Applied Econometrics GARCH Models - Extensions. Roman Horvath Lecture 2

Transcription

1 Applied Economerics GARCH Models - Exensions Roman Horva Lecre

2 Conens GARCH EGARCH, GARCH-M Mlivariae GARCH

3 Sylized facs in finance Unpredicabiliy Volailiy Fa ails Efficien markes Time-varying (rblen vs. ranqil periods) Volailiy clsering» Wen volailiy is ig, likely o remain ig» Clsering of informaion arrivals, price discovery Exremely large rerns (bo posiive and negaive) more likely an sandard normal disribion

4 A Sample Financial Asse Rerns Time Series Daily S&P 500 Rerns for Janary 1990 December 1999 Rern /01/90 11/01/93 Dae 9/01/97

5 ARCH eqaions model Mean eqaion (example AR(1)): y = ρy -1 + Variance eqaion: E( ) = α 0 + α 1 * -1 Tis is o be esimaed joinly be maximm likeliood In principle, one can also esimae i eqaion by eqaion by OLS, b is is less efficien Again, all specificaion cecks o be addressed like in ARIMA modeling

6 ARCH(q) ARCH (1) simply generalizes ino ARCH (q), q specifies e nmber of lagged sqared error erms = e (α 0 + α 1 * α q * -q ) 1/ Ten, Var(y ) = α 0 + α 1 * α q * -q Yo may inclde dmmies or any variable ino is eqaion: say day of e week dmmy, deviaion of excange rae from cenral pariy, ec.

7 Problem wi ARCH Someimes e large nmber of sqared lagged residals ms be inclded o specify e model correcly Bolerslev (1986) exends ARCH model o allow more flexible lag srcre e inrodces GARCH, i is o some exen analogy o ARMA models ARCH is like MA model and now we inrodce AR

8 GARCH (1,1) Noe a E(y ) = E( ) = α 0 + α 1 * -1 Inclde e lags of E( ) o e RHS of e eqaion E( ) = α 0 + α 1 * -1 + β 1 *E( -1 ) Typically, we wrie i is way: = α 0 + α 1 * -1 + β 1 * -1 So now, wa is e bes predicion for e nex period variance of y? Weiged average (wi e weigs in brackes) of: long-erm variance (α 0 ), is period acal variance yo may call i new informaion, no capred anywere else (α 1 ), e variance prediced for is period (β 1 )

9 GARCH (p,q) Narally, yo may generalize GARCH(1,1) o GARCH(p,q) assming is model for residal = e (α 0 + α 1 * α q * -q + β 1 * β p * -p ) 1/ Noe a e ~IID(0,1), α 0 >0, α 1 0,, α q 0, q>0, p 0, β 1 0,, β p 0 GARCH (p,q) is saionary, if e sm of α s all and β s is sricly smaller an 1 Condiional variance E(y ) = E( ) = = α 0 + α 1 * α q * -q + β 1 * β p * -p

10 How o esimae GARCH - sraegy 1. Fi correc ARIMA model for y, and ceck if residals are wie noise ec.. If yo sspec (G)ARCH residals, esimae = α 0 + α 1 * α q * -q + v (simply regress residal on eir lags, es significance, ARCH-LM es) 3. If ARCH errors no presen, all explanaory variables sold be joinly insignifican from zero 4. If joinly significan, is ells yo a is some ARCH or GARCH srcre of residals 5. Examine PACF and ACF of sqared residals, reasoning same as for ARIMA modelling, (b noe a very ofen yo find a GARCH(1,1) fis e daa bes) 6. Evalae e model

11 Exensions: TARCH Leverage effec in finance: Bad news are more imporan an good news for e beavior of sock In or case, bad news = negaive residal ( <0) Good news = posiive residal ( >0) TARCH accons for is effec I incldes dmmy*bad news Ts, TARCH (1,1): y = ρy -1 + = α 0 + α 1 * -1 + β 1 * -1 + γ 1 *I -1 * -1 Were I =1, if <0 and 0 oerwise GARCH is s special case of TARCH If γ 1 =0, no asymmeric effecs ( GARCH=TARCH ) TARCH someimes called GJR model

12 Exponenial GARCH (EGARCH) Sggesed by Nelson (1991). Te variance eqaion is given by Advanages of e model - Since we model e log( ), en even if e parameers are negaive, will be posiive. - We can accon for e leverage effec: if e relaionsip beween volailiy and rerns is negaive,, will be negaive. ) log( ) log(

13 GARCH-in Mean We expec a risk o be compensaed by a iger rern. So wy no le e rern of a secriy be parly deermined by is risk? Engle, Lilien and Robins (1987) sggesed e ARCH-M specificaion. A GARCH-M model wold be y = , N(0, ) = can be inerpreed as a sor of risk premim. I is possible o combine all or some of ese models ogeer o ge more complex ybrid models - e.g. an ARMA- EGARCH(1,1)-M model.

14 Forecasing Volailiy GARCH can model e volailiy clsering effec since e condiional variance is aoregressive. Sc models can be sed o forecas volailiy. We sowed a Var (y y -1, y -,...) = Var ( -1, -,...) So modelling will give s models and forecass for y as well. Variance forecass are addiive over ime.

15 Forecasing Variances sing GARCH Models Prodcing condiional variance forecass from GARCH models ses a very similar approac o prodcing forecass from ARMA models. I is again an exercise in ieraing wi e condiional expecaions operaor. Consider e following GARCH(1,1) model: y, N(0, ), Wa is needed is o generae are forecass of T+1 T, T+ T,..., T+s T were T denoes all informaion available p o and inclding observaion T. Adding one o eac of e ime sbscrips of e above condiional variance eqaion, and en wo, and en ree wold yield e following eqaions T+1 = T, T+ = T+1, T+3 = T+

16 Wa Use Are Volailiy Forecass? 1. Opion pricing Opion price is a fncion of volailiy of nderlying asse.. Condiional beas ( im, covariance beween marke rerns and rerns of sock i, m, variance of marke rerns a ) i, im, m, 3. Dynamic edge raios Te Hedge Raio - e size of e fres posiion o e size of e nderlying exposre, i.e. e nmber of fres conracs o by or sell per ni of e spo good.

17 Wa Use Are Volailiy Forecass? Wa is e opimal vale of e edge raio? Assming a e objecive of edging is o minimise e variance of e edged porfolio, e opimal edge raio will be given by were = edge raio p = correlaion coefficien beween cange in spo price (S) and cange in fres price (F) S = sandard deviaion of S F = sandard deviaion of F Wa if e sandard deviaions and correlaion are canging over ime? Use p p s F s, F,

18 Mlivariae GARCH Models Mlivariae GARCH models are sed o esimae and o forecas covariances and correlaions. Te basic formlaion is similar o a of e GARCH model, b were e covariances as well as e variances are permied o be ime-varying. Tere are 3 main classes of mlivariae GARCH formlaion a are widely sed: VECH, diagonal VECH and BEKK. VECH and Diagonal VECH e.g. sppose a ere are wo variables sed in e model. Te condiional covariance marix is denoed H, and wold be. H and VECH(H ) are H VECH ( H ) 11 1

19 VECH and Diagonal VECH In e case of e VECH, e condiional variances and covariances wold eac depend pon lagged vales of all of e variances and covariances and on lags of e sqares of bo error erms and eir cross prodcs. In marix form, i wold be wrien VECH H C AVECH BVECH H ~ N 0, H Wriing o all of e elemens gives e 3 eqaions as 11 1 c c c a a a a a a 1 3 Sc a model wold be ard o esimae. Te diagonal VECH is mc simpler and is specified, in e variable case, as follows: 11 1 a a a b b b b b 1 b b 13 b b

20 BEKK and Model Esimaion for M-GARCH Neier e VECH nor e diagonal VECH ensre a posiive definie variance-covariance marix. An alernaive approac is e BEKK model (Engle & Kroner, 1995). In marix form, e BEKK model is H WW AH A B Model esimaion for all classes of mlivariae GARCH model is again performed sing maximm likeliood wi e following LLF: T were N is e nmber of variables in e sysem (assmed above), is a vecor conaining all of e parameers o be esimaed, and T is e nmber of observaions. TN 1 ' 1 log log H H 1 B

21 Calclaed from nresriced BEKK-GARCH. Own calclaion sing JMlTi, EViews. Daa Reers. Condiional correlaion beween sock markes M.A. Tesis of Tereza Horáková on Volailiy Transmissions beween Financial Markes Figre E5: Condiional correlaion coefficien for eqiies marke rerns (rpx, rdj) flcaions Jan 00 Jan 01 Jan 0 Jan 03 Jan 04 Jan 05 Jan 06 Jan 07 Jan 08 Condiional correlaion coefficien_ eqiies

22

23 Readings Teory: Engle, R. (001): GARCH 101: Te Use of ARCH/GARCH Models in Applied Economerics, Jornal of Economic Perspecives, pp Engle, R. (003): Risk and Volailiy: Economeric Models and Financial Pracice, Nobel lecre, December 8, 003, Te Royal Swedis Academy of Sciences (003): Time Series Economerics: Coinegraion and Aoregressive Condiional Heeroscedasiciy Applicaions: Kocenda, E. (1998): Excange Rae in Transiion (capers -3), CERGE-EI Fidrmc, J. and R. Horva (008): Volailiy of Excange Raes in Seleced new EU Members, Economic Sysems, Codry, T. (000): Day of e week effec in emerging Asian sock markes: Evidence from e GARCH model, Applied Financial Economics, SEE THE COURSE WEBSITE FOR SOME OF THESE READINGS

24 Read abo mlivariae GARCH in Brooks (008) Inrodcory Economerics for Finance

25 Te Volailiy Laboraory of R. Engle available a p://vlab.sern.ny.ed/