Page 1

DEPARTAMENTO DE ECONOMIA

PUC-RIO

TEXTO PARA DISCUSSÃO

No. 486

MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY

WITH A FLEXIBLE COEFFICIENT GARCH MODEL

MARCELO C. MEDEIROS

ALVARO VEIGA

JULHO 2004

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MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY WITH A

FLEXIBLE COEFFICIENT GARCH(1,1) MODEL

MARCELO C. MEDEIROS AND ALVARO VEIGA

ABSTRACT. In this paper a flexible multiple regime GARCH(1,1)-type model is developed to de-

scribe the sign and size asymmetries and intermittent dynamics in financial volatility. The results

of the paper are important to other nonlinear GARCH models. The proposed model nests some of

the previous specifications found in the literature and has the following advantages: First, contrary

to most of the previous models, more than two limiting regimes are possible and the number of

regimes is determined by a simple sequence of of tests that circumvents identification problems

that are usually found in nonlinear time series models. The second advantage is that the station-

arity restriction on the parameters is relatively weak, thereby allowing for rich dynamics. It is

shown that the model may have explosive regimes but can still be strictly stationary and ergodic. A

simulation experiment shows that the proposed model can generate series with high kurtosis, low

first-order autocorrelation of the squared observations, and exhibit the so-called “Taylor effect”,

even with Gaussian errors. Estimation of the parameters is addressed and the asymptotic properties

of the quasi-maximum likelihood estimator are derived under weak conditions. A Monte-Carlo

experiment is designed to evaluate the finite sample properties of the sequence of tests. Empirical

examples are also considered.

KEYWORDS: Volatility, GARCH models, multiple regimes, nonlinear time series, smooth transi-

tion, finance, asymmetry, leverage effect, excess of kurtosis, asymptotic theory.

1. INTRODUCTION

MODELING AND FORECASTING the conditional variance, or volatility, of financial time series

has been one of the major topics in financial econometrics. Forecasted conditional variances are

used, for example, in portfolio selection, derivative pricing and hedging, risk management, market

timing, and market making. Among solutions to tackle this problem, the ARCH (Autoregressive

Conditional Heteroscedasticity) model proposed by Engle (1982) and the GARCH (Generalized

Autoregressive Conditional Heteroscedasticity) specification introduced by Bollerslev (1986) are

among the most widely used, and are now fully incorporated into financial econometric practice.

OnedrawbackoftheGARCHmodelisthesymmetryintheresponseofvolatilitytopastshocks,

which fails to accommodate sign asymmetries. Starting with Black (1976), it has been observed

that there is an asymmetric response of the conditional variance of the series to unexpected news,

represented by shocks: Financial markets become more volatile in response to “bad news” (nega-

tive shocks) than to “good news” (positive shocks). Goetzmann, Ibbotson, and Peng (2001) found

evidence of asymmetric sign effects in volatility as far back as 1857 for the NYSE. They report

that unexpected negative shocks in the monthly return of the NYSE from 1857 to 1925 increase

1

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2 M. C. MEDEIROS AND A. VEIGA

volatility almost twice as much as equivalent positive shocks in returns. Similar results were also

reported by Schwert (1990).

The above mentioned asymmetry has motivated a large number of different volatility models

which have been applied with relatively success in several situations. Nelson (1991) proposed the

Exponential GARCH (EGARCH) model. In his proposal, the natural logarithm of the conditional

variance is modeled as a nonlinear ARMA model with a term that introduces asymmetry in the

dynamics of the conditional variance, according to the sign of the lagged returns. Glosten, Ja-

gannanthan, and Runkle (1993) proposed the GJR model, where the impact of the lagged squared

returns on the current conditional variance changes according to the sign of the past return. A

similar specification, known as Threshold GARCH (TGARCH), model was developed by Rabe-

mananjara and Zakoian (1993) and Zakoian (1994). Ding, Granger, and Engle (1993) proposed

the Asymmetric Power ARCH which nests several GARCH specifications. Engle and Ng (1993)

popularized the news impact curve (NIC) as a measure of how new information is incorporated

into volatility estimates. The authors also developed formal statistical tests to check the presence

of asymmetry in the volatility dynamics. More recently, Fornari and Mele (1997) generalized

the GJR model by allowing all the parameters to change according to the sign of the past return.

Their proposal is known as the Volatility-Switching GARCH (VSGARCH) model. Based on the

Smooth Transition AutoRegressive (STAR) model, Hagerud (1997) and Gonzalez-Rivera (1998)

proposed the Smooth Transition GARCH (STGARCH) model. While the latter only considered

the Logistic STGARCH (LSTGARCH) model, the former discussed both the Logistic and the

Exponential STGARCH (ESTGARCH) alternatives. In the logistic STGARCH specification, the

dynamics of the volatility are very similar to those of the GJR model and depends on the sign of

the past returns. The difference is that the former allows for a smooth transition between regimes.

In the EST-GARCH model, the sign of the past returns does not play any role in the dynamics of

the conditional variance, but it is the magnitude of the lagged squared return that is the source of

asymmetry. Anderson, Nam, and Vahid (1999) combined the ideas of Fornari and Mele (1997),

Hagerud (1997), and Gonzalez-Rivera (1998) and proposed the Asymmetric Nonlinear Smooth

Transition GARCH (ANSTGARCH) model, and found evidence in favor of their specification.

Inspired by the Threshold Autoregressive (TAR) model, Li and Li (1996) proposed the Double

Threshold ARCH (DTARCH) model. Liu, Li, and Li (1997) generalized it, proposing the Double

Threshold GARCH (DT-GARCH) process to model both the conditional mean and the conditional

variance as a threshold process. More recently, based on the regression-tree literature, Audrino

and B¨ uhlmann (2001) proposed the Tree Structured GARCH model to describe multiple limiting

regimes in volatility1.

In this paper we contribute to the literature by proposing a new flexible nonlinear GARCH

modelwithmultiplelimitingregimes, calledtheFlexibleCoefficientGARCH(FCGARCH)model,

that nests several of the models mentioned above. As most of the empirical papers in the financial

1See also Cai (1994) and Hamilton and Susmel (1994) for regime switching GARCH specifications based on the

Markov-switching model.

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MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY3

econometrics literature deal only with GARCH(1,1)-type of models, we focus our attention only

on the first-order FCGARCH specification.

Our proposal has the following advantages: First, contrary to most of the previous models in the

literature, more than two limiting regimes can be modeled. The number of regimes is determined

by a simple and easily implemented sequence of tests that circumvents the identification problem

in the nonlinear time series literature, and avoids the estimation of overfitted models. To the best

of our knowledge, the only two exceptions that explicitly model more than two limiting regimes

in the volatility are the DTGARCH and Tree-Structure GARCH models. However, in the former,

the authors did not discuss how to determine the number of regimes and only one fixed threshold

at zero is considered in the empirical application. In the latter, the proposed procedure is based

on the use of information criteria and may suffer from identification problems when an irrelevant

regime is estimated; see Hansen (1996) for a similar discussion considering threshold regression

models and Ter¨ asvirta and Mellin (1986) for the linear regression case. The second advantage is

that the stationarity restriction on the FCGARCH model parameters is relatively weak, thereby

allowing for rich dynamics. For example, the model may have explosive regimes and still be

strictly stationary and ergodic, being capable of describing intermittent dynamics. The system

spends a large fraction of time in a bounded region, but sporadically develops an instability that

grows exponentially for some time, and then suddenly collapses. Furthermore, data with very high

kurtosis can easily be generated even with Gaussian errors. This allows for a better description

of the large absolute returns of financial time series that standard GARCH models fail to describe

satisfactorily. Reproducing the above mentioned typical behavior of financial time series maybe

important in risk analysis and management. A simulation experiment shows that the FCGARCH

model is able to generate time series with high kurtosis and, at the same time, positive but low

first-order autocorrelations of squared observations, which are frequently observed in financial

time series. Furthermore, the FCGARCH model seems to be able to reproduce the so-called

“Taylor effect” (Granger and Ding 1995). Other models such as the GARCH and the EGARCH

models are not able to reproduce adequately the above mentioned stylized facts of financial time

series; see Malmsten and Ter¨ asvirta (2004) and Carnero, Pe˜ na, and Ruiz (2004) for comprehensive

discussions.

We discuss the theoretical aspects of the FCGARCH model. Conditions for strict stationarity

and for the existence of the second- and fourth-order moments; model identifiability; and the

existence, consistency, and asymptotic normality of the quasi-maximum likelihood estimators.

Consistency and asymptotic normality are proved under weak conditions. Our results are directly

applicable to other nonlinear GARCH specifications, such as the STGARCH model. Furthermore,

existing results in the literature are special cases of those presented in the paper.

A sequence of simple Lagrange multiplier (LM) tests is developed to determine the number of

limiting regimes and to avoid the specification of models with an excessive number of parameters.

Although the test is derived under the assumption that the errors are Gaussian, a robust version

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4 M. C. MEDEIROS AND A. VEIGA

against non-Gaussian errors is also considered. A Monte Carlo experiment is designed to evaluate

the finite sample properties of the proposed sequence of tests with simulated data. The main

finding is that the robust version of the test works well in small samples, and compares favorably

with the use of information criteria.

An empirical example with seven stock indexes shows evidence of two regimes for three series

and three regimes for other three series. Only for one stock index there is no evidence of regime

switching. Furthermore, for all series with three regimes, the GARCH model associated with

the first regime, representing very negative returns (“very bad news”), is explosive. The model

in the middle regime, related to tranquil periods, has a slightly lower persistence than the stan-

dard estimated GARCH(1,1) models in the literature. Finally, the third regime, representing large

positive returns, has an associated GARCH(1,1) specification that is significantly less persistent

than the others. Thus, we find strong evidence of both size and sign asymmetries. In addition,

the FCGARCH model produces normalized residuals with lower kurtosis than the GARCH and

GJR models. When a forecasting exercise is considered, the proposed model outperforms several

concurrent GARCH specifications.

Thestructureofthepaperisasfollows. Section2presentsthemodel. Itsprobabilisticproperties

are analyzed in Section 3. Estimation of the FCGARCH model is considered in Section 4. Section

5 discusses the test for an additional regime. Section 6 summarizes the modeling cycle procedure.

A Monte Carlo simulation is presented in Section 7, and empirical examples are considered in

Section 8. Finally, Section 9 concludes the paper. All technical proofs are given in the Appendix.

2. THE MODEL

In this paper, we generalize the GARCH(1,1) and the Logistic STGARCH(1,1) formulations,

introducing a general regime switching scheme. The proposed model is defined as follows.

DEFINITION 1. A time series {yt} follows a first-order Flexible Coefficient GARCH model with

m = H + 1 limiting regimes, FCGARCH(m,1,1), if

yt= h1/2

t

εt,

ht= G(wt;ψ) = α0+ β0ht−1+ λ0y2

H

?

where {εt} is a sequence of identically and independently distributed zero mean and unit variance

random variables, εt ∼ IID(0,1), G(wt;ψ) is a nonlinear function of the vector of variables

wt= [yt−1,ht−1,st]?, and is indexed by the vector of parameters

ψ = [α0,β0,λ0,α1,...,αH,β1,...,βH,λ1,...,λH,γ1,...,γH,c1,...,cH,]?∈ R3+5H,

t−1+

i=1

?αi+ βiht−1+ λiy2

t−1

?f (st;γi,ci),t = 1,...,T,

(1)

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MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY5

and f (st;γi,ci), i = 1,...,H, is the logistic function defined as

(2)

f (st;γi,ci) =

1

1 + e−γi(st−ci).

It is clear that f (st;γi,ci) is a monotonically increasing function, such that f (st;γi,ci) → 1

as st→ ∞ and f (st;γi,ci) → 0 as st→ −∞. The parameter γi, i = 1,...,H, is called the

slope parameter and determines the speed of the transition between two limiting regimes. When

γi → ∞, the logistic function becomes a step function, and the FCGARCH model becomes a

threshold-type specification. The variable stis known as the transition variable. In this paper, we

consider st= yt−1. Hence, we model the differences in the dynamics of the conditional variance

according to the sign and size of shocks in past returns, which represent previous “news”. Of

course, there are other possible choices for st; see Audrino and Trojani (forthcoming) and Chen,

Chiang, and So (2003) for some alternatives.

Thenumberoflimitingregimesisdefinedbythehyper-parameterH. Forexample, supposethat

in (1), H = 2, c1is highly negative, and c2is very positive, than the resulting FCGARCH model

will have 3 limiting regimes that can be interpreted as follows. The first regime may be related to

extremely low negative shocks (“very bad news”) and the dynamics of the volatility are driven by

ht= α0+ β0ht−1+ λ0y2

represents low absolute returns (“tranquil periods”), ht= α0+α1+(β0+β1)ht−1+(λ0+λ1)y2

as f (yt−1;γ1,c1) ≈ 1 and f (yt−1;γ2,c2) ≈ 0. Finally, the third regime is related to high positive

shocks (“very good news”) and ht= α0+α1+α2+(β0+β1+β2)ht−1+(λ0+λ1+λ2)y2

f (yt−1;γi,ci) ≈ 1, i = 1,2. As the speed of the transitions between different limiting GARCH

models is determined by the parameter γi, i = 1,2, the multiple regime interpretation of the

FCGARCH specification will become clearer the more abrupt are the transitions (γi? 0)2. In

practical applications, the restriction γ1= γ2= ··· = γHmay be imposed in order to reduce the

number of parameters and the eventual computational cost of the estimation algorithm.

It is important to notice that model (1) nests several well-known GARCH specifications, such

as:

t−1as f (yt−1;γi,ci) ≈ 0, i = 1,2. In the the middle regime, which

t−1

t−1, as

• The GARCH(1,1) model if γi= 0 or αi= βi= λi= 0, i = 1...,h.

• The LSTGARCH(1,1) model if αi= βi= 0, i = 1...,h and h = 1.

• The GJR(1,1) model if H = 1, γ1→ ∞, α1= β1= 0, and c1= 0.

• The VSGARCH(1,1) model if H = 1 and γ1→ ∞, c1= 0.

• The ANSTGARCH(1,1) model if H = 1, and c1= 0.

• The variance component of the DTARCH(1,1) model if γi → ∞ and αi = βi = 0,

i = 1...,h.

• The variance component of the DTGARCH(1,1) model if γi→ ∞ and st= ht−1.

2Representing multiple regimes with logistic functions dates back to Bacon and Watts (1971) and Chan and Tong

(1986); see also Ter¨ asvirta (1994) and van Dijk and Franses (1999).

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6M. C. MEDEIROS AND A. VEIGA

The nonlinear GARCH model proposed in Lanne and Saikkonen (2005) is a special case of

the FCGARCH model if βi = 0, i = 1,...,H, or if st = ht−1. The FCGARCH model is a

special case of the general GARCH specification presented in He and Ter¨ asvirta (1999), Ling and

McAleer (2002), and Carrasco and Chen (2002) if st= εt−1.

3. MAIN ASSUMPTIONS AND PROBABILISTIC PROPERTIES OF THE FCGARCH MODEL

We need to make the following set of assumptions:

ASSUMPTION 1. The true parameter vector ψ0∈ Ψ ⊆ R3+5His in the interior of Ψ, a compact

and convex parameter space.

ASSUMPTION 2. The sequence {εt} of IID(0,1) random variables is drawn from a continuous

(with respect to Lebesgue measure on the real line), symmetric, unimodal, positive everywhere

density, and bounded in a neighborhood of 0.

ASSUMPTION 3. The parameters ciand γi, i = 1,...,H, satisfy the conditions:

(R.1) −∞ < M < c1< ... < cH< M < ∞;

(R.2) γi> 0.

ASSUMPTION 4. The parameters γiand ci, i = 1,...,H, are such that the logistic functions

satisfy the following restrictions: f (st;γ1,c1) ≥ f (st;γ2,c2) ≥ ... ≥ f (st;γH,cH), ∀ t ∈

[0,T].

ASSUMPTION 5. The parameters αj, βj, and λj, j = 0,...,H, satisfy the following restrictions:

(R.3)?K

Assumption 1 is standard. Assumption 2 is important for the mathematical derivations in this

section and in Section 5. Assumption 3 guarantees the identifiability of the model (see Section

4.2 for details). The restrictions stated in Assumptions 4 and 5 ensure strictly positive conditional

variances. Specifically, Assumption 4 ensures that the conditions in Assumption 5 are sufficient

for the strict positivity of the conditional variance.

Defining st= yt−1, model (1) may be written as

yt= h1/2

t

εt,

ht= gt−1+ ct−1ht−1,

where

j=0αj> 0, ∀K = 0,...,H;

(R.3)?K

j=0βj≥ 0, and?K

j=0λj≥ 0, ∀K = 0,...,H.

(3)

gt−1≡ g(yt−1,εt−1) = α0+

H

?

i=1

αifi,t−1,

ct−1≡ c(yt−1,εt−1) =

??

β0+

H

?

i=1

βifi,t−1

?

+

?

λ0+

H

?

i=1

λifi,t−1

?

ε2

t−1

?

,

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MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY7

with fi,t−1≡ f (yt−1;γi,ci).

Following Nelson (1990), the next theorem states a necessary and sufficient log-moment con-

dition for the strict stationarity and ergodicity of the FCGARCH(m,1,1) model.

THEOREM 1. Suppose that yt ∈ R follows an FCGARCH(m,1,1) process as in (1), with st =

yt−1. Under Assumptions 2–5, the process ut= (yt,ht)?is strictly stationary and ergodic if, and

only if,

?

i=1

Furthermore, there is a second-order stationary solution to (3) that has the following causal ex-

pansion:

(4)

E

log

??

β0+

H

?

βifi,t−1

?

+

?

λ0+

H

?

i=1

λifi,t−1

?

ε2

t−1

??

< 0, ∀t.

yt= h1/2

t

εt,

ht= gt−1+

∞

?

k=0

k?

j=0

gt−1−kct−1−j,

(5)

where the infinite sum converges almost surely (a.s.).

The log-moment condition is important as the condition in Theorem 1 can be satisfied even in

the absence of finite second-moments of yt; see McAleer (2005) for a comprehensive discussion

of log-moment conditions for volatility models.

COROLLARY 1. Under the assumptions of Theorem 1, a sufficient condition for strict stationary

and ergodicity of ut= (yt,ht)?in terms of the parameters is

1

2(β0+ λ0) +1

2

H

?

i=0

(βi+ λi) ≤ 1.

Deriving a general sufficient condition for the existence of the moments of ytis rather com-

plicated. However, the moment condition stated in the following theorem can be used to find a

necessary and sufficient condition for the existence of low-order moments of yt. As mentioned in

the previous section, the model families of He and Ter¨ asvirta (1999), Ling and McAleer (2002),

Lanne and Saikkonen (2005), and Carrasco and Chen (2002) do not nest the FCGARCH model

without additional restrictions. Hence, the direct application of the results of these authors is not

straightforward. In the subsequent corollary, we derive sufficient conditions for the existence of

the second- and forth-order moments of yt.

THEOREM 2. Suppose that yt∈ R follows an FCGARCH(m,1,1) process as in (1), with st= yt−1

and E?ε2k

t

?

= µ2k< ∞, for k = 1,2,3,.... Under Assumptions 2–5, and assuming that the

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8

moments of order up to n = k − 1 exist, E?y2n

(6)

E

COROLLARY 2. Suppose that yt∈ R follows an FCGARCH(m,1,1) process as in (1), with st=

yt−1. A sufficient condition for the existence of the second-order moment of ytis

1

2(β0+ λ0) +1

2

i=0

Furthermore, define βU≡?H

t

0+ β0βU+β2

22

M. C. MEDEIROS AND A. VEIGA

?< ∞, the 2kth-order moment of ytexists if

β0+ λ0ε2

i=1

t

?

t−1+

H

?

?βi+ λiε2

t−1

?fi,t−1

?k

< 1.

(7)

H

?

i=1λi. Under Assumptions 2–5, the fourth-order

?= µ4< ∞, (7) holds, and

+ µ4

+ 2λ0β0+ β0λU+ λ0βU+ λUβU< 1.

(βi+ λi) < 1.

i=1βiand λU≡?H

?

moment of ytexists if E?ε4

(8)

β2

U

λ0+ λ0λU+λ2

U

?

REMARK 1. When H = 0, conditions (7) and (8) are the usual conditions for the existence of

the second- and fourth-order moments of GARCH models. When H = 1, γ1→ ∞, α1= 0, and

β1= 0, conditions (7) and (8) become the usual ones for the GJR model.

It is important to notice that, even with explosive regimes the FCGARCH(m,1,1) may still

be strictly stationary, ergodic, and with finite fourth-order moment. Furthermore, some of the

parameters of the limiting GARCH models may exceed one. This flexibility generates models

with higher kurtosis than the standard GARCH(1,1), even with Gaussian errors.

REMARK 2. The IGARCH model with Gaussian errors is also capable of generating data with

high kurtosis. However, contrary to the FCGARCH model, it does not have finite second- and

fourth-order moments.

The following examples illustrate some interesting situations.

Consider 3000 replications of the following FCGARCH(3,1,1) models with Gaussian errors,

each of which has 5000 observations.

(1) Example 1:

yt= h1/2

ht= 1 × 10−4+ 0.96ht−1+ 0.18y2

?−0.9 × 10−4− 0.60ht−1− 0.10y2

t

εt,εt∼ NID(0,1)

t−1+

t−1

?f (5000(yt−1− 0.02)).

?f (5000(yt−1+ 0.005))+

?1 × 10−4+ 0.10ht−1+ 0.05y2

t−1

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MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY9

(2) Example 2:

yt= h1/2

ht= 6 × 10−5+ 1.10ht−1+ 0.10y2

?−5 × 10−5− 0.65ht−1− 0.09y2

t

εt,εt∼ NID(0,1)

t−1+

t−1

?f (3000(yt−1− 0.005)).

?f (3000(yt−1+ 0.005))+

?1 × 10−5+ 0.10ht−1+ 0.04y2

(3) Example 3:

t−1

yt= h1/2

ht= 6 × 10−5+ 1.20ht−1+ 0.10y2

?−5.5 × 10−5− 1.20ht−1− 0.10y2

5 × 10−5f (2000(yt−1− 0.01)).

The models in Examples 1–3 have three extreme regimes, each with the first regime being

explosive as β0+ λ0> 1. However, even with an explosive regime, the generated time series are

still stationary provided that1

2

moment exists, provided that condition (8) is also satisfied. Note also that β0> 1 in Examples 2

and 3. The model in Example 3 has the interesting property that the GARCH effect is only present

in the extreme regimes. The regime associated with tranquil periods is homoskedastic.

Figure 1 shows the scatter plot of the estimated kurtosis and first-order autocorrelation of the

squared observations. The dots indicates the cases where the first-order autocorrelation of |yt| is

greater than the first-order autocorrelation of |yt|2. The crosses indicate the opposite effect. The

simulated FCGARCH models seem to reproduce some of the stylized facts observed in financial

time series. Table 1 summarizes some statistics about the estimated kurtosis and autocorrelations.

As can be seen, the minimum value of the estimated kurtosis is over 3. In addition the mean

values of the estimated first-order autocorrelations are in accordance with the typical numbers that

are found in practical applications.

t

εt,εt∼ NID(0,1)

t−1+

t−1

?f (2000(yt−1+ 0.001))+

2(β0+ λ0) +1

?2

i=0(βi+ λi) < 1. Furthermore, the fourth-order

TABLE 1. SIMULATED MODELS: DESCRIPTIVE STATISTICS.

The table shows descriptive statistics for the estimated kurtosis and first-order autocorrelation

of the squared observations over 3000 replications of Models (1)–(3).

Kurtosis

Example Min. Max.Mean Std. Dev.

1

7.18184.5113.42 7.72

2

5.37 151.508.814.75

3

7.75 434.1215.8814.99

Autocorrelation

Max. Mean

0.69

0.65

0.72

Min.

0.10

0.09

0.02

Std. Dev.

0.07

0.06

0.08

0.37

0.29

0.22

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10M. C. MEDEIROS AND A. VEIGA

0.10.20.30.40.50.6

2

0.70.8

0

20

40

60

80

100

120

140

160

180

200

Estimated first−order autocorrelation of yt

Estimated kurtosis of yt

With Taylor effect

Without Taylor effect

(a)

0 0.10.20.30.40.5

2

0.60.7

0

20

40

60

80

100

120

140

160

Estimated first−order autocorrelation of yt

Estimated kurtosis of yt

With Taylor effect

Without Taylor effect

(b)

00.10.20.30.40.50.60.70.8

0

50

100

150

200

250

300

350

400

450

Estimated first−order autocorrelation of yt

2

Estimated kurtosis of yt

With Taylor effect

Without Taylor effect

(c)

FIGURE 1. Scatter plot of the estimated kurtosis and first-order autocorrelation of

the squared observations. The dots indicates the cases where the “Taylor effect” is

satisfied and the crosses indicate the opposite effect. Panel (a) concerns Example

1. Panel (b) concerns Example 2. Panel (c) concerns Example 3.

4. PARAMETER ESTIMATION

As the distribution of εtis unknown, the parameters of the FCGARCH model are estimated

by quasi-maximum likelihood (QML). For GARCH(1,1) models, Lee and Hansen (1994) proved

that the local QMLE is consistent and asymptotically normal if all the conditional expectations

of ε2+κ

t

(1998) discussed consistency of the QMLE under weaker conditions. More recently, Ling and

McAleer (2003) proved the consistency of the global QMLE for a VARMA-GARCH model under

only the second-order moment condition. The authors also proved the asymptotic normality of the

global (local) QMLE under the sixth-order (forth-order) moment condition. Comte and Lieberman

(2003) and Berkes, Horv´ ath, and Kokoszka (2003) proved consistency and asymptotic normality

of the QMLE of the parameters of the GARCH(p,q) model under the second- and fourth-order

moment conditions, respectively.

As in Boussama (2000), McAleer, Chan, and Marinova (forthcoming), and Francq and Zako¨ ıan

(2004), we prove consistency and asymptotic normality of the QMLE of the FCGARCH(m,1,1)

underthelog-momentconditioninTheorem1; seealsoLi, Ling, andMcAleer(2002)andMcAleer

< ∞ uniformly with κ > 0. Lumsdaine (1996) required that E?ε32

t

?< ∞. Jeantheau

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MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY11

(2005). Extending the results in Jensen and Rahbek (2004) for non-stationary ARCH models to

the case of the FCGARCH model is not straightforward, and is beyond the scope of this paper.

However, this is an interesting topic for future research.

The quasi-log-likelihood function of the FCGARCH model is given by

(9)

LT(ψ) =1

T

T

?

t=1

lt(ψ),

where lt(ψ) = −1

observed, and hence they are arbitrary constants. Thus, LT(ψ) is a quasi-log-likelihood function

that is not conditional on the true (y0,h0) making it suitable for practical applications.

However, to prove the asymptotic properties of the QMLE is more convenient to work with the

unobserved process {(yu,t,hu,t) : t = 0,±1,±2,...}, which satisfies

H

?

The unobserved quasi-log-likelihood function conditional on F0= (y0,y−1,y−2,...) is

Lu,T(ψ) =1

T

2ln(2π) −1

2ln(ht) −

y2

2ht. Note that the processes ytand ht, t ≤ 0, are un-

t

(10)

hu,t= α0+ β0hu,t−1+ λ0y2

u,t−1+

i=1

?αi+ βihu,t−1+ λiy2

u,t−1

?f (yu,t−1;γi,ci).

(11)

T

?

t=1

lu,t(ψ),

with lu,t(ψ) = −1

Lu,T(ψ) is that the former is conditional on any initial values, whereas the latter is conditional on

an infinite series of past observations. In practical situations, the use of (11) is not possible.

Let

2ln(2π) −1

2ln(hu,t) −

y2

2hu,t. The primary difference between LT(ψ) and

u,t

?ψT= argmax

ψ∈Ψ

LT(ψ) = argmax

ψ∈Ψ

?

1

T

T

?

t=1

lt(ψ)

?

,

and

?ψu,T= argmax

ψ∈Ψ

Lu,T(ψ) = argmax

ψ∈Ψ

?

1

T

T

?

t=1

lu,t(ψ)

?

.

Define L(ψ) = E[lu,t(ψ)]. In the following two subsections, we discuss the existence of L(ψ)

and the identifiability of the FCGARCH model. Then, in Subsection 4.3, we prove the consistency

of?ψTand?ψu,T. We first prove the consistency of?ψu,T. Using Lemma 3 in Appendix B, we show

of both estimators is considered in Subsection 4.4. We start proving asymptotic normality of?ψu,T.

4.1. Existence of the QMLE. The following theorem proves the existence of L(ψ). It is based

on Theorem 2.12 in White (1994), which establishes that, under certain conditions of continuity

and measurability on quasi-log-likelihood function, L(ψ) exists.

that sup

ψ∈Ψ|Lu,T(ψ) − LT(ψ)|

p

→ 0, and the consistency of?ψTfollows. The asymptotic normality

Then, using the results of Lemma 5, the proof for?ψTis straightforward.

Page 13

12M. C. MEDEIROS AND A. VEIGA

THEOREM 3. If (4) is satisfied, under Assumptions 2–5, L(ψ) exists and is finite.

4.2. Identifiability of the Model. A fundamental problem for statistical inference with nonlinear

time series models is the unidentifiability of the parameters. In order to guarantee unique iden-

tifiability of the quasi-log-likelihood function, the sources of uniqueness of the model must be

examined. Here, the main concepts and results will be discussed briefly. In particular, the condi-

tions that guarantee that the FCGARCH model is identifiable and minimal will be established and

proved. First, two related concepts will be discussed: The concept of minimality of the model,

established in Sussman (1992), also called “non-redundancy” in Hwang and Ding (1997); and the

concept of reducibility of the model.

DEFINITION 2. The FCGARCH(m,1,1) model is minimal (or non-redundant) if its input-output

map cannot be obtained from an FCGARCH(n,1,1) model, where n < m.

Onesourceofunidentifiabilitycomesfromthefactthatamodelmaycontainirrelevant“limiting

regimes”. A limiting regime is represented by the functions

µi=?αi+ βiht−1+ λiy2

This means that there are cases where the model can be reduced without changing the input-output

map. Thus, the minimality condition can only hold for irreducible models.

t−1

?f (yt−1;γi,ci), i = 1,...,H.

DEFINITION 3. Define θi = [γi,ci]?and let ϕ(yt−1;θi) = γi(yt−1− ci), i = 1,...,H. The

FCGARCH model defined in (1) is reducible if one of the following three conditions holds:

(1) One of the triples (αi,βi,λi) vanishes jointly for some i ∈ [1,H];

(2) γi= 0 for some i ∈ [1,H];

(3) Thereisatleastonepair(i,j), i ?= j, i = 1,...,H, j = 1,...,H, suchthat|ϕ(yt−1;θi)|=

|ϕ(yt−1;θj)|, ∀yt−1∈ R, t = 1,...,T (sign-equivalence).

DEFINITION 4. The FCGARCH model is identifiable if there are no two sets of parameters such

that the corresponding distributions of the population variable y are identical.

Three properties of the FCGARCH model cause unidentifiability of the models:

(P.1) The property of interchangeability of the regimes. The value of the likelihood function

of the model does not change if the regimes are permuted. This results in H! different

models that are indistinct among themselves. As a consequence, in the estimation of the

parameters, we will have H! equal local maxima for the quasi-log-likelihood function.

(P.2) The fact that f(yt−1;γi,ci) = 1 − f(yt−1;−γi,ci).

(P.3) Conditions(1)–(2)inthedefinitionofreducibilityprovideinformationaboutthepresence

of irrelevant regimes, which translate into identifiability sources. If the model contains a

regime such that αi = 0, βi = 0, and λi = 0, then the parameters γiand ciremain

unidentified, for some i ∈ [1,H]. On the other hand, if γi= 0, then the parameters αi, βi,

λi, and cimay take on any value without changing the quasi-log-likelihood function.

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MODELING MULTIPLE REGIMES IN FINANCIAL VOLATILITY13

Property (P.3) is related to the concept of reducibility. In the same spirit of the results stated in

Sussman (1992) and Hwang and Ding (1997), we show that, if the model is irreducible, properties

(P.1) and (P.2) are the only forms of modifying the parameters without affecting the log-likelihood.

Hence, by establishing the restrictions on the parameters of (1) that simultaneously avoid model

reducibility, any permutation of regimes, and symmetries in the logistic function, we guarantee

the identifiability of the model.

The problem of interchangeability, (P.1), can be prevented by imposing the Restrictions (R.1) in

Assumption3. Theconsequencesduetothesymmetryofthelogisticfunction(P.2)canberesolved

if we consider Restrictions (R.2) in Assumption 3. The presence of irrelevant regimes, (P.3), can

be circumvented by applying a “specific-to-general” modeling strategy as will be suggested in

Section 5.

Corollary 2.1 in Sussman (1992) and Corollary 2.4 in Hwang and Ding (1997) guarantee that

an irreducible model is minimal. The fact that irreducibility and minimality are equivalent implies

that there are no mechanisms, other than those listed in the definition of irreducibility, that can be

used to reduce the complexity of the model without changing the functional input-output relation.

Then, the restrictions in Assumption 3 guarantee that, if irrelevant regimes do not exist the model

is identifiable and minimal.

We need an additional assumption before establishing the sufficient conditions under which the

FCGARCH model is globally identifiable.

ASSUMPTION 6. The parameters αi, βi, and λido not vanish jointly for some i ∈ [1,H].

Assumption 6 guarantees that there are no irrelevant regimes.

THEOREM 4. Under Assumptions 3 and 6, the FCGARCH(m,1,1) model is globally identifiable.

Furthermore, L(ψ) is uniquely maximized at ψ0.

4.3. Consistency. The proof of consistency of the QMLE for the FCGARCH model follows the

same reasoning given in Ling and McAleer (2003). The following theorem states and proves the

main consistency result.

THEOREM 5. If (4) is satisfied, under Assumptions 1–5,?ψu,T

4.4. Asymptotic Normality. In order to prove asymptotic normality, we define:

∂ψ∂ψ?

∂ψ

∂ψ?

p

→ ψ0and?ψT

p

→ ψ0.

A(ψ0) = E

−∂2lu,t(ψ)

T∂Lu,T(ψ)

?????ψ0

and

?????ψ0

B(ψ0) = E

∂Lu,T(ψ)

?????ψ0

≡1

T

T

?

t=1

E

∂lu,t(ψ)

∂ψ

?????ψ0

∂lu,t(ψ)

∂ψ?

?????ψ0

.

Page 15

14M. C. MEDEIROS AND A. VEIGA

Consider the additional matrices:

AT(ψ) =1

T

T

?

T

?

t=1

?

∂lt(ψ)

∂ψ

1

2h2

t

∂ht

∂ψ

∂ht

∂ψ?

?y2

t

ht

?

1

4T

−

?y2

1

h2

t

ht

− 1

?y2

?

∂

∂ψ?

?

1

2ht

∂ht

∂ψ

??

, and

BT(ψ) =1

T

t=1

∂lt(ψ)

∂ψ?

=

T

?

t=1

t

t

ht

− 1

?2∂ht

∂ψ

∂ht

∂ψ?.

(12)

The following theorem states the asymptotic normality result.

THEOREM 6. If (4) is satisfied and E?ε4

(13)

t

?= µ4< ∞, under Assumptions 1–5,

T1/2(?ψT− ψ0)

D

→ N?0,A(ψ0)−1B(ψ0)A(ψ0)−1?,

where A(ψ0) and B(ψ0) are consistently estimated by AT(?ψ) and BT(?ψ), respectively.

REMARK 3. Under Assumption 2, it is clear that B(ψ0) =1

to the information matrix equality when E?ε4

2

?E?ε4

t

?− 1?A(ψ0), which reduces

t

?= 3.

5. DETERMINING THE NUMBER OF REGIMES

The number of regimes in the FCGARCH model, as represented by the number of transition

functions in (1), is not known in advance and should be determined from the data. One possibility

is to begin with a small model (such as GARCH(1,1) or white noise) and add regimes sequentially.

The decision to add another regime may be based on the use of model selection criteria (MSC) or

cross-validation. For example, Audrino and B¨ uhlmann (2001) used Akaike’s Information Crite-

rion (AIC) to select the number of regimes in their Tree-Structured GARCH model. However, this

has the following drawback. Suppose the data have been generated by an FCGARCH model with

m regimes (m − 1 transition functions). Applying MSC to decide whether or not another regime

should be added to the model requires estimation of a model with m logistic functions. In this

situation, the larger model is not identified and its parameters cannot be estimated consistently3.

This is likely to cause numerical problems in quasi-maximum likelihood estimation. Even when

convergence is achieved, lack of identification causes a severe problem in interpreting the MSC.

The FCGARCH model with m regimes is nested in the model with m + 1 regimes.

A typical MSC comparison of the two models is then equivalent to a likelihood ratio test of m

against m + 1 regimes; see Ter¨ asvirta and Mellin (1986) for a discussion. The choice of MSC

determines the (asymptotic) significance level of the test. When the larger model is not identified

under the null hypothesis, the likelihood ratio statistic does not have an asymptotic χ2distribution

under the null.

3In the case of the tree-structured GARCH model of Audrino and B¨ uhlmann (2001), the identification issue is related

to the location of the threshold. When an irrelevant regime is added, the location of the split cannot be estimated

consistently; see Hansen (1996) for a discussion.