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North American Journal of Economics

and Finance

journal homepage: www.elsevier.com/locate/najef

Empirical modeling of high-income and emerging stock and Forex

market return volatility using Markov-switching GARCH models

Miguel Ataurima Arellano

a,b,1

, Gabriel Rodríguez

a,c,

⁎

,2

a

Pontiﬁcia Universidad Católica del Perú, Peru

b

Development Bank of Latin America, Peru

c

Fiscal Council of Peru, Peru

ARTICLE INFO

Keywords:

MS-GARCH models

GARCH models

Returns

Volatility

Latin American countries

High-income countries

Stock

Forex

JEL Classiﬁcation:

C22

C52

C53

ABSTRACT

Using weekly data for stock and Forex market returns, a set of MS-GARCH models is estimated for

a group of high-income (HI) countries and emerging market economies (EMEs) using algorithms

proposed by Augustyniak (2014) and Ardia et al. (2018, 2019a,b), allowing for a variety of

conditional variance and distribution speciﬁcations. The main results are: (i) the models selected

using Ardia et al. (2018) have a better ﬁt than those estimated by Augustyniak (2014), contain

skewed distributions, and often require that the main coeﬃcients be diﬀerent in each regime; (ii)

in Latam Forex markets, estimates of the heavy-tail parameter are smaller than in HI Forex and

all stock markets; (iii) the persistence of the high-volatility regime is considerable and more

evident in stock markets (especially in Latam EMEs); (iv) in (HI and Latam) stock markets, a

single-regime GJR model (leverage eﬀects) with skewed distributions is selected; but when using

MS models, virtually no MS-GJR models are selected. However, this does not happen in Forex

markets, where leverage eﬀects are not found either in single-regime or MS-GARCH models.

1. Introduction

Financial market volatility plays an important role in economic performance and ﬁnancial stability. In particular, the speciﬁcation

of conditional volatility is essential for constructing risk measures; see Ardia (2008). Furthermore, modeling time-varying volatility

has been widely used in the literature on ﬁnancial time series, as the demand for monitoring volatility has increased as a means of

assessing ﬁnancial risk. Two approaches that have proved useful are the autoregressive conditional heteroskedasticity (ARCH) family,

including the ARCH model developed by Engle (1982); the generalized ARCH (GARCH) model by Bollerslev (1986); and the

https://doi.org/10.1016/j.najef.2020.101163

Received 23 October 2018; Received in revised form 3 December 2019; Accepted 23 January 2020

This document is drawn from the Master.s Thesis in economics by Miguel Ataurima Arellano, Master’s Program in Economics, Pontiﬁcia

Universidad Católica del Perú (PUCP). It is a substantially revised version of Ataurima Arellano, Collantes, and Rodriguez (2017), which was

produced with the valuable participation of Erika Collantes (PUCP). We thank the useful comments by Paul Castillo and Fernando Pérez-Forero

(Central Reserve Bank of Peru, BCRP, and PUCP), Jorge Rojas (PUCP), and participants in the 33rd BCRP Meeting of Economists (Lima, October 27-

28, 2015). The useful comments provided by the editor and two anonymous referees are gratefully acknowledged. The views expressed in this paper

are those of the authors and do not necessarily reﬂect the positions of the Development Bank of Latin America (CAF) and the Fiscal Council of Peru.

Any remaining errors are our responsibility.

⁎

Corresponding author.

E-mail addresses: miguel.ataurima@pucp.edu.pe (M. Ataurima Arellano), gabriel.rodriguez@pucp.edu.pe (G. Rodríguez).

1

Department of Economics, Pontiﬁcia Universidad Católica del Perú, 1801 Universitaria Avenue, Lima 32, Lima, Peru.

2

Mailing address: Department of Economics, Pontiﬁcia Universidad Católica del Perú, 1801 Universitaria Avenue, Lima 32, Lima, Peru.

North American Journal of Economics and Finance 52 (2020) 101163

1062-9408/ © 2020 Elsevier Inc. All rights reserved.

T

stochastic volatility (SV) model introduced by Taylor (1982) and further developed by Taylor (1986).

3

Multiple extensions of these

models have been proposed to capture additional stylized facts observed in ﬁnancial series, such as non-linearities, asymmetries, and

long memory.

4

Another characteristic of the return distribution of ﬁnancial (stock) series is the asymmetric response of volatility,

known as the leverage eﬀect, ﬁrst noted by Black (1976) and modeled by Nelson (1991) and Glosten, Jagannathan, and Runkle

(1993)-GJR, among others.

However, many ﬁnancial series exhibit structural changes in the dynamics of volatility. In these circumstances, volatility mod-

eling and predictions made using GARCH-type models fail to fully capture volatility movements; see Lamoureux and Lastrapes

(1990), Danielsson (2011) and Bauwens, De Backer, and Dufays (2014). One way to deal with this problem is allowing the parameters

of the GARCH model to vary according to a latent variable that follows a Markov process (see Hamilton, 1989), which in turn gives

rise to the MS-GARCH model. This speciﬁcation allows a diﬀerent GARCH behavior in each regime; i.e., it is possible to capture the

diﬀerence in variance dynamics both in periods of low and high volatility.

Initial studies about MS models applied to ﬁnancial time series focused on ARCH-type speciﬁcations; see Cai (1994) and Hamilton

and Susmel (1994). Excluding lagged values of the conditional variance in the variance equation allows the likelihood function to be

computationally treatable. When using a GARCH-type speciﬁcation, since there is a Markov chain with K regimes, assessing the

likelihood requires the integration of all

K

T

possible paths, which makes the estimation unfeasible. Gray (1996) and Dueker (1997),

and Klaassen (2002) ﬁrst attempted to address this issue, known as the path-dependence problem. Essentially, they tackle the

problem by collapsing the past regime-speciﬁc conditional variances using particular schemes. For instance, Gray (1996) suggests

that the conditional distribution of returns is independent of the regime path; and integrates the path of the regime not observed in

the GARCH equation through the conditional expectation of the past variance. Others suggest alternative estimation methods to face

the problem of path dependence without modifying the MS-GARCH model.

5

Recently, Augustyniak (2014), hereinafter AGK, estimates an MS-GARCH model using Monte Carlo Expectation Maximization

(MCEM) and Monte Carlo Maximum Likelihood (MCML) algorithms, and obtains an approximation of the asymptotic standard errors

of the maximum likelihood estimates. AGK ﬁnds that the MCEM-MCML algorithm is eﬀective in the simulation of the posterior

distribution of the state vector in empirical results using daily and weekly S&P500 price index returns. Another recent contribution is

Ardia, Bluteau, Boudt, and Catania (2018), hereinafter ABBC, who consider an alternative approach, suggested by Haas, Mittnik, and

Paolella (2004), which consists in letting the GARCH process of each regime evolve independently from the other states. While this

approach avoids the path-dependence problem, it has the additional advantage of allowing more clarity in the interpretation of

variance dynamics in each regime. ABBC implement their diﬀerent models using the MS-GARCH R Package of Ardia, Bluteau, Boudt,

and Catania (2019a) and Ardia, Bluteau, Boudt, Catania, and Trottier (2019b). Thus, they estimate a wide variety of models that

support several speciﬁcations (e.g., GARCH and Glosten et al. (1993)) with diﬀerent types of innovations. They apply these models to

the prediction of diﬀerent risk measures; e.g., value-at-risk (VaR) and expected shortfall; and ﬁnd that the MS-GARCH models oﬀer

better results compared to diﬀerent single-regime GARCH/GJR speciﬁcations. See also Iqbal (2016) and Abounoori, Elmi, and

Nademi (2015) about forecasting volatility and risk measures in the Karachi and Tehran stock markets, respectively.

The literature includes other contributions in addition to the empirical applications of AGK, ABBC, and Ardia et al. (2019a,

2019b). Moore and Wang (2007) analyze stock market volatility in ﬁve new states of the European Union (the Czech Republic,

Hungary, Poland, Slovenia, and Slovakia) in 1994–2005. The results reveal a tendency to low market volatility in these markets when

they joined the European Union compared with the previous high-volatility period. Liang and Yongcheol (2008) apply MS-GARCH

models to weekly data from ﬁve emerging market economies (EMEs) in East Asia; and other similar stock market studies are Marcucci

(2005), Wang and Theobald (2008), Visković, Arnerić, and Rozga (2014), Lolea and Vilcu (2018) and Korkpoe and Howard (2019).

Ardia, Bluteau, and Rüede (2019c) ﬁnd strong evidence of regime changes in the GARCH dynamics of volatility in the bitcoin market;

i.e., MS-GARCH models outperform single-regime speciﬁcations when predicting VaR. López-Herrera and Mota (2019) analyze the

relationship between Mexican stock market yields and USD yields (i.e., the appreciation rate), as well as the relationship between

their volatilities using MS-GARCH models. They ﬁnd evidence suggesting an association between stock market returns and the

appreciation/depreciation of the domestic currency and a positive association when volatility is high in both markets. Oseifuah and

Korkpoe (2018) use an MS-GARCH model with skewed Student-t innovations to examine the exchange rate dynamics in South Africa

(relative to the dollar) and ﬁnd evidence of secular changes in the South African economy that push the domestic currency into a

dominant high-volatility regime. Other studies on the Forex market are Klaassen (2002), Sopipan, Intarasit, and Chuarkham (2014),

Caporale and Zekokh (2019), and Hamida and Scalera (2019). Other documents with empirical applications of MS-GARCH models

are Gray (1996), who analyzes changes in short-term interest rates; Billio, Casarin, and Osuntuyi (2018) and Günay (2015), who

focus on the dynamics of the energy futures markets and on modeling the volatility of oil returns, respectively; and Allen et al. (2013),

who study the dynamics of hedging in energy futures markets.

3

For extensive reviews, see Bollerslev, Engle, and Nelson (1994) and Engle (1995) for the ARCH family models, and Shephard (2005) for a

comprehensive explanation of the SV models.

4

For comprehensive reviews, see Franses and van Dijk (2000), Engle (2004) and Teräsvirta (2009). For stylized facts about Peru’s stock and Forex

markets, see Humala and Rodríguez (2013). For the long-memory property and other stylized facts in Latam stock and Forex markets, see Rodríguez

(2016) and Rodríguez (2017) and the references mentioned therein.

5

Francq and Zakoian (2008) use the generalized method of moments (GMM) with the analytical expressions of Francq and Zakoian (2005),

whereas Bauwens, Preminger, and Rombouts (2010) are the ﬁrst to use Bayesian MCMC techniques to estimate the MS-GARCH model, providing

suﬃcient conditions for geometric ergodicity and the existence of moments in the process.

M. Ataurima Arellano and G. Rodríguez

North American Journal of Economics and Finance 52 (2020) 101163

2

This paper seeks to contribute to the empirical literature by modeling and analyzing volatility in stock and Forex markets for a

group of high-income (HI) countries and Latam EMEs. The HI countries chosen are Canada, the U.S., Denmark, Norway, Australia,

Switzerland, the UK, Japan, and Europe. The Latam EMEs are Argentina, Brazil, Chile, Colombia, Mexico, and Peru. Taking into

account the sample of markets and countries, as well as a broad set of single-regime GARCH/GJR and MS-GARCH/MS-GJR models,

this document has the following objectives: estimating and analyzing the behavior of a high-volatility regime while identifying the

events associated with stress periods; calculating the persistence of this regime; and identifying the presence of biases, heavy tails,

and leverage eﬀects according to the distributions selected for the estimations. Selection of the best models is done taking into

account several criteria: value of the log-marginal likelihood; signiﬁcance of the parameters; and evaluation of the smoothed curve of

probabilities for the high-volatility regime associated with the correct identiﬁcation of the main domestic and international events

that create stress in volatility episodes.

To our best knowledge, this is the ﬁrst comparative work between a diverse group of HI countries and EMEs, as well as a

comparison between the stock and Forex markets using a wide variety of single-regime GARCH/GJR and MS-GARCH/MS-GJR models

with diﬀerent innovations. The main results are: (i) the models selected using Ardia et al. (2018) have a better ﬁt than those estimated

by Augustyniak (2014), contain skewed distributions, and often require the main coeﬃcients to be diﬀerent in each regime; (ii)

estimates of the heavy-tail parameter in Latam Forex markets is smaller than in HI Forex markets and in all stock markets; (iii) the

persistence of a high-volatility regime is high and more evident in stock markets (especially in Latam EMEs); (iv) in (HI and Latam)

stock markets, a single-regime GJR model (leverage eﬀects) with skewed distributions is selected; but when using MS models,

virtually no MS-GJR models are selected. However, this does not happen in Forex markets, where leverage eﬀects are not found either

in single-regime or MS-GARCH models.

The rest of the paper is organized as follows. Section 2 presents the diﬀerent models used in this paper. Section 3 describes and

analyzes the data and shows the empirical results of the models. The conclusions are presented in Section 4.

2. Methodology

In order to abbreviate and simplify the presentation, we assume that the log-returns have a zero mean and are not autocorrelated,

6

and this variable is denoted by

r

t

. Four types of models are presented below: the single-regime GARCH(1,1) model with Normal

innovations; the MS-GARCH(1,1) model by AGK; the MS-GARCH(1,1) and MS-GJR(1,1) models used by ABBC; and the single-regime

GARCH(1,1) and the single-regime GJR(1,1) models with alternative distributional speciﬁcations. The estimation of an MS-GARCH

model suﬀers from the so-called path dependence problem, which causes serious estimation diﬃculties. The ﬁrst attempt to solve this

problem was Gray (1996), but we follow the more eﬃcient approach of AGK. Notice that the approach of ABBC, following Haas et al.

(2004), does not address this problem.

2.1. The Generalized ARCH (GARCH) Model

The GARCH(

1, 1

) model of Bollerslev (1986) can be written as:

=r h

t t

t

(1)

= + +h r h

t

t

t0 1

1

2

1

1

(2)

where

i i d~ . .

.

N

>(0, 1), 0, 0

0 1

and

0

1

to ensure a positive conditional variance

h

t

, and

+ < 1

1

1

to ensure that the

unconditional variance

=h /(1 )

t 0 1

1

is deﬁned.

2.2. The MS-GARCH model of Augustyniak (2014)

Based on Bauwens et al. (2010) and Francq, Roussignol, and Zakoian (2001) and using AGK notation, the MS-GARCH model can

be deﬁned by the following equations:

=r s h s( ) ( ) ,

t t t t

t

1:

(3)

= + +h s r s h s( ) ( ) ( ),

t t s s

t

t

s

t t1: 0, 1,

1

2

1

1,

1 1: 1

t t

t

(4)

where

i i d~ . .

.

N

(0, 1)

. At each point in time, the conditional variance is

=h s r r s( ) var( , )

t t t t t1: 1: 1 1:

, where

r

t1: 1

and

s

t1:

are

shorthand for the vectors

…r r( , , )

t1 1

and

…s s( , , )

t1

, respectively. The process

s{ }

t

is an unobserved ergodic time-homogeneous Markov

chain process with K-dimensional discrete state space (i.e.,

s

t

can take integer values from 1 to K). The

×K K

transition matrix of the

Markov chain is deﬁned by the transition probabilities

= = =

=

s j s i p{Pr[ ] }

t t

ij

i j

K

1

, 1

. The vector

=

= =

p({ , , } , { } )

i i

i

i

K

ij

i j

K

0, 1,

1,

1 , 1

denotes

the parameters of the model. In order to ensure positivity of the variance, the following constraints are required:

> 0, 0

i i0, 1,

and

= …i K0, 1, ,

i1,

. Conditions for stationarity and the existence of moments are studied by Bauwens et al. (2010), Francq et al.

(2001) and Francq and Zakoian (2005).

It is worth highlighting that the notation used in (3–4) emphasizes the dependence of the conditional variance at time t on the

6

In practice, it means that we apply the models using demeaned log-returns, as explained in Section 3.

M. Ataurima Arellano and G. Rodríguez

North American Journal of Economics and Finance 52 (2020) 101163

3

entire regime path

s

t1:

, which is the path dependence issue. Furthermore, AGK imposes the restrictions

=

1,1 1,2

and

=

1,1 1,2

on

two alternative MS-GARCH models to ﬁnd results consistent with empirical evidence.

2.3. The MS-GARCH model of Ardia et al. (2018)

We follow the notation of Ardia et al. (2018), Ardia et al., 2019a and Ardia et al., 2019b, where the process of conditional

variance is regime-switching dependent. Denote by

I

t 1

the information set observed up to

t 1

, that is,

I

>r i{ , 0}

t t i1

. In

general terms, ABBC express the MS-GARCH model using the following speciﬁcation:

I D

=r s k h( , )~ (0, , ),

t t t k t

k

1 ,

(5)

where

D

h(0, , )

k t

k

,

is a continuous (conditional) distribution with zero mean,

h

k t,

is the time-varying variance, and the vector

k

includes additional shape parameters (e.g., asymmetry, kurtosis). The latent variable

s

t

is deﬁned in the discrete space

… K{1, , }

, and

evolves according to an unobserved ﬁrst-order ergodic homogeneous Markov chain with a

×K K

transition probability