standard arguments for the efficiency of GMM no longer apply under weak identification, raising the question of whether GMM estimators should be used in such settings and, if not, what alternatives we should prefer.…”

“…This approach was initially proposed by Chernozhukov and Hong (2003) for settings where minimization is computationally intractable, and they showed that quasi-Bayes is asymptotically equivalent to GMM under strong identification. More recently, in Andrews and Mikusheva (2022) we showed that quasi-Bayes arises as the limit of a sequence of Bayes decision rules under weak identification. In the present paper, we show that quasi-Bayes posterior means are Lipschitz in the GMM objective function.…”