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Markov Switching in GARCH Processes and Mean-Reverting Stock-Market VolatilityMichael J. Dueker aa Federal Reserve Bank of St. Louis, St. Louis, MO, 63166

Version of record first published: 02 Jul 2012.

To cite this article: Michael J. Dueker (1997): Markov Switching in GARCH Processes and Mean-Reverting Stock-MarketVolatility, Journal of Business & Economic Statistics, 15:1, 26-34

To link to this article: http://dx.doi.org/10.1080/07350015.1997.10524683

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b~larkov Switching in GARCH Processes and Mean-Reverting Stock-Market Vo

Michael J. DUEKER Feds:al Reserve Bank of St. Louis, St. Louis, MO 631 66

This article introduces four models of conditional heteroscedasticitj that contain Marko\'-switching parameters to examine their multiperiod stock-market volatility forecasts as predictions of options- implied volatilities. The volatil~ty model that best predicts the behavior of the options-implied volat~lit~es allous the Student-t degrees-of-freedom parameter to switch such that the conditional variance and lcurtosis are subject to discrete shifts. The half-life of the most leptolcurtic state is estilnated to be a week, so expected market vulatility reverts to near-normal levels fairly quickly following a spike.

~ ' d WORDS: Asset-price volatility; Conditional heteroscedasticity; iCurtosis.

Volatility clustering is a well-documented feature of fi- nancial r-ates of return: Price changes that are large in mag- nitude ten& to occur in b~inches rather than with equal spacing. A natural question is how long financial mar- kets will remain volatile because volatility forecasts are central to calculating optimal hedging ratios and options prices. Indeed we can study the behavior of options- implied stock-market vob.tilities to find stylized facts that paramexis volatility models should aim to capture. Two stylized fzcts that con\7entiona2 volatility models, notably generalized autoregressive conditional heteroscedasticity [GARCH, Bollerslev (1986)], find hard to reconcile are that (1) conditional volatility can increase substantially in a short amount, of time at the onset: of a turbulent period and (2) the rate of mean reversion in stock-market volatility appears to vary positively and noniineariy with the level of volaril- ity. In ofher words, stock-market volatility does not remain persistently two to three times above its normal level in the same way it can persist at 3040% above normal.

Kamiltoi~ and SusmeP (1994) and Lamoureux and Las- trapes (3993) highlighted the forecasting dificuities of conve~~tional GARCH models by showing that they can provide worse multiperiod volatility forecasts than constant- variance models. In particular. multiperiod GARCH fore- casts of the volatility are too high in a period of above- normal vola~iliey. Friedman and Laibson (1989) addressed the forecasting issue by no? allowing the conditional vari- ance in a GARCH model to respond proportionately to "iarge" and "smaS1" shocks, %a this way: the conditional variance is restrained from increasing to a level from which volatility forecasts would be undesirably high. One draw- back of this approach is that in such a model the condi- tional volatility might understate the true variance by not responding sulfificiently to large shoclis and thereby never

ity to experience discrete shifts and discrete changes in the persistence parameters.

Partly in response to Lamoureux and Lastrapes (19901, who observed that structural breaks in the variance could account for the high persistence in the estimated conditional variance, Hamilton and Susmel(1994) and Cai (1994) intro- duced Markov-switching parameters to autoregressive con- ditional heteroscedasticity (ARCH) models, and 1 extend the approach to GARCH models because the latter are more flexible and widely used. Section 1 presents tractable meth- ods of estimating GARCH madeis with Marl<ov-switching parameters. Section 2 describes four specifications that are estimated and provides in-sample and out-of-sample goodness-of-fit test results. Section 3 uses the estimated models to generate multiperiod forecasts of stock-market volatility and compares the forecasts with options-implied volatilities to see which of the GARCH/Markov-switching models best explains the two stylized facts described pre- viously.

Each of the volatility-model specifications will assume a student-t error distribution with nt df in the dependent variable y:

zt - student-t (mean = 0. nt. ht), nt > 2. In all of the models. the conditional mean. pt, is allowed to switch ac- cording to a Markov process governed by a state variable, st:pt=plsr+ph(l-Si). stE{(O,l) v'c.

be pressed to display much mean re~~eusion. Thus, such "threshoPd" models do not necessarily address the two siyl- The unconditional probability of SI = 0 equals (I - q ) / ( 2 -

p - q). The variance of E~ is denoted uf and is a func- ized facts listed previously-sharp upward jumps in volatil- fion of nt and ht in all of the models coasidered such that ity, followed by fairly rapid reversion to near-normal lev-

els. This article endeavors to craft a volatility model that can address these two stylized facts from within the class

rameters. of GARCH models with Markov-switching pa, 4897 American Statistical Association

Journal of Business & Economic Statistics BAarltov-switching pasaneters ought to enable the volatil- January 1997, Vo!. $5, No. 1

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Duaher: Markov 31rvtichiiqg it? GPRGiN Processes 27

a: - f ( ~ 2 ~ . ii 1 ) : u/here ;h'3 specific fuszrcl!ion ,f varies ecross the n~cd.els. 1:; a31 cases, howez'e:, h is ?ssriinesT 'LO "cr a h&&:cl3( i , process -jvrltla Mcrkos-s~,~~~itching pxameters aiso g e ~ e r l e d by 5' s~ that a general form for h is

Note that the presence cf lagged i? on :i?e righr side of ( 3 ) causes tlse CAKCH variable :a be a ft1nc:lon oh' the enlire history sf the slate xrariable. If h Tp,Tere an AWCY3(p) process, ther; h ml31-1r.id depepd sl:lji oa the p most 1-ecect vnh-,s 3f the s~aee <!ariable, in the ~ ; o r i < of Cai (1994.) and Hamilton an6 Susmei (I 9 4 ) . Eere I discuss h ~ v ~ methods described by Kim (199" can be applied [a make eslimalion feasible

7 . for c!~wcE processes suDjeci f.9 l\/Jarkvv s~h~ikcilini;. Ckariy it is not pict ical to exaalnine all of tile si;ssibl&

seGnej:ceij of ~ a s t values ,3f the stak \reri&!e x,.vhm piduat- ing the IikeEhooi funciicn Laor e sa:i~pk~ 01 n:o~.e than 1,000 obscrvaciaxis bccausr, il-ue nnrnber of cases :o cocsicle~- ex- ceeds 1,000 by ;he Idme i = 1::. !<in? (19941 address66 this problem 5- introducing a collapsP.~",g procedure that greatly facilitates evaluadcn srf :he ii!celihood fiL1nction at the cost of . . introdscil~g a degree of zpprcx~~._alim~ that does aot ,rppear tc distoyt "ixe ca8ogjn*,eb likeiihoobd. b-q much. 'Bll;ie sdsllaps- ing Q"GC&LK~. when ipplied! to a GAiRCH process, calls for treai:ii~p :ke czadiaionai clispersio:?, ht. ss a f~s~~ctioal of at most the most recent 31 values of the state variable S. For the kliering to be acccraie, IClm iiored [her, when i'i is p- order a.;toregressive, then ;\I shoerld be at least p -t 1. ?n the GARGY(1, 1) case. u = 1. so we would I~a\.e to keep track of 212 oi . . feu- cases, based 011 the t v t ~ most rt:cc,nt vaines

of a binary state variable. Thus, ht is t:-dated as a func:ion 3f only ,Ti and hli. ' ' = hi [,St = i. St I = j).

Denatkg st as the i c f ~ r ~ ~ ~ a d o n available :hcougl~ ~.irne i , I k e p the number of clases to four by ineegrad:lg out St_1 before ?lugging lagged il a.to the GARCH equation

7-1 t 21s . .??e?iaod of collapsi:~g i~f iij"." cizto hi1' at every ob-

servation gives r:s a tirlc,ctai3ie CARCE forixrrla, v;.:leich is zpproximetcly cqcal to the exect GARC1-H eyuatio:~ f ~ o m Equation $3):

- - !Vole thatthe c.o;lapsing prozedure Integrates oat the first lag of the state variable* Sip,. from the GARCH f~~nct icn . h i . at the r i h r point in the Zltering process to ?revc:nt :he zondif onal de.csir y from becsmi~.g a fvnceion of a growi?i.g nr~~nber of past values of the stake vkaj-i~able.

From "ilk general framc.ijiork, I choose specificati~ns that differ accodllng to tile pe-e~~eters that switc?: and t",e re- latioxship bet.n;een the GARCA process. h . and ch:; ~al-i-

ance o'. i n severai speci5caticns. !h: GARCIi, processes are f~~ncrtions of Iaggef. 1;alues of the stale variable but not ;he contemporaneous value, St. For these, 1 seat ht as a fu~~ct ion o:' only so 1 only need to keep uack of ; ~ 3

,. ~ cases: i & ) = h (St-i = jj. Fu::hermore. arter ~ntegratiilg out I am left with a scals;r in the collapsing Frocess:

. .~ A iracable GARCH eqva.:non 1s rnes an even simpler ver- sion of Equation (5):

Ancther feature of this GARCB/Markov-switcE-ii:g f~-aaner?iork is thaf the state variable implies a co;~nectio.ji be:ween the mean stock return and the variance and pos- sibly ki-~riosis. lf the mean stock return is lower in rke ln!g5--;oiatility state, then the nodel can explain negai~i~eiy skewed distributio,~s, both unconditi3naT a d conditiionaI on a.viliieble inforrna~ion. Thc studcnt-i disrribdt;ons have reyo slte;vtless oniy when conditional on particr-lar i~alueq of the scale -,~sri&Ies, ;.;,i-*ich are unobser.c;hlt:

- i h s srst specifica:Eon is a GARCK afia.log to @ai*s (1994) ARCF model with A4arkov switching ra ? . Thwvariarxe is assuraed to follow a GAWC9 process so that o: = !if and tlhe orlly parameter i~ ht silbjec~ to i.?/iarko\ switching is 7 . -. ?lais type of switching is tantamocn: ;o allowing shifts in the unconditionail vaxiance because the i~nconditionzel vari- ance of the ocdknarj, cinsieni-psrarne~el- GAWCM(1 1) Yro- cess is - / (1 - a - 3). For: this madel, the GARCH variance ta2aes the form

with constant a and ;ir. '$be denote this model as the GARCH-UV model for CARCK w7itl-i switching ic tis,r, un- conditionai variance. in practice, we pa:-ameterize - ( S t ) es g(St)-,. where g ( S = 1) is norr-alized to unity. - r he secoxd specificatioi~ is a GARCH analog ~o 19amiT'co-I and Sa;:sme!'s (ZY9b) A R H model uvS:h Marliov switching in a normalirario-r, factor g. where the !,ariance = &hi. In illis case. the GkRCH Eqnatiox ( 5 ) ialtes the form

where : and 3 are cal-istani and g j,S = 1) is nori~.alimed to unity. 1 denote this model as the GkRCB-NF model for GAMCE xi:ii-iR s:vitching in the ~:o:naliration factor. g . Note that in ihe @ARCH-NF mode! rhe SARCH process in E ~ L T F ~ - tion 691 is not a C'snction of St" so escia~ation is somewhx si~rplifie~d~ - rhe c h i r ~ specification is e Markov-s~witclaii~g analog lo Xansen (1994). in which the variance follows a. GARZiS process (a: - i t t ) and she st.dent-i degrees-of-heedom ?a- remei-eii 1s ~~~~~~~d to switch. Marrsen (1994) iratroduc~,d a model .I! ~ih ich the student-i degrees-of-freedom parame-

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Journal of Business & Economic Statistics, January 1997

ter, nt , is allowed to vary over time as a probit-type func- tion of variables dated up to time t - I. Because Hansen's (1 994) specification is not conducive to muitiperiod fore- casting, however. 1 chose to make nt follow a Maskov pro- cess governed by St: nt = nlSt + nf,(l - St). Although nt does not enter the GARCH Equation (7) in this specifi- cation, the GARCH process is still a function of the state variable because state-switching in the mean implies that E

is a function of the state variable:

Because the kurtosis of a student-t random variable equals 3(n,t - 2) / (n t - 4) and is uniquely determined by 72,. we call this the GAWCR-M model for GARCH with switching in the conditional kurtosis,

The fourth specification is similar to the GARCH-M model except the variance is assumed to be

rather taan cr: = ht. En this model, the GAWCH process ht scales the variance of et for a given value of the shape parameter nt. Here it is convenient to define vt = l,'nt so that ( 1 - 2ct) = ((nt - 2) /n t ) and the GARCH Equation (7) becomes

I denote this specification as the GARCH-DF model for asam- GARCH with switching in the degrees-of-freedom p-

eter. As in the GAWCH-NF and GARCH-13 models, h is a function of Stp i , but not St , in the GARCH-DF model. The GARCM-DE model shares two features with the GARCH- NF rnodel: The variance is subject to discrete shifts, and the lagged squared residuals are endogenously downweighted in states in which 02 /h is large. With the GARCH-K model: the GARCH-DF model shares the feature of time-varying conditional kurtosis so that conditionai fourth moments are not assumed to be constant.

I also report results on the usual GARCH(1, 1) nlodei with Markov switching in the mean and a model of switch- ing ARCH with a Beverage eflect (SWARCH-L), as in the work of Hamilton and Susmel (1994). The SwARGH-L model has switching in a normalizing factor in the vari- ance: o% = gth,t. where ht follows an ARCH(%) process with a leverage effect:

where LIfjl is a dummy variable that equals 1 when e (SZp1 = j)t-l < 0. The leverage effect posits that neg- ative stock returns increase debt-to-equity ratios, maicing firms riskier initially. Hence the leverage-effect parameter S is expected to have a positive sign.

The log-likelihood ful~ction for the GARCH-DF model. for example, is

where 2 E (0.1) corresponds with S+ E (0. l ) . ~ E {O,1) corresponds mith St-, E (0. I}, and r is the gamma func- tion. The function maximized is the log of the expected likelihood or

as in the work of Hamilton (1990).

Estimali~n Results

The four GARCBjMarkov-swhching volatility models, the asual GARCB(I, 1) modei, and the SWARCM-E model are applied to daily percentage changes in the S&P 500 in- dex from January 6, 1982, to December 31, 1991. Observa- tions from the post- 199 1 period are reserved for evaluation of the out-of-sample fit.

Some interpretation of the parameter estimates in Table 1 follows. The GARCH-DF model shows switching in the student-t degrees-of-freedom parameter between the values of 2.64 and 8.28. This implies that conditional fourth mo- ments do not exist in one state, whereas the conditional kur- tosis is 3(n- 2)/ ' (n-3) = 4.4 in the other state. The weight given to lagged squared residuals in the GAWCH process is show11 to be ru(l - ~ U ( S ~ . . ~ ) ) in Fqua t io~ (12). and this weight shifts with the state variable between .009 and .027. In this way, shocks drawn from the Tow degree-of-freedom state do not affect the persistent GARCH dispersion process proportionately. !!/lost importantly, shifts in the degrees-of- freedom parameter bring potentially Large discrete shifts in the variance. A shift out of the low degree-of-freedom state causes the variance to decrease by about 68%, holding the dispersion constant:

The ~~nconditional probability of being in the low degree-of- freedom state is about 10% with a half-life of five trading days. The unconditional value for the degrees-of-freedom parameter is about 6.8. The GARCH-DF rnodel also sug- gests chat stock returns are negatively skewed because the mean stock return is below normal in the high-volatility state when St = 0. In fact. all of the models find negative slcewness except the conventional GARCH model.

The GARCM-WF model finds ail estimate of the variance inflation factor g ( S = 0) = 12.59 with a large standard error. The effective sample from which to estimate this pa- rameter is small because the unconditional probability of St = 0 is only about 9%.

The factor g ( S = 0) raises :. by a significant rn~iltiple of 5.7 in the GARCW-UV model. but the unconditional prob-

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- iab!e I , GARGH4ii/aj%;o:~-Sv~itchii1g Wi0d8?/s Applied tc Dail:/ Percentage Changes in S&P 502 inaex

~ - - .- . .

isara6:e:ei. ;/: R C , ~ - ~ : C C;ARC!+-NF ~ , q F~c,&;-~JI/ GARCH-K GAlqCfi S~#ARCM-L

Log-lilislihood ,u :st = 0 )

.ii (Sf = I j

" !,St = 0;

v ( S t = :!

cz

3

6 (S, = 0)

(' d t = 7 )

P

4

INOTE' Standarc' elrois a-=. n aarpniheses

aeijii;y being 4 . - h a t :;Care is o d y 11%. The slate with ..T y [S = l) is extrerizelii persisten-, wlln y = .995.

The GARtR-I( :~:&,-l esti~~~.rra:ss that the degrees-of- freedom pziamerer s~witches beithieen 10.9 a d 4.2, l>iith a2

.. ~ uncafidjl~onaj -q?Ji.e of &3ut 6. 3 5 h staxes al-e highly per- . - sigteni w:tll fieally ide~,[ic~.fij-aasition ~7:obabilities. Two

stares f 3 ~ p fim~l;ean s:ocl,: :eCu~-n are Setter defined in the cT ~..t,~..;-;i< -'P '- .- ,c,,l,: , om - -. jhnn Jn the rcnveniionai GARCH ixode! -- wiih ibvi;chi::g in iht ::ieac. ;kib/e s i l o ~ ~ thai iii !.he iisikal GAWCB(I, 1; mcdel :he meall stock yeturn, f i . is bLi.lualiy idenriel ill both states. Hence t iye i;wo slates are not well idlent:Lfied and the i:sis:letian, -?of s;endard errors for :he tran-

. . smon p--&bi3iti,:s (a:led. Usi-rg &a.ily d;,ta. ;he :;;eights zttached !o :agged scjuared

rcsldrrals ixc iloc sigrsificant in the ST?/AK@Y-L model, i ~ ~ i t k the horde:-line excepiio~ cf :he leverage-eCccl pxxm:ter, 8.

,. . The nnrlna":1z1;-g facicr, g . is estimated hc raise rht. wari- mce by a *i?~Pllp;o of 3.78 in tire hjgh-volaii'iry state, .- * * , , - , - - ,,,llich ii.ricrr.diri31~ai probability . r 2 7%~ big;- c-nrcc -ti

c:^ persi?jte~% of i;crPl states suggests that 4cv.:- and high- voietiiity siafes csns:bt.rrte regimes, as ~pposed to shori- lar;ci:ag episodes. The CAWCX-DP model, on :he otY~e:. hard. finds rc!aci\!c?y short-lasting i o w - d e g r c c - o f - f e d states,

{,,/e \ve:e certain tha: ;--I- * ig,~i5;icant slate-switching oc- cur;-ed in the iiilear,. tken Iikeihocd ratio tesrs of state- i:v;tching in fhz degrees-of'-freedom para~~e te r s ;and g would be 8.p;3iop;-iare. B J ~ , the GARGH -1anode1 rnggests :ha!; switching in ihe mean cannot be taken for grx~ied, so like- Ji!lcocP ratio tests C ~ I I Z D ~ E S S U I ~ ~ :;hat the rransitiorr probz- biiiities are i:i,eniii?eCi ennder the null of no state switckiap in L' or y . bansen i 1992) discussed sinaalaticn methods to &i.iy cratq~ll] values f ~ r silch liite'lhood 1-atio test:; wi:h nonstandal.d 2iiri--bii;%ocs. 'The C I - ~ G C ~ ~ values compu- tayionaily bcr&nsol;:e ;o :e?-uiate, hsij./ever, 53 1 :io llot p:;;sue iilal; srxetegjt her-, I~lsica,?., 3 folc..;~ 7.Jlaa; a,nd ?aim

(1993) by ~ ~ s i ; ~ g a goodness-of-fit tes: that is valid for data that are not identicelly distributed. 1 ~ ~ e r f o r ~ the test o\,e;c :he in-saxple perioci (1982-1991) and en out-of-sanqple pe- riad (I 992-September 1994). 1 divide the observations into 100 groups based on the probabi!ity cf observing a value 51naller than the actual residual. If the model's time-varying densicy function fits the data well., these probabilities sY!mld be s:niform?y distributed between 8 and. 1 . Following VZaas 2nd Palm (1993).

= o otherwise. (16)

--'1 4n2 ,;xpccted value of the cumniiitive density func~ion, T. is talten across the sraies that might ha\-e held at each kxe. The goodness-of-fit :esi staistic equals 100,/T C:: ( 1 2 , -

2/16?0j%aild is distributed X& under the null. T&?e ? provides results hoin the goodness-of-fit rests.

3nEy the GAiZE1-I-DF model Is not rejected on an in-sample basis, will? a .57 probability value. Ail six models are rs- jected out of scmpie, however. 'To examine the ssLIrce of f a i l i ~ ~ e in models other than

GARCH-DF in the goodness-of-fit tcst, Figures 1 alzd 2 plot the distribution of the En-sample observations acrosh the 10.0 groups. Figure I shows that the GAXCH-DF obser- tatiocs are roughly itailormly distributed across the grcups. whereas the GARCH-NF observations have a I.,rrmp-shaped dlstd~~i.tiori 3a Figure 2. Too many GAWCH-WF resid~r- kills are sea:- the center of the cuinulative densiiy. ,ih!krich

~. implies that <;he rnodel~s conditiorial deilsitses are overly peaked-iklet is, are too ieptoksrrtic. By mot ailowii~g :he

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30 Journa! of Business 3; Economic Statistics, January 4997

Tabie 2. Chi-squarzd Gooa'ness-of-f/t Tests for GARCH/Markov- 45 , Sv/iiching fI/iooleis: !IT-Sample Period: 7982- 7 99 7. Gut-of-Sampie: I

1'992-September 1994

12/ladel In-sam,ole Out-of-sample

GA3CH-DF 96.65 7 52.1 (4.8E-4) (577)

GARCH-NF 193.7 208.1 i.000) (.000)

GARCW-UV 136.6 i 88.4 (.007) (.00C)

GARCH-K 235.4 294.3 (.000) (.000)

GARCH 140.0 228.4 (.004) (.000)

SWARCH-L 234 .0 307.9 i.000) i.000)

NOTE Probabll$ty values are In parentheses.

c'onditioaa: :turtosis to change. the GARCM-NF model ap- parently 5ts a constant conditional k~lrtosis that is too high. i o 20 30 40 50 60 70 80 90 T O O

If time-varying kurtosis is an in?portant feature of stuck FfYure Db lrioun on of GARC ,bI-, \+- Res!dua,s ,nro Groups

returns, then it worth studying the distribution of the ob- Based on Ccmu/aiii,e Density ,~uiict;on. servations in the GARGH-K model also. Fieure 3 shovfs thet the GARCK-K model also provides condiiional den- ~~~~d ti^^^ ~ ~ ~ h ~ ~ ~ ~ , The xdlx is derived from an sities that are too Ieptolcurtic on average. despite its provi- optioris-i3ricmo is a of * - v-"

sion for time-varying kurtosis. The reasen might be thar the Never'sheless. Inany GARCH-K model has a very persistent state in wi:ich focrrh pzrilcipanrs are ii~terested in options-implied voiatilities ~norzmenis do not exist because q = ,9981;. 11 i s possible t k t .

In :heir owr. righi. Tlhe VIX attempts to represent, as the GAWCE-K model overstates the ppersisteirsce of periods closely as possible. the implied volatility on a hypother- of fat-tailed stock-retwrns distributions: 'They might be ber- .

ical ai-the-naocey opiion on the Standard & Poor (S&P) ter described as episodes than regimes. as the GARCY-DF 10C vt-iii 33 caltndci- d ~ y s (22 trading days) to expira- model suggests,

:ic.n. Details on ihe consir,Jct-Aoc of the ViX from near- [he-mcan.; opiicns ori::es were given by V!Ihaiey (1993).

3. FREDlcTING OP%IONS-[lLApLiEB \/(-JL,&T?1_!T!ES -- ~ h e ircplled volatilliy on an option re8ec:s beliefs about

As an economic test of the GAR~~~/hfark.3ij-qwi~cl.ling zvcrage vda:ilit17 I over rhc iifc of the optima. Thus, the con- models. I use 'them to predict t3.e next daq's opning leliel of s t a t ?'-day horizon of !he VIX implies that we must use the volatility index !VIX) market compiled by the Chicago the GhRCT-i/Markov-sixiitching ~:~laii l i iy models to create

Figure :. Distribufi'cn of GABCH-DF Residuals h f o 100 Groups F ! ~ u r s 3. 3istribidtion of GAt4CY-I-K R e s i d ~ ~ a ; ~ into 700 Groups Based Based on Cumulative Density F~~nction. on C ~ m ~ ! a i i : / e i3ans:iy Funct~cr;.

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D~~eke: Markov Smi~ching in GARCH Processes

multiperiod forecas~s of volatiliey for ail periods between one arid 22 days ahead. In other woa-ds, to predict the VIX miell, the GA.RCI%/fi4anarkov-s\vi'tchIr~g i~~ode i s need to pro- s-ide good ~nulriperiod forecasts for a full range from B to 22 trading days ahead.

Daily data on the k71X were available from 1986--1992. Because the VIX data are bascd on the S&P 100 and ;he stock-marltee data are S&P 500 returns. the mean of :he VKX index is slignrly ?;ig1her that the average volatility fore- cast from the GARCH =odeis. The broader S&P 500 index is somewhat less voiatile than the S&P 100, For this rea- son, I normalize each volatility measure with its 1986-1992 sal-aple mean. Hence a i:aikie 0f 1.5 means that voiaf lity is expec~ed to be one-acd-a-half times ~s normal Ieve? in the coinkg month. @etai!s on rile consdruciion of multi~l~eriocl forecasts from the 6ARCH/$,farko~-switching models are in :he Appecdix.

I use a m i n i ~ i u m - f ~ e c a s t - e ~ o r varia.nce criterion to mea- sure rhe closeness 96' the model-imnplied and options-innplied montlrij volatilities. if I denote the options-implied voiatii- ity es VIX and tile monthly average of the model-preldicted ~~,slatiYit?es as ?f. :hen the crrterlon is

.- Note tPla,i at for a M;ed.nesday, for example, is calculated us- . . mg ~nforrnar;ion a.iiaildb2e through Fdesslay, whereas VIXt is the data from 14I~daesday's opening quotes. In this st:nse. I am asing the GA.RCH/Mar&ov--switching models is predict the options-implied volatilities.

Table 3 sl~oivs that only the GARCH-DF and GARCH-M modeis predict she ophons-implied volatility index better than the conventional GARCH model and eh GARCH-DF model aci~ielies a rofable 14% ~-ed;iction i3 the forecast- e r r x variance.

Figures 4-6 depict ihe ZT-day average volatility fonzcasts for all the models and the \iBX ~roiatility in the aftermath of the October 1987 s'roclc-market crash. As described by Schas/~rt (1993B9 f9r severai days after October 19, 4987, options marlceis becane very thin and the options writ- h e ~ contained extremely large risk p~ernia---hat is, implied

Tabie 3. Preo'icriicring Opi'ions-implied Volaiilitj~ Index With GARCH/ fi/Jarkov-S~/ifching !Wode!s: 1986-1 992

- - -- flAoL;el Forecast-error variance

GARCH-CF

GARCH-[\IF

GAF;Ci-'-UC

CiARCii-K

GARCH

S1flJ.AI;Gti-i

NOTE. Size or fcrecast zrro! var:aiicc relailve to GARCH model in parentheses

0 / I

0c t 12, a7 Nov 9, 87 Dec 8.87 .Jan 7 , 88

Figure 4. i//X Options-Implied and 3 IWodei-Implied 'Jobtil~ties, Au- tumn 1987.

voiaiilities. Figures 4-6 show that the V9X rezched about eight times its norlnal level immediately following the crash but returned to less than two times norma: by the end of October 1987. The GARCH-DE model best predicts the VBX throughout November arrd early December 1987. The switch ro n (St = 0) = 2.64 led to a downweigh'ting, from .027 to .009, of the iagged squared residuals in the per- sistent GARCH process. Furthermore. the conditional vari- ance temporarily shifted discretel~y upwar-d for as long as n (St = 0) was expected to persist.

The volatility implied by the GARCH-K model in Fig- ure 4> in contrast, overpredicts the VZX for about six weeks, beginning at the end of October 1987. The variance in the

Oct 12 87 NOV 9, 87 Dec 8,87 Jan 7,88

Figure 5. VVY Options-implied and 2 Model-implied Voiaiilities, A:!- tumn 1987.

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32 Journal of Business B Economic Statistics. January 1997

I I 0 5 1 I 1

Oct 12, 87 Nov 9, 89 Dec 8,87 Jan 7,88 Sep 28, 89 Oct 26, 89 Nov 24,89 Dec 22,89

Figure 6. VlX Options-Implied and ivlodel-Implied Volatilities With Figure 8, VIX Options-Implied and 2 Mgdel-//??plied Volatilities, Au- SWARCH Autumn 1987. tumn 1989.

@ARCH-K model is a GARCH process. so it displays the same overpersistence that characterizes the conventional GARCB model, shown in Figure 6 . In fact, the forecasts from the conventional GARCH model and the GARCH-M model look very similar. The GARCH-NP model in Figure I; on the other hand, underpredicts volaiilily following the crash. Tlae GARCH-NF model quickly switched to the stale in which g (St = I) = 12.50, so the squared residuals were given little weight in the GARCH process and h d:d not increase much. The variance a: = gtht did increase with g = 12.59, but the increase was never projected to last long with p = .7.5. Conseqaently, the forecasted average volaril- ley for the month never increased to rriore than three fmes

the normal level in the GARCB-NF model. In Figure 5, the GARCH-UV rnodel badly underpredicts the VPX in late October 1987 but does fairly well in November and Decem- ber 1987. The GARCH-UI7 model estimates a constant and relatively Isw weight, n, on the lagged squared residuals in the GARCH process. so the conditional variance never in- crea-ses to more than t h e e times normal, in contrast to the spilte in the VTX. In this sense, the GARCH-UV does not neeessmi!y describe the rate of mean reversion in stock- market volatility we!! because it does not captu-re the initial volatility spilte. in Figure 6: the SVJARCB-E model shows a good deal of persistence but does ro t put enough weight on lagged squared residuals to lift the conditional variance

0.5 / I I

Sep 28,89 Oct 26, 89 Nov 24.89 Dec 22,89 0.5 / I

I I

Sep 28,89 Oct 26, 89 Nov 24 89 Dec 22, 89

Figure 7. VIX 5ptions-ln7plied and 3 Model-Implied Volatilities, Au- Figure 9. ViX Options-Implied and Model-!mplied Voiatilities With Iumn 1989. SVVARCH, Autumn 1989.

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Dueker: Markov Switching in GARCH Processes

to the levels necessary to match the spike in the VIX index either.

Figures 7-3 focus oil a milder volatility spike in Octo- ber 1989: whela the VIX peaked at a.bout 2.5 times its nor- mal level. Again, the GAWCH-DF tends to split through the middle of oscillations of the VlX index better than the other model-implied volatilities, although the improvement is less marked than in Figures 4-6. The same general patterns hold in Figures 7-9 as in Figures 4-6, with the GAWCH-K and GARCH models tending to overpredict volatility and the GARCB-UV and SWARCH-L models showing persistence but failing to yield suficientiy dramtic initial increases in volatility.

P.. CONCLUSIONS

This article introduces a trzctable framework for adding WIarkov-switching parameters to conditional-variance mod- els. Four diEerent specihcations of Markov-switching volatility models are estimated, and the addition of Markov- switching parameters is found to have a variety of effects on the behavior of the conditional volatility, relative to the model without s~vifching. The specification found to predict options-inplied expectations of stock-market volatili~:y best is the one in which the student-i degrees-of-freedom param- eter switches so as 10 Bn&oce substantial discrete shifts in the conditional variance. This model allows for two sources of mean reversion in the wake of large shocks that are not available in a standard model: A switch out of the fat-tailed state is estimated to induce a 68% decrease In voliatility for a given level of dispersion, and the weight given to the most recent shock decreases by two-thirds when the fat-tailed state pertains, thereby reducing the influence and persistence of iasge shocks.

Another novel feature of this model i s that it relates stock returns to the degree of leptoklartosis in the condil:ional- returns distribution. Traditional models, ill contrast, assume constant conditional Sturtosis and relate expected returns to the col~dilional variance. The point estimates support the hypothesis that stock retmrns are generally lower in the more fat-tailed state.

1 also draw economically relevant comparisons between the behaviors of options-in~plied volatilities and the condi- +. ilunal - variances from the volatility models studied. Because options-inplied volatilities serve as useful proxies for mar- ket expectations of 1:oiaeiliey. it is interesting to obsenie that the conditional variance from one of the switching-in-the- variance models reverts to normal about as quickly as the options-implied voiatility fo1lowing large shocks, such as the stoclc-market crash of October 1987. The coniies.tional volatility model, in contrast. has a conditional variance that remains above normal with considerably greater g~ersis- tencc. Thus, Marhov switching in the variance is shown to add a realistic degree of mean reversion to <he condi- tional variance. I n addition, the description of time-varying stock-return skewness and kurtosis provided by these mod- els could prove usefy~l irL analyzing options prices on the S&P 500 index.

An icteresting extension would be to mode! the transition probabilities of the Markov process as time-varying func-

tions of conditioning variables to test whether transitions into and out of fat-tailed states could be better predicted using more information.

ACKNOWLEDGMENTS

The content is my responsibility and does not represent oficial positions of the Federal Reserve Bank of St. Louis or the Federal Reserve System. I thank James Hamilton and Bruce Haazsen for sharing their programs and Barbara Osrdiek and Robert Whaley for the VIX data.

APPENDIX: ON MULTIPEWIO3 VOLATILITY FORECASTS

Forecasts of the volatility m periods ahead are based os the well-known relationship between CARCM models and autoregressnve moving average (ARMA) representations of the squared disturbances. A GARCH(1, 1) process.

implies that the squared residuals obey an ARMA(1, B 3 pro- cess,

where E? - ht 1s a mean zero error that 1s uncorrelated with past ~nforrnatlon In forecastmg the squared resrduals m periods ahead w ~ l h the GARCH-DF model, for example, 4 define Ht = E ; ( I - 2vt) In thrs case EP has an ARMA(I 1) represectatnon,

where

Because the sample size is large, longer-range forecasts car?; be built from the asymptotic forecasting equation for first- order autoregressive processes so that, for rn > 1 .

It remains to integrate out the unobserved states:

where Ht (St = j ) = (~;'))~(1 - 2~1:')). The expected ail- erage variance over the next 22 trading days is then taicen as

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Journal of Business 8 Economic Stat~slics. January 1997

Similar forecasts are drawn for the oth.el- models with H defined such that Hz = c ; l g t in the GARCM-NF model and i'i, = E: - :t i~ the GARCH-UV model.

For the SVdABCH-L model. the mtlltiperiod forecasts are derived by recursive serbseit~~tion as in the work of Hamilton and Susmel (1994).

[Received .Tune 1994. Revised Septeii~bei. !YY5.]

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