Sampling Properties of the Bayesian Posterior Mean With an Application to WALS Estimation

Tinbergen Institute Discussion Paper 2020-015/III

d/SEAS Working Paper Forthcoming

38 Pages Posted: 9 Mar 2020

See all articles by Giuseppe De Luca

Giuseppe De Luca

University of Palermo - d/SEAS

J.R. Magnus

Vrije Universiteit Amsterdam, School of Business and Economics

Franco Peracchi

Georgetown University - Department of Economics

Date Written: March 7, 2020

Abstract

Many statistical and econometric learning methods rely on Bayesian ideas, often applied or reinterpreted in a frequentist setting. Two leading examples are shrinkage estimators and model averaging estimators, such as weighted-average least squares (WALS). In many instances, the accuracy of these learning methods in repeated samples is assessed using the variance of the posterior distribution of the parameters of interest given the data. This may be permissible when the sample size is large because, under the conditions of the Bernstein--von Mises theorem, the posterior variance agrees asymptotically with the frequentist variance. In finite samples, however, things are less clear. In this paper we explore this issue by first considering the frequentist properties (bias and variance) of the posterior mean in the important case of the normal location model, which consists of a single observation on a univariate Gaussian distribution with unknown mean and known variance. Based on these results, we derive new estimators of the frequentist bias and variance of the WALS estimator in finite samples. We then study the finite-sample performance of the proposed estimators by a Monte Carlo experiment with design derived from a real data application about the effect of abortion on crime rates.

Keywords: Normal Location Model, Posterior Moments and Cumulants, Higher-Order Delta Method Approximations, Double-Shrinkage Estimators, WALS

JEL Classification: C11, C13, C15, C52, I21

Suggested Citation

De Luca, Giuseppe and Magnus, Jan R. and Peracchi, Franco, Sampling Properties of the Bayesian Posterior Mean With an Application to WALS Estimation (March 7, 2020). Tinbergen Institute Discussion Paper 2020-015/III, d/SEAS Working Paper Forthcoming, Available at SSRN: https://ssrn.com/abstract=3551156 or http://dx.doi.org/10.2139/ssrn.3551156

Giuseppe De Luca

University of Palermo - d/SEAS ( email )

Viale delle Scienze, edificio 13
Palermo, 90124
Italy

Jan R. Magnus (Contact Author)

Vrije Universiteit Amsterdam, School of Business and Economics ( email )

De Boelelaan 1105
Amsterdam, 1081HV
Netherlands

Franco Peracchi

Georgetown University - Department of Economics ( email )

Washington, DC 20057
United States

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