OPTIMAL PREDICTION UNDER LINLIN LOSS: EMPIRICAL EVIDENCE Yasemin Bardakci St. Cloud State University

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1 OPTIMAL PREDICTION UNDER LINLIN LOSS: EMPIRICAL EVIDENCE Yasemin Bardakci S. Cloud Sae Universiy Absrac: I compare e forecass of reurns from e mean predicor (opimal under MSE), wi e pseudo-opimal and opimal predicor for an asymmeric loss funcion under e assumpion a agens ave asymmeric LINLIN loss funcion. I consider bo univariae and mulivariae cases. For e mulivariae case, I generalize e LINLIN loss funcion o a mulivariae LINLIN loss funcion and use a normal and -diagonal-bekk GARCH(1,1) model o predic e ime varying variances. Te resuls srongly sugges no o use e condiional mean predicor under any kind of asymmery. In general, forecass can be improved considerably by e use of opimal predicor versus e pseudo-opimal predicor suggesing a e loss reducion due o using e opimal predicor can acually be imporan for a praciioner as well. Keywords: Loss funcion; GARCH models; Volailiy forecasing; Time series; Mulivariae ime series.

2 An assumpion of symmery abou e condiional mean is likely o be an easy one o accep an assumpion of symmery for e cos funcion is muc less accepable. Granger, Newbold (1986, p.125) 1. Inroducion In e lieraure, a widely used forecas evaluaion crieria is e MSE, wic is a symmeric quadraic loss funcion. MSE penalizes e posiive errors and negaive errors of e same magniude equally. However, in finance we know a realisically, forecasers do no necessarily ave a quadraic cos nor a symmeric loss funcion i. Over predicion can be more (less) cosly an under predicion depending on e siuaion. For example, if we consider e posiive relaionsip beween e volailiy of underlying sock prices and call opion prices (Brailsford, Faff, 1996), an under predicion of volailiy will resul in a downward bias o e esimaes of e call opion price wic is probably of more concern o a seller an a buyer wile e converse olds for e over predicions of volailiy (Mc Millian, Speig, Apgwilym, 2000). Despie is fac, MSE as become a very popular loss funcion mosly due o maemaical convenience. Symmeric loss funcions are convenien because e analyical expression for e opimal predicor is sraigforward: i is e condiional mean. However, if we use a general loss funcion, deriving e closed form for e opimal predicor analyically can be very callenging, and even no viable. Sudies ave avoided using general asymmeric loss funcions mainly because mos of e ime e closed form for e opimal predicor does no exis. Under asymmeric loss, e opimal predicor is no longer e condiional mean. Granger (1969) sowed a e opimal predicor under asymmeric loss is e 2

3 condiional mean plus a consan bias erm. Granger (1969) assumed a consan condiional predicion error variance. Cirisoffersen and Diebold (encefor abbreviaed CD) (1996, 1997) considered e same problem, and generalized Granger s resul. Tey sowed a for condiionally Gaussian processes if an agen as an asymmeric loss funcion, adding a consan bias erm is no sufficien and a ime varying second order momens become relevan for opimal predicion. Tey derived e analyical expression for e opimal predicor for wo specific asymmeric loss funcions, e LINLIN and e LINEX. Troug a Mone Carlo simulaion for a condiionally Gaussian GARCH(1,1) process under LINLIN loss CD (1996) sow a 35% loss reducion is aainable wen e opimal predicor is used. Aloug CD (1996, 1997) ave imporan pracical implicaions, ere is no empirical sudy a illusraes e gain in adding a ime-varying bias erm o e condiional mean o obain e opimal predicor wen agens ave asymmeric loss. In is paper I bridge is gap by illusraing e loss associaed wi using e opimal, e pseudo-opimal and e condiional mean predicor under asymmeric loss. One of e moivaions beind is sudy is e fac a e realized average loss reducion mig be considerably smaller an e eoreically expeced losses in a world of parameer uncerainy and may no be of ineres o a praciioner. I demonsrae a in pracice incorporaing ime varying second momens can acually be of ineres o a financial forecaser reducing eir loss significanly. Te second conribuion of is paper o e exising lieraure is deriving e opimal, pseudo-opimal predicors for a mulivariae asymmeric loss funcion, and using em o evaluae mulivariae forecass from mulivariae GARCH models. 3

4 I consider reurns on ree represenaive excange raes and reurns on five represenaive marke indices a are commonly used in empirical work. I consider differen predicors of e variance since ere is no unanimous agreemen on e bes variance model in e lieraure. I coose e mos popular models a are believed o caracerize condiional volailiy in asse reurns. I consider normal-garch(1,1), - GARCH(1,1), EGARCH(1,1) and a nonparameric model for univariae variance esimaion and forecasing and normal and -diagonal BEKK GARCH (1,1) models for e mulivariae case. I empirically illusrae a e loss associaed by using e opimal predicor wic incorporaes ime varying second order momens, is muc smaller an e loss associaed wi using e condiional mean predicor wen agens ave asymmeric loss funcion for forecas orizons of one week and four weeks. Moreover, wen I compare e difference in loss beween e opimal predicor and e pseudoopimal predicor, in 5 ou of e 7 series e opimal predicor ouperforms e pseudoopimal predicor. However, e magniude of loss reducion is very sensiive o e degree of asymmery, e condiional variance parameers being used, and e forecas orizon. Te mulivariae case suppors e use of opimal predicor especially for moderae and ig degrees of asymmery as well. Wen agens ave asymmeric loss funcion, e gain using e opimal predicor versus e pseudo-opimal predicor is clearer compared o e univariae case. Secion 2, is on univariae forecasing under asymmeric loss were I inroduce e LINLIN asymmeric loss funcion, and e univariae variance models I use. Secion 3 is on mulivariae forecasing under asymmeric loss were I derive e opimal and e pseudo-opimal predicors for a mulivariae asymmeric generalized loss funcion and 4

5 use i o compare forecass from normal and -diagonal-bekk GARCH(1,1) mulivariae variance models. Secion 4 describes e daa. Secion 5 presens e resuls and Secion 6 concludes. 2. Univariae Forecasing wi asymmeric Loss 2.1. LINLIN Loss Funcion: Te LINLIN loss funcion was used by Granger (1969). I is linear on eac side of e origin; owever, posiive errors are penalized differenly an e negaive errors because e lines ave differen slopes in eac side of e origin. Te LINLIN loss funcion is { ( a y yˆ if ( y yˆ ) > 0 L y yˆ ) =, b y yˆ if ( y yˆ ) <= 0 were (.) is a loss funcion defined on -sep-aead predicion error, is e realized L value of y, + periods aead and ˆ is e prediced value of. Te raio a/b y + measures e cos of under predicing relaive o e cos of over predicing. If a/b=2, i means a e loss associaed wi a posiive error is wice as muc e loss associaed wi negaive error of e same magniude. If a=b, e average loss is idenical o MAE. CD (1997) derived e opimal predicor, a pseudo-opimal predicor and e expeced losses associaed wi eac predicor for e LINLIN asymmeric loss funcion under e assumpion of condiional normaliy. Given y Ω ~ N ( µ, σ ), ey 1 sowed e opimal predicor for y +, is µ + + σ Φ ( a /( a )), and e pseudo- opimal predicor is 1 2 µ + + σ Φ ( a /( a )), were is e -sep-aead + b + b σ y + y + omoscedasic predicion error variance and Φ(z) is e N-(0,1) c.d.f ii. Under 5

6 asymmeric loss, if e condiional eeroscedasiciy is ignored, e associaed condiionally expeced loss will be greaer en e case wen opimal predicor is used. 2 2 Te pseudo-opimal predicor coincides wi e opimal predicor wen σ = σ. Noice a for a given series, and for cerain degree of asymmery e difference in mean losses associaed wi using e opimal versus e pseudo opimal predicor, depend on e deviaion of e square roo of condiional variance from is mean ( σ σ ). Tus e difference beween e mean losses is proporional o is variaion. If is variaion is large, en incorporaing e ime varying second order momens can be very crucial in erms of reducing e losses. In order o mimic forecasing in real ime, esimaion is done in rolling windows of fixed leng of one week and one-sep-aead ( = 1) and four-sep-aead ( = 4 ) forecass are consruced iii. For e series I consider, e condiional means are consan. I obain e condiional variance using differen models. I do no limi e analysis only o condiionally Gaussian processes. Aloug CD assumed condiional normaliy, i is a common fac a financial reurn series ave excessive kurosis. To ake is fac ino accoun I also esimae a GARCH(1,1) model wi -disribued innovaions and an EGARCH(1,1) model wi a generalized error disribuion. Te EGARCH(1,1) model wi generalized error disribuion allows bo for asymmery in e condiional variance and condiional non-normaliy. In order o preven any possible misspecificaion problems due o parameerizaion, I also use a nonparameric model. I compue e value of e opimal, e pseudo-opimal and e condiional mean predicors and corresponding losses over all e rolling windows of = 1 and ake e simple average o calculae e average losses associaed wi em. Aloug, e compuaion is sraig 6

7 forward for = 1, some disribuional and second momen relaed problems ave o be considered for = 4. Wen forecasing ou-of sample wi GARCH models, aloug e Gaussian assumpion olds for = 1, e disribuion of e predicion error is no known for > 1. For is case I used Cornis Fiser Asympoic Expansion o approximae e predicion error disribuion [See Bailie and Bollerslev (1992)] wen e reurns are assumed o ave condiional normal disribuion. Te approximaed four-sep-aead quanile values are used in e opimal and pseudo-opimal predicor formulas for e expression 1 Φ ( a /( a + b)). Anoer difficuly wen forecasing ou-of sample wi GARCH models for longer orizons ( > 1) is e calculaion of e var ( ). Again e y + formulas are readily available for a general ARMA(k,l)-GARCH(p,q) model for a general [See Bailie and Bollerslev (1992) for more deail]. However, expressions of e - sep-aead condiional variance as o be analyically compued for EGARCH, and mulivariae GARCH models. Te resuls don exis for e nonparameric model as well. Tis is beyond e scope of is paper. For pracical purposes, for e -sep- aead predicion, wen > 1 e losses associaed wi eac predicor are calculaed for e GARCH(1,1)-n using e exising eoreical resuls. I consider differen degrees of asymmery o compare e loss associaed wi using differen predicors. Specifically, I fix b = 1 and cange e values of a. I consider cases up o were a / b = 20. Tis asymmeric penalizaion sceme is plausible in finance Univariae Variance Models: GARCH Models: 7

8 Te mos commonly used model for ime-varying volailiy is e G/ARCH model of Engle (1982) and Bollerslev (1986). A GARCH(1,1) model for e reurn on a financial asse, r, can be wrien as r = σ z, z ~ IID(0,1), σ = γ + α r 1 + β σ 1 were γ > 0, α 0 and β 0. I assume as finie firs and second momens. For a normal z GARCH(1,1) model, denoed n-garch(1,1), I assume an independen normal innovaion. Since financial reurns are known o be eavy ailed, and ofen e condiional normaliy assumpion is inappropriae, I also esimae GARCH(1,1) models wi condiional Suden- disribuions, denoed by -GARCH(1,1), for eac reurn series. Te one-sep-aead opimal and e pseudo-opimal predicors for e GARCH(1,1) model under condiional--disribuion assumpion are: y = µ 1 + σ F (( a /( a b), υ) and 1 y + 1 = µ 1 + σ F (( a /( a + b), υ), were υ is e daa specific degrees of freedom and 1 F is e inverse of condiional cumulaive disribuion funcion. Anoer feaure of e financial daa is e skewness ineren in asse reurn volailiy. Especially wi e sock reurn daa, i is found a an unexpeced drop in price increases e predicable volailiy more an an unexpeced increase in price of similar magniude, wic is commonly referred as e leverage effec [Black (1976)]. In order o capure is skewness ineren in asse reurn volailiy, I esimae e mos commonly used asymmeric variance model, e EGARCH(1,1)-model inroduced by Nelson (1991). I esimae an EGARCH(1,1) model wi a generalized error disribuion. Te condiional variance for e EGARCH(1,1) model can be wrien as 2 2 logσ = ω + β logσ 1 + α r 1 / σ 1 + γ ( r 1 / σ 1 ), were γ is e parameer a capures e leverage effec. If γ 0, en e condiional variance is asymmeric. Tis model 8

9 as wo advanages; firs i allows for e asymmery and capures e leverage effec. Second i does no assume condiional normaliy, us allows for eavy-ailed disribuions. Nonparameric variance Model: Nonparameric models are popular because ey do no inroduce any parameric assumpion for e underlying disribuion and us preven e bias problem due o misspecificaion. Predicing excange rae reurns nonparamerically goes back o Diebold and Nason (1990). Te idea beind using nonparameric ecniques o esimae and predic excange raes is o exploi any non-lineariies a may be presen in e financial reurn daa. Pagan and Scewer (1990) also use is ecnique for in sample predicion as well. I fi a nonparameric model for e asse reurns condiioning on e lagged reurns. I coose a Gaussian mulivariae kernel, and I consider e opimal bandwid o be fixed for a given daa se [see Nadaraya (1964) and Wason (1964)]. To predic e second order condiional momen I use e formula in Pagan and Scwer (1990). Wen forecasing ou of sample, I replace e ime varying condiional mean and condiional sandard deviaions in e CD (1997) opimal predicion formula by eir nonparameric esimaors. In e pseudo-opimal predicion formula, I replace e consan condiional sandard deviaion wi e sample sandard deviaion of e in sample nonparameric residuals. 3. Mulivariae forecasing wi asymmeric loss 3.1 Mulivariae Loss funcion: 9

10 Te eory of forecasing wi asymmeric loss as originally presened by Granger (1969), and furer developed by CD (1996, 1997), only considered e predicion of a single variable based on is own passed values. In is secion I exend e eory o a mulivariae framework in wic more an one series is o be forecased. Le Y + be an n 1 vecor of variables o be forecased a orizon. ˆ be e n 1 vecor of Y + forecass and = Y Yˆ is e n 1 vecor of forecas errors. I en ave e e following exension of CD s (1997) Proposiion 1. Proposiion 1: If Ω ~ N( µ, ) is condiionally mulivariae normal and Y Σ L ( e ) is any loss funcion defined on e vecor of -sep-aead predicion error, en e opimal predicor is of e form ˆ = α, were α depends only on e Y µ + + loss funcion and e condiional predicion error variance-covariance marix = var( Y Ω ) = var( e Ω ). Σ Proof: See Appendix. Following Zellner (1986), i may be reasonable o assume e loss funcion is addiively separable in e n predicion errors and can be wrien as n i i, i, ) i= 1 L( Y Yˆ ) = L ( y yˆ. Possible coices for L ( ) are e linlin and linex loss i funcions. For e linlin loss funcion, if y i, + is condiionally normal, en e opimal predicor vecor can be wrien as y ˆ = µ + Φ 1 ( a /( a + b )). Proof: See Appendix. i, i, σ ii, i i i 3.2 Mulivariae Variance Models: A univariae GARCH model can be generalized o a n-dimensional mulivariae GARCH model as r Ψ 1 ~ N(0, Σ ), were r is e n-dimensional zero mean random variable, 10

11 Σ is e variance covariance marix a depend on informaion se available a -1. depends on q lagged values of squares and cross producs of r and p lagged values of Te exension of a univariae GARCH model o a n-dimensional mulivariae GARCH model require some resricions on e condiional variance covariance marix Σ. Tere are differen parameerizaions of e variance covariance marix. Among e mos popular ones are e VEC, e diagonal represenaion (Engle, Granger, Kraf (1984), Bollerslev, Engle and Wooldridge(1988) and e BEKK model of Engle and Kroner (1995). Te advanage of BEKK represenaion is a i is easy o impose resricions on e condiional variance-covariance marix (Engle, Kroner, (1995)) a ensures posiive definieness. For e mulivariae case I esimae e n-diagonal -BEKK mulivariae GARCH(1,1) model. Te BEKK variance model can be wrien as: Σ = C C + A ε ε A + B Σ Σ Σ. B. By e specificaion of e model, e condiional variance-covariance marix is guaraneed o be posiive-definie. I forecas e one period aead condiional variance-covariance marix by ieraing e marix equaion Σˆ = Cˆ Cˆ + Aˆ ˆ ε ˆ Aˆ + Bˆ ˆ 1 1ε 1 Σ Bˆ and compue e values of e opimal predicor and e pseudo-opimal predicor for e mulivariae LINLIN loss funcion over a rolling window. 4. Daa To deermine weer using e opimal predicor in e presence of asymmeric loss is empirically useful, I use daa represenaive of a used in financial forecasing. I esimae models and predic reurns for ree excange raes beween e U.S. Dollar and 11

12 e Canadian Dollar, e Japanese Yen and e Briis Pound. I also esimae models and predic reurns for five major marke indexes: e Dow Jones Indusrial Average, and e S&P 500, e NASDAQ, e NIKKEI, and FTSE. Te frequency of e daa is weekly. Te daa run from January 1995, o Augus 6, Te sample is used for esimaion using a rolling window wi a fixed leng of one week and e one-sep-a ead ( = 1) and four-sep-aead ( = 4 ) rolling forecass are consruced. 5. Resuls I firs esimae n-garch (1,1), models for eac reurn series over e wole sample using a rolling window wi a fixed leng of one week. Aloug e efficien marke ypoesis suggess a reurns sould be serially uncorrelaed, I ceck for possible serial correlaion in e reurns. I find a all e reurn series on excange raes are serially uncorrelaed. Table 1 sows e esimaed GARCH (1,1) models for reurns on ree excange raes and diagnosic saisics for e sandardized residuals for e las window. Te esimaes for e condiional variance parameers are all significan. Te skewness coefficiens of e sandardized residuals sow a ey are sligly negaively skewed. Te kurosis coefficiens are ree for Canada and U.K. and six for Yen/$ series. Te Jarque-Bera saisics for Canada and UK. are no significan suggesing normaliy assumpion can no be rejeced wile for e Yen/$ e Jarque-Bera saisic is significan suggesing a e daa does no ave normal disribuions. Te Ljung-Box Pormaneau ess for e serial correlaion in e sandardized and squared sandardized residuals up o 10 lags indicae a e residuals are wie noise. For e Yen/$ series I 12

13 re-esimaed e model using -GARCH(1,1) model since e Jarque-Bera saisic suggess a ere is non-normaliy. Table 2 sows e esimaed GARCH models for e reurns on indices and diagnosic saisics for e sandardized residuals for e las rolling window. I find a e reurn series for S&P 500 are AR(1) in e mean. Te esimaes for e condiional variance parameers are all significan. For e Nasdaq series e esimaed variance parameers, wi e n-garch(1,1) model are ( α, β ) = (.11,.89) and -GARCH(1,1) model are ( α, β ) = (.07,.93). Te variance parameers sum o one, indicaing a Nasdaq reurn series ave an IGARCH propery, wic suggess a e uncondiional variance for is series is no finie. Since e compuaion of pseudo-opimal predicor requires an expression for e uncondiional variance, is series is dropped from e analysis. For e res of e series e skewness coefficiens of e sandardized residuals sow a ey are sligly negaively skewed. Te Jarque-Bera saisics are all igly significan. Te Ljung-Box Pormaneau ess for e serial correlaion in e sandardized and squared sandardized residuals up o 10 lags indicae a e residuals are wie noise. In order o accoun for eavy ails and possible asymmery in e condiional variance, I re-esimaed e models using -GARCH(1,1) and EGARCH(1,1) models. Te resuls are repored in Table 2 for e las rolling window. Given e models esimaed above, for eac rolling window wi a fixed leng of one week, I use e esimaed variances from differen models o compue e opimal, pseudo-opimal and condiional mean predicors and e losses associaed wi em for forecas orizons one and four wic corresponds o weekly and monly forecasing. 13

14 Figure 1 sows e raio of e average losses beween e opimal predicor and e condiional mean, and also e raio of e average losses beween e pseudo-opimal predicor and e condiional mean for eac of e excange rae reurn series from onesep aead forecass. For all e series, i is sriking a ere is considerable gain by using e opimal or e pseudo opimal predicor versus e condiional mean predicor. Te condiional mean predicor performs e wors, even for very low degrees of asymmery. For e excange rae series, wi e excepion of e Yen/$ series, e opimal predicor and e pseudo-opimal predicor perform almos e same suggesing ere is no or very small gain using e opimal versus e pseudo opimal predicor, regardless of e variance model. For e Yen/$ series, wi e excepion of e resuls from non-parameric model, ere is evidence of reducion in loss due o using e opimal predicor, e loss reducion is around 5% for asymmery level of five and increases as e asymmery level increases and is around 15% for e asymmery level of 20. Figure 2 sows e raio of e average losses beween e opimal predicor and e condiional mean and also e raio of e average losses beween e pseudo-opimal predicor and e condiional mean for e reurns on e five marke indices from e univariae variance models from one-sep-aead forecass. Regardless of e series and e variance models being used, again e condiional mean predicor performs e wors. Te opimal predicor generally ou-performs e pseudo opimal predicor wi e excepion of Nikkei series, and as e degree of asymmery increases, so does e loss reducion. Te EGARCH(1,1) model seems o provide e iges values for e acieved loss reducion for all e models and e non-parameric model e wors and 14

15 e resuls from -GARCH(1,1) model sugges sligly more loss reducion an e n- GARCH(1,1) models. For e DJIA e resuls from series resuls from n-garch and - GARCH (1,1) sugges a e gain from using e opimal predicor is more an 15% for moderae degrees of asymmery (a=5) and is around 20% for iger degrees of asymmery. Te resuls from e EGARCH model are furer promising; 20% loss reducion is acievable by using e opimal predicor for moderae asymmery degrees (five), and e loss reducion reaces o 35% as e asymmery level increases. Te resuls from e nonparameric model sugges only 6% reducion even wi e iges level of asymmery. For e FTSE, e resuls from e n-garch(1,1) and - GARCH(1,1) model suggess a ere is more an 10% loss reducion wen opimal predicor is used, even for moderae degrees of asymmery, and more an 20% loss reducion is acievable wen a 6. Te resuls from e EGARCH(1,1) model looks more promising, suggesing up 26% loss reducion is possible using e opimal versus e pseudo opimal predicor. Te resuls from e nonparameric model, sugges only 2% loss reducion due o using e opimal predicor. Te resuls on e Nikkei series sugges a e pseudo-opimal predicor performs a leas as good as e opimal predicor, and some imes sligly beer. For e S&P 500 once again all e models sugges a e opimal predicor ou-performs e pseudo-opimal predicor. Resuls from e parameric models imply a possible loss reducion of 20% or more even for moderae degrees of asymmery, reacing 30-35% for ig degree of asymmery. We obain e iges loss reducion value from e EGARCH(1,1) model, reacing 35% for ig asymmery levels. Tis acieved loss reducion value is very close o e Mone 15

16 Carlo resuls of CD (1996). Te nonparameric model suggess 1-5% loss reducion depending on e degree of asymmery. Figure 3 presens e resuls of e average loss raios from 4-sep-aead forecass using a n-garch(1,1) model for all e series considered. Te opimal predicor generally ou performs e pseudo-opimal predicor wi e excepion of e $/ series. Te apparen loss reducion due o using e opimal predicor is ineresing because wen we use forecas orizon four, we are subjec o more uncerainy since we are approximaing e condiional disribuion of e series as well as e condiional variance four periods a ead us as forecas orizon increases we are likely o ave larger forecas errors. Even under ese circumsances i is ineresing a e loss reducion due o using e opimal predicor is no negligible. For e mulivariae case I esimae bo a normal and -diagonal-bekk mulivariae GARCH(1,1) models for e Canadian, Japanese and U.K. FX reurn series. Consisen wi e univariae normal case, e condiional mean predicor performs e wors and e opimal predicor performs sligly beer an e pseudo-opimal predicor. Te reducion in loss is around 4% (Figure 9) from opimal predicor wen a diagonal BEKK GARCH(1,1) model is used wi n-disribued errors for a ig degree of asymmeric loss. Tis loss reducion, excep for yen/$ series, is iger an e loss reducion e univariae variance models sugges. Tis is no a surprise because for a given series mulivariae BEKK model explois e informaion presen in e res of e series considered and makes use of a larger informaion se o esimae eac variance, us is more efficien en e univariae variance models. We obain similar resuls from e diagonal -BEKK GARCH(1,1) model bu e loss reducion is sligly smaller. 16

17 6. Conclusion I consider e mean losses associaed wi using e opimal predicor, pseudoopimal predicor and e condiional mean predicor wen agens ave asymmeric loss funcion. I use bo univariae and mulivariae variance models. I also derive e opimal, and e pseudo-opimal predicor for addiively separable mulivariae loss funcion. My resuls provide srong empirical evidence o e Granger (1969) regardless of e series and e variance models being used. For all series, loss associaed wi using e condiional mean predicor versus using e pseudo-opimal predicor is considerably iger even for moderae degrees of asymmery, regardless of e variance model being used. Te resuls sugges a under any kind of asymmery e condiional predicor sould no be used a all. Te resuls also provide empirical evidence o CD (1996, 1997); similar, bu no as srong argumen olds for e comparison beween e opimal versus e pseudo opimal predicor, as e resuls are sensiive o e series, variance parameers and asymmery level being used. Te loss reducion due o using e opimal predicor versus e pseudo-opimal predicor in general seems o be iger for e index reurn series, an e excange rae series, and significanly large specially wen EGARCH(1,1) model, a incorporaes e leverage effec, is used, reacing up o 35%. Tis realized loss reducion from using e opimal versus e pseudo-opimal predicor is very close o e Mone-Carlo resuls of CD (1996) for expeced losses a sor orizons iv. 17

18 Wen we compare e realized loss reducion of all e series wi e Mone- Carlo simulaions of CD (1996), owever, realized loss reducion is smaller an e expeced. Tis is no a surprise, since in empirical work we are subjec o parameer uncerainy. Tis also explains e reason wy we ge differen resuls from differen variance models. For e indices for example we see a, wi e excepion of Nikkei, e iges loss reducion is acieved by using e opimal predicor versus e pseudoopimal predicor wen we use e EGARCH(1,1) model, wic capures e leverage effec and is around 35%, and generally e realized loss reducion from -GARCH(1,1) models a accoun for e eavy ails are sligly larger an n-garch(1,1) models. Wile for e excange rae series excep for Yen/$ series, loss reducion from using e opimal predicor can be saisically small. In a few cases we also ge, e pseudoopimal predicor very sligly ou-performing e opimal predicor. Tis mig be due o e sampling variabiliy or model misspecificaions. However, e opimal predicor does end o ou perform e pseudo-opimal predicor in mos of e cases. Te difference can be very small 3-5% or as large as a 35% reducion in loss. Clearly, e resuls sugges e use of e opimal predicor, for e index series even for moderae degrees of asymmery. Te resuls on excange raes are more sensiive o e series, e variance parameers being used and e asymmery level. Te resuls in general sugges a using e opimal predicor a incorporaes ime varying second momens resuls in considerable loss reducion and can pracically be imporan. Te forecaser for sould decide if e realized loss reducion is financially imporan for e agen as e use of e 18

19 opimal versus e pseudo-opimal predicor depends on e financial imporance of e loss reducion o e individual. Is is reducion in e mean loss financially imporan? A parallel argumen o e difference beween economic significance and saisical significance can be carried ou ere. Cleary, if you are a edge fund manager, even 1% loss reducion would be of criical imporance. However for differen uiliy funcions, or moderae degrees of asymmery, agens mig be indifferen beween using e opimal or e pseudo-opimal predicor. 19

20 Table 1. Esimaed n-garch(1,1) and -GARCH(1,1) models for reurns on excange raes for e las rolling window. Model parameers FX Canada FX Japan FX U.K. ψ n-garch(1,1) (.007) (.04) (.03) α (.016) (.03) (.005) β (.02) (.02) (.003) Skewness Kurosis Jaque-Bera Q (10) Q (10) GARCH(1,1) ψ.067 (.041) α.036 (.018).937 β (.03) υ (1.759) Skewness -7 Kurosis Jarque-Bera Q (10) Q (10)

21 Table 2. Esimaed n-garch(1,1) and -GARCH(1,1) models for reurns on marke indexes. Model parameers Dow-Jones FTSE NIKKEI S&P n-garch(1,1) ψ (.276) (.064) (.259) (.053) α (.101) (.041) (.019) (.03) β.690 (.109).833 (.04).939 (.044).898 (.03) Skewness Kurosis Jarque-Bera Q (10) Q (10) (.09).195 (.183).061 (.055) (.039) (.02) (.021) (43) (34) (.025) (5.417) (5.123) (4.074) Skewness Kurosis GARCH(1,1) ψ.540 (.28) α.147 (.05).759 β (82) υ (2.307) Jarque-Bera Q (10) Q (10) EGARCH(1,1) ψ α β (.053).158 (.061).926 (.029) (.050).181 (.062).953 (.018).373 (.172).015 (.05).824 (.086) (.042).154 (.06).954 (.02) γ (.039) (.032) (.042) (.03) Skewness Kurosis Jarque-Bera Q(10) Q (10)

22 Figure 1. Raio of rolling one-sep-aead average losses from differen variance models for excange rae series. Canada/$ n-garch(1,1) Canada/$ -GARCH(1,1) Canada/$ Nonparameric Model ps_cm $/Pound n-garch(1,1) $/Pound -GARCH(1,1) $/Pound Nonparameric Model 0.1 Yen/$ n-garch(1,1) Yen/$ -GARCH(1,1) Yen/$ Nonparameric Model 22

23 Figure 2. Raio of rolling one-sep-aead average losses from differen variance models for index reurns. DJIA, n-garch(1,1) DJIA,-GARCH(1,1) DJIA, e-garch(1,1) DJIA, nonparameric Model FTSE, n-garch(1,1) FTSE, -GARCH(1,1) FTSE, e-garch(1,1) FTSE, nonparameric Model NIKKEI, n-garch(1,1) NIKKEI, -GARCH(1,1) NIKKEI, e-garch(1,1) NIKKEI, nonparameric Model S&P500, n-garch(1,1) S&P500, -GARCH(1,1) S&P500, e-garch(1,1) S&P500, nonparameric Model 23

24 Figure 3. Raio of rolling four-seps-aead average losses from n-garc(1,1) models for index and excange rae reurn series. DJIA, n-garch(1,1),4-seps-aead FTSE, n-garch(1,1), 4-seps-aead _DOW _DOW _FTSE _FTSE NIKKEI, n-garch(1,1), 4-seps-aead _NIK _NIK S&P500, n-garch(1,1), 4-seps-aead _SP _SP Canada/$, n-garch(1,1), 4-seps-aead _CAN _CAN $/Pound, n-garch(1,1), 4-seps-aead _POUND _POUND Yen/$, n-garch(1,1), 4-seps-aead _YEN _YEN 24

25 Figure 4. Raio of Average Losses for excange raes, n-bekk GARCH(1,1). pseu/cm Figure 5. Raio of Average Losses for excange raes, -BEKK GARCH(1,1). pseu/cm 25

26 Appendix: A.1. Proof: Te opimal predicor Y ˆ + minimizes e expeced loss E [ L( Y + Yˆ + )] = L( Y Σ Ω + Yˆ ) φ ( 1 / 2 + ( Y µ ) ) dy were φ ( ) denoes e mulivariae sandard normal densiy and e inegral sign denoes an n-fold inegral over e elemens of. Le = Y µ denoe Y + X e observaions deviaion from is condiional mean. Canging variables, were deerminan of e Jacobian of e ransformaion is one, e objecive funcion can be expressed in deviaion from mean as 1/ 2 α )] = L( X α ) φ( Σ X Ω E [ L( X ) dx were α = ˆ µ is cosen o be e opimal predicor of. Te Y objecive funcion does depend on e condiional mean + X + µ, and erefore, e opimal predicor only depends on e loss funcion L ( ) and e condiional variance-covariance marix Σ +. Given a α is e opimal predicor of X +, e opimal predicor of is µ + + α. Y A.2. ai yi, yˆ i, if yi, yˆ i, > 0 L ( yi, yˆ i, ) =. bi yi, yˆ i, if yi, yˆ i, 0 Te firs order condiions are E[ L( y yˆ + i, yˆ ) = b i yˆ i, + f i ( yi, ) dy i, a f i ( yi i yˆ i, +, ) dy i, = 0 i = 1,..., n or b F yˆ ) a [1 F ( yˆ )] 0, wic can be solved as i i ( i, i i i, = y ˆ = F 1 ( a /( a + b )). If i, i i i y i, + is condiionally normal, en F i ( yi, ) = Φ(( yi, i, ) / σ ii, ) µ and e opimal predicor vecor can be wrien as y ˆ = µ + Φ 1 ( a /( a + b )). i, i, σ ii, i i i 26

27 References: Baillie, T. Ricard, Bollerslev, T. (1992), Predicion in Dynamic Models wi ime- Dependen Condiional Variances, Journal of Economerics, 52, Bollerslev, T. (1986), Generalized Auoregressive Condiional Heeroscedasiciy, Journal of Economerics, 31, Bollerslev, T., Engle, R.F., Wooldridge, J.M. (1988), A capial Asse Pricing Model wi imevarying covariances, Journal of Poliical Economy, 96, Brailsford, T. J., Faff, R. W. (1996), An evaluaion of Volailiy Forecasing Tecniques Journal of Banking and Finance, 20, Cirisoffersen, P. F. and Diebold F. X. (1996), Furer resuls on Forecasing and Model Selecion Under Asymmeric Loss, Journal of Applied Economerics, Vol. 11, No. 5, Special Issue: Economeric Forecasing (Sep.-Oc., 1996), Cirisoffersen, P. F. and Diebold F. X. (1997), Opimal Predicion Under Asymmeric Loss, Economeric Teory, 13, Diebold, F. X. and Nason J. A (1990), Nonparameric Excange Rae Predicion Journal of Inernaional Economics, 28, Engle, R. F. (1982), Auoregressive Condiional Heeroscedasiciy wi Esimaes of e Variance of U.K. Inflaion, Economerica, 50, Engle, R.F., Granger, C.W.J., Kraf, D.F. (1984), Combining compeing forecass of Inflaion using a bivariae ARCH model, Journal of Economic Dynamics and Conrol, 8, Engle, R.F., T. Bollerslev (1986), Modeling e Persisence of Condiional Variances, Economeric Reviews, 5, Granger, C. W. J. (1969), Predicion wi a generalized cos of error funcion, Operaional Researc Quarerly, 20, Granger, C. W. J. and P. Newbold (1986), Forecasing Economic Time Series, 2nd ed. Orlando: Academic Press. McMillan, D., Speig, A., Apgwilym, O. (2000), Forecasing UK Sock Marke volailiy, Applied Financial Economics, 10, Nadaraya, E. A. (1964), On Esimaing Regression, Teory of Probabiliy and is Applicaions, 9,

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29 i See Granger (1969), Granger and Newbold (1986, p.125) and Sockman (1987). ii However, as CD (1997) poin ou, e condiionally Gaussian assumpion can be relaxed. Te opimal predicor is obained by subsiuing e appropriae condiional CDF. iii However, as CD (1997) poin ou, e condiionally Gaussian assumpion can be relaxed. Te opimal predicor is obained by subsiuing e appropriae condiional CDF. iv CD (1996) finds abou 35% loss reducion in condiionally expeced loss for sor orizons, from using e opimal versus e pseudo-opimal predicors. 29