Bayesian structure selection for vector autoregression model

• Author: Ray‐Bing Chen,Chi‐Hsiang Chu,Mong‐Na Lo Huang,Shih‐Feng Huang
• Published date: 01 August 2019
• DOI: http://doi.org/10.1002/for.2573
• Date: 01 August 2019

Received: 22 May 2017 Revised: 13 August 2018 Accepted: 10 January 2019

DOI: 10.1002/for.2573

RESEARCH ARTICLE

Bayesian structure selection for vector autoregression model

Chi-Hsiang Chu1,2 Mong-Na Lo Huang2Shih-Feng Huang3Ray-Bing Chen4

1Clinical Trial Center,Kaohsiung Chang

Gung Memorial Hospital, Kaohsiung,

Taiw an

2Department of Applied Mathematics,

National Sun Yat-sen University,

Kaohsiung, Taiwan

3Department of Applied Mathematics,

National University of Kaohsiung,

Kaohsiung, Taiwan

4Department of Statistics, National Cheng

Kung University,Tainan, Taiwan

Correspondence

Shih-Feng Huang, Department of Applied

Mathematics, National University of

Kaohsiung, Kaohsiung 811, Taiwan.

Email: huangsf@nuk.edu.tw

Abstract

A vector autoregression (VAR) model is powerful for analyzing economic data

as it can be used to simultaneously handle multiple time series from differ-

ent sources. However, in the VAR model, we need to address the problem of

substantial coefficient dimensionality, which would cause some computational

problems for coefficient inference. Toreduce the dimensionality, one could take

model structures into account based on prior knowledge. In this paper, group

structures of the coefficient matrices are considered. Because of the different

types of VAR structures, corresponding Markov chain Monte Carlo algorithms

are proposed to generate posterior samples for performing inferenceof the struc-

ture selection. Simulation studies and a real example are used to demonstrate

the performances of the proposed Bayesian approaches.

KEYWORDS

Bayesian variable selection, segmentized grouping, time series, universal grouping

1INTRODUCTION

In this paper, a vector autoregression (VAR) model is

considered. Essentially,a VAR model can be used to simul-

taneously analyze multiple time series, and such models

are commonly used in economic data analysis; for

example, Litterman (1979) studied the Bayesian VAR with

a Gaussian prior based on macroeconomic theory, and

Sims (1980) considered a VAR model with a vector of

response variables. Compared with other approaches for

high-dimensional time series, such as the dynamic factor

model (Forni, Hallin, Lippi, & Rechlin, 2000, 2005), Song

and Bickel (2011) reported that VAR models would have

the following advantages: (1) the VAR approach is able to

model the high-dimensional time series in one step, and

(2) the VARapproach allows variable-to-variable (impulse

response) relationship analysis and is easy for interpreta-

tion. However,the weakness of the VAR model is related to

the substantial number of coefficients. Let pbe the num-

ber of lags and mbe the number of time series. The total

number of coefficients of a VAR model is pm2,whichisa

quadratic order of m. Thus studies of VAR have recently

been focused on reducing the coefficient dimensionality

via variable selection approaches based on some model

structure assumptions.

To address the variable selection problem, the Bayesian

framework is commonly used under the linear regres-

sion model. The stochastic search variable selection (SSVS)

method, proposed by George and McCulloch (1993), is

one of the widely used Bayesian approaches. Some related

methods have also been studied by Smith and Kohn(1996),

Chipman (1996), Chipman, Hamada, and Wu (1997),

George and McCulloch (1997), and so on. In the SSVS

framework, indicator variables are added to the linear

model, and each indicator is used to denote whether the

corresponding variable is selected or not. Based on the

proper prior assumptions, the Gibbs sampler is generally

adopted to generate posterior samples of the indicators

and coefficients, which are used to identify the relevant

variables. As for the applications of SSVS in finance and

economics, So and Chen (2003) and So, Chen, and Liu

(2006) modified the SSVS for the subset selection prob-

lems in different autoregressive type models. Yu, Chen,

Reed, and Dunson (2013) used the SSVS to deal with the

422 © 2019 John Wiley & Sons, Ltd. wileyonlinelibrary.com/journal/for Journalof Forecasting. 2019;38:422–439.

CHU ET AL.423

variable selection problems in quantile regression model.

To improve the computational efficiency, Geweke (1996)

proposed using a type of spike-and-slab prior for coef-

ficient and modified the sampling procedure with the

so-called component-wise Gibbs sampler (CGS). Chen,

Chu, Lai, and Wu (2011) came up with the stochastic

matching pursuit (SMP) algorithm, which is more com-

putationally efficient than that of the Gibbs sampler, par-

ticularly when the variables are highly correlated. Note

that for structure selection problems it is easy to extend

the SSVS-type method for group selection because the

indicators can be used to denote whether the groups

are selected or not. Later, Farcomeni (2010) introduced

a Bayesian constrained variable selection approach that

can address group selection with several prespecified con-

straints. Moreover, Chen, Chu, Yuan, and Wu (2016) rec-

ommended a groupwise Gibbs sampler (GS) that not only

works for group selection but can also identify the active

variables within the selected groups.

In addition to the Bayesian selection approaches, the

penalized least squares framework has also been widely

used in variable selection problems. The most famous

approach is the Lasso method proposed by Tibshirani

(1996). Other related works include SCAD (Fan& Li, 2001)

and MCP (Zhang, 2010). Based on a Lasso-type method,

Yuan and Lin (2006) proposed a group Lasso method to

solve group selection problems. Friedman, Hastie, and

Tibshirani (2010) and Simon, Friedman, Hastie, and Tib-

shirani (2013) proposed a sparse group Lasso method

under group sparsity with the assumption that only a

small number of its constituent variables are active within

each selected group. In particular, Lasso-type methods

have been used to reduce the coefficient dimensionality

in VAR—forexample, Lasso-VAR proposedby Hsu, Hung,

and Chang (2008). Song and Bickel (2011) extended simi-

lar ideas for structure selection and focused on three types

of coefficient matrices. Nicholson, Matteson, and Bien

(2014) generalized their works to cover more types of VAR

structures, and in addition to Lasso they also proposed

penalized least squares approaches for group selection and

sparse group selection.

In this study, we focus on Bayesian-type methods for

estimating unknown parameters in the VAR models with

different structures of the associated coefficient matrix.

In the literature, SSVS has been used in VAR analysis

without assuming structures on the coefficient matrix. For

example, George, Sun, and Ni (2008) used SSVS to select

effective restrictions for a VAR model. The SSVS prior

has recently been applied within a global VARframework

by Feldkircher and Huber (2016) and Cuaresma, Feld-

kircher, and Huber (2016) to improve the forecasts when

international linkages among countries (or economies) are

considered. In reality, some time series considered in a

VAR model should be classified into groups according to

certain properties. Therefore, besides the variable selec-

tion issue, the structure selection approach is also con-

sidered here, and we target on three different types of

VAR models mentioned by Song and Bickel (2011). Sim-

ilar to the concept of SSVS, for a given VAR structure,

we add indicators to denote whether particular struc-

tures are included in the current VAR model. Then, based

on proper prior assumptions, the corresponding Markov

chain Monte Carlo (MCMC) algorithms are employed to

generate the posterior samples for the structure selection

inference. Several simulations are used to demonstrate the

performances of the proposed Bayesian methods. In addi-

tion, we also illustrate how to identify a proper group

structure for a given VAR dataset via a Bayesian model

selection principle. Finally, a real example is included for

demonstration.

The remainder of this paper is organized as follows.

Section 2 presents the Bayesian formulation for different

structures of coefficient matrices, discussion of prespeci-

fying the priors, and the details of the MCMC algorithms.

Section 3 demonstrates the performance of our selection

approach through several simulations. An empirical study

is presented in Section 4. Finally,Section 5 summarizes our

work and discusses future directions. Technical details,

tables, and figures are in the Appendix.

2MODEL AND STRUCTURE

SETTING

Consider the following VAR model with lagp:

Yt=Yt−1B1+…+Yt−pBp+𝜀t,t=1,…,T,(1)

where Yt=(Yt,1,…,Yt,m)is an 1 ×mvector, Blis an

m×mmatrix for l=1,…,p,and{𝜀t}T

t=1is a sequence of

serially uncorrelated random vectors coming from a multi-

variate normal distribution with zero mean and covariance

matrix Σ.

Based on Equation 1, except for the covariance matrix,

the total number of coefficients in Bl,l=1,…,p,is

pm2of order m2. Thus we need to estimate a considerable

number of coefficients, particularly as more time series are

included. To reduce the size of the coefficients, we take

different coefficient matrix structures into account accord-

ing to prior knowledge. Specifically, following Song and

Bickel (2011), three types of coefficient matrices for Bl,

namely universal grouping, segmentized grouping, and no

grouping, are considered. To illustrate these three struc-

tures, first the coefficient matrix B=(B1,B2,…,Bp)is

divided into two parts: one corresponds to its own lags (the

diagonal elements of the Bls); the other part, namely the