Evaluating Risk Models with Likelihood Ratio Tests: Use with

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1 Evaluaing Risk Models wih Likelihood Raio Tess: Use wih Care! Gabriela de Raaij and Burkhard Raunig *,** March, 2002 Please do no quoe wihou permission of he auhors Gabriela de Raaij Cenral Bank of Ausria Financial Markes Analysis Division Oo-Wagner-Plaz 3 POB 61, A-1011 Vienna Ausria Phone: (+43-1) Fax: (+43-1) gabriela.raaij@oenb.co.a Burkhard Raunig Cenral Bank of Ausria Economic Sudies Division Oo-Wagner-Plaz 3 POB 61, A-1011 Vienna Ausria Phone: (+43-1) Fax: (+43-1) burkhard.raunig@oenb.co.a *) Corresponding auhor **) The opinions expressed do no necessarily reflec hose of he Ausrian Cenral Bank.

2 Evaluaing Risk Models wih Likelihood Raio Tess: Use wih Care! Absrac Mos modern approaches o measure and conrol he risks of financial porfolios are eiher direcly or indirecly based on densiy forecass. Tools o evaluae he qualiy of such forecass are herefore essenial. In his paper we examine a recenly proposed mehodology o evaluae densiy forecass from risk models ha builds on likelihood raio ess. We discuss hree cases ha are highly relevan in risk managemen where likelihood raio ess fail o deec incorrec densiy forecass. We illusrae his fac wih Mone Carlo simulaions and empirical examples. We also demonsrae ha he likelihood raio esing framework in conjuncion wih addiional diagnosic ess is an aracive ool o evaluae risk models.

3 1 1 Inroducion Tradiionally, he forecas evaluaion lieraure has primarily deal wih mehods o evaluae poin forecass. However, over he las few years ineres by he financial indusry has increased ino densiy forecass. Financial insiuions became ineresed o supplemen sandard risk measures as for example porfolio variance and correlaion wih broader informaion on porfolio risk. Especially in he area of risk managemen densiy forecass are frequenly generaed since hey provide a full picure of he uncerainy associaed wih a porfolio. Therefore, densiy forecass and measures derived from such forecass play a key role in modern risk managemen. In paricular, Value a Risk (VaR), which is defined as a cerain quanile of a forecas of he enire reurn disribuion of a financial porfolio (1% and 5% quaniles are ypically used) has become he backbone of modern risk managemen (Jorion, 1996, Duffie and Pan, 1997). Moreover, regulaory auhoriies have permied banks o use VaR esimaes o deermine heir capial requiremens o cover heir exposure o marke risk. Therefore, perhaps no surprisingly, echniques o evaluae he qualiy of such forecass are of paramoun imporance for inernal as well as regulaory purposes. Various mehods o evaluae densiy forecass have been proposed in he lieraure. Mehods ha evaluae Value a Risk esimaes direcly have been proposed and examined in Kupiec (1995), Lopez (1998), Chrisoffersen (1998) and Chrisoffersen, Hahn and Inoue (2001). More general evaluaion mehodologies ha ake a broader view and consider he whole disribuion insead of jus a single quanile have recenly been proposed in Crnkovic and Drachman (1997) and Diebold, Gunher and Tay (1998). In his paper we focus on he second kind of mehodologies ha evaluae densiy forecass via he enire forecased disribuion. We examine an ineresing exension of Diebold e all. developed in Berkowiz (2001) ha suggess saisical ess of he qualiy of densiy forecass wihin a likelihood raio (LR) framework.

4 2 Alhough he LR-framework is aracive, here are imporan cases where he uncriical use of his framework or equivalen es procedures may lead o erroneous conclusions abou he qualiy of densiy forecass. We ouline hree cases where deficien densiy forecass canno be deeced wihin he LR-framework and relae hem o he evaluaion of VaR models. In hese cases variance/covariance models and hisorical simulaion models o esimae VaR may no be rejeced even if hey deliver poor densiy forecass. Using Mone Carlo simulaions and an empirical illusraion we highligh ha in he hree cases he basic LR-framework alone as well as an exended LR es ha covers higher order dependencies and cerain kinds of nonlineariies has lile power o deec incorrec densiy forecass. However, we also demonsrae ha he LR framework in conjuncion wih addiional diagnosic ess is a consrucive and powerful framework o idenify deficien forecasing models. The res of he paper is organized as follows. Secion 2 oulines he LR densiy forecas evaluaion framework of Berkowiz (2001). The hree cases ha we consider are discussed in secion 3. The Mone Carlo experimens and he empirical examples are repored in secion 4. Some final remarks are provided in secion 5. 2 Densiy Forecas Evaluaion and he LR Framework Le {x } = 1,..., m be a ime series generaed from he condiional densiies {f(x I -1 )} = 1,..., m where I -1 denoes he informaion se available a ime -1 and le {p(x I -1 )} = 1,..., m be a series of one-sep-ahead densiy forecass for {x } = 1,..., m. 1 The qualiy of such forecass can be evaluaed wih he help of a probabiliy inegral ransformaion (PIT) suggesed in Rosenbla (1952) applied o each observed x wih respec o is prediced densiy p (x ). The probabiliy inegral ransformaion for a single x is given by 1 In wha follows, f (x ) and p (x ) are someimes used as shorhand noaions for he rue and he prediced condiional densiies, respecively.

5 3 z x = p (u)du = P (x ). (1) Diebold, Gunher and Tay (1998) show ha he ransformed series {z } = 1,...,m mus be independenly and idenically uniformly disribued (iid U(0,1)) if a series of one-sep-ahead densiy forecass {p (x )} = 1,..., m coincides wih he series of he rue condiional densiies {f (x )} = 1,..., m. 2 Hence, he qualiy of densiy forecass can be assessed by an examinaion of he properies of he z-series resuling from he PIT given by equaion (1). Such examinaions can eiher be based on descripive diagnosic ools or on saisical ess as proposed in Crnkovic and Drachman (1997). Diebold e al. advocae graphical mehods. However, here may be siuaions in which saisical esing is required. For example, wihin a financial insiuion one may have o compare he qualiy of Value a Risk forecass across differen rading books wih he help of formal es procedures. Anoher example may be a regulaory auhoriy ha wans o assess he accuracy of risk measuremen sysems of differen financial insiuions. To assure a uniform reamen across he involved insiuions he auhoriy may herefore ask hem o carry ou saisical ess for a porfolio of financial insrumens as defined by he supervision auhoriy. Berkowiz (2000) emphasizes ha saisical ess ha are direcly based on a z-series require raher large sample sizes o be reliable and suggess a furher ransformaion of he individual z 's o obain more powerful es saisics. The ransformaion for a single z is given by n 1 = Φ (z ), (2) where Φ -1 (.) denoes he inverse of a sandard normal disribuion funcion. This ransformaion produces an n-series ha is independenly sandard normally disribued (iid 2 This resul can be furher exploied o evaluae mulivariae densiy forecass- and muli-sep ahead forecass, respecively (Diebold, Hahn and Tay, 1999, Clemens and Smih, 2000). I is also worh noing ha his resul does in no way depend on how he densiy forecass were generaed. Correc densiy forecass, however obained, imply a ransformed series ha is iid U(0,1).

6 4 N(0,1)) if he rue- and he forecased condiional disribuions coincide. Berkowiz proposes likelihood-raio ess agains he firs order auoregressive alernaive n µ = ρ(n µ) + ε (3) 1 o es for iid N(0,1) daa. In his framework a join es for independence, a mean of zero and a variance of one is given by ) ) ( L( 0,1,0) L ( µ,σ 2 ),ρ ) LR = 2 χ 2 (3), (4) where σ 2 is he variance of ε and L(.) denoes a Gaussian log-likelihood funcion. In simulaion experimens he demonsraes ha he es saisic has good small sample properies. He also suggess an exended LR es ha covers he possibiliy of higher order dependence as well as nonlinear dependencies ha may be consruced from he model n = α 0 + α1n α mn m + β1n β hn h ε. (5) Accepance of he hypoheses α =... = α = β =... = β = 0, Var(ε ) 1 would indicae 0 m 1 h = correc densiy forecass. Specificaions of his ype could easily be exended o include more lags, higher powers of lagged n s or cross producs of lagged n s. Individual or join hypoheses abou models (4) or (5) could also be esed wihin a regression framework using sandard -, F- and chi square ess. 3 Three Criical Cases The LR-ess based on equaions (4) or (5) or equivalen es procedures are aracive because hey are easy o implemen. Especially ess based on a seing like equaion (5) appear o be quie general. However, he evidence from such ess mus be inerpreed wih care. We now discuss hree cases where he uncriical use of he LR-ess described above may lead o erroneous conclusions abou he qualiy of densiy forecass. We also poin ou how addiional diagnosic ess may help o idenify misspecified forecasing models. Since he discussion is

7 5 no only relevan for he evaluaion of risk models bu also imporan for he evaluaion of he forecasing abiliy of ime series models in general, we firs describe he cases in a ime series seing. Hereafer we poin ou how hese cases may arise in he evaluaion of a risk measuremen sysem. We sar by assuming ha a ime series of financial reurns is generaed by he (possibly) nonlinear ime series model x µ(i 1) + σ(i 1 )ξ =, (6) where µ(.) is he condiional mean and σ(.) is he condiional sandard deviaion. The available informaion se is denoed by I -1 and he error erm ξ is assumed o be a condiionally sandardized maringale difference sequence (i.e. E(ξ I 1) = 0 and Var(ξ I 1) = 1). As discussed in Bai and Ng (2001) his framework encompasses mos sandard linear and nonlinear ime series models wih or wihou exogeneous explanaory variables, radiional ARMA models and models wih ARCH and GARCH disurbances. This model can also be inerpreed as a represenaion for he evoluion of he reurns of a porfolio of financial insrumens over ime. Le us now describe he differen cases wihin his framework. We assume ha model (6) is he rue daa generaing process. Case 1: Suppose ha a densiy forecaser has generaed densiy forecass from a misspecified version of model (6). He has correcly specified he firs and second condiional momens. However, he incorrecly assumes ha he error erms ξ * are N(0,1) whereas he rue error erms ξ are uncorrelaed wih mean zero and uni sandard deviaion bu have a (possible ime varying) disribuion D (0,1) N(0,1) which differs from a normal disribuion for all. To assess he qualiy of his forecass he applies he ransformaions given by (1) and (2) and performs an LR es of he kind described in secion 2. Since he ransformaions simply produce n = Φ -1 (Φ(ξ * )) = ξ * he will obain an n-series ha has a zero condiional mean, is

8 6 uncorrelaed and has a uni sandard deviaion. 3 Because he LR es mainains he assumpion of normaliy, he es will no rejec alhough he n-series is no normally disribued and he densiy forecass are incorrec. Wihou addiional disribuional ess for normaliy of an n- series he inadequae densiy forecass will herefore erroneously be acceped despie he fac ha he rue densiies are definiely no normal. Case 2: Now assume ha a densiy forecaser has issued densiy forecass ha only capure he condiional mean correcly. He erroneously assumes normally disribued densiy forecass wih consan uncondiional sandard deviaion σ. Transformaions (1) and (2) produce an n- series ha is uncorrelaed wih condiional and uncondiional zero mean and a uni uncondiional sandard deviaion if he esimaed uncondiional sandard deviaion σ is correc. However, he resuling n-series is heeroskedasic if he rue condiional variance σ(i - 1) is ime varying. 4 Since LR ess based on (4) or (5) are no designed o deec heeroskedasiciy, he forecaser will no be able o idenify he inabiliy of he forecass o capure he rue volailiy dynamics wihou addiional ess for heeroskedasiciy. Of course, he negleced volailiy dynamics of he densiy forecass is likely o be refleced in he shape of he uncondiional disribuion of he resuling n-series because he ime varying volailiy ends o produce n-series wih a fa ailed disribuion. However, he presence of a fa-ailed disribuion alone does no help o discriminae beween an incorrec volailiy dynamics and an incorrec shape of he condiional disribuions. Case 3: Finally, assume ha he forecaser correcly specifies he mean dynamics bu uses he uncondiional densiy of {x } = 1,,m insead of he condiional densiies as a densiy forecass for fuure x s. If he rue process is saionary hen he inegral ransformaion (1) wih respec 3 For a similar resul in he conex of a z-series, see Diebold, Hahn and Tay (1999). 4 This can be shown by noing ha s = [x µ(i -1 )]/σ and n = Φ -1 (Φ(s )) = s. Using hese relaionships i can be shown ha E(n 2 I -1 ) = σ(i -1 )/σ, E(n I -1 ) = 0, E(n n -j ) = 0, E(n ) = 0 and E(n 2 ) = 1.

9 7 o he uncondiional disribuion will produce an uncorrelaed and almos perfecly uniformly disribued z-series. 5 The subsequen ransformaion (2) wih he inverse of he sandard normal disribuion will hen of course creae an uncorrelaed n-series ha has a disribuion quie close o a sandard normal disribuion. The disribuion of he n-series will be virually sandard normal despie he fac ha he volailiy dynamics is misspecified because he uncondiional disribuion of he original daa ignores he order in which he observaions are arranged. Given such a siuaion, neiher he LR ess nor addiional disribuional ess for normaliy will indicae incorrec densiy forecass. One way o deec an incorrec volailiy dynamics of he densiy forecass is o examine he ime series of squared n s which will display clusering if he volailiy dynamics has been negleced. The n-series will also be sandard normal if he rue condiional densiies change over ime due o oher ime dependen higher momens because his is also already refleced in he shape of he uncondiional disribuion. To idenify addiional deficiencies higher powers of he n-series would have o be invesigaed. Having oulined he hree cases in a ime series seing we now poin ou how hese cases may arise in an evaluaion of he qualiy of a VaR-model. Consider a financial insiuion ha uses a variance/covariance-model o calculae is daily Value a Risk (for a comprehensive discussion of his approach, see Jorion, 1997). In his risk model he VaR of a financial porfolio is a cerain muliple of he forecased condiional sandard deviaion of he disribuion of he porfolio reurns. I is ypically assumed ha he reurns of he porfolio follow a condiional normal disribuion. For correc VaR calculaions he correc esimaion of he porfolio volailiy and he correcness of he normal disribuion assumpion are criical. Now suppose ha he VaR model adequaely capures he volailiy dynamics of he 5 Because he uncondiional disribuion can only be esimaed (for example wih he empirical disribuion funcion), small deviaions from he uniform disribuion may resul from esimaion errors. These errors are likely o be small if a sufficienly large number of observaions are available.

10 8 porfolio bu he normal disribuion assumpion for he porfolio does no hold. 6 This may happen if he porfolio conains a significan amoun of nonlinear insrumens such as opions or if he underlying risk facors are already no normally disribued. This siuaion obviously corresponds o case 1 oulined above. Therefore, i is very likely ha he LR ess or equivalen ess do no o rejec and give he impression ha he VaR model is adequae despie he fac ha he rue VaR may be far away from he VaR esimaed wih he model. An even more drasic example ha corresponds wih case 2 is a quie naïve variance/covariance model which is again buil on he assumpion of normally disribued porfolio reurns and i is furher assumed ha he porfolio variance is simply consan. If he uncondiional porfolio variance is esimaed correcly hen he LR ess will no rejec he VaR model even if boh assumpions are clearly violaed. Case 3 may arise if a financial insiuion uses a hisorical simulaion for he purpose of VaR calculaions. In he hisorical simulaion he pas observaions of a se of risk facors are inerpreed as possible fuure realizaions of he risk facors. The reurn on a porfolio of financial insrumens is compued under each of he hisorical scenarios of he risk facors and he VaR is hen calculaed as a cerain quanile of he resuling porfolio reurn disribuion. To obain accurae VaR esimaes 500, 1000 or even more hisorical realizaions are ofen used. Since each hisorical scenario is equally weighed he resuling VaR is implicily based on an esimae of he uncondiional reurn disribuion of he porfolio (Hull and Whie, 1998 and Huisman, Koedijk and Pownal, 1998). If he uncondiional disribuion is saionary or only very slowly changing hen for a fixed porfolio he marginal disribuion of he resuling n-series used in he LR ess will be close o a sandard normal disribuion even if volailiy is ime varying. The LR ess will again no rejec he null hypohesis of a correc risk model despie he fac ha he volailiy dynamics is ignored by he model. 6 For simpliciy we assume in he discussion ha he porfolio reurns have zero mean.

11 9 4 Simulaions and Empirical Illusraions We illusrae he hree cases where he LR ess fail o idenify incorrec densiy forecass wih simulaion experimens and empirical examples for he daily reurns on he S&P 500 and he FTSE 30 sock marke indices. In he Mone Carlo simulaions we consider a GARCH(1,1)- model r = σ 2 [ν/(ν 2)] -1/2 ε ε ~ 5 σ 2 = ε σ 2-1, where r denoes he simulaed reurns, σ denoes he condiional sandard deviaion and ε denoes an innovaion drawn from a suden disribuion wih ν = 5 degrees of freedom. This model is a sandard model for financial reurns. I implies fa ailed suden disribued densiy forecass and produces he volailiy clusering ofen observed in financial reurn series. We use he simulaed ime series from he model o examine he hree cases oulined in secion 3. In case 1 we assume condiionally normally disribued densiy forecass insead of suden 5 disribued forecass, and he GARCH(1,1) model is esimaed under his incorrec disribuional assumpion. Since he esimaed model parameers are sill consisen (Bollerslev and Woooldrige, 1992, Lumsdaine, 1996) he volailiy dynamics should be adequaely capured despie he fac ha he rue disribuion is a fa ailed suden disribuion. In case 2 we incorrecly assume an uncondiional normal disribuion for he densiy forecass. We hereby, in addiion o he choice of an incorrec disribuion, also misspecify he volailiy dynamics because we use a simple esimae of he uncondiional sandard deviaion insead of an esimae of he ime dependen condiional variance. In he hird case we ake he empirical disribuion funcion as our densiy forecass and herefore again misspecify he condiional disribuions of our densiy forecass.

12 10 We perform simulaions of he model for samples of 500, 1000, 2000 and 4000 observaions. For each of he hree cases we esimae he rejecion raes of a likelihood raio es (LR1) based on (4) and a likelihood raio es (LR2) based on wo lags of n and n 2 ha considers he more general alernaive (5) when applied o he resuling n-series for a 5% significance level. Using he same significance level, we also calculae he rejecion raes of a Jarque-Bera normaliy es and an ARCH es for heeroskedasiciy (ARCH) based on an F es of he resricion γ 1 = γ 2 = = γ 5 = 0 in he regression n 2 = γ 0 + γ 1 n n ξ. The resuls of he simulaion experimens are repored in able 1. INSERT TABLE 1 ABOUT HERE From able 1 i is easily seen ha in all hree cases he LR1 and LR2 ess have lile power o deec incorrec densiy forecass. This is exacly wha we would expec from our discussion in secion 3. For example, he rejecion raes of he LR1 es are exremely low and range from 0.02 o across he differen sample sizes and cases. The rejecion raes of he more general LR2 es are similar o he LR1 rejecion raes in case 1 and slighly higher, bu sill very low, in he oher wo cases. On he oher hand, noe ha he JB es virually always rejecs he incorrec densiy forecass in case 1 and case 2 and never rejecs in case 3. This finding is again consisen wih he heoreical discussion. The same is rue for he heeroskedasiciy ess. The ARCH es virually never rejecs in case 1 because he volailiy dynamics is correcly capured by he forecass bu rejecs frequenly in he oher wo cases where densiy forecass ignore he volailiy dynamics. Taken ogeher, he resuls from he simulaions clearly show ha boh LR ess are essenially unable o idenify he incorrec densiy forecass in each case. Wihou he addiional normaliy- and heeroskedasiciy ess he deficien forecass canno be deeced. Le us now urn o he empirical example. We evaluae successive one-sep-ahead densiy forecass for daily reurns on he FTSE 30 and he S&P 500 from four differen models. The firs wo models are he simple moving average model (MA) of squared reurns

13 11 wih a rolling ime window of 250 rading days and he exponenially weighed moving average model of squared reurns (EWMA) wih a decay facor of 0.94, as suggesed by J. P. Morgan. In boh models we make he convenional assumpions ha he mean of he daily reurns is approximaely zero and ha he reurns are condiionally normally disribued. These models are ofen used o generae he variance/covariance marices used in VaR calculaions. The oher wo models are he sandard GARCH(1,1)-n model where he errors are also assumed o be condiionally normal and he GARCH(1,1)- model where condiionally disribued errors are assumed. INSERT TABLE 2 ABOUT HERE! The likelihood raio-, normaliy- and heeroskedasiciy ess for he n-series resuling from he 1,000 daily densiy forecass of he differen models over he period from 4/20/1998 o 2/15/2002 for he S&P 500 and 4/21/1998 o 2/18/2002 for he FTSE 30 are summarized in able 2. The empirical evidence again highlighs he inabiliy of he LR ess o discriminae beween he densiy forecass from he differen models. For example, he LR1 and LR2 ess do no disinguish beween he GARCH-n and he GARCH- models. 7 The ess indicae correc densiy forecass for boh models. However, he addiional JB- and ARCH ess sugges ha only he densiy forecass from he GARCH- models migh be correc. The forecass of he GARCH-n model are clearly rejeced by he JB es. Since he ARCH ess do no rejec for he GARCH-n model he normaliy assumpion appears o be incorrec. In he case of he S&P 500 he LR1 es does also no rejec he simple MA model alhough he JBand ARCH ess clearly indicae ha boh, he volailiy dynamics and he normal disribuion assumpion are incorrec. The empirical evidence in general suggess ha he widely used MA- and EWMA models combined wih he assumpion of a normal disribuion do no produce accurae densiy forecass. 7 The parameer esimaes for he GARCH models are available on reques from he auhors.

14 12 Concluding Remarks In his paper we invesigaed a recenly proposed likelihood raio framework o evaluae densiy forecass. We showed ha he uncriical use of his framework may lead o incorrec conclusions abou he qualiy of risk models. Sandard variance/covariance approaches and hisorical simulaion approaches o calculae VaR may be acceped despie he fac ha hey may provide poor VaR esimaes. We furher demonsraed ha addiional diagnosic ess including normaliy ess and ess for heeroskedasiciy help o deec incorrec models. Bu hese addiional ess do no only help o deec incorrec models, hey also provide informaion abou he kind of model failure. This leads us o he conclusion ha he LR framework of Berkowiz combined wih addiional diagnosic ess is a consrucive and powerful ool o evaluae risk models. Of course, a careful risk manager would probably also perform some of he oher ess menioned in he inroducion o asses he accuracy of his risk model. However, if one of he cases oulined in his paper arises, he may obain conflicing resuls if he compares he oucome of hese ess wih he evidence from he LR ess. Wih he help of furher graphical assessmens or he addiional diagnosic ess he may hen be able o correcly inerpre he resuls.

15 13 Bai, J. & Ng, S. (2001). A Consisen Tes for Condiional Symery in Time Series Models. Journal of Economerics, 103, Berkowiz, J. (2001). Tesing Densiy Forecass, wih Applicaions o Risk Managemen. Journal of Business and Economic Saisics, 19(4), Bollerslev, T. & Wooldridge, J. M. (1992). Quasi-Maximum Likelihood Esimaion and Inference in Dynamic Models wih Time-Varying Covariances. Economeric Reviews, 11(2), Chrisoffersen, P. (1998). Evaluaing Inerval Forecass. Inernaional Economic Review, Chrisoffersen, P., Hahn J. & Inoue A. (2001). Tesing and Comparing Value a Risk Measures. Journal of Empirical Finance, 8 (July), Clemens, M. P., & Smih, J. (2000). Evaluaing he Forecas Densiies of Linear and Nonlinear Models: Applicaions o Oupu Growh and Unimploymen. Journal of Forecasing, 19, Crnkovic, C., & Drachman, J. (1997). Qualiy Conrol. In VaR: Undersanding and Applying Value-a-Risk. London: Risk Publicaions. Diebold, F. X., Hahn, J., & Tay, A. S. (1999). Mulivariae Densiy Forecas Evaluaion and Calibraion in Financial Risk Managemen: High-Frequency Reurns on Foreign Exchange. The Review of Economics and Saisics, 81(4), Diebold, F. X., Gunher, T. A., & Tay, A. S. (1998). Evaluaing Densiy Forecass, wih Applicaions o Financial Risk Managemen. Inernaional Economic Review, 39, Duffie, D. & Pan, J. (1997). An Overview of Value a Risk. Journal of Derivaives 4 (Spring), Huisman, R., Koedijk, K. G. & Pownal, R. A. J. (1998). VaR+: Fa Tails in Financial Risk Managemen. Journal of Risk, 1 (Fall), Hull, J. & Whie, A. (1998). Incorporaing Volailiy Updaing Ino he Hisorical Simulaion Mehod for Value a Risk, Journal of Risk,1 (Fall), Jorion, P. (1996). Value a Risk: The New Benchmark for Conroling Risk. Irwin Professional. Kupiec, P. (1995). Techniques for Verifying he Accuracy of Risk Measuremen Models. Journal of Derivaives 3 (Winer), Lopez, J. (1996). Regulaory Evaluaion of Value a Risk Models. FRBNY Economic Policy Review 4 (Ocober), Lumsdaine, R. (1996). Consisency and Asympoic Normaliy of he Quasi-Maximum Likelihood Esimaor in IGARCH(1,1) and Covariance Saionary GARCH(1,1) Models. Economerica, 64(3),

16 14 Rosenbla, M. (1952). Remarks on a Mulivariae Transformaion. Annals of Mahemaical Saisics, 23,

17 15 Table 1: Rejecion Raes of Densiy Forecass from GARCH(1,1)- Model, 5% Significance Level Observaions LR1 LR2 JB ARCH Case Case Case Noes: The able repors he rejecion raes of densiy forecas ess from 10,000 simulaions of he model r = σ ε, σ 2 = ε σ 2-1 where he innovaions ε are drawn from a suden disribuion wih 5 degrees of freedom. LR1 and LR2 denoe he likelihood raio ess based on equaions (4) and (5) in he ex, respecively. JB denoes a Jarque-Bera es for a normal disribuion. ARCH denoes an F es for heeroskedasiciy of he resricion γ 1 = γ 2 = = γ 5 = 0 in he regression n 2 = γ 0 + γ 1 n n ξ.

18 16 Table 2: P-Values from Evaluaions of Densiy Forecass from MA-, EWMA-, GARCH(1,1)- n and GARCH(1,1)- Models for Daily Reurns on he FTSE 30 and he S&P 500. FTSE 30 Period: 4/21/1998 o 2/18/2002 Observaions: 1000 Model LR1 LR2 JB ARCH MA EWMA GARCH-n GARCH S&P 500 Period: 4/20/1998 o 2/15/2002 Observaions: 1000 Model LR1 LR2 JB ARCH MA EWMA GARCH-n GARCH Noes: The able repors p-values from ess of he qualiy of 1000 consecuive one sep ahead densiy forecass from a moving average volailiy model (MA) wih a rolling window of 250 rading days, an exponenially weighed moving average volailiy model (EWMA) wih decay facor 0.94, a GARCH(1,1) model wih normally disribued errors (GARCH-n) and a GARCH(1,1) model wih -disribued errors (GARCH-). LR1 and LR2 denoe he likelihood raio ess based on equaions (4) and (5) in he ex, respecively. JB denoes a Jarque-Bera es for a normal disribuion. ARCH denoes an F es for heeroskedasiciy of he resricion γ 1 = γ 2 = = γ 5 = 0 in he regression n 2 = γ 0 + γ 1 n n ξ.