South American Research Journal, 4(2), 25-44

https://sa-rj.net/index.php/sarj/article/view/61

ISSN 2806-5638

both approaches enhances volatility modeling in low-liquidity

markets.

Keywords: Volatility, ARCH-GARCH models, State space

models, Kalman ﬁlter, Stochastic volatility, Merval index.

Comparison of ARCH-GARCH and

stochastic approaches for estimating

volatility. Application to a small stock

market

RESUMEN

Las series temporales económicas suelen presentar volatilidad, lo

que implica que la varianza del error de observación ﬂuctúa con

el tiempo. Una de las metodologías más utilizadas para mod-

elar estas dinámicas es el modelo ARCH, introducido por En-

gle (1982), y sus extensiones, como los modelos GARCH. Estos

suponen que la varianza condicional depende de valores pasados

de la serie. En contraste, los modelos de volatilidad estocás-

tica (SVM), propuestos inicialmente por Taylor (1980, 1986),

asumen que la volatilidad depende de las varianzas pasadas pero

no directamente de los rendimientos previos. Este estudio com-

para ambos enfoques en la modelización de la volatilidad en un

mercado bursátil pequeño. Evaluar el desempeño de los modelos

ARCH-GARCH y de volatilidad estocástica en la estimación del

riesgo de mercado e identiﬁcación de patrones de volatilidad en

el Merval index, que representa la Bolsa de Comercio de Buenos

Aires (BASE). Se realizó un estudio longitudinal observacional

basado en datos diarios del Merval index entre el 13 de enero de

2003 y el 22 de mayo de 2015, abarcando 3006 observaciones.

Se seleccionó este período para evitar cambios en la aﬁliación

política del gobierno, eliminando posibles distorsiones exógenas

del mercado. Se aplicaron pruebas estadísticas (ADF, Phillips-

Perron) para veriﬁcar estacionariedad y los modelos fueron es-

timados mediante máxima verosimilitud y ﬁltrado de Kalman.

Los modelos GARCH con distribuciones de colas pesadas predi-

jeron mejor la volatilidad en el corto plazo, capturando el cluster-

ing de volatilidad, mientras que los modelos estocásticos fueron

más eﬁcaces en la detección de cambios de régimen. El Merval

index, con una capitalización promedio de $312 millones, con-

ﬁrma las características de un mercado bursátil pequeño, donde

la modelización de la volatilidad es clave para la evaluación del

riesgo. La elección entre modelos ARCH-GARCH y estocásti-

cos depende del horizonte de pronóstico. Los modelos GARCH

son óptimos para evaluar el riesgo en el corto plazo, mientras

que los modelos estocásticos son más adecuados para detectar

cambios estructurales a largo plazo. La combinación de ambos

enfoques mejora la modelización de la volatilidad en mercados

de baja liquidez.

Comparación de los enfoques ARCH-

GARCH y estocástico para estimar la

volatilidad. Aplicación a un pequeño mer-

cado de valores

Juan Carlos Abril¹ and María de las Mercedes Abril¹

1_U_n_i_v_e_r_s_i_d_a_d_N_a_c_i_o_n_a_l_d_e_T_u_c_u_m_á_n_a_n_d_C_o_n_s_e_j_o_N_a_c_i_o_n_a_l_d_e_I_n_v_e_s_t_i_-

gaciones Cientíﬁcas y Técnicas (CONICET). Av. Independencia 1900,

San Miguel de Tucumán, Tucumán, Argentina.

Correspondence: jabril@herrera.unt.edu.ar;

mabrilblanco@hotmail.com

Received: December 18, 2024 - Accepted: January 30, 2025 -

Published: February 10, 2025

ABSTRACT

Economic time series often exhibit volatility, where the variance

of the observational error ﬂuctuates over time. One of the most

widely used methodologies for modeling these dynamics is the

ARCH model, introduced by Engle (1982), and its extensions,

such as GARCH models. These assume that conditional vari-

ance depends on past values of the series. In contrast, stochastic

volatility models (SVM), ﬁrst proposed by Taylor (1980, 1986),

assume that volatility depends on past variances but not directly

on past returns. This study compares both approaches in mod-

eling the volatility of a small stock market. To evaluate the

performance of ARCH-GARCH and stochastic volatility mod-

els in estimating market risk and identifying volatility patterns

in the Merval index, which represents the Buenos Aires Stock

Exchange (BASE). A longitudinal observational study was con-

ducted using daily Merval index data from January 13, 2003, to

May 22, 2015, covering 3006 observations. This period was cho-

sen to avoid political shifts that could introduce market distor-

tions. Statistical tests (ADF, Phillips-Perron) were performed to

check stationarity, and models were estimated using maximum

likelihood and Kalman ﬁltering. GARCH models with heavy-

tailed distributions provided better short-term volatility predic-

tions, capturing volatility clustering, while stochastic volatility

models were more effective at identifying regime shifts. The

Merval index, with an average market capitalization of $312

million, conﬁrms the characteristics of a small stock market,

where volatility models play a crucial role in risk assessment.

The choice between ARCH-GARCH and stochastic models de-

pends on the forecasting horizon. GARCH models are optimal

for short-term risk evaluation, whereas stochastic models are bet-

ter suited for detecting long-term structural changes. Combining

Palabras claves: Volatility, ARCH-GARCH models, State

space models, Kalman ﬁlter, Stochastic volatility, Merval index.˙

1

INTRODUCTION

The study of the phenomenon of volatility has been developed

mainly from the analysis of time series related to the economy.

However, it must be emphasized that any time series may be sub-

ject to the presence of volatility.

Many economic time series do not have a constant mean and

in practical situations we often see that the variance of the ob-

servational error, conditional on past knowledge, is subject to

substantial variability over time. This phenomenon is known as

volatility.

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To take into account the presence of volatility in an economic

series it is necessary to resort to models known as conditional

heteroscedastic models. In these models, the variance of a series

at a given moment of time depends on past information and other

data available up to that time, so a conditional variance must be

deﬁned, which is not constant and does not coincide with the

overall variance of the observed series.

An important characteristic of ﬁnancial time series is that they

are not generally serially correlated, but rather dependent. Thus,

linear models such as those belonging to the ARMA model fam-

ily may not be appropriate to describe these series.

There is a very large variety of non-linear models in the litera-

ture, useful for the analysis of economic time series with volatil-

ity. An important class of them are the ARCH-type models in-

troduced by Engle (1982) and its extensions. These models are

non-linear with respect to variance.

with a small stock market.

As stated, in this paper we analyze the Merval index series.

This is a series with information corresponding to all working

days of the stock market. Speciﬁcally, we work with the returns

of the quotes of this index, which consists of the ﬁrst differences

of the logarithm of the Merval levels. The period analyzed goes

from January 13, 2003 to May 22, 2015. There are 3,006 obser-

vations. It covers a period in which there was no change in the

government’s afﬁliation. In fact, during that period the wing of

Peronism called Kirchnerism governed. This eliminates the ef-

fects that could have been introduced into the market by changes

in the governing group.

To perform the analysis we used two approaches: one based

on ARCH-GARCH type models, and another based on stochastic

volatility models. We then made comparisons between these two

approaches.

The ARCH or GARCH family of models assume that the con-

ditional variance (volatility) depends on past observations. In

2

ECONOMIC AND FINANCIAL TIME SERIES

MODELLING

2

other words, if σ is the volatility, the ARCH-GARCH family

t

assumes that it depends on the series yj for j < t. On the other

hand, the stochastic volatility model or SVM, ﬁrst proposed by

Taylor (1980, 1986), is not based on this assumption. This model

The basic idea of a time series is very simple, it consists of

the recording of any ﬂuctuating quantity measured at different

points in time.

Speciﬁcally, a time series is a set of observations {y1, ..., yn}

ordered in time. The basic and general model used to represent

any time series is the additive model, given by

2

t

2

j

assumes that the volatility σ depends on its past values (σ for

j < t) but is independent of the past of the series under analysis

(

yj for j < t).

A cursory inspection of series such as the one presented in

this paper suggests that they do not have a constant mean and

variance. A stochastic variable in which the variance is con-

stant is said to be homoscedastic as opposed to a heteroscedastic

variable. For those series in which there is volatility, the uncon-

ditional variance may be constant even though the conditional

variance in some periods is unusually large and in others small.

As an application, the Merval index series is analyzed. The

Merval is a stock market index that has been calculated in the

Buenos Aires Stock Exchange (BASE), Argentina, since June

yt = µt + γt + εt,

t = 1, ..., n,

(1)

wehere µ is a component that changes smoothly over time called

trend, γ is a periodic component called seasonality and ε is an

t

irregular component called error. As we can see, the common

feature of all records belonging to the time series domain is that

they are inﬂuenced, at least partially, by sources of random vari-

ation.

The main reason for modelling a time series is to enable pre-

diction of its future values. The distinctive feature of a time se-

ries model, as opposed to, for example, an econometric model, is

that no attempt is made to formulate a behavioural relationship

between the time series under consideration and other explana-

tory variables. Movements of the series are explained solely in

terms of its own past, or by its position relative to time or by its

structure. Predictions are made by extrapolation.

Many economic time series do not have a constant mean and

in many cases there are periods of relative calm followed by pe-

riods of signiﬁcant changes. Much of the current research in

time series and econometrics is focused on extending the clas-

sical and commonly used methodology of Box and Jenkins to

analyze this type of behaviour. However, there is a characteristic

present in time series that refer to ﬁnancial assets (or directly ﬁ-

nancial time series) and other series referring to economic activ-

ities and it is what is known as volatility, which can be deﬁned in

various ways, but is not directly observable. To take into account

the presence of volatility groups in a ﬁnancial or economic se-

ries it is necessary to resort to models known as conditional het-

eroscedastic models. In these models, the variance (or volatility)

of a series at a given time depends on its past and other infor-

mation available up to that time, so a conditional variance must

3

0, 1986. It measures the traded volume of the main shares

listed on that exchange. The index is composed of a ﬁxed nom-

inal amount of shares of different listed companies, commonly

known as “leading companies”. The shares that make up the

Merval index change every three (3) months, when this portfolio

is recalculated, based on the participation in the traded volume

and the number of operations of the last six (6) months. Those

shares that are within the accumulated 80% of market participa-

tion are selected. In addition, the selected companies must meet

the requirement of having traded in at least 80% of the trading

sessions of the period considered.

The Buenos Aires Stock Exchange was founded on July 10,

854. It is the largest stock exchange and the main business and

1

ﬁnancial centre of the Argentine Republic. Its transactions are

basically shares of important national and foreign companies,

bonds, currencies and futures contracts. It is a non-proﬁt civil

association run by representatives of the various business sec-

tors. According to a study carried out by the International Fi-

nance Corporation, which is part of the World Bank, the average

value of the companies listed on the BASE is 312 million dollars,

a ﬁgure that places Argentina in 30th place among the countries

that have stock markets. That is why we say that we are dealing

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q

2

t−1

be deﬁned, which is not constant and does not coincide with the

global or non-conditional variance of the observed series.

is non-linear in variance since g(·) = 0 and h(·) = αy

and

yt−1 depends on εt−1.

ARCH models or autoregressive with conditional het-

eroscedasticity models were ﬁrst introduced by Engle in 1982

to estimate the variance of inﬂation in Britain. The basic idea of

this model is that the price of an asset yt is not serially correlated

but depends on past prices via a quadratic function.

In conventional econometric models, the variance of the dis-

turbance is assumed to be constant. However, it can be seen that

many economic series exhibit periods of unusually large volatil-

ity followed by periods of relative calm. In these circumstances,

the assumption of constant variance, also called homoscedastic-

ity, is somewhat inappropriate. There are instances when one

wishes to predict the conditional variance of a series. Asset hold-

ers may be interested in predictions of the rates of return and their

variance for a given period. The unconditional variance (i.e. the

long-term variance) would not be important if one plans to buy

the asset at time t and sell it at t + 1.

3

THE VOLATILITY

Volatility is deﬁned as the variance of a random variable, con-

ditional on all past information. Since volatility cannot be mea-

sured directly, it can manifest itself in various ways in a time

series.

Let yt be the series under study whose dimension is p = 1.

We deﬁne

µt = E(yt |Yt−1 ) = Et−1(yt),

(2)

ꢀ

ꢁ

2

σ_t

=

var(yt |Yt−1 ) = E (yt − µt) |Yt−1

Et−1(yt − µt) = vart−1(yt),

2

(3)

as the conditional mean and variance of yt given the information

up to time t − 1 contained in Yt−1, respectively.

An ARCH(q) model can be expressed as

A common model to take volatility into account is of the form

p

yt = µt + htεt,

(4)

yt

=

µt + εtht = µt + εtσt, εt ∼ iid D (0, 1) , (6)

q

where Et−1(εt) = 0, var(εt) = 1 and typically the εt are inde-

pendent and identically distributed (iid) with distribution func-

tion F. The unconditional mean and variance of yt will be de-

X

2

t−1

^σt

h = ω +

t

αiz

,

(7)

i=1

2

noted as µ = E(yt) and σ = var(yt), respectively, and let G

be the distribution function of yt. It is clear that (2), (3) and F

determine µ, σ and G, but not the opposite. More details about

this formulation can be seen in Abril M. (2014).

where zt = yt − µt and D (·) is a probability density function

with mean equal to zero and unit variance.

2

An ARCH model adequately describes volatility clustering.

The conditional variance of yt is an increasing function of the

square of the shock occurring at time t−1. Consequently, if yt−1

4

MODELS OF THE ARCH-GARCH FAMILY

2

t

is large enough in absolute value, σ and thus yt are expected to

be large enough in absolute value as well. It should be noted that

even though the conditional variance in an ARCH-type model

There is a very large variety of non-linear models available

in the literature to deal with volatility, but we will concentrate

on the ARCH type models or autoregressive with conditional

heteroscedasticity models, introduced by R. Engle (1982) and its

extensions. These models are non-linear as far as the variance is

concerned.

ꢂ

ꢃ

2

t

varies over time, i.e., σ = E z |Yt−1 the unconditional vari-

t

P

q

i=1

ance of zt is constant and, since ω > 0 and

αi < 1, we

have

ꢀ

ꢂ

ꢃꢁ

ω

2

t

σ ≡ E E z |Yt−1

=

P

.

(8)

q

1

−

αi

i=1

ꢂ ꢃ

In the analysis of non-linear models the errors (also called

innovations, because they represent the new part of the series

that cannot be predicted from the past) εt, are generally assumed

to be iid and the model has the form

3

4

If εt is normally distributed, then E ε = 0 and E ε = 3.

t

ꢂ

ꢃ

3

Therefore, E z_t= 0 and the symmetry of the variable z will

be equal to zero. Thus, the kurtosis coefﬁcient for an ARCH(1)

p

2

1

2

1

is 3(1 − α )/(1 − 3α ) if α1 < 1/3 ≈ 0, 577. In this case,

the conditional distribution of any series will have heavy tails if

yt

=

g(εt−1, εt−2, . . .) + εth(εt−1, εt−2, . . .)

gt + εtht = µt + εtht,

α1 > 0.

(5)

In most practical applications, excessive kurtosis in an ARCH

model means that a normal distribution is not adequate enough

to explain the process generating the data. Therefore, we can

make use of other distributions. For example, we can assume

that εt follows a Student t distribution with mean 0, variance

equal to 1 and υ degrees of freedom, that is, εt is ST (0, 1, υ).

In this case, the unconditional kurtosis for the ARCH(1) is

where g(·) = gt = µt represents the conditional mean and

2

t

h (·) = h is the conditional variance. If g(·) is non-linear,

the model is said to be non-linear in the mean, on the other hand

if h(·) is non-linear, the model is said to be non-linear in the

variance. For example, the model

2

t−1

2

λ(1 − α )/(1 − λα ) where λ = 3 (υ − 2) / (υ − 4). Due to

1

2

yt = εt + αε

,

1

the additional coefﬁcient υ, the ARCH(1) model based on a t

distribution will have heavier tails than the one based on a nor-

mal distribution, which will be very useful when analyzing the

data in our study. It is important to note that other distributions

2

t−1

is non-linear in the mean since g(·) = αε

and h(·) = 1, while

the model

q

2

yt = εt αy

,

t−1

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for performing the analysis are available in some software pack-

ages.

It should be noted that GARCH-type models are not the only

extension of ARCH-type models and there are at least twelve

speciﬁcations related to them that will be the subject of future

research.

The generalized ARCH models (or GARCH models as they

are also known) are based on an inﬁnite ARCH speciﬁcation and

allow reducing the number of parameters to be estimated by im-

posing non-linear restrictions on them. The GARCH(p, q) model

is expressed as follows

2

The calculation of σ in (7) depends on the past unobserved

t

2

t

quadratic residuals, z , for t = 0, −1, . . . , −q + 1.To initialize

the process, the unobserved quadratic residuals are set to a value

equal to the sample mean of the observed ones.

The conditional mean µt can contain n1explanatory variables,

which are speciﬁed as follows

n1

X

q

p

µt = µ +

δi xi,t.

(9)

X

2

t−i

2

t−j

i=1

σ = ω +

αiz

+

βjσ

.

(11)

t

i=1

j=1

On the other hand, n2 explanatory variables can be included

in the conditional variance given in (7), as follows

Using the lag operator L, the GARCH(p, q) model is trans-

formed into

n2

X

ωt = ω +

ωi xi,t.

(10)

2

t

2

t

2

t

σ = ω + α (L) z + β (L) σ ,

i=1

2

q

where α (L) = α1L + α2L + . . . αqL and β (L) = β1L +

β2L + . . . βpL .

where the xi,t of (9) are not necessarily the same as those ap-

2

p

pearing in (10).

2

If all the roots of the polynomial |1 − β (L)| = 0 lie outside

σ must obviously be positive for all t. The sufﬁcient condi-

t

the unit circle we have

tions that ensure that the conditional variance is positive in (7)

are given by ω > 0 and αi ⩾ 0 for all i. Furthermore, when

explanatory variables enter the ARCH speciﬁcation, these pos-

itivity restrictions no longer hold, although we still require that

the conditional variance be non-negative.

A very simple device for reducing the number of parame-

ters to be estimated, which we will not develop in this paper,

is called variance orientation and was ﬁrst developed by Engle

and Mezrich (1996).

The conditional variance matrix in an ARCH model, and in

most of its generalizations, can be expressed in terms of its un-

conditional variance and other parameters. By this means, it

is possible to reparameterize the model using the unconditional

variance and replace it by a consistent estimator before maximiz-

ing the likelihood.

σ = ω |1 − β (L)| ₊_α₍_L₎_|₁₋_β₍_L₎_|−1 z ,

2

−1

2

t

(12)

t

which can be seen as an ARCH(∞) model since the conditional

variance depends linearly on all previous quadratic residuals. In

this case, the conditional variance of yt can be larger than the

unconditional variance given by

ꢂ

ꢃ

ω

2

t

σ ≡ E z

=

P

,

βj

q

i=1

p

j=1

1

−

αi −

2

if past realizations of z are greater than σ (Palm, 1996).

t

Applying the variance orientation procedure to a GARCH

ꢄ

ꢅ

P

2

q

i=1

p

j=1

model involves replacing ω by σ 1 −

αi −

βj ,

2

t

where σ is the unconditional variance of z which can be con-

Applying variance orientation for an ARCH model involves

sistently estimated by means of its sample counterpart.

On the other hand, if the explanatory variables ap-

pear in a GARCH-type formulation, ω is then replaced by

P

2

q

i=1

2

replacing ω by σ (1 −

αi), where σ is the unconditional

variance of yt, which can be consistently estimated by its sample

ꢄ

ꢅ

P

counterpart.

_σ2 1 −

q

i=1

p

j=1

n2

ωi xi, where xi is the

i=1

αi −

βj

−

If explanatory variables appear in the ARCH equation then ω

is replaced by

sample mean of the variable xi,t, assuming stationarity of the n2

explanatory variables.

!

q

n2

Bollerslev (1986) showed that for

a

GARCH(1, 1)

X

2

σ

1 −

αi

−

ωixi,

with normal innovations,

the kurtosis of

y

is

h

i.h

i

2

i=1

3

1 − (α1 + β1)

1 − (α1 + β1) − 2α₁

>

3.

The

2

t

where xi is the sample average of the variable xi,t, assuming

that there is stationarity in the n2 explanatory variables. In other

words, the explanatory variables are centered.

autocorrelations of z were derived by Bollerslev (1986). For a

stationary GARCH(1, 1)

ꢆ

ꢇꢂ

ꢃꢈ

2

1

¹⁻²^α¹^β¹⁺^β1²

ρ1

ρk

=

α1 + α β1

,

While Engle (1982) certainly made the major contribution

to ﬁnancial econometrics, ARCH-type models are rarely used

in practice due to their simplicity. A good generalization of

these models is found in the GARCH-type models introduced

by Bollerslev (1986). These models are also a weighted aver-

age of the past squared residuals, are more parsimonious than

ARCH-type models and even in their simplest form have proven

to be extremely successful in predicting conditional variances.

k−1

(α1 + β1)

ρ1, for all k = 2, 3, . . . .

In other words, the autocorrelations decline exponentially with a

decline factor equal to α1 + β1.

As in the case of ARCH models, it is necessary to impose

2

t

some restrictions on σ to ensure that it is positive for all t.

Bollerslev (1986) showed that ensuring that ω > 0, αi ≥ 0

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(

for i = 1, . . . , q) and βj ≥ 0 (for j = 1, . . . , p) are sufﬁ-

Student t distribution and the skewed-Student distribution.

cient to ensure that the conditional variance is positive. In prac-

tical situations, the parameters in a GARCH-type model are es-

timated without using positivity restrictions. Nelson and Cao

The logic of the maximum likelihood method is to interpret

the density as a function of the set of parameters, conditional on

the set of sample observations. This function is called the likeli-

hood function. It is evident from (8) that the recursive evaluation

of this function is conditional on the observed values. For this

reason, we will consider the approximate or conditional maxi-

mum likelihood and not the exact maximum likelihood.

The logarithm of the likelihood function for the standard nor-

mal distribution is given by

(

1992) stated that imposing the condition that all coefﬁcients

are non-negative is too restrictive and that some of them may

in practical situations be negative while the conditional variance

remains positive (reviewing a good number of real-life situa-

tions). They consequently relaxed this fact and established suf-

ﬁcient conditions for the GARCH(1, q) and GARCH(2, q) cases

based on the inﬁnite representation given in (12). Indeed, we

can see that the conditional variance is strictly positive provided

n

1 Xꢆ

ꢂ

ꢃ

ꢈ

2

−

1

l_n_o_r_m= −

log (2π) + log σ_t+ ε_t

,

(13)

ω |1 − β (L)| is positive and all coefﬁcients of the polynomial

2

−

1

t=1

α (L) |1 − β (L)| in (12) are nonnegative. The positivity re-

strictions proposed by Bollerslev (1986) can be ﬁxed during the

estimation. If not, they, as well as those imposed during the

ARCH(∞), representation, will be tested a posteriori if there is

no explanatory variable in the conditional variance equation.

where n is the number of observations.

For a Student t distribution this function is

ꢉ

ꢊ

ꢋ

ꢃ

ꢌ

ꢄ ꢅ

ν + 1

ν

1

2

lStud

=

n log Γ

− log Γ

−

ꢊ

log [π (ν − 2)]

2

4

.1 Distributions and estimation

There are basically three methods for estimating ARCH-

n ꢍ

ꢋꢎ

X

ꢂ

2

ε

t

1

2

−

log σ + (1 + ν) log 1 +

, (14)

2

ν − 2

t=1

GARCH type models:

where ν are the degrees of freedom, with 2 < ν ≤ ∞ and Γ (·)

1

. The most commonly used one, and the one we will work

with in this case, is the standard estimation by maximum

likelihood. This uses Newton’s quasi-maximum likeli-

hood method developed by Broyden, Fletcher, Goldfrab

and Shanno (or BFGS according to its acronym).

is the gamma function.

For a skewed Student distribution (with mean zero and vari-

ance one) this function is

ꢉ

ꢊ

ꢋ

ꢄ ꢅ

η + 1

η

1

2

lSkSt

=

n log Γ

− log Γ

−

log [π (η − 2)]

2

. An optimization technique that implements quadratic se-

quence programming to maximize a nonlinear function,

subject to nonlinear constraints similar to Algorithm 18.7

in Nocedal and Wright (1999). This is particularly useful

when imposing positivity or stationarity constraints such as

α1 > 0 in an ARCH model.

2

!

)

2

+ log

+ log(s)

ξ + ¹

ξ

ꢉ

ꢍ

ꢎꢌ

1ⁿ

(sε + m)²

2

t

−2It

−

log(σ ) + (1 + η) 1 +

ξ

,

2

t=1

η − 2

(

15)

3

. And ﬁnally a simulation algorithm to optimize non-smooth

functions with multiple possible local maxima.

where

ꢉ

m

if εt ≥ −

s

1 if εt < −

1

−

In our case, the estimation in ARCH-GARCH type models is

carried out using the quasi-maximum likelihood method, so it is

necessary to make an additional assumption about the innovation

process εt, that is, to choose the density function D (0, 1) that

has a mean equal to zero and a unitary variance.

Weiss (1986) and Bollerslev and Woolridge (1992) showed

that under the assumption of normality, the quasi-maximum like-

lihood estimator (or QML according to its acronym) is consistent

if the conditional mean and the conditional variance are correctly

speciﬁed. This estimator is, however, inefﬁcient with an increas-

ing degree of inefﬁciency as it deviates from that normality (En-

gle and González-Rivera 1991).

As stated by Palm (1996), Pagan (1996) and Bollerslev,

Chou and Kroner (1992), the use of heavy-tailed distributions

is widespread in the literature. Bolleslev (1987), Hsieh (1989),

Baillie and Bolleslev (1989) and Palm and Vlaar (1997) among

others, showed that these distributions perform better when cap-

turing higher order kurtosis.

It =

,

m

s

ξ is the asymmetry parameter, η are the degrees of freedom of

the distribution,

ꢂ

ꢃ√

ꢊ

ꢋ

η−1

Γ

η − 2

2

1

2

m =

√

ꢂ ꢃ

ξ −

,

η

πΓ

ξ

and

s

ꢊ

ꢋ

s =

ξ²+¹− 1 − m .

2

_ξ2

It is important to note that this last function is the one that will

be used when analyzing our data set.

There are other distributions that can be used to carry out the

estimation process, such as the generalized error distribution or

GED, but we will not develop them in our work since they are

not objects of our research.

For our problem, we will consider three distributions when

approaching the estimation process; the normal distribution, the

In terms of the estimation process, we can say that many au-

thors have proposed using a Student t distribution or a skewed

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Student distribution in combination with a GARCH type model

to adequately model the heavy tails in economic or ﬁnancial time

series whose data are of high frequency, which will be seen later.

distribution called “log chi square”, such that

2

t

E{log(ε )} ≈ −1, 27,

(24)

(25)

2

t

2

var{log(ε )}

=

π /2.

5

MODELS FOR STOCHASTIC VOLATILITY

From (16), (17) and (18) we get

We will say that the series yt follows a stochastic volatility

model (SVM) if

2

log(y )

=

log(σ ) + log(ε ),

(26)

t

2

t

ht

log(σ ) = α0 + α1ht−1 + ηt.

(27)

yt

=

σtεt,

(16)

(17)

ht

2 2

t

t t

Calling ξ = log(ε ) − E{log(ε )} ≈ log(ε ) + 1, 27, we

2

t

_e2

σt

,

have that E(ξt) = 0, var(ξt) = π /2 and

where εt is a stationary series with mean equal to zero and vari-

ance one, and ht is another stationary series with probability den-

sity given by a function f (h).

2

log(yt ) = −1, 27 + ht + ξt,

ξt ∼ iid (0, π /2),(28)

2

ht

=

α0 + α1ht−1 + ηt,

ηt ∼ iid N(0, σ ) (, 29)

η

2

As we can see in (17), ht is not equal to the volatility σ as is

t

where iid means that the variables are independent and identi-

cally distributed. It is also assumed that ξt and ηt are indepen-

dent of each other at all times.

From the equations (16), (17) and (18) let us calculate some

moments of the SVM. Taking expectation of (16) we have

usually the notation in the ARCH-GARCH family models.

The simplest formulation of the model assumes that the loga-

rithm of volatility is given by

ht = α0 + α1ht−1 + ηt,

(18)

E(yt) = E(σtεt) = E(σt)E(εt) = 0,

(30)

where ηt is a stationary, Gaussian series, with mean zero, vari-

2

ance σ and independent of εt. It follows from this that we must

η

given that σt and εt are independent.

have |α1| < 1.

The variance of yt is

5

.1 Other SVM formulations

2

t

2

t

2

t

2

t

2

t

2

t

var(yt) = E(y ) = E(σ ε ) = E(σ )E(ε ) = E(σ ). (31)

Other SVM formulations can be found in the literature, among

which we highlight the following:

2

Since we assume that ηt ∼ N(0, σ ) and that ht is stationary

η

with

α0

E(ht) =

= µh,

(32)

1

. Canonical form of Kim, Shephard and Chib (1998). In this

case the SVM is written as

1 − α1

and with

2

η

σ

ht

2

_β_e2 εt,

var(ht) =

= σ ,

(33)

yt

=

(19)

(20)

h

2

1

− α₁

ht

µ + α1(ht−1 − µ) + σηηt,

we have that

!

with

!

2

σ

η

2

σ

α0

η

ht ∼ N

,

.

(34)

2

ht ∼ N µ,

,

(21)

1 − α₁1 − α₁

2

1

− α₁

2 ht

Since ht is normal or Gaussian, σ = e is log-normal, then

t

where εt and ηt are both N(0, 1) and independent of each

other. If β = 1, then µ = 0.

we have

2

h

2

t

2

t

µh+σ /2

var(yt) = E(y ) = E(σ ) = e

.

(35)

(36)

2

. The Jaquier, Polson and Rossi (1994) form of the SVM is

equal to

It is not difﬁcult to show that

p

2µh+2σh²

yt

=

htεt,

(22)

(23)

4

E(y ) = 3e

,

t

log(ht)

α0 + α1 log(ht−1) + σηηt,

from which we obtain that the kurtosis of yt is

where εt and ηt are both N(0, 1) and independent of each

other.

2

h

2

µ

+2σ

h

2

σ

3

e

2

h

= 3e > 3,

K(yt) =

(37)

2

h

µ

+σ

h

e

5

.2 Properties of SVM

as expected, that is, there are heavy tails for the SVM.

Let us return to the model deﬁned in the equations (16), (17) y

The autocovariance function of the series yt is given by

(

18). Suppose that {εt} constitutes a succession of independent

2

t

random variables such that εt ∼ N(0, 1), then log(ε ) has a

γy(s) = E(ytyt+s) = E(σtσt+sεtεt+s) = 0,

(38)

30

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since εt and ηt are independent. Then yt is serially uncorrelated

the yt have no serial correlation.

2

but not independent since there is correlation in log(y ). Let us

t

2

t

denote as zt = log(y ), then the autocovariance function of the

5.3 Estimation of the SVM

series zt is given by

SVM models are difﬁcult to estimate. We can use the ap-

proach proposed by Durbin and Koopman (1997a, 1997b, 2000,

γz(s) = E[(zt − E(zt))(zt+s − E(zt+s))].

(39)

2001, 2012) which consists of using a quasi-maximum likeli-

hood procedure by means of the Kalman ﬁlter and smoother. In

this case, the model deﬁned in the equations (16), (17) y (18) can

be re-expressed in the form

As the ﬁrst term in parentheses of (39) is equal to ht −E(ht)+ξt

and ht is independent of ξt, we get

ht

yt

=

σεte ²

,

(47)

γz(s)

=

E[(ht − E(ht) + ξt)(ht+s − E(ht+s) + ξt+s)]

E[(ht − E(ht))(ht+s − E(ht+s))]

ht

σt

ht

=

σe ²

,

(48)

(49)

+

E(ξtξt+s),

(40)

α1ht−1 + ηt,

where σ = exp(α0/2) is a scale factor, α1 is a parameter, and

ηt is a disturbance term which in the simplest model is uncor-

related with εt. Literature reviews of this model were carried

out by Shephard (1996, 2005) and Ghysels, Harvey and Renault

and calling γh(s) and γξ(s) respectively to the autocovari-

ances of the right hand of (40), we have

γz(s) = γh(s) + γξ(s),

(41)

(

1996). This SVM has two main attractions. The ﬁrst is that

for all s.

it is a discrete-time natural (Euler) analogue of the continuous-

time model used in option pricing work, such as that of Hull

and White (1987). The second is that its statistical properties are

easy to determine. The disadvantage with respect to conditional

variance models of the GARCH type is that likelihood-based

estimation can only be performed by computationally intensive

techniques such as those described in Kim, Shephard and Chib

As we are assuming that (18) is satisﬁed, that is, we have a

AR(1) model, we get

2

η

σ

s

1

γh(s) = α

,

2

s > 0.

(42)

1

− α₁

Besides γξ(s) = 0 for s > 0. Then, γz(s) = γh(s) for all

s = 0. With this we can write the autocorrelation function of

̸

2

zt = log(y ) as

(

1998) and Sandmann and Koopman (1998). However, a quasi-

maximum likelihood method is relatively easy to implement and

is usually reasonably efﬁcient. The method is based on writing

t

(

47), (48) y (49) in the following equivalent form

s 2

α ση/(1 − α )

1

,

γh(0) + γξ(0)

2

γz(s)

γz(0)

1

ρz(s) =

from which we get

ρz(s) =

=

s > 0,

(43)

ꢂ

ꢃ

2

log y_t

=

κ + ht + ξt,

α1ht−1 + ηt,

(50)

(51)

ht

ꢂ ꢃ

ꢂ

ꢃ

2

s

α

1

where ξt = log ε − E{log ε } and κ = log σ

+

t

ꢂ ꢃ

,

s > 0,

2

π²

E{log ε }.

1

+ ₂_σ

2

t

η

The equations (50) and (51) are expressed in the form of state

space, as can be seen in Abril and Abril (2018). The formula (50)

is called the observation equation or measurement equation and

the formula (51) is called the state equation or transition equa-

tion. The estimation process is therefore carried out using the

Kalman ﬁlter and smoother in the same way as those developed

in the previous reference.

which tends to zero exponentially from the lag s = 2, and this in-

dicates that zt = log(y ) can be modeled using an AR(1) model.

2

t

En la práctica obtenemos valores de α1 próximos de uno, lo

que implica la aparición de altas correlaciones para la volatilidad

y consecuentes grupos de volatilidades en la serie.

A general SVM can be obtained if an AR(p) model is admitted

for ht, that is

It is important to make some observations here:

yt

=

σtεt,

ht

e ²

,

(44)

(45)

1. When α1 in (51) is close to 1, the ﬁt of an SVM is sim-

ilar to that of a GARCH(1, 1) model with the sum of its

coefﬁcients close to 1.

σt

=

2

p

(

1 − α1B − α2B − · · · − αpB )ht

α0 + ηt, (46)

2

. When α1 = 1 in (51), ht is a random walk and the ﬁt of an

SVM is similar to that of an IGARCH(1, 1) model.

j

where the lag operator is deﬁned as B ht = ht−j, the assump-

tions about the innovations εt and ηt are the same as those made

previously, but now we assume that the roots of the polynomial

3. When some observations are equal to zero, which can occur

in practice, the logarithmic transformation speciﬁed in (50)

cannot be performed. One way to avoid this problem is

to subtract the overall mean of the series yt of each of the

observations and taking this result as the series to work on;

2

p

(

1 − α1B − α2B − · · · − αpB ) are outside the unit circle.

The SVM have been extended to include the fact that volatility

has long memory, in the sense that the autocorrelation function

of zt = log(y ) slowly decays, although as we saw in this case,

2

t

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that is, taking as a working series

can be formulated as

yt − y, t = 1, . . . , n,

(52)

yt

=

Ztβt + νt,

(54)

P

β_t

t

T β

+ R ω ,

ω ∼ N(0, Q ), t = 1, . . .(,5 n5 ,)

n

t−1

t

−

1

where y = n

yt. Another solution, suggested

t=1

by Wayne Fuller and analyzed by Breidt and Carriquiry

1996), is to make the following transformation based on

a Taylor expansion

with

(

ht

2

ν_t

=

σe

ε ,

t

(56)

ht

2

cS_y²

σ_t

=

,

(57)

(58)

ꢂ

ꢃ

ꢂ

ꢃ

2

t

2

y

log y_t= log y + cS

−

,

2

y + cS

y

t = 1, . . . , n,

53)

2

ht

α1ht−1 + ηt,

t

(

2

where βt is the state vector of order m × 1, ωt are serially in-

dependent disturbances, independent of each other and indepen-

dent of νt at all times. The system matrices Zt, Tt, Rt y Qt

have dimensions 1×m, m×m, m×m and m×m respectively,

and if there are unknown elements in them, they are incorporated

into the vector ψ of hyperparameters which is estimated by max-

imum likelihood. (54) is called measurement equation or obser-

vation equation and (55) transition equation or equation of state.

The equations (54) y (55) deﬁne a state space model with all the

characteristics and properties of the same presented in Abril and

Abril (2018). In effect, there can be trends, seasonality, cycles,

explanatory variables and other important characteristics that ex-

plain the behavior of the process {yt}. The equations (56), (57)

and (58) deﬁne an SVM with structural components (seasonality

in this case) for the errors of the state space model given above,

where εt is a stationary series with mean equal to zero and vari-

where S is the sample variance of the series yt and c is

y

a small number. Versions prior to 8.3 of the STAMP pro-

gram developed by Koopman, Harvey, Doornik, and Shep-

hard incorporated the transformation deﬁned in (53) with

c = 0, 02 as a pre-speciﬁed operation that could be used

if needed. Starting with version 8.3 of that program (see

Koopman, Harvey, Doornik, & Shephard, 2010) that trans-

formation was no longer a pre-speciﬁed element, and that

or other transformations could be performed, such as the

one deﬁned in (52), by using the calculator or the availabil-

ity of Algebra within that program according to the user’s

requirements and needs.

As shown in Harvey, Ruiz and Shephard (1994), the state

space form given by the equations (50) and (51) provides the

basis for quasi-maximum likelihood estimation via the Kalman

ﬁlter and smoother and also allows for constructing smoothed

estimates of the component ht of the variance and make pre-

dictions. One of the attractions of the quasi-maximum likeli-

hood approach is that it can be applied without an assumption

about a particular distribution for εt. Another attraction of using

a quasi-maximum likelihood procedure using Kalman ﬁlter and

smoother to estimate SVM is that it can be carried out directly us-

ing standard computing packages such as STAMP by Koopman,

Harvey, Doornik, and Shephard (2010). This is a major advan-

tage compared to more labor-intensive simulation-based meth-

ods.

ance one, and ηt is a stationary, Gaussian series, with mean zero,

variance σ and independent of εt at all times.

2

η

The estimates of the state space models are performed us-

ing standard computing packages such as STAMP by Koopman,

Harvey, Doornik, and Shephard (2010), which is the one used in

this work.

In this case, the volatility is equal to

2

t

2 ht

σ = σ e .

(59)

Shephard and Pitt (1997) proposed the use of importance sam-

pling to estimate the likelihood function in the non-Gaussian

case.

Since the SVM is a hierarchical model, Jaquier, Polson, and

Rossi (1994) proposed a Bayesian analysis of the model. See

also Shephard and Pitt (1997) and Kim, Shephard, and Chib

The practical treatment in these cases is as follows: given a se-

ries {yt}, the linear components that can explain the behavior of

its mean are identiﬁed, including the explanatory variables that

may correspond, in such a way as to explicitly deﬁne the model

of the equations (54) and (55).This ﬁrst performs ﬁltering and

b

then smoothing of Kalman, obtaining the smoothed estimator βt

(

1998). An overview of the SVM estimation problem is provided

of the the state vector βt. This estimator allows to calculate the

by Motta (2001).

smoothed residuals as

5

.4 Series with errors following a SVM with struc-

tural components

The basic SVM given in (47), (48) and (49) captures only the

b

νbt = yt − Ztβt, t = 1, . . . , n.

(60)

These smoothed residuals estimate the disturbances νt. The val-

ues of νbt serve as a basis for testing the null hypothesis of lack of

serial correlation of νt. If this hypothesis is accepted, it could be

said that the model given in the equations (54) and (55) was ad-

equately identiﬁed, deﬁned and estimated. On the other hand, if

salient features of changing conditional heteroscedasticity in a

time series. In some cases the model is more accurate when the

series yt is modeled by incorporating structural components, ex-

planatory variables and other characteristics that explain its be-

havior, all of this done through a state space scheme with errors

that follow a SVM with structural components, for example with

seasonality. Based on the above, for a univariate series yt, this

2

t

log(νb) shows serial correlation, it can be said that the errors νt

follow an SVM of the form given in (56), (57) and (58). There-

fore, it is taken to νbt as the observed series and the following

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state space model is estimated

3. They can show the relationships between the various ele-

ments of a system or process and provide clues to future

correlations between two or more variables.

ꢂ

ꢃ

2

log νb

=

κ + ht + ξt,

α1ht−1 + ηt,

(61)

(62)

t

ht

ꢂ ꢃ

ꢂ

ꢃ

Furthermore, the application of these methods suggests new

research hypotheses and allows the subsequent implementation

of statistical models ranging from the simplest to those that are

much more reﬁned, thus achieving a better analysis of the data

and its ﬂuctuations over time.

In Figure 1 the daily series of the Merval from January 13,

003 to May 22, 2015 is shown. The upper left box shows the

2

log σ²

where ξt

=

^l^o^g^εt − E{log ε }, κ

=

+

t

ꢂ ꢃ

2

E{log ε }, εt is a stationary series with mean equal to zero

t

and variance one, and ηt is a stationary, Gaussian series, with

2

mean zero, variance σ and independent of εt at all times. The

η

process of estimating (61), (62), is done using the Kalman ﬁlter

and smoother.

2

As shown in Harvey, Ruiz and Shephard (1994), the state

space form given by the equations (61) and (62) provides the

basis for quasi-maximum likelihood estimation via the Kalman

ﬁlter and smoother and also allows for constructing smoothed

estimates of the component ht, of the variance and make pre-

dictions. One of the attractions of the quasi-maximum likeli-

hood approach is that it can be applied without an assumption

about a particular distribution for εt. Another attraction of using

a quasi-maximum likelihood procedure using Kalman ﬁlter and

smoother to estimate SVM is that it can be carried out directly us-

ing standard computing packages such as STAMP by Koopman,

Harvey, Doornik, and Shephard (2010). This is a major advan-

tage compared to more labor-intensive simulation-based meth-

ods.

levels, the upper right box shows the ﬁrst differences of the log-

arithm of the levels called returns, the lower left box shows the

histogram with the distribution of the returns compared to a nor-

mal distribution and the lower right box shows the QQ diagram

of the returns. As can be seen, the returns are not normal, they

have a distribution with some degree of negative asymmetry and

with kurtosis. Carrying out a careful inspection of the graph of

the series of returns, we can see that there are periods where the

volatility is less pronounced than in others, such as the one cor-

responding to the year 2009.

6

.1 Analysis using models from the ARCH-GARCH

family

We begin the study of the Merval series by focusing ﬁrst on

the models of the ARCH-GARCH family and using the Estimat-

ing and Forecasting ARCH Models Using G@RCH 7 package

developed by Laurent (2013).

6

ANALYSIS OF THE SERIES UNDER STUDY

The data we handle belong to a very special ﬁeld within sta-

tistical science, which is that of time series. The common char-

acteristic of all records belonging to the domain of time series is

that they are inﬂuenced, even if only partially, by non-observable

components that contain random variations, that is, the occur-

rence of unplanned events.

As an application, the Merval index series is analyzed. This

is a series with information corresponding to all working days

of the stock market. Speciﬁcally, we work with the returns of

the quotes of this index, which consists of the ﬁrst differences

of the logarithm of the Merval levels. The period analyzed goes

from January 13, 2003 to May 22, 2015. There are 3006 obser-

vations. It covers a period in which there was no change in the

government’s afﬁliation. In fact, during that period the wing of

Peronism called Kirchnerism governed. This eliminates the ef-

fects that could have been introduced in the market by changes

in the governing group.

It is important to note that although this is a very long period

to analyze, it is possible to carry out a very interesting study in

which the main characteristics of the series can be appreciated.

First, we proceed to graph it. Within the study of a series,

graphical methods are an excellent way to begin an investiga-

tion and then be able to dive into a detailed study of the subject

under consideration Among the functions that tables and graphs

perform are the following:

In Figure 2 we observe that in general the correlations and

partial correlations are close to zero, except for those of order

ﬁve and nineteen. This can be interpreted as the presence of two

periodic components, one that coincides with the working week,

which is ﬁve days long, and another with the working month,

which is almost twenty days long, which leads us to think of an

autoregressive model of order 20 for the conditional mean.

In Figure 3 we observe the square of the Merval returns, its

distribution compared to a normal one with zero mean and vari-

2

ance (0.00966) , the autocorrelation function of that series and

also the partial autocorrelation function of the same series un-

der study. We see that a high-degree ARCH model or a more

parsimonious GARCH model may be suitable to be applied.

Different alternatives were tested regarding the modeling of

the Merval returns series for the period between January 13,

2

003, and May 22, 2015. After analyzing them and seeing the

values of different goodness-of-ﬁt statistics, such as the Akaike

Criterion, the Schwarz Criterion, the Schibata Criterion, or the

Hannan-Quinn Criterion, we were left with a model based on the

equation (5) of our work, where yt is the series under study and

its explicit speciﬁcation is given by

yt = µt + εt ht,

(63)

where εt is independent with a skewed Student distribution

whose degrees of freedom are 5.82627 and the asymmetry is

1

2

. They make the data under study more visible, systematize

and synthesize them.

−

0.103452. The conditional mean µt is equal to a general mean

given by µ plus an autoregressive process of order 19 but with

. They reveal their variations and their historical or spatial

evolution.

all coefﬁcients equal to zero except those of order 1, 5 and 19,

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Figure 1. Exploratory Analysis of the Merval Index (2003-2015)

Note. The ﬁgure presents an exploratory analysis of the Merval index series from January 13, 2003, to May 22, 2015. The top left panel shows the

index levels, while the top right panel displays the logarithmic ﬁrst differences (returns). The bottom left panel illustrates the histogram of returns

compared to a normal distribution, and the bottom right panel presents a QQ plot for assessing normality.

which is explicitly

asymmetric Student distribution with mean zero, variance one,

whose degrees of freedom are 5.82627 and the asymmetry is

µt = µ + φ1yt−1 + φ5yt−5 + φ19yt−19 + νt,

(64)

−

0.103452. From here we see that the adjustment is adequate

due to the similarity between the distribution of the standardized

residuals and the skewed Student distribution.

where νt = εtht of (63). Furthermore, the conditional variance

in (63) is given by

At the top of the Figure 5 the last ten observations of the series

under study can be seen in blue, and the corresponding predic-

tions (in red) of the conditional mean. The vertical bars corre-

spond to the 95% conﬁdence intervals that serve to compare the

predicted value with that which is actually observed. In the lower

part, the conditional variance corresponding to the last ten obser-

vations of our series under study is predicted. As can be seen, the

predicted conditional variance, or volatility, increases smoothly

over time, which is reasonable for a small market where there are

usually no major changes in the values of ﬁnancial assets listed

on the stock exchange.

2

t−1

h = σ = ω + α(yt−1 − µt−1) + βσ

,

(65)

t

that is, it is a GARCH(1, 1) model with a constant given by ω.

The estimated values of the parameters and their corresponding

standard errors for the model we have formulated in (63), (64)

and (65) are expressed in the following Table 1.

In Table 1 we see that the values of the t statistic to test the

null hypothesis that the coefﬁcients φ1 y φ5 are equal to zero,

we are led to accept this hypothesis. Consequently, models were

tested in which one of the coefﬁcients was ﬁrst removed and the

other was left, then they were exchanged between the one that

was removed and the one that was left and ﬁnally both coef-

ﬁcients were removed. In all cases, the goodness-of-ﬁt statis-

tics or information criteria such as Akaike, Shibata, Schwarz and

Hannan-Quinn gave worse results than those obtained by includ-

ing these coefﬁcients. Therefore, it was decided to leave them in

the model to be estimated.

6.2 Analysis using SVM

We continue with the study of the Merval. Now using SVM.

To perform the analysis and the estimates we use the STAMP

computing package by Koopman, Harvey, Doornik, and Shep-

hard (2010). It should be remembered that this program performs

the estimates by quasi-maximum likelihood via the Kalman ﬁl-

ter and smoother. After analyzing the series (see Figures ?? and

??), and after studying several alternative models the following

In Figure 4 we have, at the top, the conditional variance

volatility) of the Merval returns series for the period between

January 13, 2003, and May 22, 2015, and the distribution of

the standardized residuals after the adjustment compared with an

(

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Figure 2. Autocorrelation and Partial Autocorrelation of the Merval Returns (2003-2015)

Note. The ﬁgure illustrates the autocorrelation (top panel) and partial autocorrelation (bottom panel) functions of the Merval returns series for the

period between January 13, 2003, and May 22, 2015. These functions help identify the time dependence structure of the series.

Table 1: Estimated Values, Standard Deviations, and t-Statistics for the GARCH(1,1) Model of Merval Returns (2003-2015)

Estimator

Estimated value Standard deviation t value Probability

µb

0.001164

0.027376

−0.033687

0.042393

0.105309

0.092285

0.883313

−0.103452

5.82627

0.000319

0.018400

0.017660

0.017573

0.037825

0.019477

0.025945

0.024290

0.61778

3.648

1.488

−1.908

2.412

2.784

4.738

34.05

−4.259

9.431

0.0003

0.1369

0.0565

0.0159

0.0054

0.0000

φb

1

φb

5

φb

1

9

ωb

αb

ARCH

b

β

GARCH

Asymmetry

Degrees of freedom

Note. This table presents the estimated parameters for the GARCH(1,1) model ﬁtted to the Merval returns series from January 13, 2003, to May

b

2

2, 2015. The parameters include the mean (µb), autoregressive coefﬁcients (φb), conditional variance parameters (ωb, αbARCH , βGARCH ), and the

asymmetry term. The degrees of freedom estimate corresponds to the assumed residual distribution.

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Figure 3. Analysis of the Squared Merval Returns (2003-2015)

Note. The ﬁgure presents an analysis of the squared Merval returns from January 13, 2003, to May 22, 2015. The top left panel shows the squared

2

returns, while the top right panel compares their distribution to a normal one with zero mean and variance (0.000966) . The bottom left panel displays

the autocorrelation function of the squared returns, and the bottom right panel presents the corresponding partial autocorrelation function.

Figure 4. Conditional Variance and Residual Distribution of Merval Returns (2003-2015)

Note. The top panel displays the conditional variance (volatility) of the Merval returns series from January 13, 2003, to May 22, 2015. The bottom

panel shows the distribution of the standardized residuals after the adjustment, compared with a skewed Student-t distribution. The estimated degrees

of freedom for this distribution are 5.82627, and the asymmetry parameter is −0.103452.

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was decided upon

the estimated autocorrelations are practically within the conﬁ-

dence band, which implies that the respective parameters of a

possible model do not differ from zero, and the oscillations of

the spectral density are insigniﬁcant compared to its scale. The

respective statistics also support these assertions.

To apply the state space scheme and to be able to make the

respective estimates of volatility, the series of residuals must be

squared and then logarithms must be taken. With this we arrive

at the model given in (67), (68) and (69). It is then prepared

to apply the state space scheme, that is, the following model is

estimated

yt = µ + θ1yt−1 + θ5yt−5 + θ19yt−19 + νt,

(66)

(67)

with

ht

νt

=

σe ²εt,

ht

σe ²

σt

ht

=

,

(68)

(69)

α1ht−1 + ηt,

where equations (67), (68) and (69) deﬁne a SVM, εt is a sta-

tionary series with mean equal to zero and variance one, and ηt

is a stationary, Gaussian series, with mean zero, variance σ and

ꢂ

ꢃ

2

t

2

η

log νb

=

κt + ht + ξt,

α h + η_t,

(70)

(71)

independent of εt at all times.

h

t

1 t−1

This part of the study begins by estimating the model given

in the equation (66). The ﬁrst thing to be observed is that the

Doornik-Hansen normality statistic, whose distribution under the

null hypothesis of normality of the errors is a χ , gives a value

of 652.86, which is very high and leads to rejecting the null hy-

where κt is the stochastic level. In Figure 7 the logarithm of the

squared residuals of the model given in (66) after estimation is

shown at the top left, at the top right is the estimated autocor-

relation function, then at the bottom left is the estimated partial

autocorrelation function and ﬁnally at the bottom right is the es-

timated density function which is represented by a red line, com-

pared to the normal density function which is represented by a

green line. It is clearly seen there that the model given in (61)

and (62) is the right one.

2

pothesis. This is not surprising since there is no Gaussianity

(

see Figure ??) and volatility is present. The H(994) test for

heteroscedasticity, which is distributed as an F(994, 994) under

the null hypothesis of presence of heteroscedasticity in the se-

ries, gives a value of 1.4193, which leads to rejecting the hy-

pothesis of existence of heteroscedasticity. Finally, the Box-

By estimating the model given in (61) and (62) we found

2

that σb

= 5.4272, σb = 5.12422, σb = 0.296033, the

2

t

η

Ljung q statistic, which in this case is distributed as χ₅₃gives

a value equal to 74.908, which leads to accepting the hypothesis

log(νb )

ξ

level κ is stochastic with variance equal to 0.00162343, and

κb= −0.89350 at the end of the period.

of lack of serial correlation in the residuals. On the other hand,

2

y

2

ν

σb = 0.0004, and σb = 0.00039523.

The normality statistic gives a value of 624.44 which is high.

This is inevitable because the transformed model (61) and (62)

is not Gaussian. This should not worry us. On the other hand,

the estimates of α1 is αb1 = 0.91838, which is high as expected.

In Figure 8 the logarithm of the squared residuals of the model

given in (66) is shown at the top after having been estimated

In Table 2 the estimated values of the parameters of the equa-

tion (66) are shown, the respective standard deviations, the val-

ues of the t statistic to test the null hypothesis that the respective

parameter is equal to zero and the probabilities in the tails cor-

responding to that hypothesis test. If these probabilities are less

than 0.05, the respective hypothesis is rejected at that level of

signiﬁcance. As we see in these ﬁgures, except for the coef-

ﬁcient θ1, all other coefﬁcients are signiﬁcantly different from

zero. In the case of the coefﬁcient θ1, models were tested in

which it was eliminated, but the goodness-of-ﬁt tests (Akaike

and others) always gave worse results than those obtained in this

case in which it was included. Therefore, it was decided to con-

tinue working with this proposal and this gives us an idea that

the adopted model is the appropriate one.

(

black line) and the estimated level (red line), in the central part

is the estimated AR(1) component whose equation is given in

62) and ﬁnally at the bottom are the estimates of the irregular

(

component of (61).

From (68) it follows that volatility is equal to

2

2 ht

σ = σ e ,

(72)

t

which has two multiplicative components which are: a scale con-

2

h

t

stant σ and basic volatility e . Of these two components, the

basic volatility is obviously the most important one since the

other is a multiplicative constant. The basic volatility is esti-

To study volatility, because errors νt of the model (66) are not

observable, they are estimated by the residues of the same after

the corresponding estimates and are denoted as νbt. The latter is

the series with which we work to estimate everything related to

volatility.

After being estimated the model given in (66), its standardized

residuals are shown at the the upper left part of Figure 6, in the

upper right part is its estimated autocorrelation function, then, in

the lower left part is the estimated spectral density and ﬁnally in

the lower right part is the estimated density function which is rep-

resented by a red line, compared with the normal density func-

tion which is represented by a green line. In this Figure we see

that the residuals do not differ signiﬁcantly from a series without

serial correlation and approximately normal. Thus, for example,

bht

b

mated as e where ht is

b

ht = αb1ht−1,

(73)

with αb1 = 0.91838, which is the estimated value of α1. To

2

estimate σ , the estimated series of the irregular component of

(

66) corrected for basic heteroscedasticity is calculated, i.e.

(

)

b

ht

νet = νbt exp −

,

(74)

2

and then the variance σe of νet is computed, which is an estimate

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Table 2: Parameter Estimates for the Model of Merval Returns (2003-2015)

Estimator Estimated value Standard deviation

t value

1.92877

−2.71835

3.63729

Probability

0.05385

b

θ

0.03518

−0.04962

0.06633

0.01824

0.01825

0.01824

1

b

θ

0.00660

5

b

θ

0.00028

1

9

Note. This table presents the estimated parameters for the model ﬁtted to the Merval returns series from January 13, 2003, to May 22, 2015. The

b

parameters include autoregressive coefﬁcients (θ

statistical signiﬁcance.

1

, θ

5

, θ19), their standard deviations, t-values, and the corresponding probabilities, which indicate their

Figure 5. Conditional Variance and Residual Distribution of Merval Returns (2003-2015)

Note. The top panel displays the conditional variance (volatility) of the Merval returns series from January 13, 2003, to May 22, 2015. The bottom

panel shows the distribution of the standardized residuals after the adjustment, compared with a skewed Student-t distribution. The estimated degrees

of freedom for this distribution are 5.82627, and the asymmetry parameter is −0.103452.

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Figure 6. Residual Analysis of the Merval Returns Model (2003-2015)

Note. The ﬁgure presents the residual analysis of the model deﬁned in Equation (66) for the Merval returns series from January 13, 2003, to May 22,

015. The top panel displays the estimated autocorrelation function of the residuals, while the bottom panels show the estimated spectral density and

2

the residual distribution. These diagnostics help assess the adequacy of the ﬁtted model.

Figure 7. Analysis of the Log-Squared Residuals of the Merval Returns Model (2003-2015)

Note. The ﬁgure presents the analysis of the log-squared residuals of the model deﬁned in Equation (66) for the Merval returns series from January

1

3, 2003, to May 22, 2015. The top panel displays the estimated autocorrelation function, while the middle panel presents the estimated partial

autocorrelation function. The bottom panel shows the residual distribution, helping assess the properties of the squared residuals.

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2

of σ . The calculated value in our case is σe = 0.819646. With

Autoregressive integrated moving average models (or ARIMA

models) are often considered to provide the main basis for mod-

eling any time series. However, given the current state of devel-

opment of time series research, there may be more attractive and,

above all, more efﬁcient alternatives. Many economic time se-

ries do not have a constant mean and also in most cases there are

phases where relative calm reigns followed by periods of signiﬁ-

cant changes, i.e. variability changes over time. This behavior is

what is called volatility.

To remedy this fact and to take into account the presence of

volatility in an economic series, it is necessary to resort to models

known as conditional heteroscedastic models. In these models,

the variance of a series at a given point in time depends on past

information and other data available up to that point in time, so

a conditional variance must be deﬁned, which is not constant

and does not coincide with the overall variance of the observed

series.

this, and based on (68), we have that the estimated volatility is

b

2

ht

σb = σe e .

(75)

t

Figure 9 shows the estimated conditional variance,

2

b

σe exp{ht}, for the entire period studied. It is important

to note that there is a peak corresponding to October 22, 2008.

At that time, between October 14 and 24 of that year (between

observation 1403 and observation 1411) there was a sharp fall

in the stock market, capital ﬂight, and a signiﬁcant rise in the

value of the dollar. This is also seen when working with the

ARCH-GARCH family of models, as can be seen in the upper

part of Figure 4. Also observed in Figure 9 is a minor depression

close to September 19, 2008 (observation 1387) corresponding

to the rise in stocks on Wall Street due to the US bailout plan, a

peak between June 16 and 26, 2014 (between observation 2781

and observation 2788) corresponding to an adverse ruling by the

US Supreme Court on Argentina’s debt, and others.

In order to detect possible relationships between the series

studied and volatility, Figure 10 shows the estimated conditional

standard deviation, i.e. σbt, versus the standardized residuals of

the model deﬁned in the equation (66) for the Merval returns se-

ries for the period between January 13, 2003 and May 22, 2015.

It is clear that no structure is observed that relates them.

In Figure 11 we can see, at the top, the last ten observations

of the series under study, in red, and the corresponding predic-

tions (in blue) of the conditional mean (model given in (66)), in

the middle part are the residuals of that estimated model with

the corresponding 95% conﬁdence band that serve to determine

if the residuals differ signiﬁcantly from zero, so it is observed

that they do not differ signiﬁcantly from zero. In the lower part,

these residuals are shown but standardized, that is, they are trans-

formed so that they have a mean of zero and variance of one.

With this we can say that the ﬁt is adequate.

Among the models we have presented are the models of the

ARCH family. The ARCH models or autoregressive models with

conditional heteroscedasticity were ﬁrst presented by Engle in

1

982 with the objective of estimating the variance of inﬂation.

The basic idea of this model is that yt s not serially correlated

but the volatility or conditional variance of the series depends on

the past of the series by means of a quadratic function. However,

these models are rarely used in practice due to their simplicity.

A good generalization of this model is found in the GARCH-

type models introduced by Bollerslev (1986). This model is also

a weighted average of a quadratic function of the past of the se-

ries, but it is more parsimonious than the ARCH-type models

and even in its simplest form it has proven to be extremely suc-

cessful in predicting conditional variances, so we decided to use

them when working with our data.

The old saying “A painting worths more than thousand words”

is quite true in the analysis of any set of information. Before

applying any statistical method to the data under study, it is es-

sential to observe it graphically in order to become familiar with

it. This can have numerous beneﬁts, as we have explained at the

beginning of our analysis, since this process will serve as an in-

dicator of ideas for a more detailed later study. This was the ﬁrst

step in our work in which we were able to see the main char-

acteristics of the series and it helped us to make an appropriate

adjustment to it.

Firstly, we decided to ﬁt a GARCH-type model that captures

the main characteristics of the data. We saw that it adequately

takes into account the volatility of the series. With this analy-

sis we have been able to capture some situations where volatility

has a very important signiﬁcance. We understand that the model

used is adequate to predict the series and its components, in par-

ticular the volatility.

ꢂ

ꢃ

2

t

Figure 12 shows the prediction of the log νb series where

the νbt are the residuals of the estimation of (66) in blue) versus

ꢂ

ꢃ

2

t

the observed series of log νb for the period between January

1

3, 2003 and May 22, 2015 (the ones corresponding to the last

ten observations are shown) (top). Residuals of the adjustment

of the model (70) (in blue) with a 95% conﬁdence band (middle

part) and the graph of these residuals after having been standard-

ized. As we can see, it can be concluded that the adjustment is

adequate.

7

FINAL CONSIDERATIONS

The series studied is made up of the ﬁrst differences of the log-

arithm of the level of the Merval index. This is a stock market

index calculated at the Buenos Aires Stock Exchange (BCBA),

Argentina. It is a series with information corresponding to all

working days of the stock market. The period analyzed goes

from January 13, 2003 to May 22, 2015. There are 3006 observa-

tions. It covers a period in which there was no change in govern-

ment afﬁliation. This eliminates the effects that could have been

introduced into the market by changes in the governing group.

In our research, we set out to analyze methods for dealing with

a wide variety of data with irregularities that occur in time series.

In the second part of this work it was decided to use a stochas-

tic volatility approach to analyze the series under study, which

turned out to be very useful in capturing the main characteristics

of the data

The ARCH or GARCH family models assume that the con-

ditional variance (volatility) depends on past values. In other

2

t

words, and using the notation we saw above, if σ is the volatil-

ity, the ARCH-GARCH family assumes that it depends on the

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Figure 8. Decomposition of the Log-Squared Residuals of the Merval Returns Model (2003-2015)

Note. The ﬁgure presents the decomposition of the log-squared residuals of the model deﬁned in Equation (66) for the Merval returns series from

January 13, 2003, to May 22, 2015. The top panel shows the log-squared residuals with the level estimate. The middle panel displays the estimated

AR(1) component, whose equation is given in Equation (62). The bottom panel presents the estimates of the irregular component, corresponding to

Equation (61).

Figure 9. Estimated Volatility of the Merval Returns (2003-2015)

Note. The ﬁgure illustrates the estimated volatility of the Merval returns series from January 13, 2003, to May 22, 2015. The volatility is modeled

using a conditional heteroskedasticity approach, capturing periods of high and low market ﬂuctuations.

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Figure 10. Conditional Standard Deviation and Standardized Residuals of the Merval Returns Model (2003-2015)

Note. The ﬁgure compares the estimated conditional standard deviation with the standardized residuals of the model deﬁned in Equation (66) for the

Merval returns series from January 13, 2003, to May 22, 2015. This comparison helps assess whether the standardized residuals exhibit homoskedas-

ticity, validating the adequacy of the volatility model.

Figure 11. Conditional Mean Prediction and Residual Analysis of the Merval Returns Model (2003-2015)

Note. The ﬁgure presents the conditional mean prediction from the model deﬁned in Equation (66) for the Merval returns series from January 13, 2003,

to May 22, 2015. The top panel compares the predicted conditional mean (blue) with the observed returns (red), highlighting the last ten observations.

The middle panel shows the model residuals with a 95% conﬁdence band (blue). The bottom panel displays the standardized residuals for further

evaluation of model adequacy.

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Figure 12. Prediction of the Log-Squared Residuals of the Merval Returns Model (2003-2015)

c

2

Note. The ﬁgure presents the prediction of the log-squared residuals (log(ν )) from the model deﬁned in Equation (66) for the Merval returns series

t

from January 13, 2003, to May 22, 2015. The top panel compares the predicted values (blue) with the observed series (red), highlighting the last ten

observations. The middle panel displays the residuals of the model ﬁt in Equation (70) with a 95% conﬁdence band (blue). The bottom panel presents

the standardized residuals for further diagnostic analysis.

series yj for j < t. On the other hand, the stochastic volatility

model or SVM, proposed for the ﬁrst time by Taylor (1980, 1986,

els and the SVM work very well to estimate and predict volatility.

But the SVMs have the advantage that both in the deﬁnition of

their conditional mean given in the equations (54) and (55) and

in the corresponding conditional variance given in the equations

(56), (57) and (58) it is possible to introduce non-observable but

estimable components such as trend, seasonality, cycles, struc-

tural changes, etc., and also explanatory variables, all due to the

fact that these models are put in the form of a state space, which

provides great generality when carrying out the work. Another

important advantage of this approach is that it is estimated using

a procedure based on quasi-maximum likelihood, which gives

great ﬂexibility to the procedure and a lot of robustness to the

estimators.

1

994) does not start from this assumption. This model is based

2 2

j

on the fact that the volatility σ depends on its past values (σ for

t

j < t) but is independent of the past values of the series under

analysis (yj for j < t). Shephard and Pitt (1997) proposed the

use of importance sampling to estimate the likelihood function in

the non-Gaussian case. Since the MVE is a hierarchical model,

Jaquier, Polson, and Rossi (1994) proposed a Bayesian analysis

of it. See also Shephard (2005), Shephard and Pitt (1997), Kim,

Shephard, and Chib (1998), and Ghysels, Harvey, and Renault

(

1996). An overview of the SVM estimation problem is given by

Motta (2001).

As shown in Harvey, Ruiz, and Shephard (1994), the state-

station form provides the basis for quasi-maximum likelihood

estimation via the Kalman ﬁlter and smoother and also allows

for constructing smoothed estimates of the variance component

ht and making predictions. One of the attractions of the quasi-

maximum likelihood approach is that it can be applied with-

out an assumption about a particular distribution for εt. An-

other attraction of using a quasi-maximum likelihood procedure

via the Kalman ﬁlter and smoother to estimate SVM is that it

can be carried out directly using standard computing packages

such as STAMP by Koopman, Harvey, Doornik, and Shephard

REFERENCES

Abril, J. C. (1999). Análisis de series de tiempo basado en mod-

elos de espacio de estado. EUDEBA.

Abril, J. C. (2004). Modelos para el análisis de las series de

tiempo. Ediciones Cooperativas.

Abril, M. de las M. (2014). El enfoque de espacio de estado de

las series de tiempo para el estudio de los problemas de

volatilidad (Tesis doctoral). Universidad Nacional de Tu-

cumán, Argentina.

Abril, J. C., & Abril, M. de las M. (2017). La heterocedastici-

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