FORECASTING WITH A LINEX LOSS: A MONTE CARLO STUDY

Proceedings of he 9h WSEAS Inernaional Conference on Applied Mahemaics, Isanbul, Turkey, May 7-9, 006 (pp63-67) FORECASTING WITH A LINEX LOSS: A MONTE CARLO STUDY Yasemin Ulu Deparmen of Economics American Universiy of Beiru Bliss Sree, 1107-00, Beiru LEBANON Absrac: Using Mone-Carlo Simulaion, I compare he forecass of reurns from he opimal predicor (condiional mean predicor) for a symmeric quadraic loss funcion (MSE) wih he pseudo-opimal predicor and opimal predicor for an asymmeric loss funcion under he assumpion ha agens have asymmeric loss funcions. In paricular, I use he LINEX asymmeric loss funcion wih differen degrees of asymmery. I generae GARCH(1,1) processes wih differen persisence levels boh wih normally disribued errors. The resuls srongly sugges no o use he condiional mean predicor when agens have any kind of asymmery. The reducion in mean loss by using he opimal versus he pseudo-opimal predicor however depends on he degree of asymmery, and he persisence parameers being used Keywords: Volailiy, forecasing reurns, asymmeric loss, LINEX. 1. Inroducion In he lieraure, a widely used forecas evaluaion crieria is he MSE, which is a symmeric quadraic loss funcion. MSE penalizes he posiive errors and negaive errors of he same magniude equally. However, in finance forecasers do no necessarily have a quadraic cos funcion 1. Sudies have avoided using general asymmeric loss funcions mainly because mos of he ime he closed form for he opimal predicor does no exis. Granger (1969) showed ha he opimal predicor under asymmeric loss is he condiional mean plus a consan bias erm. Chirisoffersen and Diebold (1997, 1996) showed ha for condiionally Gausian processes if an agen has an asymmeric loss funcion, adding a consan erm is no sufficien and ha ime varying second order momens become relevan for opimal 1 See Granger (1969), Granger and Newbold (1986, p.15) and Sockman (1987). predicion. They derived he analyical expression for he opimal predicor for wo specific asymmeric, LINLIN and LINEX loss funcions. The LINLIN loss funcion is, firs used by Granger (1969), and LINEX loss funcion is inroduced by Varian (1974) and is used by Zellner (1986). For more general loss funcions hey showed how o approximae he opimal predicor numerically. I inroduce he LINEX asymmeric loss funcion, and he univariae variance model I use in Secion. Secion 3 Describes he Mone Carlo simulaions and resuls. Secion 4 is a conclusion.. Univariae Forecasing wih Asymmeric Loss.1. LINEX Loss Funcion: The LINEX, convex loss funcion is inroduced by Varian (1974) and used by Zellner (1986). L(x) = b[exp(ax) - ax -1], a R \{0}, b R + The LINEX loss funcion is approximaely

Proceedings of he 9h WSEAS Inernaional Conference on Applied Mahemaics, Isanbul, Turkey, May 7-9, 006 (pp63-67) linear on negaive x-axis and approximaely exponenial on posiive x-axis when a > 0. The parameer, a, deermines he shape of he loss funcion while parameer b scales i. Chirisoffersen and Diebold (1997) derived he opimal predicor, a pseudoopimal predicor and he expeced losses associaed wih each, for he LINEX asymmeric loss funcion under he assumpion of condiional normaliy. Given y + h Ω ~ N ( µ + h, σ + h ), hey showed he opimal predicor is y ˆ + h = µ + h + ( a / ) σ + h, and he pseudoopimal predicor is yˆ µ + ( a / σ + h = + h ) h, where σ is he uncondiional h -sep ahead h homoscedasic predicion error variance The pseudo-opimal predicor coincides wih he opimal predicor when σ h = σ + h. Through ou he paper I consider h = 1, which corresponds o one-sep a-head predicion. I consider differen degrees of asymmery o compare he loss associaed wih using differen predicors. Specifically, I fix b = 1 and change he values of a. I consider cases up o where a / b = 10. This asymmeric penalizaion scheme is plausible in finance.. GARCH Models: The mos commonly used model for ime-varying volailiy is he G/ARCH model of Engle (198) and Bollerslev (1986). A GARCH(1,1) model for he reurn on a financial asse, r, can be wrien (1) r = σ z, z ~ IID(0,1) σ = γ + α r 1 + β σ 1 where γ > 0, 0 α and β 0. I assume z has finie firs and second momens. For a normal GARCH(1,1) model, denoed n- GARCH(1,1), I assume an independen normal innovaion. The reurn r is weakly saionary if is variance is finie. This will be he case if α + β < 1. Since E( r Ι 1) = σ, where Ι 1 is he informaion se available a ime -1, he condiional variance σ is he minimum mean square error predicor of he realized volailiy r. 3. Mone Carlo Simulaions: In his secion I sudy he performance of he opimal, pseudo-opimal and condiional mean predicors when agens have asymmeric LINEX loss funcion by means of a Mone-Carlo simulaion. I look a he average loss raios associaed wih each predicor for differen degrees of asymmery. In he Mone Carlo analysis I consider wo scenarios. Firs I consider normal GARCH(1,1) wih differen parameer values keeping he persisence defined as α + β (Bollerslev, 1998) consan. I normalize he uncondiional variance o one. The Mone-Carlo simulaions are based on 10,000 replicaions. I hen use he simulaed n-garch(1,1) ime-varying condiional sandard deviaions o compue he opimal, pseudo-opimal and condiional mean predicors for each of he series over he forecas period. I use sample size of 50. This would represen a siuaion in which one is forecasing weekly daa an ou-ofsample period of one year. This is ypical in empirical work. In he Mone Carlo sudy, when I compue he opimal, pseudo-opimal and condiional mean predicors, I use he acual value of h raher han an esimaed value. In empirical work, ofen he majoriy of he daa is used o esimae he parameers in he condiional variance, and abou en percen of he daa is used for he ou-ofsample forecasing exercise. The GARCH (1,1) coefficiens, alpha and he bea are α = {0.3,0.15,0.5,0.10}, β = {.65,.75,.80,.85,.90} respecively. This case presens he performance of he predicors as he alpha and bea coefficiens vary for a given persisence level (.95) and differen degrees of asymmery. This high persisence level indicaes ha marke volailiy is

Proceedings of he 9h WSEAS Inernaional Conference on Applied Mahemaics, Isanbul, Turkey, May 7-9, 006 (pp63-67) predicable. I hen compue he average loss associaed wih each predicor and compue he average loss raios and % reducion in loss by using one ype of predicor versus he oher. Secondly, I change he degree of persisence and repea he same exercise. This case shows he sensiiviy of he performance of he predicors o he level of persisence presen in he daa. For his case I increase he persisence level o.99 by keeping β = {.65,.75,.80,.85,.90} he same bu increasing he ARCH coefficiens. Fig1. α =., β =. 75, loss raios from n- GARCH(1,1). 1. 1.0 0.8 0.6 0.4 0. 0.0 OPT/CM POPT/CM 10 0 30 40 50 60 70 80 90 The Figure 1 presens he raio of he average losses beween opimal and condiional mean predicors and he pseudoopimal and condiional mean predicors when α =. and β =. 75 for a persisence degree of 0.95. The condiional mean predicor performs he wors. The opimal predicor ou performs he pseudo opimal predicor even when a=1. The reducion in loss is around 30% for values of a equal o wo and greaer. We see ha a a equal o five, boh lines approach o zero. This is because he condiional mean predicor performs so poorly and he average loss associaed wih he condiional mean predicor is very large pulling he average loss raio boh wih he opimal and pseudoopimal predicors o zero. Similar resuls hold for he oher simulaed series wih differen GARCH(1,1) parameers, he condiional mean predicor performs he wors, driving he corresponding loss raios o zero. To avoid his spurious equal performance of opimal and pseudo-opimal predicors only he average loss reducion using he opimal versus he pseudo-opimal predicors will be demonsraed in he proceeding figures, for differen GARCH(1,1) models. Fig. Average % Loss reducion for differen values of alpha and bea coefficiens. % Loss Reducion 50 40 30 0 10 PCNT65 PCNT75 PCNT80 PCNT85 PCNT90 0 0 1 3 4 5 6 7 8 9 10 X Figure presens he average loss reducion using he opimal versus he pseudo-opimal predicor wih differen n- GARCH(1,1) parameers. The percen average loss reducion around 8% is minimum for β =. 90 and α =. 05 for an asymmery level of wo and greaer. The percen average loss reducion is maximum for α = 0.30 and β =. 65. The percen average loss reducion is maximum for α = 0.30 and β =. 65. I is abou 0% for a=1 and reaches o 50% for values of a equal o and greaer han hree. We see ha he percen average loss reducion increases wih an increasing level of asymmery and decreasing GARCH(1,1) bea coefficien. The percen average loss reducion is around 30% for α =. and β =. 75 when he degree of asymmery is wo or greaer. Even for low degree of asymmery he average loss reducion is around 0%. These parameer values are very represenaive of financial daa. So a praciioner migh expec around 0% loss reducion using he opimal versus he pseudo-opimal predicor for he given parameer values. 3

Proceedings of he 9h WSEAS Inernaional Conference on Applied Mahemaics, Isanbul, Turkey, May 7-9, 006 (pp63-67) Fig.3, Average % Loss reducion for α =.99, β = {.65,.75,.80,.85,.90}. % Loss Reducion 400 300 00 100 0 PCNT65 PCNT75 PCNT80 PCNT85 PCNT90 10 0 30 40 50 60 70 80 90 x Figure 3 presens he percen loss reducion for he n-garch(1,1) for a persisence level of.99 by increasing he values of alpha and keeping β = {.65,.75,.80,.85,.90} as before. As he persisence increases we see ha he percen loss reducion increases as well. The percen average loss reducion ha was 8% for β =. 90 and α =. 05 for a persisence level of.95 before increases o 70 % for an asymmery level of wo and more, for β =.90 and α =. 09. This is very ineresing because i shows ha if you have high persisence using he opimal versus he pseudo-opimal predicor reduces he loss considerably even for low degrees of asymmery and for any GARCH(1,1) coefficiens ypical of empirical work. Again he average percen loss reducion is maximum for β =. 65 and α =. 34 reaching more hen 00% even for low degrees of asymmery. 4. Conclusion I consider he mean losses associaed wih using he opimal predicor, pseudo-opimal predicor and he condiional mean predicor when agens have asymmeric LINEX loss funcion. My resuls provide srong empirical evidence o he Granger (1969) and CD(1997). The condiional mean predicor performs very poorly compared o he opimal and he pseudo- opimal predicors. For all series, loss associaed wih using he condiional mean predicor versus using he pseudo or he opimal predicor is considerably higher even for moderae degrees of asymmery, regardless of he differen n-garch(1,1) parameers. This resul suggess ha if agens have any kind of asymmery, he condiional mean predicor should no be used a all. The opimal predicor ou-performs he pseudo-opimal predicor in all he n- GARCH(1,1) series considered. However, he percenage reducion in loss is very sensiive o he n-garch(1,1) parameers being used. The average percen reducion in loss from using he opimal versus he pseudo-opimal predicor increases wih increasing asymmery, increasing persisence and decreasing bea coefficien. For an empirical represenaive n- GARCH(1,1) model wih parameers alpha=0., bea=0.75, he average percen reducion is around 0%. This resul suggess ha when agens have LINEX ype asymmeric loss funcion, even for low degrees of asymmery he opimal predicor ha incorporaes he ime varying second momens mus be used. References: [1] Bollerslev, T. (1986), Generalized Auoregressive Condiional Heeroscedasiciy, Journal of Economerics, 31, 307-37. [] Bollerslev, T., Engle, R.F., Wooldridge, J.M. (1988), A capaial Asse Pricing Model wih imevarying covariances, Journal of Poliical Economy, 96,116-131. [3] Chirisoffersen, P. F. and Diebold F. X.(1996), Furher resuls on Forecasing and Model Selecion Under Asymmeric Loss, Journal of Applied Economerics, Vol. 11, No. 5, Special Issue: Economeric Forecasing (Sep.-Oc., 1996), 561-571. [4] Chirisoffersen, P. F. and Diebold F. X. (1997), Opimal Predicion Under Asymmeric Loss, Economeric Theory, 13, 808-817. 4

Proceedings of he 9h WSEAS Inernaional Conference on Applied Mahemaics, Isanbul, Turkey, May 7-9, 006 (pp63-67) [5] Diebold, F. X. and Nason J. A (1990), Nonparameric Exchange Rae Predicion Journal of Inernaional Economics, 8, 315-33. [6] Engle, R. F. (198), Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of U.K. Inflaion, Economerica, 50, 987-1008. [7] 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, 151-165. [8] Engle, R.F., T. Bollerslev (1986), Modelling he Persisence of Condiional Variances, Economeric Reviews,5, 1-50. American Saisical Associaion, Vol. 81, No. 394. [9] Granger, C. W. J. (1969), Predicion wih a generalized cos of error funcion, Operaional Research Quarerly, 0, 199-07. [10] Granger, C. W. J. and P. Newbold (1986), Forecasing Economic Time Series, nd ed. Orlando: Acedemic Press. [11] Nadaraya, E. A. (1964), On Esimaing Regression, Theory of Probabiliy and is Applicaions, 9,141-14. [1] Pagan, A. R. and Schewer G. W. (1990) Alernaive Models for Condiional Sock Volailiy, Journal of Economerics, 45, 67-90. [13] Pagan A. and Ullah, A. (1999), Nonparameric Economerics Cambridge. [14] Silverman, B.W. (1986), Densiy Esimaion for Saisics and Daa Analysis, Chapman and Hall. [15] Sockman, A.C. (1987), Economic Theory and Exchange Rae Forecass, Inernaional Journal of Forecasing, 3, 3-15. [16] Wason, G. S. (1964), Smooh Regression Analysis, Sankhya, Series A, 6,359-37. [17] Varian, H.(1974), A Bayesian Approach o Real Esae Assessmen. In S.E. Feinberg and A. Zellner(eds.), Sudies in Bayesian Economerics and Saisics in Honor of L.J. Savage, 195-08. [18] Zellner, A. (1986), Bayesian Esimaion and Predicion Using Asymmeric Loss Funcions, Journal of 5