Estimating Stock Market Volatility with Markov

Regime-Switching GARCH Models

Eduardo de Oliveira Horta
Abstract
We estimate the Markov Regime-Switching GARCH model for the Bovespa stock index returns,
and compare it with the GARCH, the EGARCH and the GJR models, which are also estimated.
Each model is parametrized with the Normal, t, and GED distributions for the errors. The
comparison between models is in terms of their AIC, BIC, and in-sample goodness-of-t, as given
by a set of statistical loss functions.
Contents
1 Introduction 1
2 Models 2
2.1 Single regime GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Markov Regime-Switching GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Methodology 4
4 Empirical Results 5
5 Concluding Remarks 10
References 10
1 Introduction
In this paper, we estimate the Markov Regime-Switching GARCH model for the Bovespa stock index
returns, and compare it with the GARCH, the EGARCH and the GJR models, which are also esti-
mated. Each model is parametrized with the Normal, t, and GED distributions for the errors. The
comparison between models is in terms of their AIC, BIC, and in-sample goodness-of-t, as given by
a set of statistical loss functions.
As pointed out by Marcucci [1], in the last few decades a growing number of scholars have focused
their attention on modeling and forecasting volatility. So far in the literature, a myriad of volatility
models have been proposed. The most successfull among them seems to be the Generalized Autore-
gressive Conditionally Heteroscedastic model (hereafter GARCH) by Bollerslev[2], which was in turn a
generalization of Engels [3] ARCH model, and the many generalizations that added asymmetries, long
memory or structural breaks to their specications. These models became popular above all because of
their relative simplicity and their ability to capture many of the stylized facts associated with nancial
time series, such as time-varying volatility, persistence and volatility clustering.
A possibly interesting extension of the traditional GARCH models is to incorporate them into a
Markovian regime-switching framework. The literature on Markov Regime-Switching GARCH (MRS-
GARCH) models begins with Cai [4] and Hamilton and Susmel [5], while in their work these authors
1
consider a ARCH rather than a GARCH specication, in order to overcome the problem of innite
path-dependence implied by the latter. Subsequently, a few suggestions have been made as to allow
a GARCH specication with changing regimes. That by Gray [6], where the author proposes to
integrate out the unobserved regime path using the conditional expectation of the past variance, can
be considered the rst MRS-GARCH model de facto. Thereafter, Klaassen [7] suggests a similar
solution, however using a broader information set.
We thus follow Marcucci [1] and Klaassen [7] and estimate the MRS-GARCH model with both
Normal and fat-tailed distributions, such as the Students t and the GED. We compare these estimated
models against single regime GARCH, EGARCH and GJR models, in terms of their AIC, BIC, and
in-sample goodness-of-t, as given by a set of statistical loss functions.
The paper is structured as follows: in Section 2, we present the models which we are to estimate.
Subsequently, in Section 3, we discuss the methodology, which is essentially that of Marcucci [1]. In
Section 4, we present the estimation results. We conclude in Section 5.
2 Models
In this section we present the models which we estimate. Those fall into two classes: single-regime
and Markov Regime-Switching models.
2.1 Single regime GARCH Models
Let r
t
= 100 (log p
t
log p
t1
) denote the rate of return of a stock market index p
t
. The GARCH(1,1)
model for the returns process is given by
1
r
t
= +
t
(1)
h
t
=
0
+
1

2
t1
+
1
h
t1
, (2)
where
0
> 0,
1
0 and
1
0, and where E
t
= 0 and E
2
t
= h
t
. Note that h
t
is the conditional
variance of the returns. Note yet that this model is symmetric regarding positive and negative shocks.
The EGARCH(1,1) model is dened by (1) and
log h
t
=
0
+
1

t1
h
1
/2
t1

t1
h
1
/2
t1
+
1
log h
t1
. (3)
Such a specication for the conditional variance process is more general than (2), since it allows
for asymmetry in response to shocks of dierent signs. This may better capture the skewness often
encountered in the empirical distribution of nancial returns. One sees easily that the term following
the
1
coecient measures how the size of a shock may impact the future conditional variance, whilst
the term following the coecient measures how this variance reacts to shocks of dierent sign. For
instance, if < 0, a negative shock increases future volatilities (see, for example, Ziegelmann [8]).
Another model which incorporates asymmetric responses to shocks of dierent signs - the so called
leverage eect - is the GJR(1,1), dened by (1) and
h
t
=
0
+
_
1 I
{t>0}
_

2
t
+I
{t>0}

2
t
+
1
h
t1
, (4)
where I is the indicator function.
Aside from autocorrelation, conditional heteroscedasticity and asymmetry, leptokurtosis is also
an important stylized fact often found in the empirical distribution of nancial returns. To capture
this feature, it is a common practice to assume, in the models afore presented, that
t
is distributed
according to the t or the GED distributions, instead of the standard Normal.
1
We shall present those specications from the GARCH, EGARCH and GJR models which were used in our estimation
procedure. A formal exposition of the models is not our goal.
2
2.2 Markov Regime-Switching GARCH Models
The Markov Regime-Switching class of models incorporates the possibility of the parameters to change
depending on which state, or regime, the world is in at time t. This implies that the model encompasses
a mixture of distributions, and that the value of the dependent variable at any time is drawn according
to the concurrent state. The state of the world is governed by a discrete (unobservable) stochastic
process s
t
which follows a Markov chain (meaning that P (s
t
|s
t1
, s
t2
, . . . ) = P (s
t
|s
t1
)). The
transition probabilities are dened as
P (s
t
= j|s
t1
= i) = p
ij
, (5)
which denote the probabilities of switching from state i ate time t 1 to state j at time t, for all i, j
in the state space. It is convenient to group these probabilities into a matrix P, called the transition
matrix. Assuming, for simplicity
2
, the existence of two regimes across which the Markov chain evolves,
then the transition matrix is given by
P =
_
p
11
p
21
p
12
p
22
_

_
p (1 q)
(1 p) q
_
. (6)
The ergodic probability - that is, the unconditional probability - of being at state s
t
= 1 at time t is
given by

1
=
1 q
2 p q
, (7)
and that of being at state 2 is easily seen to be 1
1
.
A very general specication for such a process is
r
t
|
t1

_
_
_
f
_

(1)
t
_
w. p. p
1,t|t1
f
_

(2)
t
_
w. p. 1 p
1,t|t1
(8)
where
t
is the information set at t, f is a probability density function,
(i)
t
is the vector of parameters
in the i-th regime, and
p
j,t|t1
P (s
t
= j|
t1
) (9)
is the ex-ante probability, which obeys
p
j,t|t1
=
2

i=1
p
ij
_
f (r
t1
|s
t1
= i) p
i,t1|t2

2
k=1
f (r
t1
|s
t1
= k) p
k,t1|t2
_
. (10)
Typically, we have

(i)
t
=
_

(i)
t
, h
(i)
t
,
(i)
t
_
, (11)
where
(i)
t
= E (r
t
|
t1
, s
t
= i), h
(i)
t
= V ar (r
t
|
t1
, s
t
= i) and
(i)
t
is the shape parameter of the
conditional distribution. The superscripts refer to the state of the world to which the parameter
belongs.
The Markov Regime-Switching GARCH model, namely the MRS-GARCH(1,1) model, is therefore
dened as
r
t
=
(i)
+
t
(12)
h
(i)
t
=
(i)
0
+
(i)
1

2
t1
+
(i)
1
h
t1
, (13)
2
In our estimation procedure, we have assumed a model for the conditional variance with two states - likely one of
high and one of low volatility.
3
with
(i)
=
(i)
t
, E
t
= 0, E
2
t
= h
(i)
t
, and where h
t1
is a state-independent average of past conditional
variances
3
. A little digression is called upon regarding (12) and (13). Firstly, one must note that it is
imposed that the conditional mean of r
t
is time-independent inside each regime. Second, the conditional
variance is modeled in a way that it depends only on the current state and on the past observable
variables, but not on the entire regime path; this is achieved by dening the state-independent variable
h
t
. In fact, estimation of a model which depends on past unobservable state variables would be
infeasible. We follow Klaassen [7] end set
h
t
E
t
_
h
(i)
t
|s
t+1
_
, (14)
where
E
t
_
h
(i)
t
|s
t+1
_
=
2

j=1
p
ji,t
_
_

(j)
t
_
2
+h
(j)
t
_

_
_
2

j=1
p
ji,t

(j)
t
_
_
2
, (15)
and
p
ji,t
P (s
t
= j|s
t+1
= i,
t1
) =
p
ji
P (s
t
= j|
t1
)
P (s
t+1
= i|
t1
)
=
p
ji
p
j,t|t1
p
i,t+1|t1
. (16)
Such a specication has the advantage of allowing higher exibility in capturing the persistence of
shocks to volatility. The conditional variance equation of the MRS-GARCH model given by (12) and
(13) thus becomes
h
(i)
t
=
(i)
0
+
(i)
1

2
t1
+
(i)
1
E
t1
_
h
(i)
t1
|s
t
_
. (17)
When of estimation, we specify the above model with
t
distributed according to the Normal, the t and
the GED densities. We allow the shape parameter to switch across regimes only for the t specication,
as in Marcucci [1].
3 Methodology
Given a sample {r
t
}
T
t=1
, we estimate by quasi-maximum likelihood both the single- and the multi-
regime models. That is, for each model, the vector of parameter estimators is dened as

= arg max

log f (r
T
, r
T1
, . . . , r
1
|s
0
, ) . (18)
After estimation, each model implies a sequence
_

h
t
_
T
t=1
of estimated conditional standard deviations,
which we compare against a proxy {
t
}
T
t=1
of the actual (unobserved) conditional standard deviations
4
.
The comparison is carried out according to the both the Akaike and the Bayesian information criteria,
3
As with the single regime GARCH models, we present the specication which was used in the estimation procedure
4
One could use, for instance, intra-daily data to construct reasonably accurate proxies of the real volatilities. In the
absence of such data, we follow the common practice in the literature and use the squared returns
_
rt T
1

T
t=1
rt
_
2
instead.
4
as well as to the following statistical loss functions:
MSE
1
= T
1
T

t=1
_

t

h
1
/2
t
_
2
(19)
MSE
2
= T
1
T

t=1
_

2
t

h
t
_
2
(20)
QLIKE = T
1
T

t=1
_
log

h
t
+
2
t

h
1
t
_
(21)
R2LOG = T
1
T

t=1
_
log
_

2
t

h
1
t
__
2
(22)
MAD
1
= T
1
T

t=1

h
1
/2
t

(23)
MAD
2
= T
1
T

t=1

2
t

h
t

(24)
HMSE = T
1
T

t=1
_

2
t

h
1
t
1
_
2
. (25)
The criteria in (19) and (20) are the usual mean squared error metrics. Criterion (21) has the
property of penalizing asymmetrically forecasts in low and high volatility periods. The QLIKE criterion
corresponds to the loss implied by a Gaussian likelihood. The loss functions in (23) and (24) have the
advantage of being more robust to the presence of outliers than their mean squared error counterparts,
although they impose the same penalty on over- and under-predictions, and are not invariant to scale
transformations. At last, the HMSE criterion is a heteroscedastic adjusted mean squared error.
4 Empirical Results
The data consists of the Bovespa stock market daily closing price index
5
. The sample ranges from
January 2, 2007 to April 30, 2010, and the corresponding plot is shown in Figure 1. The analyzed data
is the returns series, in percent, which ranges from January 3, 2007 to April 30, 2010. This series is
shown in Figure 2. Table 1 brings some descriptive statistics regarding the returns series. Some stylized
facts appear clearly, as the close-to-zero mean and the signicantly high kurtosis. Furthermore, the
null hypothesis of the tests for normality, absence of ARCH eects and absence of serial correlations
on squared returns are all rejected.
The normality test is the Jarque-Bera test, which has a
2
distribution with 2 degrees of freedom
under the null of normally distributed errors. The 5% critical value is 5.99, whilst the test statistic
is 833.99. The ARCH eects test is the Engel test for residual heteroscedasticity, which is also
2
(d)
under the null of no heteroscedasticity, but with d degrees of freedom, where d is the number of lags.
The test statistic for the test up to the 36th lag is equal to 334.22. The Q
2
test is the Ljung-Box
test on the squared residuals of the conditional mean regression up to the 36th order, which is as well
distributed as a
2
(d) under the null hypothesis of no serial autocorrelation.. The test statistic is
1460.79. The 5% critical value of the
2
(36) is 50.99.
We estimated the parameters of all models from Section 2 using quasi-maximum likelihood. Both
the conditional mean and the conditional variances are estimated jointly by maximizing (18). The
maximization procedure is the Broyden-Fletcher-Goldfarb-Shanno quasi-Newton algorithm, and was
5
The series was obtained from http://nance.yahoo.com/.
5
Figure 1: Ibovespa price index - Jan/2007 - Apr/2010
Jan/2007 Jan/2008 Jan/2009 Jan/2010
2
3
4
5
6
7
8
x 10
4
Date
I
b
o
v
e
s
p
a

p
r
i
c
e

i
n
d
e
x
Figure 2: Ibovespa returns - Jan/2007 - Apr/2010
Jan/2007 Jan/2008 Jan/2009 Jan/2010
-15
-10
-5
0
5
10
15
Date
I
b
o
v
e
s
p
a

r
e
t
u
r
n
s
Table 1: Descriptive statistics
Mean Std. deviation Min Max Skewness Kurtosis Normality test ARCH test (36) Q
2
(36)
0.0485 2.3492 -12.0961 13.6766 -0.0060 7.9406 833.9932 (p=0) 334.2155 1460.7900
6
Table 2: Maximum likelihood estimates of single regime models.
GARCH EGARCH GJR
N t GED N t GED N t GED
0.1265** 0.1663*** 0.1772*** 0.0441 0.1024** 0.1183** 0.0666 0.1197** 0.1344***
(0.0635) (0.0588) (0.0566) (0.0621) (0.0579) (0.0570) (0.0636) (0.0590) (0.0572)

0
0.2010*** 0.1932*** 0.1975*** 0.0750*** 0.1026*** 0.0938*** 0.2099*** 0.1977*** 0.1999***
(0.0445) (0.0616) (0.0652) (0.0167) (0.0302) (0.0290) (0.0453) (0.0590) (0.0625)

1
0.1141*** 0.1255*** 0.1206*** 0.1490*** 0.1759*** 0.1640*** 0.2030*** 0.2218*** 0.2093***
(0.0201) (0.0324) (0.0324) (0.0247) (0.0418) (0.0414) (0.0369) (0.0534) (0.0533)

1
0.8362*** 0.8289*** 0.8309*** 0.9708*** 0.9740*** 0.9730*** 0.8380*** 0.8278*** 0.8316***
(0.0238) (0.0336) (0.0358) (0.0067) (0.0089) (0.0094) (0.0260) (0.0341) (0.0366)
- - - 0.1216*** 0.1247*** 0.1204*** 0.0135 0.0200 0.0189
(0.0213) (0.0291) (0.0302) (0.0212) (0.0300) (0.0321)
- 6.3225*** 1.3107*** - 7.4959*** 1.3929*** - 7.1097*** 1.3637***
(1.5096) (0.0893) (1.9444) (0.0932) (1.7746) (0.0904)
L() 1726.2438 1709.7641 1706.9563 1708.6286 1696.7843 1695.7853 1714.0724 1700.3779 1698.8550
carried out through slight modications of the MATLAB routines provided by Marcucci [1]. We shall
suppress the ^ from notation for the estimators as to keep notation clearer.
Table 2 shows the estimation results for the single regime models. Asymptotic standard errors are
in parentheses. Almost all parameter estimates are highly signicant. Furthermore, the estimated
degrees of freedom for those models with a t distribution are always greater than 6, as in Marcucci [1],
suggesting that all conditional moments up to the sixth order exist; the estimated conditional kurtosis
for the GARCH, EGARCH and GJR models with this distribution is respectively 5.5834, 4.7163, and
4.9294. This feature is in conformity with the typical fat-tailed behavior of nancial returns. Moreover,
regarding the models with GED innovations, these conditional kurtosis are 3.9533, 2.8917, and 3.2129,
respectively. While the GARCH and the GJR further point towards a fat-tailed distribution, the
EGARCH suggests a thinner distribution
6
.
Table 3, for instance, shows the estimation results for the MRS-GARCH models
7
. Of particular
interest are the standard deviations conditional on each regime, dened as

(i)
=
_

(i)
0
1
(i)
1

(i)
1
_1
2
. (26)
These statistics support the assumption of existence of one regime with high volatilities and one with
low ones, though the volatility gap between the two regimes greatly varies depending on which model
one looks at. For the specication with GED innovations, this magnitude is about 1.5%, while it is
almost 8% for the model with a Normal distribution. Also, the transition probabilities, p and q, are all
highly signicant and close to 1, except for the probability of remaining in the high volatility regime
for the t distribution, which is neither signicant nor relatively large. One sees as well that the overall
persistence of regimes, given by
i
=
(i)
1
+
(i)
1
, is very high. The ergodic probabilities dont show any
strong pattern besides that the unconditional probability of being in a low volatility regime is higher
than that of being in one of high volatility for every model specication.
Table 4 shows the AIC, the BIC and the in-sample goodness-of-t statistics given by equations
(19) through (25), for the estimated models. These statistics suggest that the best overall model
among those evaluated is the EGARCH with the GED distribution. A similar result was found by
Marcucci [1]. It is followed by the EGARCH with Normal and t errors, respectively. Curiously, the
6
The conditional kurtosis of the t and the GED distributions are given, respectively, by 3 ( 2) / ( 4) and
(
1
/) (
5
/) (
1
/)
2
, where is the gamma function.
7
We point out that, while estimating the model with GED innovations, we used the data rt T
1

T
t=1
rt instead
of rt itself, since for some reason we were getting bad conditioned matrices using the original data.
7
Table 3: Maximum likelihood estimates of MRS-GARCH models.
MRS-GARCH-N MRS-GARCH-t2 MRS-GARCH-GED

(1)
0.3070*** 0.0522 0.0565
(0.0779) (1.2501) (0.1187)

(2)
0.2788* 0.2186*** 0.1739**
(0.1733) (0.0838) (0.0803)

(1)
0
0.1552** 2.3316 0.1936
(0.0929) (11.7971) (0.1519)

(2)
0
0.4456*** 0.0000 0.2808**
(0.1356) (0.5430) (0.1687)

(1)
1
0.0000 0.1988 0.0871***
(0.0604) (0.7446) (0.0358)

(2)
1
0.0641** 0.1156* 0.0750*
(0.0351) (0.0808) (0.0458)

(1)
1
0.7785*** 0.7294 0.8879***
(0.0588) (1.1841) (0.0448)

(2)
1
0.9296*** 0.8409*** 0.7777***
(0.0491) (0.0799) (0.1053)
p (0.9070)*** 0.3157 0.9961***
(0.0327) (2.4628) (0.0045)
q 0.8987*** 0.9317*** 0.9972***
(0.0400) (0.3614) (0.0036)

(1)
- 7.1203 1.353***
(27.2525) (0.0922)

(2)
- 5.9771*** -
(2.2626)
L() 1704.3353 1.7096109 1700.8443
#Params. 10 12 11

(1)
0.8369 5.6998 2.7797

(2)
8.3867 0.0005 1.3806

1
0.5214 0.0907 0.4171

2
0.4786 0.9093 0.5829

1
0.7785 0.9282 0.9749

2
0.9937 0.9565 0.8527
8
Table 4: In-sample goodness-of-t statistics.
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9
Figure 3: EGARCH-GED and MRS-GARCH-GED estimated conditional standard deviations.
Jan/2007 Jan/2008 Jan/2009 Jan/2010
0
10
20
30
40
50
Date
E
s
t
i
m
a
t
e
d

c
o
n
d
i
t
i
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n
a
l

s
t
a
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d
a
r
d

d
e
v
i
a
t
i
o
n


EGARCH-GED
MRS-GARCH-GED
MRS-GARCH models fare quite badly. This is particularly unexpected due to the fact that these
models are over-parametrized and should t the data well
8
. If we disregard the AIC and the BIC, the
situation doesnt change much. In that case, the best model is the EGARCH with Normal innovations,
and the second best is the EGARCH with GED innovations.
The results obtained here suggest that, at least in terms of in-sample goodness-of-t, the EGARCH
class of models is more adequate than other model specications for the conditional variance. As a
means of visual comparison, Figure 3 plots the estimated path of the conditional standard deviations
obtained from the EGARCH and the MRS-GARCH estimated models. There one sees that these are
quite similar. Figure 4 shows the path of the estimated probability of being in the high volatility
regime.
We point yet that the corrected errors, dened as

t
=
r
t

h
t
,
has sample variance close to unity, and fails to reject null hypotheses of both the ARCH and the Q
2
tests.
5 Concluding Remarks
We estimated the Markov Regime-Switching GARCH model for the Bovespa stock index returns.
These estimation results were compared against the estimation results for the GARCH, the EGARCH
and the GJR models using the same data. The comparison between models was according to their
AIC, BIC, and to their in-sample goodness-of-t, as given by a set of statistical loss functions. We
found, as in Marcucci [1], that the EGARCH model, specied with the GED distribution, fare better
than its competitors. However, we found also that the MRS-GARCH class of models didnt provide
a good in-sample goodness-of-t. This is in disagreement with both the ndings by Marcucci [1] and
with what is expected from a highly over-parametrized model.
8
A possible explanation to this fact is that our proxy for the actual volatility, the squared demeaned returns, is too
noisy
10
Figure 4: MRS-GARCH-GED estimated path of probability of being at regime 1.
Jan/2007 Jan/2008 Jan/2009 Jan/2010
0
0.2
0.4
0.6
0.8
1
Date
p
1
,
t
|
t
References
[1] MARCUCCI, J. Forecasting Stock Market Volatility with Regime-Switching GARCH Models.
Studies in Nonlinear Dynamics & Econometrics, v. 9, n. 4, p. 6, 2005.
[2] BOLLERSLEV, T. Generalized autoregressive conditional heteroskedasticity. Journal of econo-
metrics, v. 31, n. 3, p. 307327, 1986.
[3] ENGLE, R. Autoregressive conditional heteroscedasticity with estimates of the variance of United
Kingdom ination. Econometrica: Journal of the Econometric Society, v. 50, n. 4, p. 9871007,
1982.
[4] CAI, J. A Markov model of switching-regime ARCH. Journal of Business & Economic Statistics,
v. 12, n. 3, p. 309316, 1994.
[5] HAMILTON, J.; SUSMEL, R. Autoregressive conditional heteroskedasticity and changes in regime.
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[6] GRAY, S. Modeling the conditional distribution of interest rates as a regime-switching process.
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[7] KLAASSEN, F. Improving GARCH volatility forecasts with regime-switching GARCH. Empirical
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[8] ZIEGELMANN, F. A. Estimation of Volatility Functions: Nonparametric and Semi-Parametric
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(UK), 2002.
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