Forecasting under model uncertainty: Non‐homogeneous hidden Markov models with Pòlya‐Gamma data augmentation

• Date: 01 July 2020
• Author: Ioannis Vrontos,Loukia Meligkotsidou,Constandina Koki
• Published date: 01 July 2020
• DOI: http://doi.org/10.1002/for.2645

Received: 4 June 2019 Revised: 10 December 2019 Accepted: 13 December 2019

DOI: 10.1002/for.2645

RESEARCH ARTICLE

Forecasting under model uncertainty: Non-homogeneous

hidden Markov models with Pòlya-Gamma data

augmentation

Constandina Koki1Loukia Meligkotsidou2Ioannis Vrontos1

1Department of Statistics, Athens

University of Economics and Business,

Athens, Greece

2Faculty of Mathematics, National and

Kapodistrian University of Athens,

Panepistimioupolis, Athens, Greece

Correspondence

Constandina Koki, Department of

Statistics, Athens University of Economics

and Business, 76 Patission Street, Athens.

Email: kokiconst@aueb.gr

Abstract

We consider finite state-space non-homogeneous hidden Markov models for

forecasting univariate time series. Given a set of predictors, the time series

are modeled via predictive regressions with state-dependent coefficients and

time-varying transition probabilities that depend on the predictors via a logis-

tic/multinomial function. In a hidden Markov setting, inference for logistic

regression coefficients becomes complicated and in some cases impossible due

to convergence issues. In this paper, we aim to address this problem uti-

lizing the recently proposed Pólya-Gamma latent variable scheme. Also, we

allow for model uncertainty regarding the predictors that affect the series both

linearly — in the mean — and non-linearly — in the transition matrix. Predic-

tor selection and inference on the model parameters are based on an automatic

Markov chain Monte Carlo scheme with reversible jump steps. Hence the pro-

posed methodology can be used as a black box for predicting time series. Using

simulation experiments, we illustrate the performance of our algorithm in vari-

ous setups, in terms of mixing properties, model selection and predictive ability.

An empirical study on realized volatility data shows that our methodology gives

improved forecasts compared to benchmark models.

KEYWORDS

non-homogeneous hidden Markov models, model selection, forecasting, Pólya-Gamma data aug-

mentation, realized volatility

1INTRODUCTION

Discrete-time finite state-space homogeneous hidden

Markov models (HHMMs) have been extensively studied

and used to model stochastic processes that consist of

an observed process and a latent (hidden) sequence of

states which is assumed to affect the observation sequence

(see, e.g., Billio et al., 1999; Cappé et al., 2005). Bayesian

inference, using Markov chain Monte Carlo (MCMC)

techniques, has enhanced the applicability of HHMMs

and has led to the construction of more complex model

specifications, including nonhomogeneous hidden

Markov models (NHHMMs). Initially, Diebold et al.

(1994) studied 2-state Gaussian NHHMMs, in which

the time-varying transition probabilities were modeled

via logistic functions. Their approach was based on

the expectation-maximization (EM) algorithm. Filardo

and Gordon (1998) adopted a Bayesian perspective to

overcome technical and calculation issues of classical

approaches. Since then, various Bayesian methods have

been proposed in the literature. For example, Spezia

(2006) modeled time-varying transition probabilities via

© 2019 John Wiley & Sons, Ltd. Journal of Forecasting. 2020;39:580–598.

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a logistic function depending on exogenous variables and

performed model selection based on the Bayes factor.

In the same spirit, Meligkotsidou and Dellaportas (2011)

considered an m-state (m2) NHHMM and assumed

that the elements of the transition matrix are linked

through exogenous variables with a multinomial logistic

link, whereas the observed process conditional on the

unobserved process follows an autoregressive model

of order p. They accommodated and exploited model

uncertainty within their Bayesian model — by allowing

covariate selection only on the transition matrix — to

improve the predictive ability of NHHMMs on economic

data series.

Based on experimental evidence, the algorithm of

Meligkotsidou and Dellaportas (2011; M&D) faces

convergence issues when there exists model uncertainty,

due to the data augmentation scheme of Holmes and Held

(2006). Polson et al. (2013) confirm the efficiency issues in

the Holmes and Held scheme and propose a Pólya-Gamma

data augmentation strategy that significantly improves

over various benchmarks (e.g., Frühwirth-Schnatter &

Frühwirth, 2010; Fussl et al., 2013; O'Brien & Dunson,

2004). Furthermore, the recent work of Holsclaw et al.

(2017) confirms that using Pólya-Gamma data aug-

mentation to parametrize the transition probabilities of

NHHMMs results in an algorithm that mixes well and

provides adequate estimates of the model parameters.

Motivated by this, we revisit the work of Meligkotsidou

and Dellaportas (2011) by employing the recent method-

ological advances on the Pólya-Gamma data augmentation

scheme of Polson et al. (2013). We consider NHHMMs in

which the time series are modeled via different predictive

regression models for each state, whereas the transition

probabilities are modeled via logistic regressions. Given

an available set of predictors, we allow for model uncer-

tainty regarding the predictors that affect the series both

linearly — directly in the mean regressions — and

non-linearly — in the transition probability matrix.

The resulting model is a nonhomogeneous

Pólya-Gamma hidden Markov model, which we will

denote by NHPG. Bayesian inference is performed via an

MCMC scheme that overcomes difficulties and conver-

gence issues inherent in existing MCMC algorithms. To

this end, we exploit the missing data representation of hid-

den Markov models and construct an MCMC algorithm

based on data augmentation, consisting of several steps.

First, we sample the latent sequence of states via the

scaled forward–backward algorithm of Scott(2002), which

is a modification of the forward–backward algorithm of

Baum et al. (1970), who used it to implement the classical

EM algorithm. We then use a logistic regression repre-

sentation of the transition probabilities and simulate the

parameters of the mean predictive regression model for

each state, via Gibbs sampling steps. Finally, we incorpo-

rate variable selection within our MCMC scheme by using

the reversible jump (RJ) algorithm of Green (1995) and

Hastie and Green (2011).

Different approaches have been used in the literature

to cope with the model selection problem. The use of

information criteria, such as Akaike's information crite-

rion (AIC; Akaike et al., 1973), the Bayesian information

criterion (BIC) of Schwarz (1978), the deviance informa-

tion criterion (DIC; Spiegelhalter et al., 2002) or the widely

applicable Bayesian information criterion (WBIC; Watan-

abe, 2013), is another approach to variable selection. A

study for comparing variable selection methods is well pre-

sented in O'Hara and Sillanpää (2009), while Dellaportas

et al. (2002) study the variable selection methods in the

context of model choice. Holsclaw et al. (2017) consider an

NHHMM similar to ours for modeling multivariate mete-

orological time series data. In that paper, the transition

probabilities are modeled via multinomial logistic regres-

sions affected by a specific set of exogenous variables. The

authors use the BIC for choosing the best model among a

prespecified class of models. We extend this work by con-

sidering the problems of statistical inference and variable

selection jointly,in a purely Bayesian setting. The proposed

model is flexible, since we do not decide a priori which

covariates affect the observed or the unobserved process.

Instead, we have a common pool of covariates {X}and

within the MCMC algorithm, we gauge which covariates

are included in subset X(1)affecting the mean predic-

tive equation of the observed process, and which covariates

are included in subset X(2)affecting the time-varying

transition probabilities.

Our probabilistic approach is based on the calculation

of the posterior distribution of different NHPGs. Posterior

probabilities can be used either for selecting the most prob-

able model (i.e., making inference using the model with

the highest posterior probability), or for Bayesian model

averaging (i.e., producing inferences averaged over differ-

ent NHPGs). Barbieri and Berger (2004) argue that the

optimal predictive model is not necessarily the model with

the highest posterior probability. Specifically, they show

that the optimal predictive model for linear regression

models is the median probability model, i.e. the model that

consisted of those covariates which have overall posterior

inclusion probabilities greater or equal to 0.5. We calcu-

late both the posterior probabilities of the models and the

probabilities of inclusion.

Weuse our model for predicting realized volatility. Accu-

rate forecasting of future volatility is important for ass

et allocation, portfolio construction, and risk manage-

ment (see Gospodinov et al., 2006). A review of the real-

ized volatility literature can be found in McAleer and

Medeiros (2008). The relationship between the volatility

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