Efficient selection of hyperparameters in large Bayesian VARs using automatic differentiation

• Published date: 01 September 2020
• Author: Liana Jacobi,Dan Zhu,Joshua C. C. Chan
• DOI: http://doi.org/10.1002/for.2660
• Date: 01 September 2020

Received: 10 September 2019 Accepted: 12 January 2020

DOI: 10.1002/for.2660

RESEARCH ARTICLE

Efficient selection of hyperparameters in large Bayesian

VARs using automatic differentiation

Joshua C. C. Chan1,2 Liana Jacobi3Dan Zhu4

1Department of Economics, Purdue

University, WestLafayette, Indiana

2Economics Discipline Group, University

of Technology Sydney, Sydney, NSW,

Australia

3Department of Economics, University of

Melbourne, Australia

4Department of Business Statistics and

Econometrics, Monash University,

Clayton, Victoria, Australia

Correspondence

Dan Zhu, Department of Business

Statistics and Econometrics, Monash

University, Clayton,3168 Victoria,

Australia.

Email: dan.zhu@monash.edu

Abstract

Large Bayesian vector autoregressions with the natural conjugate prior are now

routinely used for forecasting and structural analysis. It has been shown that

selecting the prior hyperparameters in a data-driven manner can often substan-

tially improve forecast performance. We propose a computationally efficient

method to obtain the optimal hyperparameters based on automatic differentia-

tion, which is an efficient way to compute derivatives. Using a large US data set,

we show that using the optimal hyperparameter valuesleads to substantially bet-

ter forecast performance. Moreover, the proposed method is much faster than

the conventional grid-search approach, and is applicable in high-dimensional

optimization problems. The new method thus provides a practical and sys-

tematic way to develop better shrinkage priors for forecasting in a data-rich

environment.

KEYWORDS

automatic differentiation, forecasts, marginal likelihood, optimal hyperparameters, vector autore-

gression

1INTRODUCTION

Since the seminal paper of Banbura, Giannone, and

Reichlin (2010) showed that it is feasible to estimate

large Bayesian vector autoregressions (BVARs) with over

100 variables, there has been a lot of interest in using

large BVARs for forecasting and structural analysis. A

few prominent examples include Carriero, Kapetanios,

and Marcellino (2009), Koop (2013), Koop and Korobilis

(2013), and Banbura et al. (2013). One key aspect of these

large BVARs is the use of shrinkage priors that formally

incorporate sensible nondata information, and one popu-

lar way to do so is the Minnesota-type natural conjugate

prior that gives rise to a range of analytical results, includ-

ing closed-form expressions of the marginal likelihood.1

1Early seminal works of shrinkage priors were developed by Doan, Lit-

terman, and Sims (1984) and Litterman (1986). Similar shrinkage priors

for structural VARsare formulated in Leeper et al. (1996) and Sims and

Zha (1998).

These analytical results are later used in Carriero, Clark,

and Marcellino (2016) and Chan (2020) to develop efficient

sampling algorithms to estimate large BVARs with flexible

error covariance structures, such as stochastic volatility,

serially correlated and non-Gaussian errors.

The natural conjugate prior depends on a few hyperpa-

rameters that control the degree of shrinkage, and they

are typically fixed at some subjectively chosen values.

Alternatively, a data-based approach to select these hyper

parameters might be more appealing as it reduces the

number of important subjective choices required from

the user. For example, Del Negro and Schorfheide (2004),

Schorfheide and Song (2015), and Carriero et al. (2015)

obtain the optimal hyperparameters by maximizing the

marginal likelihood over a grid of possible values.2This

2Giannone, Lenza, and Primiceri (2015) show that a data-based approach

of selecting the hyperparameters—comparedto the convectional method

of fixing them to some ad hoc values—can substantially improve the

forecast performance of large BVARs.

Journal of Forecasting. 2020;39:934–943.

wileyonlinelibrary.com/journal/for© 2020 John Wiley & Sons, Ltd.

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