The Generalized AutoRegressive Conditional Heteroskedasticity Parkinson Range (GARCH-PARK-R) Model for Forecasting Financial Volatility.

Transcription

1 The Generalized Aoegressive Condiional Heeroskedasiciy Parkinson ange (GACH-PAK- Model for Forecasing Financial Volailiy. Dennis S. Mapa School of Saisics Universiy of he Philippines Diliman, Qezon Ciy Philippines April 5, 004 Insrcor, School of Saisics, Universiy of he Philippines, Diliman, Qezon Ciy, Philippines, Phone: ( ,

2 ABSTACT A new varian of he ACH class of models for forecasing he condiional variance, o be called he Generalized Aoegressive Condiional Heeroskedasiciy Parkinson ange (GACH-PAK- Model, is proposed. The GACH-PAK- model, ilizing he exreme vales, is a good alernaive o he ealized Volailiy ha reqires a large amon of inra-daily daa. The esimaes of he GACH-PAK- model are derived sing he Qasi-Maximm Likelihood Esimaion (QMLE. The Parkinson ange is also sed o evalae he o-ofsample forecasing performance of 68 ACH models sing he iner-daily Philippine Peso-U.S. Dollar exchange rae. Key Words: Volailiy, GACH-PAK-, QMLE

3 I. INTODUCTION Since he inrodcion of he seminal paper on he ACH model by ober Engle in 98, varios works on he financial and ime series economerics have been dominaed by he exensions of he ACH process. This area of research has been growing very fas over he years and while one migh hink ha he fronier of his research program has already been reached and opics already exhased, new ineresing papers relaed o he sbec are sill being pblished in rapid sccession. One pariclar difficly experienced in evalaing he varios ACH-ype of models is he fac ha volailiy is no direcly measrable he condiional variance is nobservable. The absence of sch a benchmark ha we can se o compare forecass of he varios models makes i difficl o idenify he good models from he bad ones. Anderson and Bollerslev (998 inrodced he concep of realized volailiy from which evalaion of he ACH volailiy models are o be made. ealized volailiy models are calclaed from high-freqency inra-daily daa, raher han iner-daily daa. In heir seminal paper, Anderson and Bollerslev colleced informaion on he DM-Dollar and Yen-Dollar spo exchange raes for every fivemine inerval, resling o a oal of 88 5-mine observaions per day! The 88 observaions were hen sed o compe for he variance of he exchange rae of a pariclar day. Alhogh volailiy is an insananeos phenomenon, he concep of realized volailiy is by far he closes we have o a model-free measre of volailiy. While he concep of realized volailiy does provide a highly efficien way of esimaing he nknown condiional variance, he problem of generaing informaion on he price of an asse every five mines or so is simply enormos. An alernaive measre is o se exreme vales, he highes and lowes prices of

4 an asse, o prodce wo inra-daily observaions. The range, he difference beween he highes and lowes log prices, is a good proxy for volailiy. The range has he advanage of being available for researchers since high and low prices are available daily for a variey of financial ime series sch as price of individal sock, composie indices, Treasry bill raes, lending raes, crrency prices and he like. This paper proposes he se of he ange, in pariclar he Parkinson ange (Parkinson, 980, as a benchmark from which o compare forecass of he differen volailiy models. The range as a proxy for he sandard deviaion is raher poplar in saisics, especially in he area of qaliy conrol. The advanage of sing he range is ha we only need o record he exreme vales of he daa se he lowes and highes and hese vales are readily available for mos financial ime series. Moreover, he range can also be sed o model volailiy direcly. In his sdy, he ahor proposes a new varian of volailiy model o be called he Generalized Ao-egressive Condiional Heeroskedasiciy sing he Parkinson ange or GACH-PAK- model for he condiional variance. This model is similar o he Condiional Aoegressive ange Model of Cho (003. This paper is organized as follows: secion serves as inrodcion, secion discsses he esimaion procedres cenering on he Maximm Likelihood and Qasi-Maximm Likelihood Esimaion Procedres, secion 3 inrodces he concep of ealized Volailiy and he GACH-PAK- is discssed in secion 4. Secion 5 provides he empirical discssion and secion 6 concldes... The Aoegressive Condiional Heeroskedasiciy (ACH Process Le { (θ, (,-,0,, } denoe a discree ime sochasic process wih he condiional mean and variance fncions having parameerized by he finie dimensional vecor θ Θ m and le θ o denoe he re vale of he parameer.

5 Le E[( Ι - ] or E - ( denoe he mahemaical expecaion condiioned on he informaion available a ime (-, Ι -. Definiion. In he relaionship, Z, he sochasic process { (θ, (-, } follows an ACH process if: a. E ( (θ o Ι - 0, for,, b. Var ( (θ o Ι - (θ o depends non-rivially on he sigma field generaed by he pas observaions, { - (θ o, - (θ o, }. (θ o is he condiional variance of he process, condiioned on he informaion se Ι -. The condiional variance is cenral o he ACH process. Alhogh he concern of he ACH model is on he process { (θ, (-, }, he same idea can be applied, and in fac being sed, when oher variables are involved sch as in he regression model, y x β +,, K,T in which case, he model above can be re-wrien as, y + wih x β. Leing Z (θ o (θ o / (θ o,,, we have he sandardized process { Z (θ o (-, } and i follows ha,

6 (i E [Z (θ o Ι - ] 0 (ii Var [Z (θ o Ι - ] Ths, he condiional variance of Z (. is ime invarian. Moreover, if we assme ha he condiional disribion of Z (. is ime invarian wih a finie forh momen, i follows from Jensen s ineqaliy ha, E( 4 E(Z 4 E( 4 E(Z 4 [E( ] E(Z 4 [E( ] wih he las eqaliy holding only when he condiional variance is consan. Assming ha Z (. is normally disribed, i follows ha he ncondiional disribion of is lepokric. The ACH (q process can be defined as, ω L + q q ( For his model o be well defined and have a posiive condiional variance almos srely, he parameers ms saisfy ω > 0 and,, q 0... Exensions of he ACH Process This secion discsses some of he common exensions of he ACH process originally proposed by ober F. Engle. The moivaions behind hese new models range from finding a more parsimonios model like he Generalized ACH or GACH process o explaining some of he sylized facs in he financial ime series ha are no capred by he original ACH process, like he Exponenial GACH (EGACH process. In he very firs empirical applicaions of he ACH model sing he level of inflaion in he Unied Kingdom and he Unied Saes, Engle (98,983

7 realized he need o se a large lag q o correcly specify he condiional variance. This resls o a large nmber of parameers o be esimaed in he model, ω q -q, sbec o he ineqaliy resricion ha ω > 0 and Σ < for all, o assre a nonnegaive condiional variance almos srely. In order o circmven his compaional brden wih respec o he parameers, Engle parameerized he condiional variance sing weighed coefficiens. His new formlaion for he condiional variance is as follows, where, w i ω + w (q + i q i ( q (q + i i Under his new formlaion, a large lag becomes possible b he esimaion reqires only wo parameers, ω and. However, he se of declining weighs imposes nnecessary resricions on he ACH process.... The Generalized ACH (GACH Process Following he naral exension of he AMA process as a parsimonios represenaion of a higher order A process, Bollerslev (986 exended he work of Engle o he Generalized ACH or GACH process. In he GACH (p,q process defined as, ω > 0, i p ω + β 0, β q + i i i ( 0 i, K,q, K,p

8 he condiional variance is a linear fncion of q lags of he sqares of he error erms ( or he ACH erms (also referred o as he news from he pas and p lags of he pas vales of he condiional variances ( or he GACH erms, and a consan ω. The ineqaliy resricions are imposed o garanee a posiive condiional variance, almos srely. Ofen, he GACH (, process, ω β -, is sfficien enogh o explain he characerisics of he ime series and is a poplar model in economerics and financial ime series (Hansen and Lnde, The Exponenial GACH (EGACH Process The GACH process being an infinie or a higher order represenaion of he ACH process capres he empirical reglariies observed in he ime series daa sch as hick-ailed disribions and volailiy clsering. However, he GACH process fails o explain he so-called leverage effecs ofen observed in financial ime series. The concep of leverage effecs, firs observed by Black (976, refers o he endency for changes in he sock prices o be negaively correlaed wih changes in he sock volailiy. In oher words, he effec of a shock on he volailiy is asymmeric, or o p i differenly, he impac of a good news (posiive lagged residal is differen from he impac of he bad news (negaive lagged residal. The GACH process, being symmeric, fails o capre his phenomenon since in he model, he condiional variance is a fncion only of he magnides of he lagged residals and no heir signs. A model ha accons for an asymmeric response o a shock was credied o Nelson (99 and is called an Exponenial GACH or EGACH model. The specificaion for he condiional variance sing he EGACH (p,q is,

9 log( p ω + β log( q + i i i i r + γ k k k k The log of he condiional variance implies ha he leverage effec is exponenial raher han qadraic. A commonly sed model is he EGACH (, given by, log( β log( + γ (3 The presence of he leverage effecs is acconed for by γ, which makes he model asymmeric. The moivaion behind having an asymmeric model for volailiy is ha i allows he volailiy o respond more qickly o falls in he prices (bad news raher han o he corresponding increases (good news...3. The Threshold GACH (TACH Process Anoher model han accons for he asymmeric effec of he news is he Threshold GACH or TACH model de independenly o Zakoïan (994 and Glosen, Jaganahan and nkle (993. The TACH (p,q specificaion is given by,

10 I ω + β where, k p if < 0 0 oherwise q + i i i r + γ k k k I k (4 In he TACH model, good news, -i > 0 and bad news, -i < 0 have differen effecs on he condiional variance. When γ k 0, we conclde ha he news impac is asymmeric and ha here is a presence of leverage effecs. When γ k 0 for all k, he TACH model is eqivalen o he GACH model. The difference beween he TACH and he EGACH models is ha in he former he leverage effec is qadraic while in he laer, he leverage effec is exponenial...4. The Power ACH (PACH Process Mos of he ACH-ype of models discssed so far deal wih he condiional variance in he specificaion. However, when one alks of volailiy he appropriae measre is he sandard deviaion raher han he variance as noed by Barndorff-Nielsen and Shephard (00. A GACH model sing he sandard deviaion was inrodced independenly by Taylor (986 and Schwer (989. In hese models, he condiional sandard deviaion as a measre of volailiy is being modeled insead of he condiional variance. This class of models is generalized by Ding e al. (993 sing he Power ACH or PACH model. The PACH specificaion is given by,

11 δ where, δ > 0, γ i p ω + β δ q + i ( i γ for i,, K,r and γ i i i i δ 0 for (5 i > r,and r p. Noe ha in he PACH model, γ 0 implies asymmeric effecs. The PACH model redces o he GACH model when δ and γ i 0 for all i. II. ESTIMATION POCEDUE Since he variance is nobservable, he parameers of he ACH models are esimaed from he disrbance erm of he mean specificaion via he maximm likelihood esimaion procedre. Le he mean of y be specified sing he regression model, y x β +,, K,T where x denoes he (px vecor of predeermined independen variables, inclding possibly lagged vales of y and AMA erms, β is a (px vecor of nknown parameers. The disrbance erm saisfies he following condiions: (a (b Z ω + where E(Z 0 and + + K + m m E(Z Le I, he informaion se, denoes he vecor of observaions obained hrogh ime. Normally, we condiion he process on he firs n observaions and se he observaions,,,t for esimaion. In his case, I {y, y -,,y,y 0,,y -n+, x, x -, x -, x,x o,,x -n+ }.

12 .. Gassian Disribion for Z Le Z be idenically and independenly disribed as Gassian wih mean 0 and variance. Moreover, if Z is independen of x and I -, hen he condiional disribion of y is Gassian wih mean x β and variance. Following he argmens of Hamilon (994, We can combine all he parameers and express as, The vecors of nknown parameers o be esimaed consiss of β and δ and can be represened as θ (β, δ. The sample log likelihood condiioned on he firs n observaions is given by, ( m m m x (y x (y where, (6,,...,T x y exp,i x f(y β + + β ω + β π K [ ] [ ] ( ( [ ] m m m x y,, x y, ( w and,,,, ( where, ( w β β β ω δ δ β K K ( ( β π θ θ λ T T T x y log( log( T ;,I x logf(y

13 Taking he derivaive wih respec o θ, Noe however ha, Moreover, Plgging in (8 and (9 ino (7 will give s, ( ( ( ( ( (7 x y x y log ;,I x y logf θ β θ β θ θ θ ( (8 0 x x y δ β θ β ( (9 w x ( m m m m β θ + θ + θ ω θ ω + θ ( ( ( (0 0 x w x ;,I x y f log m + β θ θ

14 The likelihood fncion in (0 can be maximized sing he Marqard or he Bernd, Hall, Hall and Hasman (BHHH algorihm... Non-Gassian Disribion The ACH processes are fond o be considerably sefl in explaining reglariies in financial ime series. However, financial ime series exhibi excess krosis in empirical sdies. Therefore, he assmpion of a Gassian disribion in generaing he MLE is deemed o be inappropriae for financial ime series. Aside from he Gassian disribion, wo oher disribions have been proposed as alernaives in esimaing he MLE of he parameers of he ACH process. These are he -disribion wih v degrees of freedom as proposed by Bollerslev (987 and he Generalized Error Disribion (GED as proposed by Nelson (99. The wo disribions have faer ails han he Gassian disribion and can be sed o explain excess krosis.... The Sden s -disribion If follows a Sden s -disribion wih v degrees of freedom and scale parameer M, hen he densiy fncion is given by, f( v + Γ ( πv ( v / Γ M / + M v ( v+ ( where Γ(. is he gamma fncion and he degree of freedom v > conrols he ail s behavior. The -disribion is symmeric a zero and when v > 4, he

15 condiional krosis is eqal o 3(v-/(v-4 > 3. The -disribion approaches he Gassian disribion as v. If v >, hen Z has a mean zero and variance E( M v/(v- (Hamilon, p.66. We can derive M in erms of v and, where M [(v- ]/v. If we plg in M in he densiy fncion in (, we have, f( v + Γ / π Γ ( v / (v / ( / + ( (v v+ The sample log likelihood condiioned on he firs n observaions is given by, λ ( θ T logf(y x,i T log( Γ[(v + ; θ Tlog / π Γ / ] ( v / v + T (y log + (v x β (v / (3 where, ω + ( y x β + K + m (y m x mβ The log likelihood in (3 can be maximized nmerically wih respec o v and θ sbec o he consrain ha v >. The approach is similar o he Gassian disribion given in (7 o (0.

16 ... The Generalized Error Disribion (GED Nelson (99 sggesed he se of he Generalized Error Disribion (GED wih densiy fncion given by, v v exp 0.5 λ (, v > 0 (+ / v Γ(v λ f (4 where v is he ail hickness parameer and λ Γ(v Γ(3v / v When v, follows he sandard Gassian disribion. For v <, he disribion of has hicker ails han he Gassian disribion. In pariclar, when v, follows he doble exponenial disribion. When v >, he disribion of has hinner ails han he Gassian disribion. When v, he disribion of is niform on he inerval [-3 /, 3 / ] (Angelidis, Benos and Degiannakis, 003. The log likelihood fncion of he GED condiioned on he firs n observaions is given by, λ ( θ T logf(y x,i T log(v / λ ; θ λ v ( + v log( logγ(/ v log (5 The log likelihood in (5 can be maximized nmerically wih respec o θ sbec sing he approach similar o he Gassian disribion given in (7 o (0.

17 .3. Qasi-Maximm Likelihood Esimaion (QMLE When he disribion of he re daa generaing process (DGP is known o be disribed as Gassian, Sden s or Generalized Error, he maximm likelihood esimaion procedres discssed in he previos secion exhibis desirable properies sch as consisency, asympoic normaliy and asympoic efficiency (Weiss, 986. However, we know ha for ACH processes he assmpion of Gassian disribion of he DGP, for insance, is seldom saisfied (if a all becase of he hick-ails characerisics of he ime series. Working on he alernaive forms of he probabiliy densiy fncions sch as he Sden s or he GED presens some compaional brden in he esimaion. In his conex, i wold be advanageos o consider only minimal assmpions on he ACH process by avoiding perhaps he specificaion of he probabiliy densiy fncion and assme only condiions on he lower-order momens sch as he mean and co-variances srcres. This ype of esimaion procedre is known as he Qasi-Maximm Likelihood Esimaion (QMLE (someimes referred o in he lierare as Psedo-MLE and was firs inrodced by Whie (98. The idea behind he QMLE is ha even if he re probabiliy densiy fncion family is misspecified, i is possible for an exremm esimaor based on he likelihood fncion associaed wih he misspecified probabiliy densiy fncion o possess good asympoic properies sch as consisency and asympoic normaliy. Moreover, he resling covariance marix of he vecor of esimaors is also a consisen esimaor of he nknown variance-covariance marix (Mielhammer, Jdge and Miller (000. When he re probabiliy densiy fncion is correcly specified, he QMLE is eqivalen o he MLE and he esimaors are also asympoically efficien. The assmpion in he QMLE is o correcly specify he mean and variance of he random variable Z in definiion, ha is, E (Z x,i 0 and E(Z x,i (6

18 and se he Gassian log likelihood fncion in (7 as a vehicle o esimae he parameers. The approach in esimaion of he parameers is similar o hose in (7 o (0. The esimaion procedre is known as Qasi-Maximm Likelihood since he likelihood fncion need no be correcly specified. Bollerslev and Wooldridge (99 firs derived he large sample properies of he QMLE for a wide range of he ACH models. Lee and Hansen (994 and Lmsdaine (996 derived he consisency propery of he esimaors of he GACH (, process. Berkes, Horvah and Kokoszka (003 exended he work of Lee and Hansen and Lmsdaine o he case of he GACH (p,q process. Berkes e al made se of he fac ha he GACH (p,q process can be represened as an ACH ( process for any p > 0 and q > 0. They find i sfficien o show consisency and asympoic Gassian disribion for he ACH ( and hs for any GACH (p,q process. The specificaion for he mean, y, is imporan before one can esimae he parameers of he ACH process. Mos of he ime, only he consan is sed o describe he mean eqaion, y c +. If his is an incorrec specificaion for he mean, he esimaes can be inconsisen (Bsch, 003. Some ahors find he consan in he mean eqaion as sfficien becase financial ime series, while showing srong dependency on he condiional variance of he pas observaions, have raher weak dependency on he firs momen. Lee and Hansen and Lmsdaine specified only he consan in he mean eqaion for he GACH (, process, while Berkes e al focsed only on he ACH componen, wih he mean specificaion y.

19 III. EALIZED VOLATILITY Difficly in evalaing and comparing volailiy models is de o he fac ha volailiy is no direcly observable. Since here is no benchmark from which we can compare he forecass of he differen volailiy models, idenifying bad models from good ones is qie difficl. An earlier approach o solve his pariclar problem is o compare he volailiy forecass, ˆ wih û for T +,T +, K,T + n However, his resled in poor o-of-sample forecasing performance for mos, if no all, of he volailiy models leading o he criicism as o he relevance of he exercise of forecasing he condiional variance. In defense o varios criicisms, Anderson and Bollerslev (998 inrodced he concep of realized volailiy from which evalaion of he ACH volailiy models are o be made. ealized volailiy models are calclaed from high-freqency inra-daily daa, raher han iner-daily daa. The wo ahors arged ha he sqared residals conain a lo of noise and do no represen he nderlying volailiy srcre of he daa. In heir seminal paper, Anderson and Bollerslev colleced informaion on he DM- Dollar and Yen-Dollar spo exchange raes for every five-mine inerval, resling o a oal of 88 5-mine observaions per day! The 88 observaions were hen sed o compe for he variance of he exchange rae of a pariclar day. Alhogh volailiy is an insananeos phenomenon, he concep of realized volailiy is by far he closes we have o a model-free measre of volailiy. Le P n, denoe he price of an asse (say US$ in Philippine Peso a ime n 0 a day, where n,,,n and,,,t. Noe ha when n, P is simply he iner-daily price of he asse (normally recorded as he closing price. Le p n, log(p n,, denoe he naral logarihm of he price of he asse. The observed discree ime series of coninosly componded rerns wih N observaions per day is given by,

20 r n, p n, p n, (7 When n, we simply ignore he sbscrip n and r p p - log(p log(p - where,,t. In his case, r is he ime series of daily rern and is also he covariance-saionary series. In (7, we assme ha: (a E(r n, 0 (b E(r n, r m,s 0 for n m and s (c E(r n, r m,s < for n,m,s, Assmpion (a implies ha he mean rern which follows from he fac ha he log prices, p, follow an i.i.d. random walk process wiho a drif, p n, pn, + εn, where εn, I ~ i.i.d.(0, (8 Following (8, r n, p n, p n-, ε n, and hs, E(r n, E(ε n, 0. Assmpion (b follows from he fac ha ε n, are i.i.d. and from (a which gives s E(r n, r m,s E(ε n, ε m,s 0. Assmpion (c saes ha he variances and co-variances of he sqared rerns exis and are finie. This follows from he fac ha E(r n, r m,s E(ε n, ε m,s < for n,m,s,. From (7, he coninosly componded daily rern (Campbell, Lo, and Mackinlay, 997 p. is given by,

21 r N r n, n (9 and he coninosly componded daily sqared rerns is, r N r n n, N rn, n N rn, n N N + N + r n, n m N n m n+ r r m, r n, mn, (0 Noe ha Var(r E(r since E(r 0. From (0 and sing assmpion (b of (7 above, we have, E(r where s ( E s N rn, n ( Ths, he sm of he inra-daily sqared rerns is an nbiased esimaor of he daily poplaion variance. The sm of he inra-daily sqared rerns is known as he realized volailiy (also called he realized variance by Barndorff-Nielsen and Shephard (00. Given enogh observaions for a given rading day, he realized volailiy can be comped and is a model-free esimae of he condiional variance. The properies of he realized volailiy are discssed in Anderson, Bollerslev, Diebold and Labys (999. In pariclar, he ahors fond ha he realized volailiy is a consisen esimaor of he daily poplaion variance,.

22 While he concep of realized volailiy does provide a highly efficien way of esimaing he nknown condiional variance, he problem of generaing informaion on he price of an asse every five mines or so is simply enormos. An alernaive measre is o se exreme vales, he highes and lowes prices of an asse, o prodce wo inra-daily observaions. The range, he difference beween he highes and lowes log prices, is a good proxy for volailiy. The range has he advanage of being available for researchers since high and low prices are available daily for a variey of financial ime series sch as price of individal sock, composie indices, Treasry bill raes, lending raes, crrency prices and he like. The log range,, is defined as, log(p (N, log(p (, p (N, p (,,K,T ( where P (N, is he highes price of he asse a day (or ime and P (, is he lowes price of he asse a day. The log range is sperior over he sal measre of volailiy based on daily daa, he sqare rern r log(p log(p -. Alizadeh, Brand and Diebold (00 noed ha he log range is a beer measre of volailiy in he sense ha he log range has fewer measremen errors compared o he sqared-rerns. For insance, on a given day, he price of an asse flcaes sbsanially hrogho he day b is closing price happens o be very close o he previos closing price. If we se he iner-daily sqared rern, he vale will be small despie he large inra-daily price flcaions. The log range, sing he highes and lowes vales, reflecs a more precise price flcaions and can indicae ha he volailiy for he day is high. Moreover, he Gassian disribion can approximae

23 he disribion of he log range qie well. The disribion of he range was firs derived by Feller (95 sing a drif-less Brownian moion process. As compared o he realized volailiy, he log range has he advanage of being robs o cerain marke microsrcre effecs. These microsrcre effecs, sch as he bid-ask spread, are noises ha can affec he feares of he imeseries. The log range, on he oher hand, is no seriosly affeced by he bid-ask spread. Parkinson (980 was he firs o make se of he range in measring volailiy in he financial marke. Parkinson developed he PAK daily volailiy esimaor based on he assmpion ha he inra-daily prices follow as Brownian moion. This sdy will make se of he PAK ange in modeling ime-varying volailiy. The model will be called he Generalized Ao-egressive Condiional Heeroskedasiciy Parkinson ange (GACH-PAK- model. IV. THE GACH-PAK- 4.. The GACH-PAK- Model Consider he covariance-saionary ime series { p } where, P log(p (N, log(p 4log( (,,, K,T (3 P is he PAK-ange of he asse a ime. Moreover, le P 0 for all and ha P( P < δ P-, P-, > 0 for any δ > 0 and for all. This condiion saes ha he probabiliy of observing zeros or near zeros in P is greaer han zero. Le,

24 E[ P I - ] be he condiional mean of he PAK range and Var[ P I - ] be he condiional variance of he PAK range The moivaion behind he GACH-PAK- model is he Ao-egressive Condiional Draion (ACD model of Engle and ssel (998 sed o model observaions ha arrive a irreglar inervals. Le, P ε where ε I ~ iid(, φ and q ω + P p + β (4 The model in (4 is known as he GACH-PAK- process of orders p and q. The mean and variance of he PAK range are given by, (a (b E( P Var( P E( E ( φ P ( ε [E( + P ] φ (5 The GACH-PAK- model is similar o he Condiional Ao-egressive ange (CA model sggesed by Cho (003. Two differences beween his sdy and ha of Cho s are o be clarified. Firs, his sdy ses he Parkinson range o sdy volailiy insead of he sal log range (Cho s measre. The Parkinson range has been fond o be a beer esimaor of volailiy (sandard deviaion. The second difference is he se of he daa, his sdy makes se of he daily daa while Cho s paper sed weekly daa. Weekly daa may have disored

25 esimaes de o he presence of aggregaion effec ha is why he ahor of his paper sed he daily daa insead. 4.. Esimaion Procedres 4... Maximm Likelihood The parameers of he model given in (4 can be esimaed sing he maximm likelihood procedre. For he densiy fncion of ε, his sdy follows he sggesion of Engle and ssel (998 and Engle and Gallo (003 of sing he gamma densiy, Since E(ε β (by assmpion in (4, i implies ha /β. Ths (6 now becomes, ( (6 exp I ( f β ε ε Γ β ε ( { } ( ( ( ( ( ( (7 exp exp exp I f I f( exp I f( P P P P P P P P Γ Γ Γ ε ε Γ ε

26 From (7, he condiional mean and variances of P are, The densiy fncion in (7 approaches he Gassian densiy as increases. Moreover, he likelihood fncion is given by, If he parameer of ineres are only hose ha define in (4, denoed by (θ, hen he log likelihood can be simplified ino, ( ( P P I Var( I E ( ( ( (8 exp L P P T Γ, ( where (9 ( ( log( log( L log P T P T T P γ γ θ + θ γ γ

27 Taking he derivaive of he log likelihood fncion wih respec o θ, we have, The parameer vecor θ in (30 can be esimaed nmerically sing some ieraive algorihms sch as he Marqard or he BHHH Qasi Maximm Likelihood An easier way of esimaing he parameer vecor θ is o apply he mehod of esimaing he parameers of a GACH (p,q process. ecall ha in he GACH (p,q process discssed secion.. (30 0 ( ( ( 0 ( ( ( 0 ( ( ( ( log(l T P T P T P θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ + β ω + q i i i p and N(0, ~ Z Z

28 For he GACH-PAK- process le, E ( P I E( 0 ν Var( I E( ν and wih P q ν ω + P P p ν + β ~ iid(0, given in (4 Ths, an analogos mehod of esimaing he parameer vecor θ is o esimae he variance eqaion for he posiive sqare roo of he PAK sing GACH (p,q specificaion wih zero in he mean specificaion. The Qasi-Maximm Likelihood esimaors are consisen and disribed as Gassian asympoically even if he probabiliy densiy fncion of he error is mis-specified following he resls of Lee and Hansen and Lmsdaine for he GACH (, and Berkes e al for he GACH (p,q process. Obviosly, if he correc specificaion is saisfied, for insance sing he gamma disribion, he QMLE is he MLE and he esimaors are asympoically efficien. Therefore, a rade-off has o be made. This sdy will make se of he QMLE and aspire for consisency and asympoic normaliy of he esimaors. The daa and he resls of he empirical analysis are discssed in he nex secion.

29 V. DATA ANALYSIS 5.. Model Specificaions This secion discsses he resls of forecasing he condiional variance sing he differen ACH and GACH-PAK- models. In his sdy, a oal of 77 models were esimaed: 68 ACH-ype models and 9 GACH-PAK- models. The model specificaions are provided in Tables A and B below. Each specificaion is esimaed sing he Maximm Likelihood, wih Gassian, Sden s and Generalized Error disribions and he Qasi-Maximm Likelihood. The sofware sed in he esimaion is Eviews 5.0 (Bea Version of Qaniaive Micro Sofware. Table A. Specificaion for ACH-ype Models * Model Specificaion Model Specificaion ACH ( 0 TACH (, GACH (, TACH (, 3 GACH (, TACH (, 4 GACH (, 3 TACH (, 5 GACH (, 4 PACH (, 6 EGACH (, 5 PACH (, 7 EGACH (, 6 PACH (, 8 EGACH (, 7 PACH (, 9 EGACH (, * The 7 models are esimaed via he MLE sing he Gassian, Sden s and he Generalized Error Disribion and sing he QMLE resling o 68 models.

30 Table B. Specificaion for GACH PAK Models* Model Specificaion Model Specificaion ACH ( 6 EGACH (, GACH (, 7 EGACH (, 3 GACH (, 8 EGACH (, 4 GACH (, 9 EGACH (, 5 GACH (, * The models are esimaed sing QMLE. These models were esimaed o fi he daily rerns of he peso-dollar exchange rae from Janary 0, 997 o December 05, 003, a oal of 730 observaions. The ime plos of he peso-dollar exchange rae and he daily rerns, defined as he difference in he log prices, are provided in Figres 3 and 4, respecively Figre 3. Daily Peso-Dollar Exchange ae (Janary 0, 997 o December 5,

31 Following he approach of Hansen and Lnde (00, he ime series was divided ino wo ses, an esimaion period and an evalaion period. 443 T +, K,0 esimaion period,, 443 K,n evalaion period The parameers of he volailiy models are esimaed sing he firs T iner-daily observaions and he esimaes of he parameers are sed o make forecass of for he remaining n periods. The esimaion period made se of daily rerns from Janary 0, 997 o December 7, 00, a oal of 493 observaions. In he evalaion period he daily volailiy is esimaed sing he sqare of he Parkinson, defined in (3. The sqare of he PAK serves as he proxy for he nknown condiional variance. The evalaion period makes se of daily rerns from Janary 0, 003 o December 05, 003, a oal of 37 observaions.. Figre 4. Time Plo of he Daily Peso-Dollar erns

32 5.. Measres o Evalae he Forecasing Performance The main obecive of bilding volailiy models is o forecas fre volailiy. Given a nmber of compeing models, here is a need o evalae he forecasing performance of he models o segregae good models from bad ones. This secion discsses some of he commonly sed measres o evalae he forecasing performance of he volailiy models. Le h denoe he nmber of compeing forecasing models. The h model provides a seqence of forecass for he condiional variance, ˆ,, ˆ,, K, ˆ,, K,h,n ha will be compared o he sqare of he Parkinson range, he proxy of he inradaily calclaed volailiy, P, K, P n The forecas of h model leads o he observed loss, L ( ˆ,,, P,,...,77 and,,...,37 In his sdy, five (5 differen loss fncions are sed o evalae he forecasing performance of he differen models. The loss fncions are:

33 MAD n P n ˆ (3 MAD n P n ˆ (3 MSE n ( ˆ P n (33 MSE n ( ˆ P n (34 LOG n [ log( / ˆ ] n P (35 The crieria (3 o (34 are he sal mean absole deviaions and mean sqare errors sing he forecass of he condiional sandard deviaion and he condiional variance. Crierion (35 is eqivalen o he crierion sing he regression eqaion, log( P a + blog( ˆ + ε,, K,37 discssed in Engle and Paon (00 and Taylor (999.

34 5.3. Empirical esls The op 0 models are shown in Table below. The bes over-all ACH model is he TACH (, model wih he Sden s as he nderlying disribion. The second bes model is he PACH (, model, also sing he Sden s disribion. The parameer esimaes of he bes model are given in Table 3. Table. Forecasing Performance of he Top 0 ACH Models MAD MAD MSE MSE LOG Model Mean Model Mean Model Mean Model Mean Model Mean E E E E- 3.05E E E E-06 GED E E+00 GED.030E-03 GED 6.700E-06 GED.00E E-0 GED E+00 GED E-03 GED E-06 GED 4.800E-06 GED.00E-0 GED E+00 GED E-03 GED E-06 GED 5.300E-06 GED 4.500E-0 GED E+00 GED E-03 GED E-06 GED 7.300E-06 GED 5.00E-0 GED E+00 GED.0570E-03 GED E-06 GED 5.600E-06 GED 5.600E-0 GED.356E+00 GED E-03 GED E-06 GED 0.300E-06 GED 6.800E-0 GED.3590E+00 GED E-03 GED E-06 GED.300E-06 GED E-0 GED E+00 GED E-03 GED E-06 GED 4.300E-06 GED 0.300E-0 GED E+00 From Table, i is ineresing o noe ha models sing he Generalized Error Disribion performed relaively well sing he five forecasing crieria, wih 8 o of 7 models landing in he op 0 models. In general, he models wih relaively sperior forecasing performance, sing he peso-dollar exchange rae, are hose ha accommodae he leverage effecs sch as he TACH, PACH and EGACH. However, while he correc specificaion of he volailiy is imporan, one ms also consider he disribion sed in esimaing he parameers of he model.

35 Table 3. Esimaed Coefficien of he TACH (, Model Coefficien Esimaed Vale Sandard Error ω 6.93 E-0 5. E β β γ The resls of he empirical analysis showed ha volailiy models ha assmed he Gassian disribion or hose ha sed he QMLE performed wors compared o models ha assmed he Sden s or Generalized Error disribions. Therefore, i is imporan o correcly specify he enire disribion and no only o focs on he specificaion of he volailiy, even if i is he obec of ineres. A similar observaion was made in he sdy of Hansen and Lnde (00. Using he five crieria discssed above he forecasing performance of he GACH PAK models are given in Table 4. The op hree models are GACH (,, (, and (,. I shold be noed ha while he GACH (, and he GACH (, have operformed, albei slighly, he GACH (,, he laer is preferred since he coefficiens and β are significanly differen from zero.

36 Table 4. Forecasing Performance of he GACH-PAK- Models PAK MAD MAD MSE MSE LOG Model GACH (, 9.600E E E E-.0403E+00 GACH (, E E E E-.050E+00 GACH (, E E E E-.0540E+00 GACH (, E E E E-.0544E+00 EGACH (,.0600E E E E-.04E+00 EGACH (,.060E E E E-.E+00 EGACH (,.060E E E E-.3E+00 ACH (.760E E E E E+00 EGACH (,.450E E E E E+00 As expeced, he GACH-PAK- models performed beer han mos of he ACH-ype models. This is expeced since he proxy for he condiional variance in he evalaion period is he sqare of he Parkinson range. However, i is ineresing o noe ha he forecasing performance of he bes ACH-ype model, he TACH (, model wih a sden s disribion, is relaively near he bes GACH-PAK- model. The resls somewha provide an assrance ha volailiy models sing iner-daily daa can forecas he condiional variance raher well (a leas sing he Parkinson range. The plo of he esimaed condiional variance sing he GACH-PAK- (, and TACH (, models are provided in Figre 5 below Figre 5. Esimaed Condiional Variance sing he GACH-PAK-(, and he TACH (, GACH-PAK- TACH (,

37 VI. CONCLUSIONS This sdy inrodced a relaively simple, ye efficien, model o describe he variaion in volailiy of he peso-dollar exchange rae sing inra-daily rerns. The Generalized Ao-egressive Condiional Heeroskedasiciy Parkinson ange (GACH-PAK- model can acally prodce volailiy esimaes ha are relaively sperior han he ACH class of models sing iner-daily rerns. The GACH-PAK- model is a good alernaive o he so-called ealized Volailiy ha makes se of large qaniy of inra-daily daa, somehing ha is difficl o obain in emerging markes sch as he Philippines. This sdy also compared a large nmber of ACH-class of volailiy models sing iner-daily rerns of he peso-dollar exchange rae. The esimaed models are compared in erms of heir o-of-sample forecasing performance o characerize he variaion in he volailiy. The Parkinson ange is sed as he esimae of he daily volailiy where comparison of he differen volailiy models was made. The empirical analysis showed ha i is imporan o correcly specify he enire disribion of he volailiy model and no only focs on he specificaion of he volailiy. New direcions in he consrcion of volailiy models, relaed o he GACH- PAK- model, are promising. In he heoreical fron, one can exend he Maximm Likelihood Esimaion procedre o inclde more disribions, noably he Gamma densiy. I is noed ha idenificaion of he correc densiy is as imporan as specifying he volailiy model correcly. Inclding addiional densiy fncions will enail he developmen of sofware for comping. This is anoher area ha can be prsed, especially by researchers who are inclined in programming.

38 In he empirical fron, one can compare he forecasing performance of he GACH-PAK- model wih he ealized Volailiy sing inra-daily rerns. One can perhaps consrc, hrogh eiher he PSE or he PDS, one year of inra-daily rerns in he crrencies or eqiies marke. ealized Volailiy esimaes can hen be comped from he inra-daily rerns (he arge series where he o-ofsample forecass of he GACH-PAK- model can be compared. Finally, if inra-daily rerns are no possible hen one can perhaps simlae inradaily observaions sing boosrapping or Mone Carlo procedres. Comparison of he forecasing performance of he GACH-PAK- models and he ealized Volailiy sing simlaed inra-daily daa will srely be ineresing. I is graifying o noe ha, years afer he original ACH paper of Engle and a year afer Engle won he Nobel Prize, doing research in he area of volailiy esimaion is sill boh dynamic and challenging.

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