金融数据分析04 chap4_h

1、TIME SERIES ANALYSISChapter FourUnivariate GARCH ModelsReference: “Analysis of Financial Time Series”, 2010, Tsay, Chapter 3.Univariate GARCH Models c SEMChapter FourCNU14.1MotivationEmpirical regularities (? 5?5)Many financial time series have a number of characteristics in common. Asset prices pt=

2、log Pt are generally non-stationary.Returnsareusually egrated.Some financial time series are fractionally Return series t usually show no or little autocorrelation.2 Serial independence between the squared values of the series o tis often rejected, implying the existence of non-linear

3、relationshipsbetween subsequent observations.Univariate GARCH Models c SEMChapter FourCNU2 Volatility of the return series appears to be clustered. Normality has to be rejected in favor of some thick-taileddistribution.often stationary. Volatility seems to react differently to a big price drop, refe

4、rred to as the leverage effect.priceincreaseorabigUnivariate GARCH Models c SEMChapter FourCNU3Why do we need volatility models?Woldsdecompositiontheoremestablishesthatanycovariancestationary yt may be written asyt = t + ut,where t is linearly deterministic and ut is a linearly regular covariancesta

5、tionary stochastic process, given byXXiLi,2 ,ut = (L)t,(L) =0 = 1,ii=0i=0(20, 0, zt is an iid centered process with zero mean and unitvariance, i.e., zt i.i.d.(0, 1), andt() = Eyt|Ft1,2() = E(yt t() 2|F2, = Et1t1ttwhereUnivariate GARCH Models c SEMChapter FourCNU7 Et1 = E|Ft1 denotes the conditional

6、 expectation; Ft is the information set available at time t. t() may be an ARMA(p, q) process or could consist of seasonality features.Univariate GARCH Models c SEMChapter FourCNU8(1) Model properties The process t is a martingale difference sequence (MDS) and it is(conditionally) serially uncorrela

7、ted. SinceEt1t = Et1tzt = tEt1zt = 0,(MDS)we haveEtht = EthEt1(t) = Eth0 = 0.This further shows that t is conditionally uncorrelated:Covtht, t+k = Ethtt+k EthtEtht+k= Ethtt+k = EthEt+k1(tt+k)= EthtEt+k1t+k= 0.Univariate GARCH Models c SEMChapter FourCNU92ois unbiased estimator of 2().ttzt NID(0, 1)

8、and thus independent of (), we Lets supposehave2t2 = E2Ez2 = E2 = 2,Et1t1t1t1ttttt| Ft1because z2 2(1).t Lopez (2001) shows that 2 is an unbiased but extremely imprecisetestimator of 2 due to its asymmetric distribution.t Moreover, the absolute return |t| is a biased estimator for t, sincep 2/.| Ft1

9、2if t| | = tN (0, ), then Et1ttUnivariate GARCH Models c SEMChapter FourCNU10 If the conditional distribution of zt is time invariant with a finite fourth moment, the fourth moment of t is444422422E = Ez E Ez E= Ez E ttttttt4422E Ez E tttby Jensens inequality.1 The equality holds true for a constant

10、 conditional variance only. NID(0, 1), then Ez t4 = 3, the unconditional distribution for If ztt is therefore leptokurtic42 2E 3E ttK(t) = E4t/E22 3.t1Jensens inequality: Eg(x) g(Ex) if g(x) is concave; Eg(x) g(Ex) if g(x) is convex.Univariate GARCH Models c SEMChapter FourCNU11 The kurtosis can be

11、expressed as a function of the variability of the conditional variance.| Ft12 N(0, ) If t, thent4 = 3E4 = 3E( 2) 2EEt1t1t1t1ttt2E 4 = 3E( 2) 2( 2) 3= 3E(2)2EEEt1t1ttttand we have2422( 2) 2( 2)E 3E( )= 3E 3EEE ttttt1t12( 2) 2( 2)E4 = 3E(2)2 + 3E E 3EEt1t1ttttTherefore, the kurtosisUnivariate GARCH Mo

12、dels c SEMChapter FourCNU122( 2) 2( 2)EEEEE4= 3 + 3 t1tt1t t =E(2)2E(2)2tt2 Var2EVartt1t= 3 + 3= 3 + 3E(2)2E(2)2ttThe existence of Var2 ensures the existence of the kurtosis .tUnivariate GARCH Models c SEMChapter FourCNU13(2) Types of models Conditionally heteroscedastic (or GARCH-type) process. 2 i

13、s generated by the past of t.t The volatility is here a deterministic function of past oft. Differentchoiceofthisdeterministicfunctionproducesdifferent processes of this GARCH class. ThestandardGARCHmodelsarecharacterizedbyavolatilityspecified as a linear function of the past values of 2,tXqXpj 2j 2

14、2z iid(0, 1),t = tzt,= +tjtjttj=1j=1They will be studies in detail in our course.Univariate GARCH Models c SEMChapter FourCNU14 Stochastic volatility processes2 t is generated by vt, where vt is a strong white noise and is independent of t. In these models, volatility is a latent process. The most p

15、opular model in this class assumes thattheprocesslog t or log 2follows an AR(1) of the formtzt iid(0, 1),t= tzt,log t= + log t1 + vtorlog 2= + log 2+ vt,t1twhere the noise vt and t are independent.2Note, however, that the volatility is also a random variable in GARCH-type processesUnivariate GARCH M

16、odels c SEMChapter FourCNU15 Regime switching models The conditional variance is specified as t = (t), where t is a latent (unobservable) integer-valued process, independent of zt. The state of the variable t is here interpreted as a regime. The process t is generally supposed to be a finite-state M

17、arkov chain,pM 1.pMM . . . p11.p1MP =,where pij = Prt = j | t1 = i denotes the transition probabilityfrom the state= i to the statet = j, and it satisfies thatt1PMj=1pij= 1, i = 1, . . . , M .Univariate GARCH Models c SEMChapter FourCNU164.2The ARCH modelsARCH(q) modelAn ARCH(q) model, originally in

18、troduced by Engle (1982), is a linearfunction of past squared disturbances: i.i.d.(0, 1),2(3)ztt= tzt,= + 122+ + .(4)qtt1tq The conditional volatility is a moving average of squaredinnovations. The 0, andparameters must satisfy the following constraints:i 0, i = 1, , q.22 Defining vt , where Ev = 0,

19、 we can rewrite (4) asan=t1tttUnivariate GARCH Models c SEMChapter FourCNU17AR(q) model for the squared innovation 2:t2= + (L)2 + vt,ttwhere (L) = 1L + 2L2 + + qLq. The process t is covariance stationary if and only if the sum of thePq autoregressive parameters is less than one, i.e., 1.ii=1We leave

20、 the proof of covariance stationarity to the GARCH process. In this case, the unconditional variance of innovation is22= E = /(1 ).1tqPqi=1 Given the stationary condition that is a white noise. 1, the process of itUnivariate GARCH Models c SEMChapter FourCNU18Kurtosis of an ARCH modelARCHmodelsareab

21、letogenerateexcesskurtosis.Indeed,evenifthestandardized innovationsztisassumedtobenormal, theunconditional distribution forthasfattertailsthanthenormaldistribution. Fortance, consider an ARCH(1) model2() = + 12,tt1where t = tzt. The unconditional variance is given by 2should have 0 1 1 for 2 to be f

22、inite./(1 1) so that we=Univariate GARCH Models c SEMChapter FourCNU19 Under normality, the conditional fourth moment is givenby4 = E4z 4 = 3E4 = 34 = 3( + 12)2,Et1t1t1tttttt1whereas the unconditional fourth moment is2m4 = E 4 = EE4z 4 = 3E( + 1)2t1t1ttt2 + 2 2132(1 + 1)2=.= 3+ 1m421 1(1 1 )(1 31) 0

23、 and 32 0,1, . . . , q andi 0,Remark 1. The parametersi = 1, . . . , p.ii=0,Pq i=12= Et | FRemark 2. If E| Ft1 = 0 and t 2 = +2tt1itiPp j=1j2, the t is a semi-strong GARCH process.tjIn our course, we mainly focus on the Strong-GARCH case.Univariate GARCH Models c SEMChapter FourCNU29Inlagpolynomialf

24、orm,theconditionalvarianceofaGARCH(p, q)process can be rewritten as:2= + (L)2 + (L)2,(6)tttwhere (L) = 1L + + qLq, (L) = 1L + + pLp.The GARCH(1,1) is the most popular model in the empiricalliterature:2= + 12+ 12(7).tt1t1Univariate GARCH Models c SEMChapter FourCNU30The implied condition of ARCH() mo

25、delRewriting the GARCH(p, q) model as an ARCH():2t212= 1 (L) + (L) = + (L)ttX 2,=+itii=1where (L) = 1L + 2L2 + . Nonnegativity:Here we have 2 0, if 0 and all i 0. Thetnon-negativity of and i is also a necessary condition for the nonnegativity of 2.t Invertibility of (1 (L):defined, assume that :and

26、In order to makewellii=1Univariate GARCH Models c SEMChapter FourCNU31 The roots of the polynomial (z) = 1 lie outside the unit circle, and that 0, this is a condition for to be finite and positive. (z) and 1 (z) have no common roots.2 However, these conditions are establishing nor that 0, i 0, i = 1, . . . , q, j 0, j = 1, .