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A frequentist approach to Bayesian asymptotics

A frequentist approach to Bayesian asymptotics
A frequentist approach to Bayesian asymptotics

Ergodic theorem shows that ergodic averages of the posterior draws converge in probability to the posterior mean under the stationarity assumption. The literature also shows that the posterior distribution is asymptotically normal when the sample size of the original data considered goes to infinity. To the best of our knowledge, there is little discussion on the large sample behaviour of the posterior mean. In this paper, we aim to fill this gap. In particular, we extend the posterior mean idea to the conditional mean case, which is conditioning on a given vector of summary statistics of the original data. We establish a new asymptotic theory for the conditional mean estimator for the case when both the sample size of the original data concerned and the number of Markov chain Monte Carlo iterations go to infinity. Simulation studies show that this conditional mean estimator has very good finite sample performance. In addition, we employ the conditional mean estimator to estimate a GARCH(1,1) model for S&P 500 stock returns and find that the conditional mean estimator performs better than quasi-maximum likelihood estimation in terms of out-of-sample forecasting.

Bayesian average, Conditional mean estimation, Ergodic theorem, Summary statistic
0304-4076
Cheng, Tingting
59d6a97b-57cd-4f2c-b9dd-11d5c92f06bb
Gao, Jiti
fb907009-eef0-4e30-aca7-b484324f4955
Phillips, Peter C.B.
f67573a4-fc30-484c-ad74-4bbc797d7243
Cheng, Tingting
59d6a97b-57cd-4f2c-b9dd-11d5c92f06bb
Gao, Jiti
fb907009-eef0-4e30-aca7-b484324f4955
Phillips, Peter C.B.
f67573a4-fc30-484c-ad74-4bbc797d7243

Cheng, Tingting, Gao, Jiti and Phillips, Peter C.B. (2018) A frequentist approach to Bayesian asymptotics. Journal of Econometrics. (doi:10.1016/j.jeconom.2018.06.006).

Record type: Article

Abstract

Ergodic theorem shows that ergodic averages of the posterior draws converge in probability to the posterior mean under the stationarity assumption. The literature also shows that the posterior distribution is asymptotically normal when the sample size of the original data considered goes to infinity. To the best of our knowledge, there is little discussion on the large sample behaviour of the posterior mean. In this paper, we aim to fill this gap. In particular, we extend the posterior mean idea to the conditional mean case, which is conditioning on a given vector of summary statistics of the original data. We establish a new asymptotic theory for the conditional mean estimator for the case when both the sample size of the original data concerned and the number of Markov chain Monte Carlo iterations go to infinity. Simulation studies show that this conditional mean estimator has very good finite sample performance. In addition, we employ the conditional mean estimator to estimate a GARCH(1,1) model for S&P 500 stock returns and find that the conditional mean estimator performs better than quasi-maximum likelihood estimation in terms of out-of-sample forecasting.

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Accepted/In Press date: 31 January 2018
e-pub ahead of print date: 28 June 2018
Keywords: Bayesian average, Conditional mean estimation, Ergodic theorem, Summary statistic

Identifiers

Local EPrints ID: 422647
URI: http://eprints.soton.ac.uk/id/eprint/422647
ISSN: 0304-4076
PURE UUID: 874cd852-d6df-41ac-b0df-8040e6c7985b
ORCID for Peter C.B. Phillips: ORCID iD orcid.org/0000-0003-2341-0451

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Date deposited: 27 Jul 2018 16:30
Last modified: 16 Mar 2024 06:53

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Contributors

Author: Tingting Cheng
Author: Jiti Gao

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