OPTIMAL PREDICTION UNDER ASYMMETRIC LOSS 1. By Peter F. Christoffersen and Francis X. Diebold 2 1. INTRODUCTION

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1 OPTIMAL PREDICTION UNDER ASYMMETRIC LOSS 1 By Peter F. Cristoffersen and Francis X. Diebold Cristoffersen P. and Diebold F.X. (1997) "Optimal Prediction Under Asymmetric Loss" Econometric Teory Keywords: Forecasting loss function asymmetric loss nonlinear eteroskedasticity 1. INTRODUCTION A MOMENT'S REFLECTION yields te insigt tat prediction problems involving asymmetric loss structures arise routinely as a myriad of situation-specific factors may render positive errors more (or less) costly tan negative errors. Te potential necessity of allowing for asymmetric loss as long been acknowledged. Granger and Newbold (1986) for example note tat altoug "an assumption of symmetry about te conditional mean... is likely to be an easy one to accept... an assumption of symmetry for te cost function is muc less acceptable" (p. 15). Practitioners routinely eco tis sentiment (e.g. Stockman 1987). In tis paper we treat te prediction problem under general loss structures building on te classic work of Granger (1969). In Section we caracterize te optimal predictor for non- Gaussian processes under asymmetric loss. Te results apply for example to important classes of conditionally eteroskedastic processes. In Section 3 we provide analytic solutions for te optimal predictor under two popular analytically-tractable asymmetric loss functions. In Section 4 we provide metods for approximating te optimal predictor under more general loss functions. We conclude in Section 5.. OPTIMAL PREDICTION FOR NON-GAUSSIAN PROCESSES Granger (1969) studies Gaussian processes and sows tat under asymmetric loss te optimal predictor is te conditional mean plus a constant bias term. Granger's fundamental result owever as two key limitations. First te Gaussian assumption implies a constant

2 -- conditional prediction-error variance. Tis is unfortunate because conditional eteroskedasticity is widespread in economic and financial data. Second te loss function must be of predictionerror form; tat is L(y t ) L(y t ) L(e t ) were y t is te -step-aead realization is te -step-aead forecast (made at time t) and is te corresponding forecast error. More general functions of realizations and predictions are excluded. Let us begin ten by generalizing Granger's result to allow for conditional variance dynamics. We acieve tis most simply by working in a conditionally-gaussian but not necessarily unconditionally-gaussian environment wit prediction-error loss. Subsequently we sall allow for bot conditional non-normality and more general loss functions. e t PROPOSITION 1: If y N(µ ) is a conditionally Gaussian process and L(e t ) is any loss function defined on te -step-aead prediction error e t ten te optimal predictor is of te form µ were depends only on te loss function and te conditional prediction-error variance var(y ) var(e ). PROOF: See Appendix. Te optimal predictor under conditional normality is not necessarily just a constant added to te conditional mean because te conditional prediction-error variance may be time-varying. Conditionally Gaussian GARCH processes for example fall under te jurisdiction of Proposition 1. Tus under asymmetric loss conditional variance dynamics are important not only for interval prediction but also for point prediction. If loss is asymmetric but conditional eteroskedasticity is ignored te resulting point predictions will be suboptimal and may ave dramatically greater conditionally expected loss in consequence. Te result of Proposition 1 tat te "adjustment factor" depends only on te conditional variance depends crucially on conditional normality. We can dispense wit conditional

3 -3- normality and still obtain a sarp result owever wic is a straigtforward extension of Proposition 1. PROPOSITION : If y as conditional mean µ and a vector of (possibly time varying) conditional moments of order two and iger and L(e t ) is any loss function t+ t defined on te -step-aead prediction error e t ten te optimal predictor is of te form µ were depends only on te loss function and t+ t. PROOF: See Appendix. Note owever tat altoug Propostion does not require a Gaussian process it does require prediction-error loss. In Section 4 we will relax tat assumption as well. 3. ANALYTIC SOLUTIONS UNDER LINEX AND LINLIN LOSS Here we examine two asymmetric loss functions ("linex" and "linlin") for wic it is possible to solve analytically for te optimal predictor. To maintain continuity of exposition we work trougout tis section wit te conditionally Gaussian process y N(µ ). 3 For eac loss function we caracterize te optimal predictor µ and we compare its conditionally expected loss to tat of two competitors te conditional mean µ and te pseudo-optimal predictor µ were depends only on te loss function and te unconditional prediction-error variance var(e t ). Te optimal predictor acknowledges loss asymmetry and te possibility of conditional eteroskedasticity troug a possibly time-varying adjustment to te conditional mean. Te conditional mean in contrast is always suboptimal as it incorporates no adjustment. Te pseudo-optimal predictor is intermediate in tat it incorporates only a constant adjustment for asymmetry; tus it is fully optimal only in te conditionally omoskedastic case t Linex Loss

4 -4- Te "linex" loss function introduced by Varian (1974) and used by Zellner (1986) is L(x) b exp(ax) ax 1 a {0} b. It is so-named because wen a>0 loss is approximately linear to te left of te origin and approximately exponential to te rigt and conversely wen a<0. Te optimal -step-aead predictor under linex loss solves min E t b exp(a(y t )) a(y t ) 1. Differentiating and using te conditional moment-generating function for a conditionally a Gaussian variate we obtain µ. Similar calculations reveal tat te pseudooptimal predictor is were a µ var(e t ) is te unconditional -step-aead prediction-error variance. Proposition 1 sows tat te optimal predictor under conditional normality is te conditional mean plus a function of te conditional prediction-error variance. Under linex loss te function is a simple linear one depending on te degree of asymmetry of te loss function as 4 captured in te parameter a. Te reason is simple--wen a is positive for example positive prediction errors are more devastating tan negative errors so a negative conditionally expected error is desirable. Te optimal amount of bias depends on te conditional prediction-error variance of te process; as it grows so too does te optimal amount of bias in order to avoid large positive prediction errors. Effectively optimal prediction under asymmetric loss corresponds to conditional-mean prediction of a transformed series were te transformation reflects bot te loss function and te iger-order conditional moments of te original series.

5 For example te optimal predictor of y under conditional normality and linex loss t+ µ a -5- x a is te conditional mean of x t+ were t y t. 5 Inserting te optimal pseudo-optimal and conditional mean predictors into te conditionally expected loss expression we see tat te conditionally-expected linex losses are ba / b[exp(a ( )/) a / 1] b[exp(a and /) 1] respectively. By construction te conditionally expected loss of te optimal predictor is less tan or equal to tat of any oter predictor. Interestingly owever it is not possible to rank te pseudo-optimal as superior to te conditional mean predictor. Tedious but straigtforward algebra reveals tat for sufficiently small values of (depending non-linearly on te values of a and ) te conditionally expected loss of te conditional mean will be smaller tan tat of te pseudooptimal predictor. In very low volatility times te conditionally optimal amount of bias is very small resulting in a lower conditionally expected loss for te conditional mean tan for te pseudo-optimal predictor te bias of wic is optimal in "average" times but too low in lowvolatility times. Te situation is illustrated in Figure 1 in wic we plot conditionally expected linex loss as a function of for eac of te tree predictors. Te conditionally expected loss of te optimal predictor is linear in and is of course always lowest. Te losses of te pseudooptimal and te optimal predictors coincide wen 1. As falls below te loss of te conditional mean intersects te loss of te pseudo-optimal predictor from above. As gets close to zero te optimal predictor incorporates progressively smaller corrections to te conditional mean so te conditionally expected losses of te optimal and conditional mean predictors coincide. 3.. Linlin Loss

6 -6- Te "linlin" loss function L(y t ) a y t if (y t ) > 0 b y t if (y t ) 0 so-called because of its linearity on eac side of te origin was used by Granger (1969) and is te loss function underlying quantile regression. Te optimal predictor solves min a (y t )f(y )dy t b (y t )f(y )dy t. Te first-order condition is F( t ) conditional density of y. t+ a(1 F( t )) b F( t ) 0 a a b were F(y ) is te conditional c.d.f. of y and f(y ) t+ is te In te conditionally Gaussian case we ave from Proposition 1 tat wic is equivalent to F( t ) Pr (y t (µ t )) t Pr y t µ t t t a a b were (z) is te N(01) c.d.f. It follows tat te conditionally optimal amount of bias is a b ŷ so tat t µ a b. tat te pseudo-optimal predictor is µ a b. Similar calculations reveal Now let us compute conditionally expected linlin loss for te optimal pseudo-optimal and conditional mean predictors. Recall te formulae for te truncated expectation 6 y t f(y t E t {y t (y t > )} y t f(y )dy t 1 F( t ) E t {y t (y t < )} F( t ) t )dy t

7 and substitute tem into te expected loss expression to obtain -7- {L(y t )} a(1 F( t ))[E t y t (y t > ) ] bf( t )[E t y t (y t < ) But under conditional normality ( E t (y t (y t > )) µ ) 1 ( ) E (y t t (y <ŷ )) µ µ were and ( ) is te N(01) p.d.f. Substituting into te conditionally expected loss expression we obtain (after some algebraic manipulation) ( ) ( ) E t (L(y t )) (a b) ( ) a( t µ ) (a b) ( )( t µ ). For te optimal predictor (a b) a b yielding an expected loss of (a b) a b a b. For te pseudo-optimal predictor For te conditional mean predictor a a b 0 yielding an expected loss of (a b) a b yielding an expected loss of a b a b. E t (L(y t )) (a b) /. Qualitatively te situation is identical to tat sown in Figure 1 for te linex case. 4. APPROXIMATING THE OPTIMAL PREDICTOR Te analytic results above rely on simple loss functions. In general owever it is not possible to solve analytically for te optimal predictor. Here we develop an approximately optimal predictor via series expansions. Te approac is of interest because it frees us from two potentially restrictive assumptions -- conditional normality and prediction-error loss. For te moment maintain te conditional normality assumption and assume tat te optimal predictor exists and is unique G(µ ) were G( ) is at least twice

8 -8- continuously differentiable. Ten we can take a second order Taylor series expansion around te unconditional (and time invariant) moments µ and G(µ µ ) G (µ ) µ 1 (µ µ ) G (µ ) µ µ. Rewrite tis as 0 1 µ 3 (µ ) 4 ( ) 5 (µ ) y t ( ) were ( ) and i H i (µ ) i Because te function G( ) is generally unknown so too are te H( ) functions. But µ and are known and te ˆ N ˆ N minimization tat defines can be done over a very long simulated realization of lengt N argmin N L(y t y t ( )). Under regularity conditions given in te Appendix te t 1 following proposition is immediate. PROPOSITION 3: As N y t (ˆ N) y t ( 0 ) were y t ( 0 ) is te best predictor witin te y t ( ) family wit respect to te metric L( ). PROOF: See Appendix. A number of remarks are in order. First te -step-aead conditional expectation and te corresponding conditional variance may be computed conveniently using te Kalman filter recursions. Second if loss is in fact of prediction-error form L(e ) one may set = nd = t = 0 a priori due to Proposition 1. Tird it is clear tat iger-order expansions in and µ may be entertained and may lead to improvements. Fourt conditional non-normality may be andled wit expansions involving more tan te first two conditional moments (e.g. involving conditional skewness and kurtosis). Fift and related parametric economy can be

9 -9- acieved in conditionally non-gaussian cases using te autoregressive conditional density framework of Hansen (1994). Hansen's framework exploits parametric conditional mean and variance functions but allows for iger-order conditional dynamics by letting te normalized variable z t ( ) y t µ ( ) / ( ) follow a distribution wit possibly time varying "sape" parameters suc as a t-distribution wit time-varying degrees of freedom (and variance standardized to 1). Sixt in bot te conditionally Gaussian and conditionally non-gaussian cases one is of course not limited to series expansions; oter nonparametric functional estimators may be used. 5. SUMMARY AND CONCLUDING REMARKS Tis paper is part of a researc program aimed at allowing for general loss structures in estimation model selection prediction and forecast evaluation. Recently a number of autors ave made progress toward tat goal including Weiss (1994) on estimation Pillips (1994) on model selection and Diebold and Mariano (1995) on forecast evaluation. Here we focused on prediction and analyzed te optimal prediction problem under asymmetric loss. We computed te optimal predictor analytically in two leading tractable cases and sowed ow to compute it numerically in less tractable cases. A key teme is tat te conditionally optimal forecast is biased and tat te conditionally optimal amount of bias is time-varying in general and depends on iger-order conditional moments. Tus even for models wit linear conditional-mean structure te optimal predictor is in general nonlinear tereby providing a link wit te broader nonlinear time series literature. Interestingly some important recent work in dynamic economic teory is very muc linked to te idea of prediction under asymmetric loss discussed ere. Building on Wittle (1990) Hansen Sargent and Tallarini (1993) set up and motivate a general-equilibrium economy

10 -10- wit "risk sensitive" preferences resulting in equilibria wit certainty-equivalence properties. Tus te prediction and decision problems may be done sequentially--but prediction is done wit respect to a distorted probability measure tat yields predictions different from te conditional mean. University of Pennsylvania APPENDIX PROOF OF PROPOSITION 1: We seek te predictor tat solves min E t L(y t ) min L(y t ) f(y ) dy t. (Here and trougout E t (x) denotes E(x t ). ) Witout loss of generality we can write µ and y t µ x t so tat argmin E t L(y t ) µ argmin L(x t ) f(x ) dx t. Because f(x ) depends on but not µ so too does te tat solves te minimization problem depend on but not µ. Q.E.D. PROOF OF PROPOSITION : Precisely parallels tat of Proposition 1. Q.E.D. PROOF OF PROPOSITION 3: Following Amemiya (1985) we require tree conditions: k (1) 0 a compact subset of. N () L N ( ) L(y t y t ( )) is continuous in for all y=(y...y ) and is a t 1 1+ N+

11 uniformly in as N and L( ) attains a unique global minimum at. argmin Under te conditions LN ( ) converges in probability to by te argument of ˆ N -11- measurable function of y for all. -1 (3) N L N( ) converges to a nonstocastic continuous function L( ) in probability Amemiya (1985 p. 107). Tus y t (ˆN) converges in probability to y t ( 0 ) by continuity of 0 0 y t (ˆN). Q.E.D. REFERENCES Amemiya T.: Advanced Econometrics. Cambridge Mass.: Harvard University Press Cristoffersen P.F. and Diebold F.X. (1994): "Optimal Prediction Under Asymmetric Loss" National Bureau of Economic Researc Tecnical Working Paper No. 167 Cambridge Mass. Diebold F.X. and R.S. Mariano (1995): "Comparing Predictive Accuracy" Journal of Business and Economic Statistics Granger C.W.J. (1969): "Prediction wit a Generalized Cost of Error Function" Operational Researc Quarterly Granger C.W.J. and P. Newbold: Forecasting Economic Time Series (Second edition). Orlando: Academic Press Hansen B.E. (1994): "Autoregressive Conditional Density Estimation" International Economic Review Hansen L.P. T.J. Sargent and T.D. Tallarini (1993): "Pessimism Neurosis and Feelings About Risk in General Equilibrium" Manuscript University of Cicago. Pillips P.C.B. (1994): "Bayes Models and Macroeconomic Activity" Manuscript Yale

12 -1- University. Stockman A.C. (1987): "Economic Teory and Excange Rate Forecasts" International Journal of Forecasting Varian H.: "A Bayesian Approac to Real Estate Assessment" in Studies in Bayesian Econometrics and Statistics in Honor of L.J. Savage ed. by S.E. Feinberg and A. Zellner. Amsterdam: Nort-Holland Weiss A.A. (1994): "Estimating Time Series Models Using te Relevant Cost Function" Manuscript Department of Economics University of Soutern California. Wittle P.: Risk-Sensitive Optimal Control. New York: Jon Wiley Zellner A. (1986): "Bayesian Estimation and Prediction Using Asymmetric Loss Functions" Journal of te American Statistical Association

13 -13- FOOTNOTES 1. Tis paper is a eavily-revised and sortened version of parts of Cristoffersen and Diebold (1994) wic may be consulted for additional results discussion and examples.. We benefitted from constructive comments from te Co-Editor and two referees as well as from Clive Granger Hasem Pesaran Enrique Sentana Bob Stine Jim Stock Ken Wallis and numerous conference and seminar participants. Remaining inadequacies are ours alone. We tank te National Science Foundation te Sloan Foundation te University of Pennsylvania Researc Foundation and te Barbara and Edward Netter Fellowsip for support. 3. As will be made clear owever altoug conditional normality is crucial to our derivation of te optimal predictor under linex loss it may readily be discarded under linlin loss. 4. Note tat as a 0 te conditionally optimal amount of bias approaces zero. Quadratic loss obtains as a 0 because if a is small one can replace te exponential part of te loss function by te first two terms of its Taylor series expansion yielding te approximation L(x) x. 5. Because y is conditionally normal wit E[y ] µ x is conditionally normal wit t+ a E[x ] µ ŷ t. 6. Note tat wit linlin loss (in contrast to linex loss) it is very easy even for non-gaussian conditional distributions to find te optimal predictor -- just draw te conditional c.d.f. and read te value on te x-axis corresponding to a/(a+b). More formally is simply te (a/(a+b))t conditional quantile. Wen a=b of course median. t+ F a b t so is te conditional

14 Figure 1 Conditionally Expected Linex Loss of Conditional Mean Pseudo-Optimal and Optimal Predictors Notes to Figure: Te Linex loss parameters are set to a=nd b=. Te unconditional variance is fixed at 1.