1

PhD Program in Business Administration and Quantitative Methods

FINANCIAL ECONOMETRICS

2007-2008

ESTHER RUIZ

CHAPTER 3. GARCH Models

3.1 Empirical properties of financial time series.

Financial econometrics is fundamentally empirical. That is why we will start by

describing the main empirical characteristics often observed when analysing high

frequency series of financial returns.

3.1.1 Marginal distribution

The marginal distribution of financial returns depends on the frequency of observation.

Considering high frequency data as, for example, daily or weekly, the returns are

characterized by non-Normal distributions with heavy tails. The distributions are

usually heavy tailed although symmetric.

2

3.1.2 Temporal dependency

Financial returns are characterized by being uncorrelated (efficient market hypothesis).

However, there are non-linear transformations that are serially correlated. In particular,

powers of absolute returns are correlated with the largest autocorrelations often

observed for absolute returns (Taylor effect).

It is very usual to focus the analysis primarily on the autocorrelations of squares.

Remember that, if the series is linear, the autocorrelations of squares are equal to the

squared autocorrelations of the original observations. Given that, as we mentioned

above, financial returns are usually uncorrelated, if their squares are correlated, means

that the volatility appears in clusters and has predictable components.

3

4

3.1.3 Basic model

The basic model able to represent non-correlated series with excess kurtosis and

autocorrelated squares is given by

ttty σε=

where tε is an i.i.d process with zero mean and variance 1, and tσ is the volatility that

evolves over time. There are a plethora of models proposed in the literature to specified

the dynamic evolution of tσ . In this chapter, we will describe the most basic models

that have been seminal in the literature. In particular we focus on the GARCH models

proposed by Engle (1982) and Bollerslev (1986) and the Stochastic Volatility models

proposed by Taylor (1982) and popularized by Harvey, Ruiz and Shephard (1992).

Modelling volatility is important for several reasons:

· It is a fundamental component of many financial models for derivatives valuation.

Consider, for example, the Black and Scholes formula for valuation of an European

option given by

llKrP

x

lxKrxPc

tt

lt

tl

tt

σσ

σ

21)/log(

)()(

+=

−Φ−Φ=−

−

where tP is the current price of the underlying stock, r is the risk-free interest rate,

tσ is the conditional standard deviation of the log return of the stock, and )(xΦ is the

cumulative distribution function of the standard normal random variable evaluated at x .

· Volatility is also important for risk management.

· Modelling volatility can improve the efficiency in parameter estimation and the

accuracy in interval forecast.

3.2 Properties of GARCH models

3.2.1 The ARCH model

The volatility, 2tσ , in the basic ARCH(1) model is given by

21

2−+= tt yαωσ

where 0>ω and 0≥α for 2tσ to be positive at every t . Furthermore, notice that ω

needs to be strictly positive for the process not to degenerate. Suppose, for example, that

5

0=ω and 0=ty , in this case, 021 =+tσ and 01 =+ty . The process is zero for ever. We

know that returns can be zero in a given day, because the price of the corresponding

stock does not change, but then the price can change again. Therefore, the parameter ω

has to be different from zero. As we will see later, this restriction has important

practical implications in prediction of future volatilities.

The parameter ω is related with the scale (the marginal variance) of the process while

the parameter α models the evolution of the volatility. If 0=α , the volatility is

constant over time, the process of returns, ty , is homoscedastic, while if 0≠α , 2tσ

evolves depending on past returns, when the market has a large return in a given day,

the volatility next day is going to be large while if the returns is small, the next day

volatility is also small. This behaviour generates the clustering of volatility observed in

real time series.

To analyse the marginal distribution of returns generated by an ARCH(1) model, we

next obtain, the marginal mean, variance and kurtosis of ty .

0)()()(11

=⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡=

−−tttttt EEyEEyE εσ

αωσαω

σεσσ

−=⇒+

==⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡==

−

−−

1)(

)()()()(

221

22

1

22

1

2

yt

ttttttyt

yE

EEEyEEyVar

[ ])1)(1(

)1()()( 2

2444

1

4

ακααωκσκσε

εεε −−

+==⎥⎦

⎤⎢⎣⎡=−

ttttt EEEyE

2

2

4

4

11)(

)(ακακ

σκ

εε

−−

===y

tyt

yEyKur

Note that the weak stationarity condition is that 1<α regardless of the distribution of

tε . If this condition is satisfied, the marginal variance is constant although the

conditional variance evolves over time. However, weak stationarity is not necessary for

strict stationarity; Milhoj (1985). The condition for strict stationarity is

[ ] 0)log( 2 <tE αε

This condition depends on the distribution of tε . If, for example, it is Gaussian, then the

ARCH(1) model is strictly stationary if 56.3<α . The necessary and sufficient

condition for strict stationarity was established by Bougerol and Picard (1992) and

6

Nelson (1990). The regions of strict stationarity are, in general, much larger than those

of weak stationarity.

Furthermore, the ARCH(1) model can generate excess kurtosis. However, note that the

condition for the fourth order moment to be finite depends on the distribution of tε . If it

is Gaussian then, the kurtosis of ty is finite if 5774.0<α . As we commented before,

when 0=α , the process is homoscedastic and, consequently, the kurtosis is 3 (the

process is Gaussian). Under Gaussianity of the errors, Engle (1982) shows that the 2mth

order moment of ty exists if

∏=

<−m

j

m j1

1)12(α .

Alternatively, it is possible to derive the moments of the ARCH(1) by noticing that 2ty

is an AR(1) model

tttttt yy υαωεσσ ++=−+= −2

12222 )1(

where )1( 22 −= ttt εσυ . The process tυ is a white noise process

0))1(())1(()( 2

1

222

1=⎥⎦

⎤⎢⎣⎡ −=⎥⎦

⎤⎢⎣⎡ −=

−−ttttttt EEEEE εσεσυ

[ ]4222

1)1())1(()( ttttt EEEVar σκεσυ ε −=⎥⎦

⎤⎢⎣⎡ −=−

0)1()1())1()1((),( 211

21

2221

21

221 =⎥⎦

⎤⎢⎣⎡ −−=−−= −

−−−−− ttttttttttt EEECov εσεσεσεσυυ

However, it is conditionally heteroscedastic 22

14222

1

2

1))(1()1())1(()( −

−−+−=−=−= ttttttt

yEE αωκσκεσυ εε

The dynamic dependence of returns can be analysed by deriving the autocorrelation

function (acf) of ty and that of 2ty . We derive first, the autocovariances of ty :

0)()()()(1111 =⎥⎦

⎤⎢⎣⎡===

−−−− ttttttttt EyEyEyyEh εσσεγ

Therefore, the series of returns are uncorrelated. However, the squared returns are

serially correlated. Taking into account that 2ty follows an AR(1) model with parameter

α , its autocorrelation function is given by hh αρ =)(2

The acf of general transformations δty is not known for 2≠δ .

7

Finally, it is straightforward to see that the conditional distribution of ty is the same as

the conditional distribution of the noise tε . If we assume that it is normal then,

),0(,...,| 211 ttt Nyyy σ⎯→⎯−

Note that the volatility, 2tσ , coincides with the conditional variance of ty , and,

consequently, it is observable one step ahead.

The properties of the ARCH(1) model described before can be easily extended to the

ARCH( q ) model that specifies the volatility as

∑=

−+=q

iitit y

1

22 αωσ

3.2.2 The GARCH(1,1) model

Early applications of ARCH models needed many lags to adequately represent the

dynamic evolution of the conditional variances. In some applications, q could be even

50. To avoid computational problems when estimating such a large number of

parameters, the parameters were restricted in an ad hoc manner. For example, Engle

(1983) assume that )1(

)1(+−+

=qq

iqi

αα .

Later, Bollerslev (1986) implemented the same kind of restriction used to approximate

the infinite polynomial of the Wald representation by the ratio of two finite

polynomials, usually of very low orders. As a result, he proposed the GARCH(p,q)

model given by

∑∑=

−=

− ++=p

iiti

q

iitit y

1

2

1

22 σβαωσ

In practice the most useful GARCH(p,q) model is the GARCH(1,1) model so, we

describe its properties in detail. The conditional variance of the GARCH(1,1) model is

given by 2

12

12

−− ++= ttt y βσαωσ

The parameters have to be restricted to guarantee the positiveness of the conditional

variance. In particular, 0>ω , 0≥α and 0≥β . The positivity restrictions for the

general GARCH(p,q) model have been derived by Nelson and Cao (1992).

Alternatively, the GARCH(1,1) model can be written as an ARMA(1,1) model for

squared residuals as follows:

8

12

12

12

12

1

21

21

21

21

21

2

)()1()(

)()(

−−−−−

−−−−−

−+++=+−−++

=+−−++=+++=

ttttttt

tttttttt

yy

yyyy

βυυβαωυεβσβαω

υσββαωυβσαω

The ARCH parameter,α , has to be strictly positive for the process to be conditionally

heteroscedastic. If 0=α , then there is a common root between the AR and MA

components of the model and the parameter β is not identified. The sum of the

parameters, βα + , is related with the persistence of shocks to the volatility.

The weak stationarity condition of the GARCH(1,1) model is 1<+ βα . However, the

condition for strict stationarity is

[ ] 0)log( 2 <+ βαε tE

Bai et al. (2004) derive the properties of the GARCH(p,q) model using the ARMA

representation of 2ty . In particular,

βαω−−

=1

)( 2tyE

1

2

2

)(1)1(

1−

⎥⎥⎦

⎤

⎢⎢⎣

⎡

+−

−−=

βαακ

κκ εεy

9

This expression shows that persistence and kurtosis are highly tied up in GARCH

models. If the errors are Normal, the condition for the existence of the fourth order

moment is 12)( 22 <++ αβα . From a given value of the persistence, the values of α

that guarantee a finite fourth order moment should decreases as the persistence

increases. In the limit, if α+β=1, α=0 and the process is homoscedastic. Allowing tε

having a leptokurtic distribution, reduces the space of values of the ARCH parameter, α,

that guarantee the existence of the fourth order moment. Consequently, in GARCH

models, the dynamics of the volatility are severely restricted to guarantee that the fourth

order moment is finite.

The dynamics of returns appear in the acf of squares given by

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

>+

=++−

+++−

=− 1,))(1(

1,)(1

))()(1(

)(1

2

22

2

2

h

h

hhβαρ

αβαβααβαα

ρ

For a given persistence, )1(2ρ increases with α. Therefore, α measures the dependence

between squared observations and can be interpreted as the parameter leading the

volatility dynamics.

Looking at the relationship between kurtosis, persistence and )1(2ρ , it is possible to

conclude that the GARCH model is very rigid to represent simultaneously series with

high kurtosis and small autocorrelations of squares. Only when the persistence is very

close to one, the GARCH model is able to represent both characteristics.

10

3.2.3 The IGARCH(1,1) model

In practice, when the GARCH(1,1) model is fitted to real financial returns, it is often

observed that 1ˆˆ ≈+ βα . For this reason, Bollerslev and Engle (1986) proposed the

IGARCH(1,1) model which is given by

)()1( 21

21

21

21

21

2−−−−− −++=−++= tttttt yy σασωσααωσ

The IGARCH model is not weakly stationary although, it is strictly stationary if the

errors are Normal.

Note that the volatility is modelled as a random walk plus drift model. However,

IGARCH processes have a rather regular dynamic behaviour. In this sense, Kleiberger

and van Dijk (1993) have shown that the probability of an increase in the variance is

smaller than the probability of a decrease and, consequently, shocks to the volatility are

not usually very persistent.

3.3 Testing for ARCH effects

Testing for ARCH effects is usually based on the fact that observations generated by a

GARCH model are uncorrelated although the autocorrelations of squares are not zero.

11

Consequently, it is possible to test for conditional heteroscedasticity by testing whether

the autocorrelations of squared returns are significantly different from zero, i.e.

0)(...)2()1(: 2220 ==== MH ρρρ

In this sense, McLeod and Li (1983) proposed to implement the Box-Ljung statistic to

the squared residuals of an ARMA model fitted to returns to remove the sample mean

and any serial correlation. Therefore, the McLeod-Li statistic is given by

∑=

=M

kkrTMQ

1

22 ))(~()(

where )(2)(~22 kr

kTTkr−+

= and )(2 kr is the sample autocorrelation of order k of 2ty .

If the eighth order moment of ty is finite, the McLeod-Li statistic is asymptotically

distributed as a 2)(Mχ variable.

However, it is important to note that the sample autocorrelations of squares are usually

rather small and have very large finite sample negative biases, especially in the more

persistent cases. Consequently, in these cases, the McLeod-Li test may have low power.

To overcome this problem, Rodríguez and Ruiz (2005) have proposed a new statistic

that takes into account that under the null hypothesis, the sample autocorrelations are

not only not significantly different from zero but also mutually uncorrelated. The

statistic is given by

∑ ∑−

= =⎥⎦

⎤⎢⎣

⎡+=

iM

k

i

li lkrTMQ

1

2

0

* )(~)(

If the eighth order moment of ty is finite, the asymptotic distribution of )(* MQi can be

approximated by a Gamma distribution, ),( τθG with parameters b

a2

2=θ and

ba2

=τ

where ))(1( iMia −+= and ∑+

=−+−−++−=

1

1

22 )1)((2)1)((i

jjijiMiiMb .

Finally, note that the two previous tests can be alternatively implemented to the

autocorrelations of absolute returns instead of squares. This alternative has several

advantages. First, the asymptotic distribution only requires finite fourth order moment

of ty . Furthermore, the autocorrelations of absolute returns are larger and have smaller

biases than that of squares. Consequently, the power of both tests is larger when they

are implemented with absolute returns.

12

3.4 Maximum Likelihood estimation

The GARCH parameters are usually estimated by maximizing the Gaussian log-

likelihood given by

∑∑==

−−=T

t t

tT

tt

yL

12

2

1

2

21)log(

21log

σσ

Since the conditional distribution of tε is not assumed to be Normal, the corresponding

estimator is a Quasi-Maximum-Likelihood (QML) estimator. If the second order

moment of ty is finite, the QML is consistent and, if the 6th order moment is finite, it is

asymptotically Normal; see Ling and McAleer (2002). The asymptotic distribution of

the QML estimator is then given by

),0()ˆ( 11 −−⎯→⎯− IJJNT dT θθ

where

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂∂

−= −

');|(log 1

2

θθθtt Yyl

EJ

⎥⎦⎤

⎢⎣⎡

∂∂

∂∂

= −−

');|(log);|(log 11

θθ

θθ tttt YylYyl

EI

Both matrices coincide when the errors are conditionally Normal. Note that, in this case,

Tθ̂ is the ML estimator.

Consistency and asymptotic normality of the QML estimator do not require that he

parameters satisfy the stationarity condition α+β<1 but they continue to hold for the

ICARCH(1,1) model; see Lumsdaine (1996).

It is important to note that when the autocorrelations of the original series are different

from zero, the two step estimator is efficient as far as the conditional mean does not

depend on the parameters of the conditional variance and the errors are conditionally

Normal. In this case, the asymptotic covariance matrix of the ML estimator is block-

diagonal and the estimation of the parameters can be undertaken separately without

loosing asymptotic efficiency; see Engle (1982) and Gourieroux (1997). Therefore, it is

possible to estimate efficiently by fitting first an ARIMA model to the original data to

filter any dependencies in the conditional mean and then fitting a GARCH model to the

residuals of the linear model. However, in empirical applications, the existence of a risk

13

premium implies that the parameters of the volatility equation are likely to appear in the

conditional mean.

The main problem estimating GARCH models by QML is that the likelihood function is

rather flat and, consequently, the estimates very imprecise; see Shephard (1996).

Example: IBEX35

The diagnosis is based on the standardized observations given by t

tt

yσ

εˆ

ˆ = where

21

21

2 ˆ904.0088.00000014.0ˆ −− ++= ttt y σσ . If the fit is appropriate then they should be

distributed as N(0,1) and their squares should be uncorrelated. For the IBEX35 data, the

correlogram of squared standardized observations is given by

14

15

The model also allows to obtain estimates of the one-step ahead volatilities within-

sample by 21

22

2 ˆ859.0141.00000025.0ˆ −− ++= ttt y σσ given by

16

3.5 Prediction

The predictions of the volatility k steps ahead depend on the model fitted to the

data. For example, if the model is an ARCH(1) model, then 222

|1 )( TTTTT yyE αωαωσ +=+=+

( ) 22221

21

2|2 )1( TTTTTTTT yyEyE ααωαωαωσαωαωσ ++=++=+=+= +++

( ) 2322222

22

2|3 )1()1( TTTTTTTT yyEyE αααωααωαωσαωαωσ +++=+++=+=+= +++

22| 1

1T

kk

TkT yαααωσ +−−

=+

Note that, as expected the predictions of future volatilities tend to the marginal

variance as the forecast horizon tends to infinity. However, in the short run the

predictions of volatility can be over or under the marginal variance depending on the

value of 2Ty , the squared return at the end of the sample.

Similar expressions can be obtained for the GARCH(1,1) model. In this case, 22222

|1 )()( TTTTTTTT yEyE βσαωσβαωσ ++=++=+

))(()()( 2221

2|2 TTTTTT yE βσαωβαωσβαωσ ++++=++= ++

17

)()()(1

)(1 2211

2| TT

kk

TkT y βσαωβαβα

βαωσ +++++−

+−= −

−

+

Finally, in the IGARCH(1,1) model

)()()( 2222222|1 TTTTTTTTTT yyEE σασωσασωσ −++=−++=+

)(2)( 22221

2|2 TTTTTTT yE σασωσωσ −++=+= ++

)( 2222| TTTTkT yk σασωσ −++=+

As we have mentioned before, if the errors are conditionally Normal, the one

step ahead distribution of ty is given by

),0(,...,| 211 ttt Nyyy σ⎯→⎯− .

Therefore, it is possible to obtain one step-ahead prediction intervals of 1+Ty by

1ˆ96.1 +± Tσ

When the errors are conditionally Normal, the Normal distribution is also an

adequate approximation to the distribution of TkTy |+ although this is not Normal.

In any case, it is possible to use bootstrap procedures to obtain the distribution of

TkTy |+ without assuming any particular distribution of the errors and incorporating the

uncertainty due to parameter estimation. Furthermore, using bootstrap procedures it is

also possible to obtain prediction densities for future volatilities; see Pascual, Romo and

Ruiz (2005).

3.6 Extensions

The main advantage of GARCH models is that they are conditionally Gaussian and,

consequently, inference can be carried out by standard procedures. However, the

analysis of the restrictions needed to guarantee stationarity and the existence of the

fourth order moment are very complicated and, in high order models difficult to check:

“Bollerslev (1986) provided the necessary and sufficient condition for the existence of the 2mth moment of the GARCH(1,1) model, and the necessary and sufficient condition for the fourth-order moments of the GARH(1,2) and GARCH(2,1) models. Using a similar method in Bollerslev (1986), He and Tërasvirta (1999a) provided the moment conditions for a family of GARCH(1,1) models. Ling and McAleer (2002d) derived the sufficient condition for the existence of the statioanru solution for this family of GARCH(1,1) models,

18

showed that He and Tërasvirta (1999a) condition is necessary but not sufficient, and provided the sufficient moment condition. He and Tërasvirta (1999b) and Karanasos (1999) examined the fourth moment structure of the GARCH(r,s) process. From the proof in Karanasos (1999), it can be seen that the condition is necessary but not sufficient. He and Tërasvirta (1999b) stated that their condition is necessary and sufficient. Ling and McAleer (2002c) showed that the existence of the fourth moment is incomplete, that the condition is not sufficient for the existence of the fourth order moment, and also derived the necessary and sufficient conditions of all the moments.”

Li, Ling and McAller (2003)

On top of this difficulty, there are also practical problems when the estimated

parameters do not satisty the restrictions to guarantee the positivity of the conditional

variance. Even if the parameters satisfy all the previous restrictions, we have seen that

GARCH models are not flexible enough to represent series which simultaneously have

large kurtosis and small although significant autocorrelations of squares. Finally, it has

been often observed in real time series of financial returns that the response of volatility

to positive and negative returns of the same magnitude can be asymmetric (leverage

effect).

Next, we describe some models that try to solve one or more of the previous limitations.

3.6.1 GARCH in mean

In finance, the mean returns of a shock may depend on its volatility. Larger uncertainty

is associated with larger expected returns. The GARCH-M model, proposed by Engle,

Lilien and Robinns (1987), is given by

21

21

2

2 )(

−− ++=

=++=

ttt

ttt

ttt

y

aagy

βσαωσ

σεσδμ

The parameter δ is called the risk premium parameter. The inclusion of σt in the returns

equation is an attempt to incorporate a measure of risk into the returns generating

process and is an application of the “mean-variance hypothesis” underlying many

theoretical asset pricing models such as the intertemporal CAPM. Under this hypothesis

the parameter of the volatility in the returns equation should be positive indicating that

the expected return is positively related to its past volatility.

There are alternative specifications of )( 2tg σ . For example, Merton (1987) in the

intertemporal CAPM model, assumes that 22 )( ttg σσ = but other functional forms for

19

g(·) can be specified, possibly allowing the response to depend upon the sign and level

of volatility.

It is straightforward to see that the conditional distribution of returns is given by

)),((,...,| 2211 tttt gNyyy σσδμ +⎯→⎯−

Using this distribution, it is possible to estimate by ML the parameters of the

ARCH-M model. However, it is not known whether the model satisfies the regularity

conditions for the asymptotic normality of the ML estimator.

Analysis of ARCH-M models is much more complex than was true of pure GARCH

models. For example, the conditions for the series to be stationary have still to be

worked out. The returns are correlated in this case. Hong (1991) has derive the

autocorrelations of squares of the GARCH(1,1)-M model.

Notice that the volatility, 2tσ , can also be specified as other alternative heteroscedastic

processes that will be described later.

Example: GARCH-M model fitted to S&P 500 returns.

1)1.17()01.7(148.00695.0 −−+= tttt aay σ

22)83.2(

21)60.3(

21)90.7()17.6(

62 357.0502.0139.01020.1 −−−− +++= tttt ax σσσ

The conditional standard deviation is included as an explicative variable in the mean

equation that also contains a MA(1) component. Large values of the conditional

variance are expected to be associated with large returns. The MA(1) error may capture

the effect of nonsynchronous trading and is highly significant. As it is usual, the

GARCH parameters sum to almost unity indicating high persistence in the conditional

variance.

3.6.2 Leverage effect

Another stylised fact of many financial time series is the asymmetric response of

volatility to positive and negative movements in prices. This is known as leverage effect

and was originally described by Black (1986). The first model proposed in the literature

to represent the leverage effect was the Exponential GARCH (EGARCH) of Nelson

(1991). In the simplest case, the EGARCH(1,1) model is given by

ttty σε=

( )[ ] 1112

12 ||||)log()log( −−−− +−++= ttttt E γεεεασβωσ

20

If, tε is Gaussian, then π

ε 2|| 1 =−tE if, for example, it has a standardized

Student-ν distribution, then ( )πνν

ννε)2/()1(

2/)1(22|| 1Γ+

+Γ−=−tE . There is no need to restrict

the parameters to guaranty the positivity of the conditional variance given that the

model is formulated for the logarithmic volatility. Furthermore, the stationarity

condition is ⏐β⏐< 1.

The asymmetric response of volatility is represented by the parameter γ. If the

return at time t-1 is positive then the response of volatility is given by γ+α. However, if

the return is negative, then the response is γ-α. Usually, the parameter γ is negative, and

consequently, the effect on volatility is bigger when the return is negative than when is

positive.

The expressions of the marginal variance, kurtosis and autocorrelations of

squares are complicated; see He, Teräsvirta and Malamsten (2002) and Karanasos and

Kim (2003). It is interesting to note that the autocorrelations of squares of EGARCH

series may be negative. Therefore, EGARCH models can represent the dynamic

behaviour of series with cycles in the squares. The persistence of shocks to volatility is

measured by the parameter β.

It seems that the properties of EGARCH models are similar to the ones of

GARCH models; see Carnero et al. (2004).

21

22

There are many other models proposed in the literature to represent the leverage

effect. Just to name a few:

i) GJR-GARCH of Glosten, Jagannathan and Runkle (1993)

( ) 211

2121

211

2 )0()0(1 −−−−− +>+>−+= tttttt II βσεεαεεαωσ

ii) VS-GARCH of Fornari and Mele (1997)

( )( ) ( ) )0()0(1 12

122

12212

112

1112 >+++>−++= −−−−−− ttttttt II εσβεαωεσβεαωσ

iii) Q-GARCH of Sentana (1993) 2

112

12

−−− +++= tttt βσγεαεωσ

In this case, the impact of 21−tε on 2

tσ is equal to αεγ +−1/ t . Therefore, if 0<γ ,

the effect of a negative shock is larger than the effect of a positive shock of the same

size. Furthermore, in the Q-GARCH model, the effect of 1−tε is symmetric around

αγ 2/ . The properties of the Q-GARCH model are also very similar to the properties of

GARCH models.

The asymmetries generate correlations between 2ty and hty − .

iv) Hentschel (1995) proposed a Family-GARCH model where the conditional

variance is given by

λσ

βεασωλ

σ λνλ

λ 1)(

1 111

−++=

− −−−

ttt

t f

||||)( bcbf ttt −−−= εεε || bt −ε

The parameter λ represents a Box-Cox transformation and, consequently, when

it tends to zero, we obtain the logarithmic transformation. The model proposed by

Hentschel (1995) encompases many other models with leverage effect as, for example,

the EGARCH model.

3.6.3 Long memory

As we have seen in the empirical examples, the autocorrelations of squares are usually

small although they persist during very long periods. Consequently, the behaviour of

these autocorrelations is not in concordance with the pattern expected if the series of

squares returns were stationary. On the other hand, the forecasting implications of

models with unit roots in the conditional variances are not satisfactory. Therefore, it has

been often postulated that the volatility may have a long-memory behaviour.

23

Baillie et al. (1996) proposed the Fractionally Integrated GARCH model. Consider, for

example, the ARMA(1,1) expression of the GARCH(1,1) model, given by

12

12 )( −− −+++= tttt yy βυυβαω

tt LyL υβωφ )1()1( 2 −+=−

Then, if there is a fractional difference in the autoregressive component,

ttd LyLL υβωφ )1()1)(1( 2 −+=−−

))(1()1)(1( 222ttt

d yLyLL σβωφ −−+=−−

2112 ))1)(1()1(1()1( td

t yLLL −−−−+−= −− φββωσ

The restrictions needed to guarantee that the conditional variance is positive are

complicated and difficult to check. Furthermore, Baillie, Bollerslev and Mikkelsen

(1996) argue that the model is strictly stationary and ergodic, the IGARCH model is

only weakly stationary when d=0. Moreover, Davidson (2004) shows that the dynamic

properties of the FIGARCH model are somehow unexpected in the sense that the

persistence of the volatility is larger the closer the long memory parameter is to zero.

The properties of the FIGARCH model have not been established jet. The

autocorrelation function of squares is not known, although Karanasos, Psaradakis and

Sola (2004) and Palma and Zevallos (2004) have derive the autocorrelations of squares

of long memory GARCH models closely related to the FIGARCH model.

Finally, Bollerslev and Mikkelsen (1996) proposed the Fractionally Integrated

EGARCH (FIEGARCH) model that represents simultaneously long memory and

asymmetric volatilise. In the FIEGARCH(1,d,1) model, the volatility equation is given

by

[ ] )(1)log()1)(1( 12

−−+=−− ttd gLLL εδωσφ

The asymptotic properties of the ML estimator of the parameters of IGARCH and

FIGARCH models are still without being established.

References

Bai, X., J.R. Russell and G.C. Tiao (2004), Kurtosis of GARCH and Stochastic Volatility Models with Non-normal innovations, Journal of Econometrics, 114, 349-360. Baillie, R.T., T. Bollerslev and H.O. Mikkelsen (1996), Fractionally integrated autoregressive heteroskedasticity, Journal of Econometrics, 74, 3-30.

24

Bollerslev, T. (1986), Generalized autoregressive conditional hetersokedasticity, Journal of Econometrics, 31, 307-327. Bollerslev, T. and H.O. Mikkelsen (1996), Modeling and pricing long memory in stock market volatility, Journal of Econometrics, 73, 151-184. Bougerol, P. and N. Picard (1992), Stationarity of GARCH processes and some nonnegative time series, Journal of Econometrics, 52, 115-128. Carnero, M.A., D. Peña and E. Ruiz (2004), Persistence and kurtosis in GARCH and Stochastic Volatility Models, Journal of Financial Econometrics, 2, 319-342. Carrasco, M. and X. Chen (2002), Mixing and moment properties of various GARCH and SV models, Econometric Theory, 18, 17-39. Engle, R.F. (1982), Autoregressive conditional heteroskedasticity with estimates of the variance of UK inflation, Econometrica, 50, 987-1007. Engle, R.F. (1983), Estimates of the variance of U.S. inflation based on the ARCH model, Journal of Money, Credit and Banking, 15, 286-301. Engle, R.F. and T. Bollerslev (1986), Modelling the persistence of conditional variances, Econometrics Reviews, 5, 1-50. Harvey, A.C., E. Ruiz and N.G. Shephard (1994), Multivariate Stochastic Variance Models, Review of Economic Studies, 61, 247-264. He, C., T. Teräsvirta and H. Malmsten (2002), Moment structure of a family of first-order exponential GARCH models, Econometric Theory, 18, 868-885. Hentschel, L. (1995), All in the family: nesting symmetric and asymmetric GARCH models, Journal of Financial Economics, 39, 71-104. Hong, E.P. (1991), The autocorrelation structure of GARCH-M processes, Economics Letters, 37, 129-32. Karanasos, M. and J. Kim (2003), Moments of the ARMA-EGARCH model, Econometrics Journal, 6, 146-166. Karanasos, M., Z. Psaradakis and M. Sola (2004), On the autocorrelation properties of long-memory GARCH processes, Journal of Time Series Analysis, 25, 265-281. Kleibergen, F. and H.K. van Dijk (1993), Non-stationarity in GARCH models: A Bayesian Analysis, Journal of Applied Econometrics, 8, 41-61. McLeod, A.I. and W.K. Li (1983), Diagnostic checking ARMA time series models using squared-residual autocorrelations, Journal of Time Series, 4, 269-273. Nelson, D.B. (1991), Conditional heteroskedasticity in asset returns: a new approach, Econometrica, 59, 347-370.

25

Nelson, D.B. and C.Q. Cao (1992), Inequality constraints in the univariate GARCH model, Journal of Business and Economic Statistics, 10, 229-235. Palma, W. and M. Zevallos (2004). Analysis of the correlation structure of square time series, Journal of Time Series Analysis, 25, 529-550. Pascual, L., J. Romo and E. Ruiz (2005), Forecasting returns and volatilities in GARCH processes using the bootstrap, Computational Statistics and Data Analysis, forthcoming. Rodríguez, J. and E. Ruiz (2005), A powerful test for conditional heteroscedasticity for financial time series with highly persistent volatilities, Statistica Sinica, 15, forthcoming. Taylor, S. J. (1982), Financial returns modelled by the product of two stochastic processes – a study of daily sugar prices 1961-79. In O.D. Anderson (ed.), Time Series Analysis: Theory and Practice, 1, 203-226. North-Holland, Ansterdam. Teräsvirta, T. (2007), Univariate GARCH models, in Andersen, T.G., R.A. Davis, J.-P. Kreiss and T. Mikosch (eds.), Handbook of Financial Econometrics, Springer, New York. Exercises

1. Represent graphically the relation between the ARCH parameter α and the kurtosis of a GARCH model. Observe that the autocorrelation of order one increases as the kurtosis increases.

2. Derive the relationship between the degrees of freedom, ν , the ARCH parameter, α , and the kurtosis of returns, yκ , in an ARCH(1) model with Student-t errors with ν degrees of freedom.

3. Exercises 1, 2, 3, 4 and 7 (with any series of financial returns you like) of chapter 3 of Tsay (2002).