Article

Stochastic Volatility Journal of Emerging Market Finance

17(3s) 1–32

in the Peruvian © 2018 Institute for Financial

Management and Research
SAGE Publications
Stock Market and sagepub.in/home.nav
DOI: 10.1177/0972652718800560
Exchange Rate http://journals.sagepub.com/home/emf

Returns: A Bayesian
Approximation*

Willy Alanya1
Gabriel Rodríguez1

Abstract
This study is one of the first to utilize the stochastic volatility (SV)
model to modelling the Peruvian financial times series. We estimate and
compare this model with generalized autoregressive conditional heter-
oscedasticity (GARCH) models with normal and t-student errors. The
analysis in this study corresponds to Peru’s stock market and exchange
rate returns. The importance of this methodology is that the adjustment
of the data is better than the GARCH models, using the assumptions of
normality in both models. In the case of the SV model, three Bayesian
algorithms have been employed where we evaluate their respective
inefficiencies in the estimation of the model’s parameters—the most

* This article is drawn from the thesis of Willy Alanya (2014) at the Department of
Economics, Pontificia Universidad Católica del Perú (PUCP). We thank useful comments
of Paul Castillo B. (Central Reserve Bank of Peru and PUCP), Rodolfo Cermeño
(CIDE-Mexico), Luis García (PUCP), Pierre Perron and Z. Qu (Boston University).
We also appreciate comments from one anonymous referee and the editor of the journal.
Any remaining errors are our responsibility.
1
Pontificia Universidad Católica del Perú, Lima, Peru.

Corresponding author:
Gabriel Rodríguez, Department of Economics, Pontificia Universidad Católica del Perú,
Av. Universitaria 1801, Lima 32, Lima, Peru.
E-mail: gabriel.rodriguez@pucp.edu.pe
2 Journal of Emerging Market Finance 17(3s)

efficient being the integration sampler. The estimated parameters in the

SV model under the various algorithms are consistent, as they display
little inefficiency. The figures of the correlations of the iterations sug-
gest that there are no problems at the time of Markov chaining in all
estimations. We find that the volatilities in the exchange rate and stock
market volatilities follow similar patterns over time. That is, when eco-
nomic turbulence caused by the economic circumstances occurred, for
example, the Asian crisis and the recent crisis in the USA, considerable
volatility was generated in both markets.

JEL Classification: C22

Keywords
Stochastic volatility model, Bayesian estimation, Gibbs sampler, mixture
sampler, integration, stock market, forex market, GARCH models, Peru

1. Introduction
Many financial series, especially stock market and exchange rate market
returns, are characterised by their volatile behaviour, where periods
of major stability and times of high uncertainty can be appreciated—a
phenomenon known in the literature as clustering. These clusters of
volatility can be caused by economic or political factors that affect the
perception of investors about the stock market and the agents of the
economy with regard to the exchange rate. Stock markets offer to investors
a more liquid alternative of investment relative to other instruments in
the financial system (Levine, 1996). Moreover, stock markets contribute
to the economic growth in emerging market economies; see Enisan and
Olufisayo (2009). However, in the short run, stock markets can amplify
external shocks as shareholders reallocate their investment portfolios,
for instance, they may opt to sell their shares. Hence, it is important to
study this market. With respect to the forex market, since 2002, the
Central Bank of Peru has aimed to smoothen the exchange rate volatility
in line with the inflation-targeting scheme. A greater exchange rate
volatility implies a higher risk in a dollarised Peruvian economy as
individuals may take loans or save in the foreign currency and are
vulnerable to exchange rate depreciations (Grippa & Gondo, 2006).
Despite that the stock and forex market are crucial for the dynamics of
the Peruvian economy, there is a paucity of studies dedicated to stock and
Alanya and Rodríguez 3

exchange rate returns.1 To understand the dimension of the Lima stock

market in Peru, during 2010, Lima stock market traded around US$6748.5
and reported an equity level equivalent to 104.6 per cent of the Peruvian
GDP. In addition, the Peruvian economy is characterised by a high level
of dollarisation. Around a third of the total credits and deposits in the
financial system was denominated in US dollars by 2010. Thus, one of the
academic impacts that can be taken into consideration is the determina-
tion of periods of volatility in these markets; these estimates are used as
risk aversion analysis for listed companies, as well as the possible effects
on the process of economic growth, arbitrage and decision-making of
economic agents (e.g., the export sector). Two branches have emerged in
the literature to model volatilities in these markets: generalised autore-
gressive conditional heteroscedasticity (GARCH) models and stochastic
volatility (SV) models. In GARCH models, it is assumed that the variance
follows a single process between the return and the volatility (a single
shock or error term component); in contrast, SV models admit greater
flexibility, given that their variance has a process that is independent
from that of return.
We use the SV model of Kim, Shephard and Chib (1998) to identify the
main periods of volatilities over a decade in the Peruvian exchange rate
and stock market returns. Kim et al. (1998) show the best performance
of the SV model against the conditional heteroscedasticity models under
three statistical tests. The first test is the likelihood ratio statistic that
measures the model’s degree of adjustment to the data. The second test
is based on the Atkinson criterion (1986) that consists of simulations of
series with SV and GARCH structures using the estimated parameters
and then contrast with the estimation performed on the models studied.
The last criterion is that of Chib (1995), who includes both the posterior
and prior estimations in the likelihood ratio statistic.
We find that according to these statistics, in general, the SV model
exhibits a better performance, in terms of their adjustment to the data,
than GARCH specifications. Furthermore, the integration sampler algo-
rithm is the most efficient for the estimation of parameters as this method
approximates properly the error term in the volatility equation and displays
relatively less correlated iterations across parameters. The estimated
volatilities reveal the main episodes of uncertainty in the Peruvian
economy such as the Asian crisis and the recent US crisis; it is also
observed that the Stock and Forex markets reacted at a similar pace to
internal shocks, for instance, at times of electoral processes.
The structure followed by this article is as follows. Section 2 explains
the main contributions of modelling volatility in the literature. Section 3
presents the structure of the SV models. Section 4 shows and analyses
4 Journal of Emerging Market Finance 17(3s)

the empirical results for the series studied, and the performance of the
SV model is compared with the traditional N-GARCH and t-GARCH
models. The conclusions are set out in Section 5.

2. Literature Review
An initial model for conditional variance was developed by Engle
(1982), called autoregressive conditional heteroscedasticity (ARCH),
which was applied to inflation in the United Kingdom, showing high
persistence of the variance. Bollerslev (1986) presents the GARCH
model whose conditional variance groups together the extensive lags of
Engle’s ARCH model (1982), developing an autoregressive moving
averages (ARMA) structure for the variance. Along these lines, the
model proposed by Nelson (1991), known as exponential GARCH
(EGARCH), allows the leverage effect to be studied, that is, the asym-
metrical relationship between returns and variance, which occurs, for
example, when there is bad news in the stock market, thus generating
volatility that is more than proportional to the shock that originally
occurred. Another model along the same lines is that put forward by
Glosten, Jagannathan and Runkle (1993), which analyses the leverage
effect. When the variance is included in the equation for the mean, the
model is denoted by GARCH-M.
Further contributions to the literature include those by Baillie,
Bollerslev and Mikkelsen (1996) consisting of GARCH models with
fractional integration, which allow the long-term dependencies of the
conditional variances to be modelled. The authors apply these models to
the US stock market returns and present results that are highly significant
for the integration parameter by rejecting the extreme cases of GARCH
and IGARCH.2
Another strand developed in the literature is provided by the SV models
that also establish the mean equation, a stochastic process inherent to the
volatility or variance that determines the values realised from the vari-
ance conditioned to the data. It is an unobservable process and one that
changes over time.3 These models arise in the modelling of share prices.
In a continuous version, Hull and White (1987) and Wiggins (1987) model
option pricing where variance follows a stochastic process. Hull and White
(1987) find that the Black-Scholes model overestimates the price of an
option in relation to the SV model and that this problem worsens if the
option’s time to maturity is greater.4 In Wiggins (1987), this relationship
Alanya and Rodríguez 5

is analysed for different market options with similar results of those of

Hull and White (1987), as only for options with an average duration of
more than 6 months does the volatility model proposed offer further
advantages for price valuation.
The SV model does not have analytical representation for the
likelihood function. Therefore, a number of parameter estimation methods
have been proposed. A first approach is the estimation through the method
of moments analysed by Wiggins (1987), and this estimation method was
given further prominence by Melino and Turnbull (1990). They consider
the selection of moments in accordance with familiarity, identification
and efficiency and apply their methodology to the exchange rate between
the Canadian and US dollars. Harvey, Ruiz and Shephard (1994) employ
the quasi-maximum likelihood model for the estimation of the likelihood
function, which is based on procedures filtered using the Kalman filter.
Moreover, Jaquier, Polson and Rossi (1994) develop a discrete version
of the SV model and compare the estimation methodologies proposed
with a new Bayesian approach. To do so, the authors simulate a series
and assess the efficiencies of these methods. Efficiency implies a smaller
correlation in the iterations performed and a rapid convergence to the true
values of the parameters, in pursuit of the model’s objective distributions.
The authors conclude that the Bayesian approximation is the best in terms
of efficiency and generates better predictions due to the filtering procedure
for estimating volatilities.
However, Kim et al. (1998) show the poor performance of the estima-
tion in small samples, caused by poor approximation of the error term to
a normal distribution, and the parameters are bounded to a predetermined
range of values. For this same model, in Kim et al. (1998), new Bayesian
algorithms are established that help to improve the quality of the estima-
tors, starting off with the classic Gibbs sampler method, to later develop
new algorithms such as the mixture sampler and the integration sampler.
These algorithms serve to improve the quality of the estimations and
their efficiency.
On the other hand, to estimate the logarithm of likelihood that will
allow us to ascertain the adjustment of the model to the data and estimate
the filtered volatilities, it is necessary to use the so-called particle filter.
This sequential Monte Carlo algorithm generates approximate samples
on the distribution of latent variables at each point of time, using a similar
methodology to the Kalman filter, where a state-space structure is used for
the model. Gordon, Salmond and Smith (1993) propose a bootstrap filter in
the state-space framework by utilising approximations or samples on the
state vector. Moreover, Pitt and Shephard (1999) employ discrete auxiliary
6 Journal of Emerging Market Finance 17(3s)

variables whose function consists of having a better sample on the densi-

ties of the stochastic volatilities. Meanwhile, Kim et al. (1998) establish a
particular algorithm of Pitt and Shephard’s auxiliary filter (1999), where
they perform approximations on the objective distributions by way of
Taylor’s expansions. Kim et al. (1998) show the best performance of the
SV model in relation to the conditional heteroscedasticity models under
three statistical tests: the likelihood ratio statistics, the Atkinson criterion
(1986) and the marginal likelihood of Chib (1995).
In Peru, there are no works that aim to model the volatility of different
financial series. Humala and Rodríguez (2013) present and analyse the
stylised facts of Peru’s stock market and exchange rate returns and volatili-
ties. On the basis of that paper, different lines of research are proposed, to
which this article seeks to contribute. We apply the SV model as per the
discrete version employed by Kim et al. (1998), which consists of a first-
order autoregressive SV model. For the estimation of parameters, the prior
distributions by Kim et al. (1998) and Jaquier et al. (1994) are assumed.
The samples for exchange rate returns cover the months from January
1994 to December 2010 and the stock market returns from January 1992
to December 2010. The frequency is daily for both time series.

3. Methodology
The SV models assume that financial series is generated under a stochastic
process, both for the mean equation and for the variance. Moreover, at
each point in time this process determines the volatilities realised, which
follow a latent process, that is, they are not observable. For a yt financial
series corrected by the mean of each of the observations {t = 1, …, T},
the representation of a general canonical SV model has the following
structure:

y t = exp (h t /2) e t,(1)

h t + 1 = n + z (h t - n) + v n h t,

b = exp (n/2),

h 0 ~ N e n, o,
v2
1 - z2
Alanya and Rodríguez 7

e t ~ N (0, 1),

h t ~ N (0, 1),

where ht is the volatility of the return at moment t. It is assumed that ht

follows an AR(1) stationary process and vn represents the volatility of
the process ht, and the parameter b represents a factor of scale for the
equation of volatility. Moreover, for the equation of volatility, we assume
that the initial volatility h0 follows a normal distribution with the above-
mentioned characteristics. Finally, the shocks et and ht are i. i. d.; thus,
the cov(et ,ht) = 0.5
One of the advantages of the SV model is that it allows a linear
representation, and thus the use of estimation methods is feasible.
The linearisation is for the first equation in system (1),6 which corresponds
to the mean-corrected return:

y 2t = [exp (n/2 + h t /2)] 2 (e t) 2,(2)

log ( y 2t ) = n + h t + log (e 2t ),

h t + 1 = n + z (h t - n) + v nh t.

Nonetheless, under this linearised scheme, another difficulty related to

the approximation of the term log (f 2t ) is presented, as it is now a variable
that is distributed as |2.
A first proposal is given by Harvey et al. (1994), using the Kalman filter
under the estimation scheme by quasi-maximum likelihood. Nonetheless,
Kim et al. (1998) conclude that this approximation performs very badly in
small samples. Kim et al. (1998) establish an approximation to this new
error term by way of a combination of normal distributions. First, they
perform a normalisation technique that allows a linearisation that is appro-
 *t ~ | i = 1 q i N (m i, v 2i ), where each distribution of
k
priate to the error term: e
the new error term e  *t has a probability of a mean mi and a variance v 2i .
The values that are determined by Kim et al. (1998) and which approxi-
mate the distribution |2 as much as possible are when k = 7. In this way,
a new return is considered, from which the expected value of the logarithm
of |2 is subtracted to maintain equality in the mean equation in system (2):

x 2t = log (y 2t + c) - E [log (e 2t )],(3)

8 Journal of Emerging Market Finance 17(3s)

xt = n + ht + e
 *t ,

h t + 1 = n + z (h t - n) + v n h t.

where c is a constant (offset = 0.001) used to avoid the logarithm values

close to zero. Model (3) is used for the estimations. The estimation of the
SV models entails the estimation of two groups of variables. First, the
parameters of the model i = {n,z,vn} are estimated. Second, based on
these parameters, the filtered volatilities are obtained. In an SV model,
the estimation of the i parameters requires the construction of the like-
lihood function to then be maximised. The likelihood function is
f ( y | i) = # f ( y | h, i) f (h | i) dh. Under the SV model, the set of returns
is conditioned to the vector of unobservable volatilities. This expression
has to be integrated into each point of time in the sample t = 1, …,T.
Jaquier et al. (1994) argued that the likelihood function does not have
an analytical representation. Nonetheless, in the literature, three approxi-
mations have been developed with the aim of overcoming this difficulty:
approximation through the Kalman filter using techniques of quasi-
maximum likelihood, method of moments, and the Bayesian approach.
The first approach does not adequately estimate the non-linear approxi-
mation of the error, log (f 2t ), as well as a marked bias in finite samples.
The second approach has the limitation of determining the number of
moments, which if badly specified leads to a significant loss of informa-
tion derived from the data; refer to Jaquier et al. (1994). The approach
adopted in this study is the Bayesian approach, whose advantages lie in
its efficiency in the estimation of the parameters and the filtered procedure
for the estimation of the latent processes. Jaquier et al. (1994) conclude
that the Bayesian method offers an optimum solution in identifying the
unobservable process for the variance in the context of the model (3).
In Bayesian econometrics, the aim is to find the posterior distributions.
If we have the set of parameters, i and the data y, then through Bayes’
theorem: r(i| y) ? r( y|i)r(i), where r(i| y) is the posterior distribution
conditioned by the data whose mean will be the Bayesian estimation of
the parameters. Meanwhile, r( y|i) is the likelihood function, and r(i)
is the prior distribution, which constitutes the beliefs with regard to the
parameter distributions. To calculate the likelihood function, the Markov
chain Monte Carlo (MCMC) is employed, which allows a direct estima-
tion with multiple simulations of this posterior distribution. Following on
from Kim et al. (1998), we use three methods for estimating the model’s
parameters, sampling the significant improvements in the factor of
Alanya and Rodríguez 9

inefficiency, which determines improvements in terms of the convergence

of estimated values.
Efficiency in the simulation is determined by the relationship between
the standard square error of the parameter and the variance in the itera-
tions performed in the process. Thus, this factor directly depends on
the standard errors of the MCMC, whose mathematical definition is as
follows: R BM = 1 +
2BM
| BM Kb i lt (i), where RBM represents
BM - 1 i = 1 BM
the standard MCMC error associated with the estimation of the para-
meter; BM represents the bandwidth, which is the range of frequencies for
performing the Parzen kernel which is denoted by K(.); and the term
t(i) is the autocorrelation in the delay i.
The first method is the MCMC Gibbs sampler algorithm, which
assumes that the error behaves under a standard normal distribution,
with consequences for the estimation of the parameters. The Gibbs sam-
pler method seeks to find the posterior distribution of parameters based
on conditional posterior distributions considering the following steps:
(a) set the initial values to volatility ht and the parameters n, v 2h, z;
(b) sample values for ht are obtained from the conditional posterior
distribution r (h t | h (- t), b, n, v h2 , z, y) for t = 1, …,T; (c) draws for the
parameter v 2h are generated based on r (v 2h | h t, b, n, z, y); (d) similarly,
sample values for z comes from the distribution r (z | h t, b, n, v 2h);
and (e) finally, the draws for n are determined under r (n | h t, v 2h, z) .
In this way, the steps (b)–(e) are repeated under multiple simulations until
the estimation of the joint posterior distribution, and thus the marginal
distributions are reached.
In the second method, called mixture sampler, the term of error of the
linearised model is approximated through this combination of normal
distributions, and the realisation of this new error term is called ~t.
In turn, xt is determined by the series transformed in Equation (3). The pro-
cess of convergence to the joint posterior distribution is performed under
the following steps: (a) the initial values are given for ~ t, z, v 2h, n, x t;
(b) the distribution is obtained for ht based on r (h t | ~ t, b, n, v 2h, z, x t);
(c) the distributions for the mixture ~t are obtained based on r(~t |xt,ht);
(d) the conditional posterior distributions are taken for steps (c)–(e) of the
Gibbs sampler method. The iteration and update process occur between
steps (b) and (d). Step (b) differs from the Gibbs sampler ratio in this case,
due to the approximation of the error in linear terms.
10 Journal of Emerging Market Finance 17(3s)

The third method is the integration sampler method: It is an exten-

sion of the mixture sampler algorithm that consists of the integration or
separation of the volatilities in the sampling process with the aim of
improving randomness and providing less correlation between the
parameters and the volatilities. The process contains the following steps:
(a) the initial values are given for ~ t, z, v 2h , n t, x ; (b) the distributions are
obtained for z, v 2h based on r (z, v 2h | ~ t, x t) ; (c) the distributions of ht and
n are obtained from r (h t, n | ~ t, v 2h, z, x t) ; (d) finally, the distributions
of ~t are obtained on the basis of r(~t |xt,ht). The iteration and update
follow the steps sequentially (b)–(d). What is new about this procedure
is the obtention of posterior distributions at the margin of the volatilities
in step (b). This is possible as a consequence of the Metropolis-Hastings
algorithm, which consists of whether or not to accept the values for
the distribution, based on the probability of rejection g (z, v 2h). Step (c)
is similar to the mixture sampler method.
The volatilities are estimated after the SV model parameters. Given
that volatility is a latent variable, its estimation can be achieved through
methods known as particle filters. These methods allow the subsequent
density of the volatilities at each point of time to be estimated. In this study,
the filters of Kim et al. (1998) and Pitt and Shephard (1999) are employed.

4. Empirical Results
In this section, we present the results of the SV model estimations: the
parameters and the volatilities. Moreover, we describe the efficiency
gain in the estimation of parameters in utilising the three algorithms
presented in the previous section, as well as the filtered and smoothed
estimations of the SV model. Finally, we compare the adjustment to the
SV model data, with which the N-GARCH and t-GARCH models are
obtained.

4.1 The Data

The data used for stock market and exchange rate returns are end of day
and comprise the period of analysis from January 1992 to December
2010 in the first case, while for the exchange rate data from January 1994
to December 2010 are utilised.7 The returns are calculated as rt = [log(Pt)
– log(Pt–1)] × 100, where Pt represents the closure price that takes the
Alanya and Rodríguez 11

Table 1. Descriptive Statistics

Stock Forex
Returns Volatility Returns Volatility
Values (1992:01–2010:12) (1994:01–2010:12)
Mean 0.001 –4.890 0.000 –6.189
Median 0.006 –4.863 0.000 –6.287
Maximum 0.128 –2.046 0.022 –3.728
Minimum –0.114 –6.907 –0.023 –6.907
Standard 0.015 0.911 0.002 0.554
deviation
Skewness 0.012 –0.015 0.243 0.926
Kurtosis 10.179 2.591 15.820 3.648
Jarque–Bera 9831.276 31.960 23205.82 543.796
Observations 4577 3384
Source: Authors’ calculation.

variable in its original form in the period t. Following the literature in

general, we use the squared returns logarithm as a proxy of volatility.
For practical effects, x t = [log (r 2t + 0.001)] is used with the aim of
correcting the returns close to zero.
The main descriptive statistics for both financial series is presented in
Table 1. The mean for stock market returns is 0.001; for exchange rate
returns it is practically a value of zero. This implies that there are clusters
of data around zero. On the other hand, the standard deviation for exchange
rate returns is 0.015, greater than the deviation of 0.002 for exchange rate
returns,8 which implies that the stock market returns display more volatile
behaviour than the exchange rate returns (see Figure 1).
Moreover, the asymmetry of stock market and exchange rate returns
are 0.012 and 0.243, respectively, from which it is inferred that the
observations of the exchange rate returns are biased to one side of density.
The fourth moment, or kurtosis, provides evidence of expected results
for the financial series, as the observations group together and extend
at the tails of the densities. For stock market returns, it is 10.179 and
for exchange rate returns it is 15.820.
The initial values and the prior distributions are based on studies
by Kim et al. (1998). Thus, we have v 2h ~ IG (vr /2, S v /2), where IG
denotes the inverse-gamma distribution. It is assumed that vr = 5 and
Sv = 0.01 × vr. For the case of the parameter of persistence z it is
specified that z = 2z* – 1, where z* is distributed in accordance with
a beta distribution with parameters(z(1),z(2)). In this way, the prior for
12 Journal of Emerging Market Finance 17(3s)

Figure 1. Stock and Forex Returns Series

Source: Authors' estimation.
Disclaimer: T
 his image is for representational purposes only. It may not appear well
in print.

(1 + z) z - 1 (1 + z) z - 1
(1) (2)

z is r (z) ? ) 3 ) 3 where z(1), z(2) > 1/2 and has

2 2
support on the interval (–1, 1) with a prior mean of {2z(1)/(z(1) + z(2)) – 1}.
This study utilises z(1) = 20 and z(2) = 1.5.
Alanya and Rodríguez 13

4.2 Gibbs Sampler Estimation

Table 2 shows the estimation of the parameters z, vh and b for the series
employed. The initial values and the prior distributions are the same
as in Kim et al. (1998); nonetheless, the results presented are robust to
the different initial iteration values.9 We have employed 1,000,000
iterations and discard the first 50,000 iterations. The associated number
for determining the inefficiency statistic is determined with the values
established in Kim et al. (1998), that is, a bandwidth of 2,000 for
the parameters z and b and a value of 4,000 for vh, determining the
standard errors on the estimation of these parameters.
The results of the estimations for stock market returns are as follows.
The mean of z is 0.957. Moreover, the estimation of the parameter
b, which represents the scale factor, shows a mean of 1.082. Finally, the
parameter vh has a posterior mean distribution of 0.322. The estimation
of the parameters by way of the Gibbs sampler is quite inefficient, in the
sense that convergence is very slow, and thus there are probabilities of a
biased estimation. In this case, the estimation of the parameter z has an
inefficiency of 97.655, the estimate of vh has 183.86 and b has 3.921.
Another statistic that is used to determine whether the estimation con-
verges adequately is the randomness of the iterations. In this case, Figure 2
(top panel) presents the autocorrelation functions of the parameters. It is
appreciated that the autocorrelations decay more slowly in the parameters
z and vh, which at the same time shows the highest levels of inefficiency.
On the other hand, the result for the exchange rate returns is that the
mean of z is 0.969; the scale factor, b, obtains a value of 0.140 as a mean
of marginal posterior distribution. Moreover, the mean of the parameter
vh is 0.372. The inefficiency is 53.503, 133.14 and 2.021 for z, vh and b,
respectively. Just like the exchange rate returns, the inefficiency statistics
associated with the estimation of these parameters is large on average and
for the majority of the parameters. In addition, it is evidenced in Figure 2
(bottom panel) that the autocorrelations of the parameters with greater
inefficiency show problems of correlation in a greater lag horizon.
The estimations of the parameter of persistence z show that the mean
life of the shocks is 15.77 and 22.01 days in the stock market and exchange
rate markets, respectively.

4.3 Mixture Sampler Estimation

This method approximates the errors of the system variance in Equation
(3) to the correct distribution corresponding to |2, and the approximation
Table 2. Estimations

MCMC
Mean Standard Error Inefficiency Var–Cov Matrix
Gibbs Sampler
Stock Returns
z|y 0.957 6.7652e-005 97.655 4.4523e-005 −0.00010263 1.8442e-005
vh|y 0.322 0.00029937 183.86 −0.00010263 0.00046308 −8.6000e-005

b|y 1.082 0.00012745 3.9206 1.8442e-005 −8.6000e-005 0.0039360

Forex Returns
z|y 0.969 4.2009e-005 53.503 3.1335e-005 −8.7707e-005 4.7550e-006
vh|y 0.372 0.00029696 133.14 −8.7707e-005 0.00062922 −2.0438e-005
b|y 0.140 2.2889e-005 2.0213 4.7550e-006 −2.0438e-005 0.00024624
Mixture Sampler
Stock Returns
z|y 0.957 5.6888e-005 54.303 4.4101e-005 −0.00010267 1.6253e-005
vh|y 0.316 0.00024641 97.452 −0.00010267 0.00046105 −7.9943e-005
b|y 1.086 8.8757e-005 1.4751 1.6253e-005 −7.9943e-005 0.0039519
Forex Returns
z|y 0.976 3.1152e-005 30.542 2.3513e-005 −5.9172e-005 3.7994e-006
vh|y 0.293 0.00020558 76.073 −5.9172e-005 0.00041110 −1.3642e-005
b|y 0.152 2.1593e-005 1.1825 3.7994e-006 −1.3642e-005 0.00029179
Integration Sampler
 Stock Returns
z|y 0.957 4.4575e-005 11.398 4.3538e-005 −9.9104e-005 1.7526e-005
vh|y 0.315 0.00017581 17.351 −9.9104e-005 0.00044488 −6.7189e-005
b|y 1.086 0.00030554 5.8851 1.7526e-005 −6.7189e-005 0.0039619
Forex Returns
z|y 0.976 2.8332e-005 8.613 2.3276e-005 −5.7679e-005 1.6331e-006
vh|y 0.293 0.00015557 14.964 −5.7679e-005 0.00040390 −1.0475e-005
b|y 0.152 7.8126e-005 5.236 1.6331e-006 −1.0475e-005 0.00029114
Source: Authors’ calculation.
Note: Cov=Covariance; Var=Variance; Var-Cov=Variance_Covariance
16 Journal of Emerging Market Finance 17(3s)

Figure 2. Estimation by Gibbs Sampler. Stock Markets (top panel) and Forex
Market (bottom panel). Inside of Each Panel (from left to right): (a) Iterations for
z|y, (b) Iterations for vh|y, (c) Iterations for b|y, (d) Density of z|y, (e) Density
of vh|y, (f) Density of b|y, (g) Correlogram of z|y, (h) Correlogram of vh|y and
(i) Correlogram of b|y
Source: Authors’ estimation.
Disclaimer: T
 his image is for representational purposes only. It may not appear well in
print.
Alanya and Rodríguez 17

occurs under a combination of normal densities. The results (Table 2)

show that this approximation to the model proposed provides better
levels of inefficiency overall. In this case, we use 750,000 iterations and
discard the first 10,000. The bandwidth associated with the standard
parameter estimates is 2,000 for the parameters z and vh and 100 for the
parameter b.
In the case of stock market returns, the parameter z has a mean
(of its marginal posterior distribution) of 0.957; moreover, vh has a mean of
0.316 and finally, the scale factor b has a mean of 1.086. The inefficiency
of the parameters are 54.303, 97.452 and 1.475 for the parameters z,
vh and b, respectively. The improvement in the efficiency of all para-
meters, in relation to the levels presented by the Gibbs sampler estimation,
is important. Moreover, Figure 3 (top panel) shows that the correlation
is significantly smaller in all cases; in other words, the autocorrelation
decays more rapidly than in the results obtained by the Gibbs sampler.
This algorithm is more useful, as it provides better convergence in the
pursuit of marginal subsequent distributions.
Meanwhile, the results for exchange rate returns have the same impli-
cations. The persistence for the variance process has a mean of 0.976,
moreover its scale factor and volatility are estimated at 0.152 and 0.293,
respectively. Unlike the stock market returns, the estimates are slightly
different in relation to the Gibbs sampler. Nonetheless, the efficiency gains
are similar, in that the inefficiencies are 30.542, 76.073 and 1.183 for the
parameters z, vh and b, respectively. Moreover, a significant improve-
ment can be noted in the autocorrelations of the iterations performed in
Figure 3 (bottom panel).
Although the mean life of stock market shocks is similar to that found
in the Gibbs sampler method, in the case of the volatility of exchange rate
returns, this duration increases to 28.53 days.

4.4 Integration Sampler Estimation

The integration sampler process entails integrating or separating the
convergence process of the volatility from that of the parameters. In this
way, an improvement in the quality and speed of the estimation and
convergence of the estimated value is expected. In this case, 25,0000
iterations were used, with the first 250 discarded. Equivalently, the
quantity BM is used by Kim et al. (1998), who determine a value of
1,000 for all parameters.
The estimations of the parameters for the stock market returns by the
integration sampler are very similar to those obtained by the mixture sampler.
18 Journal of Emerging Market Finance 17(3s)

The estimates imply a mean of 0.957 for z; moreover, the scale factor b
has a mean of 1.086. Finally, the mean for the volatility of the equation for
the variance is 0.315. The efficiency continues to improve and this time
posts 11.398, 17.351 and 5.885 for c, vh and b, respectively. Nonetheless,
a degree of efficiency is lost for the parameter b with relation to the results
by way of the Gibbs sampler and mixture sampler, but the level of inef-
ficiency for this and the other parameters is low. Moreover, in Figure 4
(top panel), the improvement in the autocorrelation of the iterations is
evident; thus, for the persistence z and the scale factor b, the correlation
decays rapidly in delay 50 in both cases and in delay 25, approximately
for vh, unlike the previous autocorrelations where these decay with a
delay of 200 and 250 for b and vh, respectively.
For the case of exchange rate returns, the mean of persistence z is
0.976, for the factor b it is of 0.152 and for vh it is 0.293. Efficiencies also
improve significantly and are 8.613, 14.964 and 5.236 for the parameters
z, b and vh, respectively. Equivalently, in the case of the stock market,
there is also an improvement of the efficiency of the two first estimates
with relation to the Gibbs and mixture sampler algorithms. In Figure 4
(bottom panel), the autocorrelations of the parameters improve and decay
rapidly in lag 40 for the parameters z and vh, which previously, for the
mixture sampler, were prolonged up to 100, approximately.
Based on the estimates of the z parameter, we can say that the mean
life of a shock (or the persistence of the volatility) in the stock market
has a duration of 16 days. In the case of the exchange rate market, the
duration of these shocks rises to 28 days.

4.5 Model Volatility Estimations

The unobservable stochastic volatilities of the model are estimated
through sequential Monte Carlo simulations or particles filters. Because
of the greater efficiency presented, only the integration sampler para-
meters are considered.10 Two filtration methods are employed: the algo-
rithm proposed by Kim et al. (1998) and that of Pitt and Shephard (1999).
Figure 5 shows the estimated volatilities of both returns for the two
algorithms by utilising 12,500 iterations.11 Using the filter of Kim et al.
(1998), in both cases, the presence of atypical observations can be
appreciated, and this is because this approximation is quite sensitive to
observations very close to zero; in particular, this problem is heightened
for exchange rate returns. In consequence, the logarithm of likelihood as a
result of this filter is not well estimated. In return, the estimations due to the
Pitt and Shephard (1999) filter are robust to the data of returns, reflecting
Alanya and Rodríguez 19

the patterns of volatility of our data and generating an unbiased estimate

of the SV model’s logarithm of likelihood (see Figure 5).
There is an alternative for viewing the ht volatilities of the model, which
is known as smoothed volatility. The smoothed estimation is a simple
average | t = 1 h t /H , where H is the quantity of iterations. This process
H

determines the value of the volatility in time t using the information

from the entire sample and the estimated parameters in the H iterations.
The smooth estimation of the volatility is presented in Figure 6 (top panel)
for the Gibbs, mixture and integration sampler methods for the series of
stock market returns. Analogously, in Figure 6 (bottom panel), the smooth
estimations of the volatility for the exchange rate returns are presented.
The results presented are realised with 12,500 iterations for all cases. For
stock market returns, the estimations for the volatility using these methods
are practically the same, which signifies that there is a strong convergence
towards the real values of the parameters. In the case of the exchange rate
returns, it is appreciated that the Gibbs sampler method provides greater
values for volatility, especially in the peaks of high uncertainty in relation
to the mixture and integration sampler, which almost do not differ in the
estimation of the volatility across the sample.
Likewise, Figure 7 shows the filtered and smoothed estimations of
the volatilities (upper and lower panels). The filtered estimations tend to
reflect greater volatility because the filtered estimation uses information
up to the period t, while the smoothed estimation utilises the informa-
tion from the entire period (T ). For purposes of the following analysis,
we only consider the estimated volatilities by using the integration sampler
process in both the series. Figure 8 establishes a visual comparison of
the evolutions of volatility and the absolute value of the returns. In both
cases, the series shows similar patterns at times of low, medium and high
uncertainty, reflected in greater values for the estimated SV.
Based on these estimations, the integration sampler method is used
to perform a brief analysis of the main economic and financial facts that
had repercussions on the volatility of both series. The variances of stock
market and exchange rate returns are clearly affected by the international
crises between 1997 and 1999. Both volatilities reflect high volatility due
to the economic problems that occurred in the countries of Asia and in
Russia. Equally, the recent US financial crisis has had severe repercus-
sions in both financial markets, while in 2010 there were no major scares
in the markets; indeed, a period of stability was established. On the other
hand, internal circumstances have also been a source on uncertainty, such
as the political instability of the 1990s. In particular, the stock market of
20 Journal of Emerging Market Finance 17(3s)

Figure 3. Estimation by Mixture Sampler. Stock Markets (top panel) and Forex
Market (bottom panel). Inside of Each Panel (from left to right): (a) Iiterations
for z|y, (b) Iterations for vh|y, (c) Iterations for b|y, (d) Density of z|y,
(e) Density of vh|y, (f) Density of b|y, (g) Correlogram of z|y, (h) Correlogram
of vh|y and (i) Correlogram of b|y
Source: Authors’ estimation.
Disclaimer: T
 his image is for representational purposes only. It may not appear well in
print.
Alanya and Rodríguez 21

Figure 4. Estimation by Integration Sampler. Stock Markets (top panel)

and Forex Market (bottom panel). Inside of Each Panel (from left to right):
(a) Iterations for z|y, (b) Iterations for vh|y, (c) Iterations for b|y, (d) Density
of z|y, (e) Density of vh|c, (f) Density of b|y, (g) Correlogram of z|y,
(h) Correlogram of vh|y and (i) Correlogram of b|y
Source: Authors’ estimation.
Disclaimer: T
 his image is for representational purposes only. It may not appear well in
print.
22 Journal of Emerging Market Finance 17(3s)

Figure 5. Estimation of Filtered Stochastic Volatility. Stock Market (top panel)

and Forex Market (bottom panel)
Source: Authors’ estimation.
Disclaimer: T
 his image is for representational purposes only. It may not appear well in
print.
Alanya and Rodríguez 23

Figure 6. Different Estimates of Stochastic Volatility. Stock Market (top panel)

and Forex Market (bottom panel)
Source: Authors’ estimation.
Disclaimer: T
 his image is for representational purposes only. It may not appear well in
print.
24 Journal of Emerging Market Finance 17(3s)

Figure 7. Filtered and Smoothed Stochastic Volatility. Stock Market (top panel)
and Forex Market (bottom panel)
Source: Authors’ estimation.
Disclaimer: T
 his image is for representational purposes only. It may not appear well in
print.
Alanya and Rodríguez 25

Figure 8. Estimates of Stochastic Volatility and Absolute Value of Returns.

Stock Market (top panel) and Forex Market (bottom panel)
Source: Authors’ estimation.
Disclaimer: T
 his image is for representational purposes only. It may not appear well in
print.
26 Journal of Emerging Market Finance 17(3s)

that decade shows permanently erratic behaviour, initially because of a

market reaction to the structural reforms (since 1991) and later due to
somewhat complex electoral processes (e.g., 2001 and 2006). Between
both variables there is a coefficient of correlation of 0.4, which confirms
the similar relative dynamic of financial markets in the face of systematic
shocks in the economy.

4.6 Comparing SV and GARCH Models

Because the SV and GARCH models do not possess the same structure, it
is necessary to resort to so-called non-nested tests. In this article, we resort
to three tests: the likelihood ratio test; the second test through simulations
following Atkinson (1986); and the third test, which is the marginal like-
lihood criteria proposed by Chib (1995). The likelihood ratio, which
compares both models, is determined by LR = 2 {log f 1 (y | M 1, i *1) -
log f 0 ( y | M 0, i *0)}, where log f ( y|M,i*) is the logarithm of likelihood
of the M model conditioned to the estimated parameters i*. In this case, if
the LR statistic is greater than zero, it is an indication in favour of the SV
model denoted by M1; moreover, if it is negative it will favour the GARCH
model denoted by M0.
Given that the function of likelihood of the SV model requires simula-
tions based on an approximated density of the likelihood logarithm, Pitt
and Shephard’s auxiliary particle filter (1999) is employed. Thus, Table 3
shows the estimated likelihoods of the SV and GARCH models for both
series of returns.12 According to this test in both series of returns, the
SV model is superior to the N-GARCH model. In the case of the stock
market, the LR statistic is 208.932, and in the case of the exchange rate,
it is 354.824, which categorically favours the SV model. The results of this
test for the case of the t-GARCH model do not favour the SV model, with
the LR statistics of −68.218 and −52.998 for stock market and exchange
rate returns, respectively.13
The test of Atkinson (1986) consists of simulations of series based
on the estimated parameters of the SV and GARCH models. Based on
these series, the LR statistics are calculated exactly as is set out earlier.
These statistics are a sample of the observed LR, which is calculated by
employing the original series. Based on the value observed, its position
or ranking inside the simulated LR is determined. The criteria, according
to Atkinson (1986), is that if the ranking is close to the extreme values
of the sample, it is a result that goes against the null hypothesis model.
Meanwhile, if the ranking is relatively far from these values, the null
Alanya and Rodríguez 27

hypothesis model is superior. In our case, under the null hypothesis we

have simulated 99 series for the SV and GARCH models, that is, when
the null hypothesis is the SV model, as well as when the GARCH is.
For the series simulated under the SV model, this null hypothesis is
favoured, with the ranking located in position 64 for stock market returns
and position 36 for exchange rate returns. Moreover, when the series is
simulated using the GARCH model parameters, the rankings were 2 and
92 for stock market and exchange rate returns, respectively. Given that the
ranking is close to the limit values, this hypothesis was rejected in favour
of the SV model. Nonetheless, the case in which the null hypothesis is
the SV model and the alternative is a t-GARCH model is also studied.
The results for the stock market returns favour the SV model, but in the
case of the exchange rate, it is not possible to determine which model is
better, because the null hypotheses are rejected when the data are simulated
under the two models. This is because the ranking for the exchange rate
returns is far from the extreme values and occupies position 21 for the
SV case and position 82 in the t-GARCH case.
Finally, the marginal likelihood test of Chib (1995) consists of
estimating the Bayes factor through marginal likelihood as follows:
VM = log f ( y | M 1, i *1) + log f (i *1) - log f (i *1 | M 1, y), where the first term
of the equation is the logarithm of likelihood for the model M1, the term
log f (i *1) is the logarithm of the prior distribution evaluated in the mean
of the subsequent distribution i *1 and the final term is the logarithm
of subsequent distributions evaluated at this point through a Gaussian
kernel. The difference in the marginal likelihoods of the two models is the
Bayes factor. The analysis of this factor follows the same criterion as the
likelihood ratio test. The main input in this test is the logarithm of
likelihood, which is estimated by the filtered procedure. Therefore, the
GARCH models are estimated using Bayesian inference; more specifically,
the algorithms of Gilks and Wild (1992) and the results of this estimation
are presented in Table 4. The results are similar to those obtained by the
estimation of maximum likelihood. The marginals are −7,473.98 and
−7,312.51 for the stock market (N-GARCH and t-GARCH, respectively)
and 1,263.03 and 1,471.37 for the exchange rate market.14 The difference
in these likelihoods allows the Bayes factor to be obtained. The entire pro-
cedure has been conducted under 12,500 iterations, using the integration
sampler algorithm. By calculating the Bayes factor, it can be concluded
that the SV model is superior to the N-GARCH model. In effect, the
Bayes factor is 157.34 and 196.10 for the stock market and exchange rate
markets, respectively. On the other hand, this cannot be affirmed when the
SV model is compared with the t-GARCH model. In effect, the Bayes
28 Journal of Emerging Market Finance 17(3s)

Table 3. Maximum Likelihood of SV and GARCH Models

Models a0 a1 + a2 o Log-likelihood
Stock Returns
SV – – – −7305.667
N-GARCH 0.077 0.987 – −7410.133
t-GARCH 0.078 0.985 5.890 −7271.558
Forex Returns
SV – – – 1469.234
N-GARCH 0.0003 1.043 – 1291.822
t-GARCH 0.0003 1.053 4.523 1495.733
Source: Authors’ calculation.
Note: The GARCH models have been estimated using E-views while the SV models
used the integration sampler.

Table 4. Bayesian Estimations of GARCH Models

a0 a2 Log-
Models 1 - a1 - a2 a1 + a2 a1 + a2 o likelihood
Stock Returns
N-GARCH 0.982 0.953 0.847 – −7462.4
t-GARCH 0.975 0.953 0.847 7.335 −7299.8
Forex Returns
N-GARCH 0.169 0.997 0.809 – 1274.1
t-GARCH 0.108 0.997 0.821 5.375 1484
Source: Authors’ calculation.

factor is −4.164 and −13.23 for the stock and exchange rate markets,
respectively. Note that the differences between the SV and t-GARCH
model are not high, which suggests that an SV model with t-student
innovations would overcome a t-GARCH model. This, nonetheless,
is the topic of an ongoing investigation.

5. Conclusions
The SV model is an alternative to the conditional heteroscedasticity
models for the estimation of the volatilities of the financial series. This
study is one of the first to utilise the SV model to model Peruvian
financial series, as well as estimate and compare with GARCH models
normal and t-student errors. The analysis in this study corresponds to
Peru’s stock market and exchange rate returns. The importance of this
Alanya and Rodríguez 29

methodology is that the adjustment of the data is better than the GARCH
models using the assumptions of normality in both models.
In the case of the SV model, three Bayesian algorithms have been
employed where we evaluate their respective inefficiencies in the esti-
mation of the model’s parameters. The most efficient and used algorithm
is the integration sampler. With respect to the GARCH models, they
are all estimated by Bayesian inference, with the aim of rendering them
comparable with the SV model for the marginal likelihood test of Chib
(1995). The main input for this test is the logarithm of likelihood, which
is estimated using the auxiliary estimate of Pitt and Shephard (1999),
which also determines the filtered likelihoods of our model.
The estimated parameters in the SV model under the various algo-
rithms are consistent, as they display little inefficiency. The figures of the
correlations of the iterations suggest that there are no problems at the
time of Markov chaining in all estimations.
On making a simple correlation between the stochastic volatilities in
Stock and Forex markets, this was found to be significant, though not
highly so (0.40). We therefore find that the volatilities in exchange rate and
stock market volatilities follow similar patterns over time. That is, when
economic turbulence caused by the economic circumstances occurred, for
example, the Asian crisis and the recent crisis in the USA, considerable
volatility was generated in both markets. In fact, the volatilities of stock
market and exchange rate returns are clearly affected by the international
crises between 1997 and 1999. Both volatilities reflect high volatility due
to the economic problems that occurred in the countries of Asia and in
Russia. Equally, the recent US financial crisis has had severe repercussions
in both financial markets, while in 2010 there were no major scares in the
markets; indeed, a period of stability was established. On the other hand,
internal circumstances have also been a source on uncertainty, such as
the political instability of the 1990s. In particular, the stock market of that
decade shows permanently erratic behaviour, initially because of a market
reaction to the structural reforms (since 1991) and later due to somewhat
complex electoral processes (e.g., 2001 and 2006).

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the
research, authorship and/or publication of this article.

Funding
The author(s) received no financial support for the research, authorship and/or
publication of this article.
30 Journal of Emerging Market Finance 17(3s)

Notes
1. To our knowledge, there is only another ongoing study conducted by Lengua
Lafosse and Rodríguez (2018), who apply an stochastic volatility (SV) model
with leverage and heavy-tailed errors to Latin-American stock returns using
a GH skew student’s t-distribution. Though the algorithm applied is also
Bayesian, the proposal is different as it modifies the structure of the distribu-
tion of the errors in the model.
2. For a more complete review of the list of models in the GARCH family, refer
Andersen and Bollerslev (1998), Bollerslev (2008), Bollerslev, Chou and
Kroner (1992), Engle and Bollerslev (1986), Bollerslev, Engle and Nelson
(1994), De Arce (2004), Degiannakis and Xekalaqki (2004), Engle (2001),
Laurent et al. (2010) and Taylor (1994, 2005).
3. For more details in the definition, refer Taylor (1994).
4. Wiggins (1987), on the other hand, finds that the estimators for the SV model
parameters do not differ from the Black-Scholes model.
5. Other assumptions in the terms of error can also be assumed; refer Kim et al.
(1998). On the other hand, ht can take more complex structures as a general
autoregressive moving averages (ARMA) process.
6. It is also known as a transition equation, if we consider the structure of
a state-space model.
7. The data on the Lima stock market is provided by Bloomberg, while that for
the exchange rate comes from the Central Reserve Bank of Peru (BCRP).
8. This may be due to the frequent interventions in the exchange rate market by
the BCRP.
9. In the results reported, we have used the bias in the generation of random
numbers by default in Oxmetrics 6.0.
10. Though the results are similar when the parameters are estimated by the
Gibbs sampler and mixture sampler.
11. For the auxiliary filter, 62,500 auxiliary components were employed.
12. The estimated GARCH (1,1) model has the following structure: rt | rt - 1 ~N
(0, v 2t ), v 2t = a 0 + a 1 rt - 1 + a 2 v 2t - 1 with v 2t as the conditional variance, a1 as
a parameter related to the past values of the returns and a2 as the persistence
of variance. When a t-student specification rt|rt–1 is considered, it is distrib-
uted with a t (0, v 2t ).
13. Note that the mean life of the shocks to the variance of stock market returns
is between 53 and 46 days, according to the N-GARCH and t-GARCH
model, respectively. In the case of the volatility of exchange rate returns,
the sum of the parameters is slightly greater than one, giving evidence of an
IGARCH process where the mean life of the shocks to volatility is infinite.
14. Note, nonetheless, that unlike the difference in the estimation due to maxi-
mum likelihood (Table 3), now the mean life of shocks to the volatility of
stock market returns is approximately 15 days. In the case of the volatility
of exchange rate returns, the mean life is almost 231 days. This is evidence
of greater persistence in the exchange rate market.
Alanya and Rodríguez 31

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