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Ruprecht-Karls-Universitat Heidelberg

Fakultat fur Mathematik und Informatik

Bachelorarbeit

zur Erlangung des akademischen Grades

Bachelor of Science (B. Sc.)

GARCH(1,1) models

vorgelegt von

Brandon Williams

15. Juli 2011

Betreuung: Prof. Dr. Rainer Dahlhaus

Abstrakt

In dieser Bachelorarbeit werden GARCH(1,1)-Modelle zur Analyse nanzieller Zeitreihen unter- sucht. Dabei werden zuerst hinreichende und notwendige Bedingungen dafur gegeben, dass solche Prozesse uberhaupt stationar werden konnen. Danach werden asymptotische Ergebnisse uber rel- evante Schatzer hergeleitet und parametrische Tests entwickelt. Die Methoden werden am Ende durch ein Datenbeispiel illustriert.

Abstract

In this thesis, GARCH(1,1)-models for the analysis of nancial time series are investigated. First, sucient and necessary conditions will be given for the process to have a stationary solution. Then, asymptotic results for relevant estimators will be derived and used to develop parametric tests. Finally, the methods will be illustrated with an empirical example.

Contents

1 Introduction2

2 Stationarity4

3 A central limit theorem 9

4 Parameter estimation 18

5 Tests22

6 Variants of the GARCH(1,1) model 26

7 GARCH(1,1) in continuous time 27

8 Example with MATLAB 34

9 Discussion39

1

1 Introduction

Modelling nancial time series is a major application and area of research in probability theory and statistics. One of the challenges particular to this eld is the presence of heteroskedastic eects, meaning that the volatility of the considered process is generally not constant. Here the volatility is the square root of the conditional variance of the log return process given its previous values. That is, ifPtis the time series evaluated at timet, one denes the log returns X t= logPt+1logPt and the volatilityt, where

2t= Var[X2tj Ft1]

andFt1is the-algebra generated byX0;:::;Xt1. Heuristically, it makes sense that the volatility of such processes should change over time, due to any number of economic and political factors, and this is one of the well known \stylized facts" of mathematical nance. The presence of heteroskedasticity is ignored in some nancial models such as the Black-Scholes

model, which is widely used to determine the fair pricing of European-style options. While this leads

to an elegent closed-form formula, it makes assumptions about the distribution and stationarity of the underlying process which are unrealistic in general. Another commonly used homoskedas- tic model is the Ornstein-Uhlenbeck process, which is used in nance to model interest rates and credit markets. This application is known as the Vasicek model and suers from the homoskedastic assumption as well. ARCH (autoregressive conditional heteroskedasticity) models were introduced by Robert Engle in a 1982 paper to account for this behavior. Here the conditional variance process is given an au-

toregressive structure and the log returns are modelled as a white noise multiplied by the volatility:

X t=ett

2t=!+1X2t1+:::+pX2tp;

whereet(the 'innovations') are i.i.d. with expectation 0 and variance 1 and are assumed indepen- dent fromkfor allkt. The lag lengthp0 is part of the model specication and may be determined using the Box-Pierce or similar tests for autocorrelation signicance, where the case p= 0 corresponds to a white noise process. To ensure that2tremains positive,!;i08iis required. Tim Bollerslev (1986) extended the ARCH model to allow2tto have an additional autoregres- sive structure within itself. The GARCH(p,q) (generalized ARCH) model is given by X t=ett

2t=!+1X2t1+:::+pX2tp+12t1+:::+q2tq:

This model, in particular the simpler GARCH(1,1) model, has become widely used in nancial time series modelling and is implemented in most statistics and econometric software packages. GARCH(1,1) models are favored over other stochastic volatility models by many economists due 2 to their relatively simple implementation: since they are given by stochastic dierence equations in discrete time, the likelihood function is easier to handle than continuous-time models, and since nancial data is generally gathered at discrete intervals. However, there are also improvements to be made on the standard GARCH model. A notable problem is the inability to react dierently to positive and negative innovations, where in reality, volatility tends to increase more after a large negative shock than an equally large positive shock. This is known as the leverage eect and possible solutions to this problem are discussed further in section 6. Without loss of generality, the timetwill be assumed in the following sections to take values in eitherN0or inZ. 3

2 Stationarity

The rst task is to determine suitable parameter sets for the model. In the introduction, we considered that!;;0 is necessary to ensure that the conditional variance2tremains non- negative at all timest. It is also important to nd parameters!;;which ensure that2thas nite expected value or higher moments. Another consideration which will be important when studying the asymptotic properties of GARCH models is whether2tconverges to a stationary distribution.

Unfortunately, we will see that these conditions translate to rather severe restrictions on the choice

of parameters. Denition 1.: A processXtis called stationary (strictly stationary), if for all timest1;:::;tn;h2Z: F

X(xt1+h;:::;xtn+h) =FX(xt1;:::xtn)

whereFX(xt1;:::;xtn) is the joint cumulative distribution function ofXt1;:::;Xtn. Theorem 2.Let! >0and;0. Then the GARCH(1,1) equations have a stationary solution if and only ifE[log(e2t+)]<0. In this case the solution is uniquely given by 2t=! 1 +1X j=1j Y i=1(e2ti+) Proof.With the equation2t=!+(e2t1+)2t1, by repeated use ont1, etc. we arrive at the equation

2t=!(1 +kX

j=1j Y i=1(e2ti+)) + (k+1Y i=1(e2ti+))2tk1; which is valid for allk2N. In particular,

2t!(1 +kX

j=1j Y i=1(e2ti+)); since;0. Assume that2tis a stationary solution and thatE[log(e2t+)]0. We have logE[jY i=1(e2ti+)]E[logjY i=1(e2ti+)] =jX i=1E[log(e2ti+)] and therefore, ifE[log(e2t+)]>0, then the productQj i=1(e2ti+) diverges a.s. by the strong law of large numbers. In the case thatE[log(e2t+)] = 0, thenPj i=1log(e2ti+) is a random walk process so that limsup j!1j X i=1log(e2ti+) =1a:s: so that in both cases we have limsup j!1j Y i=1(e2ti+) =1a:s: 4

Since all terms are negative we then have

2tlimsup

j!1!jY i=1(e2ti+) =1a:s: which is impossible; therefore,E[log(e2t+)]<0 is necessary for the existence of a stationary solution. On the other hand, letE[e2t+]<0. Then there exists a >1 with log+E[log(e2t+ )]<0. For thiswe have by the strong law of large numbers: log+1n n X i=1log(e2ti+)a:s:!log+E[log(e2t+)]<0; so log(nnY i=1(e2ti+)) =n(log+1n n X i=1log(e2ti+))a:s:! 1; and nnY i=1(e2ti+)a:s:!0:

Therefore, the series

1 +1X j=1j Y i=1(e2ti+) converges a.s. To show uniqueness, assume thattand ^tare stationary: then jt^tj= (e2t1+)j2t1^2t1j= (nnY i=1(e2ti+))nj2tn^2tnjP!0:

This means thatP(jt^tj> ) = 08 >0, sot= ^ta.s.Corollary 3.The GARCH(1,1) equations with! >0and;0,have a stationary solution with

nite expected value if and only if+ <1, and in this case:E[2t] =!1. Proof.: SinceE[log(e2t+)]log(E[e2t+]) = log(+)<0, the conditions of Theorem 1 are fullled. We have

E[2t] =E[!

1 +1X j=1j Y i=1(e2ti+) =!(1 +1X j=1E[jY i=1(e2ti+)]) =!(1 +1X j=1(+)j) !(1) if this series converges, that is, if+ <1, and1otherwise.5 Remark 4.This theorem shows that strictly stationary IGARCH(1,1) processes (those where += 1) exist. For example, ifetis normally distributed, and= 1;= 0, then

E[log(e2t+b)] =E[loge2t] =(

+ log2)<0; where

0:577 is the Euler-Mascheroni constant. Therefore, the equationsXt=ett;2t=X2t1,

or equivalentlyXt=etXt1dene a stationary process which has innite variance at everyt. On the other hand,2t=2t1has no stationary solution. In some applications, we may require that the GARCH process have nite higher-order moments;

for example, when studying its tail behavior it is useful to study its excess kurtosis, which requires

the fourth moment to exist and be nite. This leads to further restrictions on the coecients and.

For a stationary GARCH process,

E[X4t] =E[e4t]E[4t]

=E[e4t]E[!2 1 +1X j=1j Y i=1(e2ti+) 2] =!2E[e4t]E[1 + 21X j=1j Y i=1(e2ti+)2+1X k=11 X l=1k Y i=1l Y j=1(e2ti+)(e2tj+)] =!2E[e4t]

1 + 21X

j=1(+)j+1X k=11 X l=1E[(e2t+)2]k^l(+)k_lk^l which is nite if and only if2:=E[(e2t+)2]<1. In this case, using the recursion2t= !+2t1(e2t1+),

E[4t] =!2+ 2!E[e2t1+]E[2t1] +E[(e2t1+)2]E[4t1]

=!2+ 2!2+1+2E[4t]; so

E[X4t] =E[4t]E[e4t] =!2E[e4t]1 ++(12)(1):

In the case of normally distributed innovations (et), the condition2<1 means E[(e2t+)2] =2E[e4t] + 2E[e2t] +2= 32+ 2+2= (+)2+ 22<1: The excess kurtosis ofXtwith normally distributed innovations is then

E[X4t]Var[Xt]23 =3(1 ++)!2(1)(122(+)2)(!1)23

= 3

1(+)2122(+)23

22122(+)2>0;

6 which means thatXtis leptokurtic, or heavy-tailed. This implies that outliers in the GARCH model should occur more frequently than they would with a process of i.i.d. normally distributed variables, which is consistent with empirical studies of nancial processes. More generally, forXtto have a nite 2n-th moment (n2N) a necessary and sucient condition is thatE[(e2t+)n]<1. Another interesting feature of GARCH processes is the extent to which innovationsetat time tpersist in the conditional variance at a later time2t+h. To consider this mathematically we will use the following denition. For the GARCH(1,1)-processX= (Xt), dene

X(t;n) =20t+nY

i=1(e2t+ni+) +!(t+n1X k=nk Y j=1(e2t+nj+)): Denition 5.The innovationetdoesnotpersist in X inL1i

E[X(t;n)]!0 (n! 1);

and almost surely (a.s.) i

X(t;n)a:s:!0 (n! 1):

If every innovationetpersists inX, then we callXpersistent.

To see how this denition re

ects the heuristic meaning of a shock innovation persisting in the conditional variance, consider that for a GARCH time series with nite variance,

E[X(t;n)] =E[(E[2t+n]E[2t+njet])1!(+)n1n1Y

i=1(e2t+ni+)] = (E[2t+n]E[2t+njet])1! which tends to zero if and only ifE[2t+n]E[2t+njet] tends to zero as well. Theorem 6.(i) Ifetpersists inXinL1for anyt, thenetpersists inXinL1for allt. This is the case if and only if+1. (ii) Ifetpersists inXa.s. for anyt, thenetpersists inXa.s. for allt. This is the case if and only ifE[log(e2t+)]0.

Proof.(i) First,

E[X(t;n)] =20t+nY

i=1E[e2t+ni+] +!t+n1X k=nk Y j=1E[e2t+nj+]: For this value to be converge to zero (that is, foretto not persist), we needE[20] to be nite, which means+ <1. On the other hand, let+ <1. Then we have

E[X(t;n)] =!1(+)t+n+!t+n1X

k=n(+)k !(+)n1!0 (n! 1); 7 soetdoes not persist. (ii) LetE[log(e2t+)]<0. By the strong law of large numbers, 1n n X i=1log(e2ti+)a:s:!E[log(e2t+)]<0; so lognY i=1(e2ti+) =n1n n X i=1log(e2ti+)a:s:! 1 and therefore nY i=1(e2ti+)a:s:!0:

This means that we have

X(t;n) =20t+nY

i=1(e2t+ni+) |{z} !0+!(t+n1X k=nk Y j=1(e2t+nj+) |{z} !0) a:s:!0: On the other hand, letE[log(e2t+)]0. Then by the argument in the proof of Theorem 1, we have limsup j!1j Y i=1(e2ti+) =1a:s:

so thatX(t;n) cannot converge to zero.It is a peculiar property of GARCH(1,1) models that the concept of persistence depends strongly

on the type of convergence used in the denition. Persistence inL1is the more intuitive sense, since it excludes pathological volatility processes such as2t= 32t1, which is strongly stationary sinceE[log(3e2t+ 0)] =( + log2) + log3<0. Denition 7.We call:=+the persistence of a GARCH(1,1) model with parameters!;;. As we have seen earlier, the persistence of the model limits the kurtosis the process can take. Since the estimated best-t GARCH process to a time series often has persistence close to 1, this severely limits the value ofto ensure the existence of the fourth moment. From the representation of2tin theorem 2.1, we immediately have Theorem 8.The autocorrelation function (ACF) offX2ngdecays exponentially to zero with rate if <1. 8

3 A central limit theorem

Having derived the admissible parameter space, we consider the task of estimating the parameters and predicting the values ofXtat future timest. SinceXtis centered at everyt, a natural estimator for its variance is the average of the squares 1n P n t=1X2t. The following theorem will show that, under a stationarity and moment assumption, this is a consistent and asymptotically normal estimator. Theorem 9.For a wide-sense stationary GARCH(1,1)-processXtwithV ar[X2t]<1,E[e4t]<1 and parameters!;;, the following theorem holds: 1pn n X t=1(X2tE[X2t])D! N(0;!21 ++(1)2

E[e4t](1 +) + 21211

where2:=E[(e2t+)2]as in section 2. Proof.By Corollary 2.2 the conditionE[X2t]<1implies that+ <1 and we have

E[X2t] =E[2t] =!1:

Similarly, we have seen thatE[X4t] is nite if and only if2<1 and in this case one has

E[X4t] =E[4t]E[e4t] =!2E[e4t]1 ++(12)(1):

DeneYt:=X2tE[X2t]. Then the variablesfY1;Y2;:::gare weakly dependent in the following sense: Lemma 10.Lets1< s2< ::: < su< su+r=tand letf:Ru!Rbe quadratically integrable and measurable. Then jCov[f(Ys1;:::;Ysu);Yt]j CpE[f2(Ys1;:::;Ysu)]r for a constantCwhich is independent ofs1;:::;su;r.

Proof.Letw=!E[X2t]. Then we have

Y t=we2t(1 +1X j=1j Y i=1(e2ti+)):

We dene the helper variable

Yt=we2t(1 +r1X

j=1j Y i=1(e2ti+)): Then ~Ytis independent of (Ys1;:::;Ysu) and by the Cauchy-Schwarz inequality: pE[f2(Ys1;:::;Ysu)]qE[(Yt~Yt)2]: 9

However, we have

E[(Yt~Yt)2] =E[w2e4t(1X

j=1j Y i=1(e2ti+)r1X j=1j Y i=1(e2ti+))2] =w2E[e4t]E[(1X j=rj Y i=1(e2ti+))2] =w2E[e4t]E[(rY k=1(e2tk+))2]E[(1 +1X j=r+1j Y i=r+1(e2ti+))2] =2rw2!

2E[e4t]E[4t]

=2rE[Y2t]; and therefore jCov[f(Ys1;:::;Ysu);Yt]j qE[Y2t]|{z} CpE[f2(Ys1;:::;Ysu)]r:Similarly, we will need an additional inequality: Lemma 11.Lets1< s2< ::: < su< su+r=tand letf:Ru!Rbe bounded, measurable and integrable. Then jCov[f(Ys1;:::;Ysu);YtYt+h]j Ckfk1r for anyh >0. Proof.Dene~Ytas earlier. Then by Holder's inequality,

2kfk1E[jYtYt+h~Yt~Yt+hj]:

Using the triangle and Cauchy-Schwarz inequalities, we have E[jYtYt+h~Yt~Yt+hj]E[jYt~YtjYt+h] +erwjYt+h~Yt+hj~Yt qE[(Yt~Yt)]qE[Y2t+h] +qE[(Yt+h~Yt+h)2]qE[~Y2t] rqE[Y2t]qE[Y2t+h] +rqE[Y2t+h]qE[~Y2t]; so that jCov[f(Ys1;:::;Ysu);YtYt+h]j 2(E[Y2t] +qE[Y2t]qE[~Y2t])|{z}

Ckfk1r:10

The theorem to be proved is now

1pn n X t=1Y tD! N(0;!21 ++(1)2(E[e4t](1 +) + 21211)): Dene 2:=1X k=1E[Y0Yk] =E[Y20] + 21X k=1E[Y0Yk] and

2n:= Var[1pn

n X t=1Y t]:

Then we have

2=E[X40]E[X20]2+ 21X

k=1(E[X20X2k]E[X20]E[X2k]):

However,

E[X20X2k] =E[202k]

=E[20(a0k1X j=1j Y i=1(e2ki+) +20k Y i=1(e2ki+))] = (+)kE[40] +!1(+)k1E[20] =!2((1 ++)(+)k(12)(1)+1(+)k(1)2); so

2=!2E[e4t]1 ++(12)(1)!21(1)2+

+ 2 1X k=1!

2((1 ++)(+)k(12)(1)(+)k(1)2)

!21(E[e4t]1 ++1211+ 2(1 ++1211)1X k=1(+)k) !2(1)2(E[e4t]1(+)2121 +2 + 2+ 21221) =!21 ++(1)2(E[e4t](1 +) + 21211): 11

SinceYtis centered for everyt, we have

j~2n2j=j(1n n X s;t=1E[YsYt])2j =jE[Y20] + 2n1X k=1(1kn )E[Y0Yk]2j 21X
k=njE[Y0Yk]j+ 2n1X k=1kn jE[Y0Yk]j

2CqE[Y20]1X

k=n k |{z} !0+2CqE[Y20]n1X k=1k kn |{z} !0!0 (n! 1); using the weak independence of the variablesfYkg, where the limit of the second sum is due to

Kronecker's lemma sinceP1

k=1k<1. This means that ~2n!2forn! 1.quotesdbs_dbs21.pdfusesText_27

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