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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,

⁎

Pontiﬁcia Universidad Católica del Perú, Peru

Development Bank of Latin America, Peru

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).

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

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

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

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.

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

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.

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.

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.

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.

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

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,

and this variable is denoted by

. 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

(1)

= + +h r h

t0 1

(2)

where

i i d~ . .

>(0, 1), 0, 0

0 1

and

to ensure a positive conditional variance

, and

+ < 1

to ensure that the

unconditional variance

=h /(1 )

t 0 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

(3)

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

t t s s

t t1: 0, 1,

1 1: 1

t t

(4)

where

i i d~ . .

(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

t1: 1

and

t1:

are

shorthand for the vectors

…r r( , , )

t1 1

and

…s s( , , )

, respectively. The process

s{ }

is an unobserved ergodic time-homogeneous Markov

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

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

i j

, 1

. The vector

= =

p({ , , } , { } )

i i

i j

0, 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

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

entire regime path

t1:

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

1,1 1,2

and

1,1 1,2

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

t 1

the information set observed up to

t 1

, that is,

>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

1 ,

(5)

where

h(0, , )

k t

is a continuous (conditional) distribution with zero mean,

k t,

is the time-varying variance, and the vector

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

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 matrix

pP { }

i j

, 1

as deﬁned above. In this context,

i i d~ . . . (0, 1)

, which appears in 2.1 and 2.2, is a special case of

(·)

. Furthermore,

deﬁning

= =E r s k h[ , ]

t t k t

1 ,

, where

k t,

is the variance of

conditional on the realization of

=s k

and

t 1

, ABBC specify the

variance of

as a GARCH type model; i.e., conditional on the regime

=s k

, we have that

h h r h( , , )

k t t k t k, 1 , 1

. Diﬀerent speci-

ﬁcations for

h (.)

are proposed by ABBC. We use two of them: (i) the ﬁrst one is an MS-GARCH (1,1) model as suggested by Haas et al.

(2004), that is,

= + +h r h ,

k t k k

k t, 0, 1,

, 1

for

=k 1, 2

an where, to ensure positivity, we require that

> >0, 0, 0

k k

0, 1,

and to ensure stationarity

we require that

+ < 1

. In this case,

( , , )

k k k

0, 1,

; (ii) the second one is an MS-GJR(1,1) model which exploits the volatility speciﬁ-

cation of Glosten et al. (1993):

= + + +

h r h( ) ,

k t k k k r

k t, 0, 1, 2, { 0}

, 1

t 1

for

=k 1, 2

, where

{·}

is the indicator function taking a value of one if the conditions hold, and zero otherwise; and where, to ensure

positivity, we require that

> >0, 0, 0

k k

0, 1,

and to ensure stationarity we require that

+ + <

E [ ] 1

k k

k t

1, 2,

{ 0}

k t,

this case,

( , , , )

k k k k

0, 1, 2,

. The parameter

k2,

controls the degree of asymmetry in the conditional volatility process. This

model accounts for the well-known asymmetric reaction of volatility to the sign of past returns, which is frequently referred to as the

leverage eﬀect; see Black (1976).

Flexibility in the approach of Ardia et al. (2019a) and Ardia et al., 2019b provides several options for

(.)

, of which we use four:

(i) the standard Normal distribution (

); (ii) the Student-t distribution (

); (iii) the skewed Normal distribution (

); and (iv) the

skewed Student-t distribution (

). The distributions in (iii) and (iv) are used to assess the beneﬁts of introducing asymmetry in the

analysis following the suggestions of Fernández and Steel (1998) and Bauwens and Laurent (2005); see Trotier and Ardia (2016) for

further details concerning the derivation of the moments of these standardized distributions.

2.4. Other models

While the single-regime GARCH (1,1)-

model presented in 2.1 is usually not a good competitor against more ﬂexible speci-

ﬁcations, such as MS-type models, other single-regime GARCH models may be better competitors if distributional ﬂexibility of

(.)

allowed. Thus, we estimate two additional models: a single-regime GARCH(1,1) and a single-regime GJR(1,1) with the distributional

speciﬁcations mentioned in (ii), (iii) and (iv) in 2.3. Therefore, we also estimate models where

= + +h r h

t0 1

and

= + + +

h r h( )

t r

t0 1 2 { 0}

t 1

, with distributions

S N

sk,

, and

, respectively.

2.5. Estimation of the MS-GARCH model

Given the predominantly empirical nature of this paper, we do not intend to explain the detail the AGK and ABBC estimation

methods. For information, a brief sketch is provided below.

To estimate the parameters of the MS-GARCH model, AGK develops a method based on the MCEM algorithm of Wei and Tanner

(1990) and the MCML method suggested by Geyer (1994) and Geyer (1996). Since the method uses the posterior distribution of the

state variables, it is usually seen as an equivalent frequentist approach to the Bayesian MCMC method proposed by Bauwens et al.

(2010). The MS-GARCH model speciﬁed by Eqs. 3,4 presents estimation challenges, because the conditional variance t depends on the

complete path

t1:

. The likelihood of the observations,

f r( )

, is calculated by integrating all possible regime paths. An accurate

estimate of the log-likelihood is obtained by Bauwens et al. (2010) by writing

= +

f r r f r rlog ( ) log( ) log ( , )

t t1

1 1:

and es-

Covariance-stationarity is a stronger requirement than that of Hass et al. (2004); see ABBC for further details.

In general, covariance-stationarity is achieved by imposing

+ + < 1

k k k

1, 2,

, where

< =r s kPr[ 0 , ]

k t t t1 1

. If

k t,

is symmetrically

distributed, then

E [ ]

k t

{

. For skewed distributions, we follow Trottier and Ardia (2016).

M. Ataurima Arellano and G. Rodríguez

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

timating

= …

f r r t T( , ), 1 , 1

t t1 1:

sequentially with the aid of particle ﬁlters. Simulation of the log-likelihood is diﬃcult to max-

imize with standard optimization routines, because this ﬁlter is not a continuous function of

The hybrid AGK approach implies a

two-step calculation. While in the ﬁrst step the MCEM algorithm obtains a good MLE estimate,

, in the second step the MCML

method replaces many potential iterations of the MCEM with a single iteration, leading to faster convergence. The MCEM-MCML

algorithm uses as starting points the approximations of the Gray (1996), Dueker (1997) and Klaassen (2002) models. To initialize the

Gibbs sampler, a smoothed inferred state vector, as proposed by Gray (1996), is taken as the ﬁrst state vector; and to generate the ﬁrst

Markov chain

, we assume the initial state

as given (i.e., it is ﬁxed instead of estimated).

On the other hand, ABBC oﬀer two techniques for estimating the parameters of the MS-GARCH models: ML and Bayesian MCMC.

In order to compare with AGK, we use the ﬁrst approach. Let

P( , )

be the vector of parameters. The log-likelihood function is

I I

f rLlog ( ) log ( , )

t t

, where

I I

= =

f r p z f r s j, , ,

t t

i t t t t1

1 1

, 1 1

denotes the conditional density of

given past observations,

t 1

; and

=z s iPr[ , ]

i t t t, 1 1 1

represents the ﬁltered probability of state i at time

t 1

obtained via the

ﬁlter proposed by Hamilton (1989) and Hamilton (1994). The ML estimator,

, is obtained by maximizing the log likelihood

function. For more details, see Trottier and Ardia (2016), and ABBC.

3. Empirical evidence

3.1. Data and preliminary statistics

Weekly series for stock and Forex market returns are built for HI countries (Canada, the U.S., Denmark, Norway, Australia,

Switzerland, the UK, Japan, and Europe) and Latam EMEs (Argentina, Brazil, Chile, Colombia, Mexico, and Peru) from daily

Bloomberg Financial Data. The weekly data are from Wednesday to Wednesday to avoid most public holidays.

11, 12

denotes a

stock market index or an exchange rate,

then the percentage log-return series is deﬁned by

= ×y p p100 [log( ) log( )],

t t 1

where

the index t denotes the weekly closing observations. Then, following ABBC, we demean the returns

using an AR(1) ﬁlter, and use

the ﬁltered returns, denoted by

, to estimate all models. The samples for all countries and markets end on July 24, 2019, but the

beginning of each sample is diﬀerent and explained by the availability of observations. In the case of HI stock markets, the U.S.,

Denmark, Switzerland, Japan, and Europe begin in 1990 (February 26, January 30, January 24, and January 17, respectively). The

starting dates of Latam stock market series are as follows: December 25, 1991 (Argentina); March 15, 1995 (Brazil); August 8, 1990

(Chile); July 25, 2001 (Colombia); March 30, 1994 (Mexico); and February 6, 2002 (Peru). All series for HI Forex markets start in

January 1990 except Canada (August 26, 1998). The starting dates of Latam Forex markets are as follows: July 2, 1999 (Brazil);

January 17, 1990 (Chile); September 2, 1992 (Colombia); May 8, 1996 (Mexico); and May 24, 1995 (Peru). The sample for Argentina

begins on March 5, 2014 given presence of ﬁxed exchange rate periods in that country.

Table 1 shows the descriptive statistics for stock and Forex returns. Panels (a) and (b) show information for (HI and Latam) stock

markets, respectively. HI markets show extreme values between a minimum of −21.23 (Japan) and a maximum of 20.38 (Norway).

Among Latam markets, Brazil and Argentina show extreme values. Chile and Colombia have the smallest maximum values, although

they are higher than for Canada and the U.S. Regarding the standard deviation, all Latam markets are more volatile than HI markets.

Overall, the most volatile markets are Norway, Japan, Argentina, and Brazil. Additionally, all asymmetry coeﬃcients are negative,

with higher values (in absolute values) in Norway, Canada, and the U.S. In general, this coeﬃcient is smaller (in absolute values) in

Latam markets. At the same time, Norway shows the widest departure from Normality. Among Latam markets, Brazil shows high

kurtosis, followed by Peru, Colombia, and Chile, with values higher than the other HI markets.

Panels (c) and (d) show information about (HI and Latam) Forex markets. In general, the observed minimum values (in absolute

values) are lower than in the stock market. The maximum values are also smaller than in the stock markets, except in Norway and

Australia among HI markets and Argentina, Brazil, and Mexico among Latam markets. The standard deviation is no greater than 1.6

in HI countries, lower than in stock markets. In Latam markets, Argentina and Brazil show higher values than HI markets, but smaller

than Latam stock markets. Peru appears to be the less volatile Forex market. It is important to mention that lower volatility in some

Latam markets may be due to central bank Forex market intervention. Additionally, positive and negative asymmetry values are

present. The asymmetry coeﬃcients for HI markets are smaller than for their respective stock markets, except in Australia,

Given this deﬁciency, Gray (1996) proposes replacing the conditional variance

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

t t t t t1 1: 1 1 1: 2 1: 1

=h r rvar[ ]

t t t1 1 1: 2

; i.e.,

collapsing all possible conditional variances at time

t 1

into a single value that does not depend on the regime path. However, AGK shows that

Gray’s method does not generate consistent estimators for the MS-GARCH.

Future research may include estimates using Bayesian MCMC methods. For this paper, given the large number of models that have been

estimated (see next Section), this possibility has been ruled out.

Given the omission of Wednesday data for any given week, we decided to choose some other “feasible day”’ of the week. The criterion for this

choice is based on the construction of a ranking of missing data (from lowest to highest) on each day of the week throughout the daily series,

selecting as the “ﬁrst feasible” the day of the week with fewer omissions. If it did not exist, we selected the following day in the ranking of omissions

as a “second feasible” day; and so on. In this way we built weekly series with no missing data.

Weekly data are used due to the presence of more noise in higher frequencies, such as daily data, which makes it more diﬃcult to isolate cyclical

variations, thereby obscuring the analysis of driving moments of switching behavior; see for instance Moore and Wang (2007).

The exchange rate is measured as domestic currency units per USD.

This allows only 282 observations to be obtained, which represents between 15% or 20% of the samples from the other Latam Forex markets.

M. Ataurima Arellano and G. Rodríguez

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

Switzerland, and the UK. In Latam markets, all asymmetry coeﬃcients are positive, indicating predominance of depreciation epi-

sodes. Argentina has the highest asymmetry value, while kurtosis is higher in Argentina, Brazil, and Peru.

Figs. 1 and 2 show the evolution of (HI and Latam) stock and Forex market returns, respectively. We can see typical stylized facts

such as clusters, higher volatility during the 2008–2009 Global Financial crisis (GFC), asymmetries, and departures from Normality.

All these characteristics are present in both markets across all countries.

3.2. Results

The following models have been estimated for each market across all countries: (i) a single-regime GARCH (1,1)-

(described in

2.1); (ii) the MS-GARCH(1,1)-

model proposed by AGK (described in 2.2); (iii) the ABBC MS-GARCH(1,1) and MS-GJR(1,1) models

(mentioned in 2.3) using innovations

N S N

sk, ,

, and

; and (iv) a single-regime GARCH/GJR model (described in 2.4) using

innovations

S N

sk,

and

. In addition to the models mentioned in (iii), three additional scenarios are contemplated: (i) imposing

restrictions

1,1 1,2

and

1,1 1,2

; (ii) imposing the restriction that the parameter is the same in both regimes (

1 2

); and (iii)

imposing the restriction that the bias parameter

is the same in both regimes (

1 2

). This implies estimating 144 models for each

market across all countries. Since we have 15 stock markets and 14 Forex markets, the total number of models to estimate is 4,437.

The single-regime GARCH model (1,1)-

appears in the ﬁrst row of Tables 2a-3b. The AGK MS-GARCH-

model appears in the

third row, while the best model obtained from the ABBC approach is shown in the fourth row. Following 2.4 above, the best single-

regime GARCH/GJR model appears in the second row.

The best models are selected using 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 a correct identiﬁcation of the main domestic and international events involving volatility stress.

Table 1

Descriptive Statistics for Stock and Forex Markets Returns.

Country Security ID Start Date End Date Obs. Std. Min Max Skew Kurt

(a) Stock – High Income Countries

Canada SPTSX 30-Jan-1991 24-Jul-2019 1487 2.06 −15.31 7.77 −0.73 6.74

USA SPX 24-Jan-1990 24-Jul-2019 1540 2.17 −16.75 9.58 −0.75 7.88

Denmark KFX 26-Dec-1990 24-Jul-2019 1492 2.61 −18.83 11.20 −0.59 6.78

Norway OSEBX 28-Feb-1996 24-Jul-2019 1222 3.08 −20.99 20.38 −0.97 9.42

Australia AS51 10-Nov-1993 24-Jul-2019 1342 1.96 −12.06 11.58 −0.49 6.17

Switzerland SMI 24-Jan-1990 24-Jul-2019 1540 2.46 −14.88 14.00 −0.71 7.31

UK UKX 17-Jan-1990 24-Jul-2019 1541 2.24 −13.05 12.80 −0.37 6.35

Japan NKY 17-Jan-1990 24-Jul-2019 1541 3.06 −21.23 14.71 −0.40 5.84

Europe SX5E 17-Jan-1990 24-Jul-2019 1541 2.80 −15.88 15.15 −0.66 6.53

(b) Stock – Emerging Countries (Latam)

Argentina MERVAL 25-Dec-1991 24-Jul-2019 1440 5.02 −22.77 26.43 −0.29 5.15

Brazil IBOV 15-Mar-1995 24-Jul-2019 1272 4.21 −27.73 31.20 −0.40 10.94

Chile IPSA 08-Aug-1990 24-Jul-2019 1512 2.76 −21.83 13.03 −0.20 8.39

Colombia IGBC 25-Jul-2001 24-Jul-2019 940 2.89 −22.11 14.84 −0.68 9.98

Mexico MEXBOL 30-Mar-1994 24-Jul-2019 1322 3.25 −19.65 17.22 −0.11 6.59

Peru SPBLPGPT 06-Feb-2002 24-Jul-2019 912 3.47 −18.36 24.96 0.07 9.99

Canada CAD 26-Aug-1998 24-Jul-2019 1092 1.20 −5.35 5.99 0.19 5.17

Denmark DKK 17-Jan-1990 24-Jul-2019 1541 1.38 −10.19 7.26 −0.01 6.09

Norway NOK 17-Jan-1990 24-Jul-2019 1541 1.57 −7.71 11.91 0.49 6.51

Australia AUD 31-Jan-1990 24-Jul-2019 1539 1.53 −7.00 16.96 1.08 13.89

Switzerland CHF 31-Jan-1990 24-Jul-2019 1539 1.55 −16.89 8.43 −0.96 14.77

UK GBP 24-Jan-1990 24-Jul-2019 1540 1.32 −4.97 9.94 0.92 8.22

Japan JPY 17-Jan-1990 24-Jul-2019 1541 1.41 −11.77 7.04 −0.57 7.88

Europe EUR 17-Jan-1990 24-Jul-2019 1541 1.37 −10.17 7.70 −0.04 6.25

(d) Forex – Emerging Countries (Latam)

Argentina ARS 05-Mar-2014 24-Jul-2019 282 2.57 −5.26 27.98 5.39 51.82

Brazil BRL 02-Jun-1999 24-Jul-2019 1052 2.41 −18.59 18.35 0.47 13.38

Chile CLP 17-Jan-1990 24-Jul-2019 1541 1.25 −6.36 9.22 0.26 6.83

Colombia COP 02-Sep-1992 24-Jul-2019 1404 1.49 −6.37 8.68 0.31 6.83

Mexico MXN 08-May-1996 24-Jul-2019 1212 1.49 −7.99 11.48 0.68 8.82

Peru PEN 24-May-1995 24-Jul-2019 1262 0.65 −4.99 4.69 0.20 13.72

Other speciﬁcations, such as the one suggested by Gray (1996), have also been considered and estimated. However, given the poor performance

in terms of the signiﬁcance of the parameters and the values of the log-marginal likelihood, these results have been ruled out. An earlier version of

this paper includes those estimates; see Ataurima Arellano et al. (2017), where an MS-mean model (change only in the mean) is also estimated.

M. Ataurima Arellano and G. Rodríguez

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

Fig. 1. Weekly High Income and Emerging (Latam) Stock Market Returns.

M. Ataurima Arellano and G. Rodríguez

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

Fig. 2. Weekly High Income and Emerging (Latam) Forex Market Returns.

M. Ataurima Arellano and G. Rodríguez

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

3.2.1. Stock markets

Tables 2a (HI) and 2b (Latam) show the results for stock markets. A common result is that, in terms of the log-marginal likelihood,

the data never validate the single-regime GARCH-

model. Furthermore, between the two alternative MS-GARCH models, ABBC

obtain the highest value for the log-marginal likelihood (see the fourth row in both tables). Other results are as follows. First, the

distribution is selected for the U.S., Denmark, Switzerland, the UK, and Europe; the Student-t (

) distribution is only chosen for

Canada and Norway; the

distribution is validated by data for Japan; and Australia is the only case where the

innovation is

selected. Second, in general the models require that the

and

parameters be diﬀerent between regimes, except for the U.S.,

Denmark, Switzerland, the UK, and Japan, where the equality restriction (

1,1 1,2

and

1,1 1,2

) is not rejected. Third, the

parameter

is small, reﬂecting the presence and identiﬁcation of heavy tails. The restriction that

= 10.770

is only used in Canada for

both regimes, while Japan shows that regime 1 does not have heavy tails (

= 99.876

), but regime 2 (high volatility) estimates that

Table 2a

Estimated Parameters for Weekly High Income Stock Market Returns.

Model

0,1

1,1

2,1

0,2

1,2

2,2

log-lik

Canada (SPTSX)

GARCH-

0.164

0.147

0.817

−3019.011

GJR-

0.173

0.058

0.146

0.829

12.012

0.780

−2974.511

MS-GARCH-

0.318

0.083

0.745

1.712

0.083

0.745

0.981

0.952

−3013.149

MS-GARCH-

0.293

0.126

0.750

10.770

0.347

0.144

0.818

10.770

0.996

0.993

−3004.626

USA (SPX)

GARCH-

0.183

0.127

0.836

−3216.280

GJR-

0.176

0.006 0.203

0.844

7.791

0.742

−3105.362

MS-GARCH-

0.093

0.015 0.907

2.947

0.015 0.907

0.966

0.664

−3171.702

MS-GARCH-

0.238

0.105

0.788

0.674

0.917

0.105

0.788

0.833

0.990

0.982

−3152.032

Denmark (KFX)

GARCH-

0.331

0.116

0.838

−3444.548

GJR-

0.358

0.057

0.089

0.846

7.628

0.917

−3407.313

MS-GARCH-

0.049

0.008 0.921

2.151

0.008 0.921

0.925

0.698

−3425.307

MS-GJR-

0.099

0.038

0.064

0.867

0.974

0.951

0.038

0.115

0.867

0.736

0.729

0.359

−3403.474

Norway (OSEBX)

GARCH-

0.327

0.145

0.819

−2925.746

GJR-

0.323

0.012 0.121

0.877

10.443

0.741

−2866.553

MS-GARCH-

0.507

0.145

0.716

1.659

0.145

0.716

0.991

0.993

−2917.138

MS-GARCH-

0.146

0.045

0.916

21.261

2.921

0.045

0.916

21.261

0.985

0.817

−2897.624

Australia (AS51)

GARCH-

0.097

0.879

−2692.522

GJR-

0.139

0.000 0.171

0.873

76.181 0.809

−2656.686

MS-GARCH-

0.275

0.099

0.844

0.133

0.099

0.844

0.996

−2693.582

MS-GARCH-

0.055

0.040

0.937

4.886

0.319 0.266 0.994

0.949

−2685.804

Switzerland (SMI)

GARCH-

0.370

0.166

0.778

−3426.322

GJR-

0.513

0.034

0.215

0.761

8.583

0.758

−3340.298

MS-GARCH-

1.444

0.155

0.321

6.324

0.155

0.321

0.980

0.961

−3398.070

MS-GARCH-

1.137

0.174

0.442

0.745

3.976

0.174

0.442

0.748

0.985

0.975

−3367.963

UK (UKX)

GARCH-

0.245

0.148

0.807

−3289.774

GJR-

0.299

0.000 0.269

0.800

11.145

0.796

−3215.772

MS-GARCH-

0.156

0.020 0.888

2.306

0.020 0.888

0.963

0.768

−3270.870

MS-GARCH-

0.173

0.115

0.833

0.788

4.740

0.115

0.833

0.788

0.984

0.290

−3246.697

Japan (NKY)

GARCH-

1.138

0.127

0.753

−3850.347

GJR-

0.913

0.033

0.173

0.780

7.897

0.868

−3791.228

MS-GARCH-

0.741

0.004 0.772

5.739

0.004 0.772

0.925

0.799

−3821.982

MS-GARCH-

0.471

0.112

0.801

99.876

0.533

0.994

0.112

0.801

6.261

0.945

0.886

0.954

−3800.883

Europe (SX5E)

GARCH-

0.309

0.148

0.814

−3590.995

GJR-

0.320

0.054

0.146

0.824

10.118

0.755

−3518.341

MS-GARCH-

0.120

0.009 0.937

2.563

0.009 0.937

0.979

0.854

−3573.478

MS-GARCH-

0.293

0.050

0.859

0.756

1.505

0.183

0.704

0.756

0.993

0.991

−3537.931

a, b, c denote signicance level at 1%, 5% and 10% respectively.

M. Ataurima Arellano and G. Rodríguez

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

= 6.261

. Fourth, regarding the bias parameter ( ), all estimates show values smaller than the unity, indicating a left bias; i.e.,

towards negative returns. This is consistent with the rational behavior hypothesis proposed by Samuelson (1970) and Lorenzo-Valdes

and Ruiz-Porras (2014). Fifth, only Denmark shows leverage eﬀects (MS-GJR), with

= 0.115

2,2

for regime 2.

The calculation of unconditional volatilities in each regime suggests that regime 2 (high volatility) is usually twice as volatile as

regime 1. There are cases where regime 2 is much more volatile relative to regime 1, like in Denmark (4 times), and Norway and the

UK (5 times). The persistence of volatility in both regimes is also calculated using the best model estimated by ABBC. All countries,

except Switzerland, show high persistence for the high-volatility regime (0.91 or higher). However, Switzerland shows the lowest

persistence (0.616). Using the estimates performed by AGK, Denmark and Europe show the highest persistence, while the lowest

persistence is observed in Switzerland (0.476) and Japan (0.773).

Table 2b (Latam) suggests the following. First, among ABBC models, the

distribution is only selected in the case of Mexico,

whereas the

distribution is selected for Argentina. The

distribution is selected for the remaining Latam countries (Brazil,

Chile, Colombia, and Peru), indicating the presence of bias and heavy tails in the distributions. Second, in general, the models require

that the

and

parameters be diﬀerent between regimes, except for Brazil and Mexico, where the equality restriction (

1,1 1,2

and

1,1 1,2

) is not rejected. Third, the parameter (only estimated for Brazil, Chile, Colombia, and Peru) is small, reﬂecting the

presence and identiﬁcation of heavy tails. However, this parameter is ﬁxed (around 11) in both regimes for Colombia and Peru. In the

case of Chile, a large asymmetry is observed in the estimation of this parameter (99.703 in regime 1 and 5.759 in regime 2). This

shows that regime 2 (high volatility) is characterized by heavy tails, which also happens in the case of Japan. Fourth, regarding the

bias parameter (

), the estimates show values smaller than the unity, indicating a left bias; i.e., towards negative returns. This

happens for all countries in the high-volatility region (regime 2). In the cases of Argentina, Chile, and Colombia, the hypothesis that

this parameter is the same in both regimes (

1 2

) cannot be rejected. Fifth, leverage eﬀects (through MS-GJR) are not detected in

any country.

The calculation of unconditional volatilities in each regime suggests that regime 2 (high volatility) is usually more volatile than

regime 1. For example, in Chile the high-volatility regime is twice as volatile as regime 1, and considerably more in Argentina (4

times), Mexico (5 times), Peru (6 times), and Colombia (10 times). Only in Brazil are unconditional volatilities similar in both

regimes. This ﬁnding is consistent with the ﬁndings shown in Table 2a (HI markets), because volatility in Latam countries is much

Table 2b

Estimated Parameters for Weekly Emerging (Latam) Stock Market Returns

Model

0,1

1,1

2,1

0,2

1,2

2,2

log-lik

Argentina (MERVAL)

GARCH-

1.627

0.117

0.816

−4264.286

GJR-

0.961

0.066

0.089

0.848

7.737

0.879

−4222.621

MS-GARCH-

0.327

0.030

0.908

7.425

0.030

0.908

0.943

0.723

−4239.723

MS-GARCH-

0.101

0.016

0.964

0.876

3.957 0.075

0.866

0.876

0.960

0.914

−4224.661

Brazil (IBOV)

GARCH-

0.793

0.124

0.828

−3486.504

GJR-

1.203

0.052

0.134

0.811

8.757

0.815

−3438.180

MS-GARCH-

1.747

0.096

0.752

168.324

0.096

0.752

0.995

0.001 −3470.578

MS-GARCH-

1.155

0.122

0.817

11.479

1.091

1.021

0.122

0.817

7.787

0.721

0.996

0.991

−3441.919

Chile (IPSA)

GARCH-

0.103

0.100

0.892

−3526.751

GJR-

0.257

0.121

0.030 0.834

6.411

0.925

−3477.233

MS-GARCH-

0.168

0.000 0.915

3.116

0.000 0.915

0.965

0.806

−3490.151

MS-GARCH-

0.887

0.188

0.452

99.703

0.915

0.726

0.111

0.819

5.759

0.915

0.994

0.997

−3468.753

Colombia (IGBC)

GARCH-

0.552

0.149

0.787

−2236.533

GARCH-

0.879

0.193

0.702

5.504

0.911

−2196.250

MS-GARCH-

0.589

0.045

0.766

9.947

0.045

0.766

0.958

0.598

−2207.241

MS-GARCH-

0.014 0.006

0.979

11.155

0.864

0.484

0.124

0.871

11.155

0.864

0.894

0.898

−2191.718

Mexico (MEXBOL)

GARCH-

0.110

0.124

0.874

−3239.792

GJR-

0.130

0.070

0.110

0.868

8.824

0.877

−3206.752

MS-GARCH-

0.314

0.082

0.842

3.012

0.082

0.842

0.984

0.933

−3240.842

MS-GARCH-

0.068

0.039

0.941

1.753

0.039

0.941

0.984

0.814

−3221.202

Peru (SPBLPGPT)

GARCH-

0.353

0.177

0.796

−2257.710

GJR-

0.266

0.101

0.050 0.846

5.416

1.016

−2224.389

MS-GARCH-

0.202

0.000 0.922

7.828

0.000 0.922

0.979

0.797

−2227.305

MS-GARCH-

0.051

0.011

0.969

11.771

1.014

2.531 0.154

0.814

11.771

0.988

0.976

0.920

−2217.607

a, b, c denote signicance level at 1%, 5% and 10% respectively.

M. Ataurima Arellano and G. Rodríguez

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

higher in regime 2 (high volatility). The persistence of volatility in both regimes is also higher in these markets compared to Table 2a

(HI markets). The most persistent are Colombia (0.995), Mexico (0.980), and Peru (0.968), while the least persistent are Brazil

(0.939) and Chile (0.930). These values indicate that the high-volatility regime (regime 2) in Latam markets is more persistent than in

HI markets. This implies that these episodes of turbulence are longer and more uncertain in Latam markets and that the adjustment is

usually slow. Using AGK estimates, the persistence values of the high-volatility regime are usually lower than the ones indicated

above, but remain higher compared to those of HI markets.

3.2.2. Forex markets

Tables 3a (HI) and 3b (Latam) show the results for Forex markets. As in Tables 2a-2b, in terms of log-marginal likelihood, the

single-regime GARCH-

model is never validated by the data, except for Canada, where a similar ﬁt as in AGK is observed. The

models estimated using ABBC yield the highest log-marginal likelihood value (see the fourth row). Other results are as follows. First,

among the models estimated by ABBC, the

distribution is never selected, which is an important diﬀerence with the results for

stock markets. The

distribution is selected for Canada and the

for Europe. For most remaining countries (six),

dis-

turbances are selected, indicating the presence of bias and heavy tails in the distributions, in contrast with stock markets (Table 2a).

Second, all countries require diﬀerent

and

between regimes; i.e., unlike in some stock markets, the hypothesis that

1,1 1,2

and

Table 3a

Estimated Parameters for Weekly High Income Forex Market Returns.

Model

0,1

1,1

2,1

0,2

1,2

2,2

log-lik

Canada (CAD)

GARCH-

0.016

0.068

0.919

−1641.517

GARCH-

0.015

0.067

0.921

97.589

1.043

−1641.028

MS-GARCH-

0.023

0.057

0.920

2.755

0.057

0.920

0.999

0.812

−1641.746

MS-GJR-

0.007

0.030

0.000

0.960

0.864

0.074

0.135

0.000

0.841

1.415

0.984

0.961

−1633.873

Denmark (DKK)

GARCH-

0.033

0.062

0.920

−2605.826

GARCH-

0.037

0.059

0.920

10.632

1.031

−2589.642

MS-GARCH-

0.021

0.000 0.970

0.250

0.000 0.970

0.979

0.886

−2594.190

MS-GARCH-

0.006

0.017

0.975

11.570

1.020

0.149

0.075

0.877

11.570

1.020

0.991

0.980

−2585.717

Norway (NOK)

GARCH-

0.117

0.093

0.858

−2788.855

GARCH-

0.118

0.085

0.863

14.729

1.120

−2774.963

MS-GARCH-

0.064

0.069

0.861

0.253

0.069

0.861

0.919

0.932

−2787.339

MS-GARCH-

0.005 0.037

0.954

18.784

0.815

0.179

0.092

0.847

18.784

1.319

0.975

0.984

−2768.436

Australia (AUD)

GARCH-

0.070

0.089

0.881

−2703.267

GARCH-

0.045

0.067

0.913

11.457

1.151

−2680.467

MS-GARCH-

0.070

0.020

0.923

1.075

0.020

0.923

0.989

0.828

−2690.934

MS-GJR-

0.038

0.000

0.936

78.338

1.138

0.877

0.000

0.901

6.932

1.101

0.994

0.924

−2679.270

Switzerland (CHF)

GARCH-

0.216

0.082

0.833

−2818.165

GJR-

0.057

0.051

0.008 0.919

7.004

0.914

−2721.105

MS-GARCH-

0.043

0.000 0.939

1.338

0.000 0.939

0.960

0.489

−2736.796

MS-GARCH-

0.007 0.008

0.978

7.714

0.916

0.268

0.041

0.854

7.714

0.916

0.990

0.997

−2710.422

UK (GBP)

GARCH-

0.044

0.073

0.902

−2498.914

GARCH-

0.045

0.057

0.915

9.400

1.111

−2463.837

MS-GARCH-

0.315

0.020 0.918

0.067

0.020 0.918

0.998

0.984

−2477.169

MS-GJR-

0.045

0.018

0.000

0.940

33.968

1.020

0.778

0.118

0.003

0.664

8.830

1.322

0.996

0.981

−2453.528

Japan (JPY)

GARCH-

0.098

0.090

0.864

−2662.570

GJR-

0.085

0.058

0.017 0.889

6.424

0.908

−2605.315

MS-GARCH-

0.061

0.000 0.913

0.699

0.000 0.913

0.952

0.768

−2621.528

MS-GARCH-

0.001 0.006

0.988

8.013

0.949

0.154

0.041

0.905

8.013

0.864

0.960

0.968

−2602.260

Europe (EUR)

GARCH-

0.035

0.066

0.916

−2586.209

GARCH-

0.037

0.062

0.917

10.526

1.014

−2570.000

MS-GARCH-

0.025 0.000 0.963

0.474

0.000 0.963

0.975

0.755

−2572.376

MS-GARCH-

0.009

0.019

0.970

11.419

0.150

0.078

0.871

11.419

0.989

0.981

−2566.737

a b c, ,

denote signicance level at 1%, 5% and 10% respectively.

M. Ataurima Arellano and G. Rodríguez

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

1,1 1,2

is always rejected. Third, the parameter (estimated for all countries except Canada) shows values that may be considered

suitable for capturing heavy tails. In Denmark and Europe, values of

are around 11. Japan and Switzerland show smaller estimates

(between 7.714 and 8.013) and in both cases the hypothesis that this parameter is the same (

1 2

) in both regimes is not rejected.

Norway shows a ﬁxed

estimate of 18.784 for both regimes. In the cases of the UK and Australia, an interesting asymmetry is

observed in the estimates for

. In regime 1, the value of this parameter is high and the existence of Normality can be suggested.

However, in regime 2 (high volatility),

= 8.830

and

= 6.932

for the UK and Australia, respectively, indicating heavy tails in the

high-volatility regime. Fourth, regarding the bias parameter (

), the estimates show values greater than unity for Canada, Norway,

Australia, the UK, and Denmark. In this case, values greater than the unity indicate a bias towards the right side (positive returns

values); i.e., a greater presence of depreciation episodes. The most notable cases are Canada, Norway, and the UK. In the cases of

Switzerland and Japan, the values are lower than the unity, suggesting a bias towards the appreciation of domestic currencies. Only

in Switzerland and Denmark the hypothesis that this parameter is the same (

1 2

) in both regimes cannot be rejected. Fifth, positive

leverage eﬀects (MS-GJR) are observed in Canada, Australia, and the UK, but the magnitudes are extremely small.

The calculation of unconditional volatilities in each regime suggests that regime 2 is usually more volatile than regime 1. In most

cases (Canada, Denmark, Norway, Switzerland, The UK, and Europe), regime 2 is twice as volatile as regime 1 (4 and 2.5 times higher

in Japan and Australia, respectively). In general, volatility is more moderate than in HI stock markets (Table 2a). The persistence of

volatility in both regimes is high. The most persistent case is Canada (0.979), with a value around 0.95 in the other countries. The

lowest levels of persistence occur in Switzerland (0.985), Australia (0.902), and the UK (0.784). Using AGK estimates, the most

persistent countries are Canada, and Europe, while the rest show lower levels. However, compared with the stock market (Table 2a),

AGK does not seem to underestimate the probability

Like previous Tables, Table 3b shows that, in terms of the log-marginal likelihood values, the single-regime GARCH-

model is

never validated by the data, Argentina being the clearest example. The models estimated using ABBC yield the highest log-marginal

likelihood values (fourth row). Other comments are as follows. First, among the diﬀerent models estimated using ABBC, the

N N

sk,

and S distributions are never selected, which is an important diﬀerence with Latam stock markets (Table 2b). In all cases,

disturbances are chosen, indicating the presence of bias and heavy tails in the distributions. The

distribution was selected for

most HI Forex markets, but in Latam countries the evidence is stronger. Second, half of the countries (Brazil, Colombia, and Mexico)

Table 3b

Estimated Parameters for Weekly Emerging (Latam) Forex Market Returns.

Model

0,1

1,1

2,1

0,2

1,2

2,2

log-lik

Argentina (ARS)

GARCH-

4.199

0.147

0.254

−658.423

GARCH-

0.000

0.033 0.967

2.467

1.356

−391.582

MS-GARCH-

0.002 0.798

0.416

44.795

0.798

0.416

0.967

0.268 −420.537

MS-GARCH-

0.000

0.065

0.934

2.187

1.512

2.732

0.221

0.779

2.187

1.268

0.986

0.988

−379.335

Brazil (BRL)

GARCH-

0.245

0.173

0.791

−2270.085

GARCH-

0.301

0.175

0.798

5.414

1.242

−2223.117

MS-GARCH-

0.183

0.064

0.841

7.056

0.064

0.841

0.974

0.505

−2241.482

MS-GARCH-

0.639

0.182

0.751

3.343

1.798

0.577

0.182

0.751

6.354

1.138

0.989

0.966

−2220.142

Chile (CLP)

GARCH-

0.036

0.115

0.863

−2383.337

GJR-

0.000 0.071 0.000 0.928

4.914

1.041

−2294.421

MS-GARCH-

0.008

0.049

0.744

0.607

0.049

0.744

0.847

0.854

−2308.896

MS-GARCH-

0.000 0.003

0.997

2.143

1.121

0.042

0.086

0.898

7.054

1.121

0.968

0.994

−2226.545

Colombia (COP)

GARCH-

0.047

0.154

0.826

−2293.218

GARCH-

0.018

0.119

0.871

5.734

1.095

−2232.589

MS-GARCH-

0.002 0.011 0.924

0.654

0.011 0.924

0.944

0.786

−2246.028

MS-GARCH-

0.003 0.085

0.902

5.785

1.094

0.165

0.085

0.902

21.396 1.267

0.944

0.757

−2218.962

Mexico (MXN)

GARCH-

0.113

0.099

0.851

−2112.664

GARCH-

0.067

0.121

0.854

5.473

1.270

−2014.534

MS-GARCH-

0.069

0.000 0.885

1.739

0.000 0.885

0.954

0.618

−2048.274

MS-GARCH-

0.114

0.102

0.826

3.875

1.307

0.226

0.102

0.826

9.470

1.226

0.994

0.993

−2008.449

Peru (PEN)

GARCH-

0.011

0.191

0.790

−979.722

GARCH-

0.015

0.229

0.757

4.104

1.192

−899.181

MS-GARCH-

0.003 0.084

0.800

0.196

0.084

0.800

0.913

0.658

−930.948

MS-GJR-

0.015

0.325

0.000

0.606

4.018

2.466

0.034

0.197

0.000

0.761

4.018

1.110

0.920

0.982

−880.529

a b c, ,

denote signicance level at 1%, 5% and 10% respectively.

M. Ataurima Arellano and G. Rodríguez

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

do not reject the hypothesis that

1,1 1,2

and

1,1 1,2

, while for the other half (Argentina, Chile, and Peru) the evidence points to

the opposite. Third, the results for the parameter

are clearly diﬀerent from those obtained for stock markets (HI and Latam, Tables

2a and 2b) and HI Forex markets (Table 3a), because the values are smaller; e.g.,

= 2.187

in Argentina for both regimes. In the other

countries the values are between 2.143 and 9.470, thereby rejecting the hypothesis that

1 2

in most countries. Only Colombia

shows

= 21.396

in regime 2 (high volatility) compared to

= 5.785

for regime 1. Fourth, regarding the bias parameter ( ), the

estimates show values greater than unity in all countries and for both regimes. In addition, the values have magnitudes greater than

those found in all other markets and countries. Values greater than the unity indicate a bias towards the right; i.e., greater presence of

depreciation episodes. The most notable cases are Argentina, Colombia, and Mexico. Fifth, leverage eﬀects (MS-GJR) are only ob-

served for Peru, the magnitude being very small in regime 1, while in regime 2 the value is 0.034, higher than for HI markets.

The calculation of unconditional volatilities in each regime suggests that regime 2 (high volatility) is usually more volatile than

regime 1. The lowest levels are for Mexico and Peru, where volatility in regime 2 is 1.5–2.0 times the volatility in regime 1. However,

in the other countries, volatility in regime 2 is much higher than in regime 1; i.e., 8, 20, and 60 times greater in Colombia, Chile, and

Argentina (the most extreme case), respectively. The values suggest a more volatile behavior in regime 2 compared with HI Forex

markets and Latam stock markets. In the case of Peru (one of the lowest), it seems that central bank intervention in the Forex market

reduces stress in the high-volatility regime. The persistence of volatility in both regimes is high: Argentina 0.999, Chile 0.983,

Colombia 0.987, and Peru 0.957. The other countries show less persistent high-volatility regimes (0.928 or 0.933). Using AGK

estimates, Argentina shows an explosive persistence (1.210), given that covariance stationarity is not imposed. The least persistent

are Chile (0.793), Peru (0.883), and Mexico (0.893). We conﬁrm that AGK always estimates lower persistence than ABBC, except in

the case of Argentina.

3.2.3. Identiﬁcation and characteristics of high-volatility periods

Figs. 3 and 4 show two panels for each country. The upper panel shows two series: squared returns (gray) and ﬁltered volatility

(blue) extracted from the best-estimated model using ABBC; see fourth row in Tables 3a through 3b. The lower panel shows two series

of smoothed probabilities associated with regime 2 (high volatility): those extracted using AGK in red and those extracted using the

ABBC approach in blue. Both panels show how well the smoothed probabilities follow the patterns of the ﬁltered volatilities, which in

turn follow the behavior of the squared returns. Figs. 3 and 4 show the results for the stock and Forex markets, respectively.

Four aspects are evidenced from Figs. 3 and 4. First, in most of cases, the smoothed probabilities obtained from the best models

estimated using ABBC reﬂect better the behavior of the ﬁltered volatilities. This is achieved by including skewed innovations such as

and

. In various cases, the smoothed probabilities obtained from the ABBC models seem to form an envelope of the

probabilities extracted by AGK. In general, the distributional ﬂexibility allowed by ABBC produces more persistent smoothed

probabilities in regime 2. This ﬂexibility is not allowed by AGK. Second, although the log-marginal likelihoods of AGK are smaller

than those obtained by ABBC, there are some cases where the smoothed probabilities are better behaved under the ﬁrst approa-

ch; i.e., the smoothed probabilities are more similar to the movements of the ﬁltered volatilities (AGK envelopes ABBC). Even

comparing with the evolution of the returns (Figs. 1 and 2) we ﬁnd the same observation. Third, there are cases where both types of

methods allow noise and seem to try to capture all movements of ﬁltered volatilities. Fourth, in some cases it is diﬃcult to argue in

favor of one of the two methods or models, given the noise in both series of smoothed probabilities under regime 2.

Fig. 3 shows the case of HI and Latam stock markets. The smoothed probabilities of regime 2 are best identiﬁed (not noisy) for

Canada, the U.S., and Europe when ABBC is used. The blue line looks like an envelope of the red line (AGK). However, it is worth

noting the following aspects: (i) the blue line appears as the envelope for 1997–2003 and 2007–2012, covering the GFC (2008–2009)

and the European debt crisis; and (ii) the small jumps on the blue line (upper panel) in 1994 are only captured by AGK. The same

happens with the 2015–2016 and 2019 episodes. In sum, ABBC MS-GARCH models capture volatility persistence in regime 2, but fail

to capture some particular jumps. Similar results are observed for the U.S. market, while the estimation made using ABBC seems to

capture well the probabilities of regime 2. The case of Europe is another good illustration of the qualities of the ABBC method

compared with AGK. However, an opposite result occurs for Norway and Australia. In both cases, the probabilities of regime 2

estimated by AGK are more persistent and appear to be an envelope of the results obtained from ABBC, although in the case of

Australia there seems to be an overestimation of the probabilities under regime 2 using AGK. In the case of Switzerland, both models

capture in the same way (almost identical) the probabilities and persistence of regime 2. In the cases of Denmark, UK, and Japan it is

diﬃcult to decide which of the two models performs better. In Denmark, both estimates of the smoothed probabilities are very noisy.

However, the red line (AGK) seems to perform relatively better. A similar observation can be made in the case of the UK. The case of

Japan is the most diﬃcult to pin down.

Fig. 3 also shows the results for Latam stock markets. In the cases of Brazil and Chile, the smoothed probabilities for the high-

volatility regime are well captured by ABBC; i.e., while AGK identiﬁes four points associated with the high-volatility regime, the

ABBC approach considers wide and persistent probabilities. In the case of Mexico, the smoothed probabilities obtained using AGK

appear as the envelope of the other method. However, ﬁltered volatilities (top panel) suggest jumps in 2011–2012 and 2019, which

are only detected by ABBC (blue line). In the case of Peru, similar smoothed probabilities are observed, but the ABBC approach oﬀers

better results. In the cases of Argentina and Colombia, noisy probabilities are obtained, although with relative advantage for the

speciﬁcation estimated using ABBC.

Fig. 4 shows the case of HI and Latam Forex markets. The ﬁrst Figures correspond to HI markets. Unlike stock markets, ABBC

show more persistence in regime 2 (high volatility) than AGK. In Canada, Switzerland, and Europe this seems to be clearly the case. In

the cases of Denmark, Australia, and the UK, both approaches yield similar results in terms of the smoothed probabilities of regime 2.

Norway appears as a diﬃcult case. Initially both methods work in a similar way, but ABBC oﬀer better results starting in 1999.

M. Ataurima Arellano and G. Rodríguez

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

(caption on next page)

M. Ataurima Arellano and G. Rodríguez

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

The rest of Fig. 4 shows the case of Latam Forex markets. Argentina is a special case, as long as it maintained a ﬁxed exchange rate

regime for a long time. Indeed, Argentina shows the highest ﬁltered volatility values across all countries in both markets. AGK selects

only a small number of weeks as belonging to the high-volatility regime. However, ABBC identify a small jump in 2015, and all

observations from 2016 until the end of the sample are cataloged as regime 2 observations. In the case of Brazil, ABBC estimates seem

like an envelope of the results obtained using the other method. Chile also shows a very persistent high-volatility regime. AGK seems

to perform relatively better in the case of Colombia. In the case of Mexico, both methods yield similar estimates, but ABBC capture

additional high-volatility observations in 2011–2012 and 2019. In the case of Peru, both methods yield similar performances, al-

though ABBC oﬀer relatively better estimates.

From the previous analysis, we can distinguish three main stress episodes which are common across countries and markets over

time: (i) 1994–2004 (Dotcom bubble, September 11 attacks); (ii) 2007–2012 (subprime mortgage crisis, GFC, European debt crisis);

and (iii) 2014–2016 (China’s economic slowdown, trade wars, oil price crisis). First, we count the number of countries experiencing

high-volatility episodes per week. A main observation is that ABBC capture more countries facing high volatility than AGK, which is

more notorious in Latam Forex markets. For instance, AGK does not identify any Latam stock market in the third stress period, but

does for Forex markets in the ﬁrst stress period.

Second, for simpliﬁcation, subsets of countries are chosen for each period of common stress, where some of the MS-GARCH

methods capture at least one high-volatility period.

For explanatory purposes, we take the Canadian stock market for the ﬁrst stress

period. In this case, AGK detects 5 high-volatility sub-periods with diﬀerent duration (172 weeks in all). However, using an MS-

GARCH-

, ABBC select only one long high-volatility period lasting 271 weeks. A similiar issue is observed for the second stress

period. Other very similar examples are the U.S. and Chile. In the case of the U.S., AGK detects 7 high-volatility sub-periods (48 weeks

in all). At the same time, using an MS-GARCH-

, ABBC select only two long sub-periods (47 and 257 weeks, respectively). In the

case of Chile, AGK identiﬁes 12 high-volatility sub-periods (111 weeks in all), whereas ABBC consider only one very long 713-week

sub-period (using an MS-GARCH-

). The results are similar for the second and third stress periods in Europe, Peru, and Colombia.

In all these cases, ABBC look like an envelope of AGK. A special case is Mexico, where a similar speciﬁcation (MS-GARCH-

with

= =,

1 2

) was estimated by AGK and ABBC, but the persistence of the-high volatility regime of the former envelopes the latter.

Findings are similar in Forex markets: a relevant case is Canada, where AGK is unable to detect any high-volatility period in the

ﬁrst and third stress periods, while ABBC do it using an MS-GJR-

3.2.4. Other models

While ABBC are better than AGK models in measuring performance in terms of the log-marginal likelihood, there are other single-

regime models that may yield better log-marginal likelihood values. These are simpler models in that they do not require non-linear

MS-type modeling, as mentioned in 2.4. These are single-regime GARCH or GJR models, but with ﬂexibility in the distribution, like

Ardia et al. (2019a) and Ardia et al. (2019b). We have selected the best models according to the values of the log-marginal likelihood

(second row of Tables 2 and 3). Many of these models are not only better than the single-regime GARCH-

model (ﬁrst row of Tables

2 and 3), but in several cases they outperform the best models selected using ABBC.

In HI stock markets (Table 2a), all models, except for Denmark, have better log-marginal likelihood values compared with ABBC.

Another important detail is that, in all cases, the selected model is a single-regime GJR-

. Other points to highlight are the

following. First, the parameter

is between 7.897 (Japan) and 12.612 (Canada). In all cases it is statistically signiﬁcant, except for

Australia, where a higher value is obtained (

= 76.181

). Second, the bias parameter ( ) is always smaller than unity, indicating

asymmetries towards negative returns. Third, there are leverage eﬀects in all cases, all statistically signiﬁcant. The values are high

and range between 0.121 (Norway) and 0.269 (UK). Fourth, the volatility persistence values are between 0.902 and 0.960.

In the case of Latam stock markets (Table 2b), only in Colombia, Chile, and Peru, ABBC modeling yields better log-marginal

likelihood values. In the remaining cases (Argentina, Brazil, and Mexico), the single-regime GJR-

model (the same model as in

the case of selected HI markets) a better ﬁt (Table 2a). Thus, we note the following: (i) the values of parameter

are smaller than

those for HI stock markets (Table 2a); (ii) the bias parameter (

) is also higher than for HI stock markets; and (iii) the leverage eﬀects

are smaller than for HI stock markets (and are not even signiﬁcant for Chile).

In the case of the HI Forex markets (Table 3a), the results obtained from ABBC are hard to beat. Only in the case of Australia a

single-regime GARCH-

model manages to have a log-marginal likelihood almost equal to that obtained using ABBC. In this case

= 11.457

, while the asymmetry parameter

= 1.151

. The latter reﬂects a bias towards the right side, i.e., a predominance of

depreciation episodes. The persistence of the high-volatility regime is very similar to that obtained with ABBC. In the cases of Japan

and Europe, the log-marginal likelihoods of these models are close to those obtained by ABBC, always using a

distribution. In

sum, for this market there are advantages in using MS-GARCH/MS-GJR models, especially those selected using ABBC.

Table 3b shows the results for Latam Forex markets. In no case can a single-regime GARCH or GJR model beat an MS-GARCH/MS-

GJR model estimated using ABBC. The only case where the log-marginal likelihoods are close is Brazil, where an

distribution is

Fig. 3. Filtered Volatility and Smoothed Probabilities for Stock Markets Returns. First panels shows ﬁltered volatility using the MSGARCH model of

ABBC (blue line) and the square of the returns (gray line). Second panels shows: smoothed probabilities of high volatility regime of MSGARCH

model of AGK (red line); and the best MSGARCH model of ABBC (blue line).

A detailed description for all countries is available upon request.

M. Ataurima Arellano and G. Rodríguez

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

Fig. 4. Filtered Volatility and Smoothed Probabilities for Forex Markets Returns. First panels shows ﬁltered volatility using the MSGARCH model of

ABBC (blue line) and the square of the returns (gray line). Second panels shows: smoothed probabilities of high volatility regime of MSGARCH

model of AGK (red line); and the best MSGARCH model of ABBC (blue line).

M. Ataurima Arellano and G. Rodríguez

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

selected again.

The following are some conclusions about this discussion: (i) there are some single-regime models with ﬂexible distributional

characteristics that yield better log-marginal likelihood values; (ii) in (HI and Latam) stock markets, a single-regime model always

yields a better log-marginal likelihood than ABBC (with the only exception of Chile); (iii) regarding (HI and Latam) Forex markets,

there is a larger number of countries where the log-marginal likelihood of ABBC MS-GARCH/MS-GJR models is higher; i.e., com-

pared with stock markets, in Forex markets it seems necessary to introduce MS-type non-linearities.

4. Conclusions

This document seeks to contribute to the empirical literature by modeling and analyzing the volatility of returns in stock and

Forex markets for HI countries and Latam EMEs. Using a sample of markets and countries, as well as a broad set of single-regime

GARCH/GJR and MS-GARCH/MS-GJR models, this document pursues the following objectives: estimating and analyzing the be-

havior of the high-volatility regime while identifying events associated with stress periods; calculating the persistence of this regime;

and identifying the presence of biases, heavy tails, and leverage eﬀects based on the distributions selected for the estimations. The

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 probability curve for the high-volatility regime associated with the correct identiﬁcation

of the main domestic and internationals events involving volatility stress.

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

stock and Forex markets, using a wide variety of single-regime GARCH/GJR and MS-GARCH/MS-GJR models with diﬀerent in-

novations. 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 more signiﬁcant and 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.

In most cases, persistence in the high-volatility regime is better captured using the ABBC method. This may be due to the fact that

it allows the use of non-Gaussian distributions and is more ﬂexible than AGK. At the same time, we have also found cases where AGK

and ABBC estimates are very similar, and in some cases the estimates of the smoothed probabilities are better using AGK. While it is

true that selecting periods with more persistence can help to identify high-volatility periods, in several cases the duration of the latter

is long and diﬃcult to validate. If this is not due to distributional ﬂexibility, then it may be linked to the estimation method; i.e.,

while AGK addresses the problem of path dependence, ABBC do not. In this regard, an avenue of future research may be to extend the

AGK method to allow ﬂexible distributions and see if the resulting estimations make high-volatility regimes more persistent.

The results show distinctively diﬀerent volatility behaviors and dynamics in stock and Forex markets and between HI countries

and Latam EMEs. Latam markets appear to be more volatile and need heavier tails and biases in their distributions. This has sub-

stantial implications for modeling and calculating risk measures in these markets and countries. Therefore, a possible extension of this

document is to evaluate the diﬀerent models in terms of forecasting. The calculation of risk measures following the ABBC approach,

as well as the Bayesian estimation of these models, is another issue for study.

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