Forecasing Daily Supermarke Sales Using Exponenially Weighed Quanile Regression James W. Taylor Saïd Business School Universiy of Oxford European Journal of Operaional Research, 2007, Vol. 178, pp. 154-167. Address for Correspondence: James W. Taylor Saïd Business School Universiy of Oxford Park End Sree Oxford OX1 1HP, UK Tel: +44 (0)1865 288927 Fax: +44 (0)1865 288805 Email: james.aylor@sbs.ox.ac.uk
Forecasing Daily Supermarke Sales Using Exponenially Weighed Quanile Regression Absrac Invenory conrol sysems ypically require he frequen updaing of forecass for many differen producs. In addiion o poin predicions, inerval forecass are needed o se appropriae levels of safey sock. The series considered in his paper are characerised by high volailiy and skewness, which are boh ime-varying. These feaures moivae he consideraion of forecasing mehods ha are robus wih regard o disribuional assumpions. The widespread use of exponenial smoohing for poin forecasing in invenory conrol moivaes he developmen of he approach for inerval forecasing. In his paper, we consruc inerval forecass from quanile predicions generaed using exponenially weighed quanile regression. The approach amouns o exponenial smoohing of he cumulaive disribuion funcion, and can be viewed as an exension of generalised exponenial smoohing o quanile forecasing. Empirical resuls are encouraging, wih improvemens over radiional mehods being paricularly apparen when he approach is used as he basis for robus poin forecasing. Keywords: Forecasing; Exponenial smoohing; Quanile regression; Inerval forecasing; Robus poin forecasing
1. Inroducion When forecasing for invenory conrol, predicions are very ofen required a frequen inervals for many differen producs or pars. This moivaes he auomaion of some or all of he forecasing process. The case ha we consider is one of forecasing daily observaions of supermarke sales. The daa is aken from an oule of a large UK supermarke chain, which ypically socks more han 10,000 differen iems in each of is sores. The 256 ime series in our sample are characerised by high volailiy, srong inraweek seasonaliy and, for mos of he series, fairly low volume. Few of he series exhibi rend. The high volailiy is eviden in Figure 1, which is a plo of one of he series. The figure also shows ha sales is posiively skewed for many of he days, which is undersandable, given ha he series is obviously bounded below by zero. ---------- Figure 1 ---------- Exponenial smoohing is a simple and pragmaic approach o poin forecasing whereby he forecas is consruced from an exponenially weighed average of pas observaions. The impressive performance of exponenial smoohing mehods, when applied across a variey of differen series (see, for example, Makridakis and Hibon, 2000), has led o heir widespread use in applicaions, such as ours, where a large number of series necessiaes an auomaed procedure. Indeed, our collaboraing company currenly uses exponenial smoohing, albei a non-sandard mehod. Inerval forecasing is imporan for invenory conrol because i enables he seing of appropriae levels of safey sock. A common approach o inerval forecasing is o use a poin forecas wih an esimae of he forecas error disribuion. For he widely used exponenial smoohing mehods, Hyndman e al. (2005) provide heoreical forecas error variance formulae, which hey derive by referring o he equivalen sae-space models. A Gaussian assumpion is hen used o deliver predicion inervals. However, he Gaussian assumpion is ofen no realisic. Indeed, he shape of he disribuion may no even be 1
consan over ime. This appears o be he case in Figure 1, where he skewness is more eviden when he series is closer o zero. The variance is very ofen also no consan over ime, and may possess a seasonal paern independen of seasonaliy in he mean. A problem wih he use of heoreical error variance formulae is ha hey are no readily available for non-sandard exponenial smoohing approaches, such as he one used by our collaboraing company. In reviewing he predicion inerval lieraure, Chafield (1993) observes ha when heoreical formulae are no available, or here are doubs abou model assumpions, empirically based mehods should be considered. Gardner (1988) and Taylor and Bunn (1999) provide such mehods, bu, unforunaely, hese mehods are no able o accommodae a disribuion wih changing shape or variance. In essence, he supermarke sales case presens he challenge of rying o produce, in an auomaed way, inerval forecass for disribuions ha have changing locaion, variance and shape. Raher han base esimaion on a poin forecas, in his paper, we address he problem by direcly forecasing he sales quaniles. The θ quanile of a variable, y, is defined as he value, Q (θ), for which P(y Q (θ))=θ. Inerval forecass can be consruced from forecass of symmeric quaniles, Q (θ) and Q (1-θ). Direcly forecasing quaniles avoids he need for assumpions regarding he spread and shape of he disribuion. In view of he success of exponenial smoohing mehods for poin forecasing wihin auomaed applicaions, here is srong appeal o developing he approach for inerval forecasing. We do his hrough he use of exponenially weighed quanile regression (EWQR), which is equivalen o exponenial smoohing of he cumulaive disribuion funcion (cdf). Our applicaion of EWQR provides new insigh ino he mehod because he only previous use of he approach was for he somewha differen applicaion of value a risk esimaion in finance (see Taylor, 2006). The non-gaussian naure of he sales series also moivaes consideraion of poin forecasing mehods ha are robus o he disribuional shape. We propose ha robus poin 2
forecass be consruced from he weighed average of EWQR forecass of several quaniles, such as he 0.25, 0.50 and 0.75 quaniles. Secion 2 reviews he exising approaches o univariae quanile modelling. In Secion 3, we discuss EWQR and is use for univariae quanile forecasing and robus poin forecasing. In Secion 4, we use he mehod for poin forecasing of he supermarke series, and in Secion 5 we consider inerval forecasing. Secion 6 provides a summary and concluding commens. 2. Quanile modelling mehods The value a risk (VaR) lieraure is concerned wih esimaing he ail quaniles of disribuions of financial reurns. By conras wih our sales series, a series of financial reurns ends o have a mean ha is almos consan. Neverheless, some of he VaR mehods can a leas provide insigh for our case. Boudoukh e al. (1998) describe heir approach as being he analogy of exponenially weighed moving averages for quaniles. Their mehod involves allocaing o he mos recen year of daily reurns, exponenially decreasing weighs, which sum o one. The reurns are hen placed in ascending order and, saring a he lowes reurn, he weighs are summed unil θ is reached. The θh quanile esimae is se as he reurn ha corresponds o he final weigh used in he previous summaion. Linear inerpolaion is used if he esimae falls beween wo reurns. Engle and Manganelli s (2004) condiional auoregressive value a risk (CAViaR) is a class of quanile modelling mehods. The following is he adapive CAViaR mehod: [ θ I( y Q ( ))] Q ( θ ) = Q ( θ ) + β 1 < 1 1 θ where y is he reurn; β is a consan parameer; and I( ) is he indicaor funcion, which akes a value of one if he expression wihin he parenhesis is rue, and zero oherwise. The indicaor funcion has he effec of reducing he esimae of he nex quanile if he curren quanile esimae is greaer han he reurn. If he reurn exceeds he quanile esimae, he 3
nex esimae is increased. If we assume ha he probabiliy of he error falling below he θ quanile esimaor is θ, hen he expeced value of he expression wihin he square parenheses is zero. I hen follows ha he muliperiod forecass are equal o he one sepahead predicion. CAViaR mehod parameers are esimaed using he quanile regression minimisaion, which we presen in Secion 3.2. The oher CAViaR mehods are unsuiable for our supermarke sales series because, for hese mehods, saionariy is assumed and muliperiod forecasing is no sraighforward. Expressions (2.1) and (2.2) presen he mehod of Gorr and Hsu (1985), which was developed for non-financial daa. The formulaion is similar o he adapive CAViaR mehod, excep ha in expression (2.2), he Gorr and Hsu mehod sofens he impac of he indicaor funcion hrough he use of exponenial smoohing. If α=1, he wo mehods are idenical. Q ( ) ( ) [ ˆ θ = Q 1 θ + β θ θ 1 ] (2.1) ˆ θ α I y < Q θ + (1 α) ˆ θ (2.2) where 1 = ( 1 1 ( )) 2 3. Exponenially weighed quanile regression 3.1. Exponenially weighed quanile regression and he cdf I is well known ha he simple exponenial smoohing esimaor is idenical o he resul of an exponenially weighed (discouned) leas squares (EWLS) regression for a model wih consan erm bu no regressors (see, for example, Harvey, 1990, p. 28). Taylor (2006) proposes EWQR as he analogous approach for quanile forecasing. Koenker and Basse (1978) presen heoreical resuls for he use of quanile regression o esimae, for a dependen variable y, linear quanile models, Q (θ) = x β, where x is a vecor of regressors and β is a vecor of parameers. For a specified value of he weighing parameer, λ, Taylor s EWQR minimisaion has he form: 4
min β λ T y Q ( θ) y < Q ( θ) ( 1 θ ) T θ y Q ( θ ) + λ y Q ( θ ) (3.1) where T is he sample size, and λ [0,1] is a weighing parameer. The case of λ=1 corresponds o he sandard form of quanile regression, which Koenker and Basse (1978) show can be modelled as a linear program, provided he quanile model is linear. In a similar way, for a linear quanile model, he EWQR minimisaion of expression (3.1) can also be formulaed as a linear program. Taylor explains ha he resuling quanile esimaor, Q ˆ ( θ ), ha minimises expression (3.1), obeys expression (3.2), which involves an exponenially weighed summaion of an indicaor funcion. T = 1 T λ T I = 1 ( y < Qˆ ( θ )) T λ = θ (3.2) The EWQR quanile esimaor, Q ˆ ( θ), is hus defined as he esimaor ha pariions he y observaions so ha he sum of he weighs on hose observaions less han he corresponding quanile esimaor, as a proporion of he sum of all he weighs, is θ. If we noe ha θ is a realisaion of F, he cdf of y, expression (3.2) can be rewrien as an esimaor of he cdf: Fˆ T ( Qˆ ( )) T T = 1 ( y < Qˆ ( θ )) T λ I θ = = 1 T (3.3) T λ If T is large, such an expression can be rewrien in recursive form: ( Qˆ ( θ) ) α I( y < Qˆ ( θ) ) + (1 α) Fˆ ( Qˆ ( )) F ˆ (3.4) T T = T T T 1 T 1 θ where α = 1-λ. This shows ha EWQR amouns o simple exponenial smoohing of he cdf. The connecion beween his finding and quanile modelling is ha he inverse of he cdf is 1 he quanile funcion, ( θ) F ( θ ) Q. T = T 5
A convenien recursive formula does no exis for updaing he quanile esimae. Therefore, he EWQR linear program mus be solved afresh as each new observaion becomes available. Wih he developmen of compuaional power, his should no be a significan obsacle o implemenaion. For he case of EWQR on a consan wih no regressors, expression (3.3) becomes expression (3.5), which provides an esimae of he cdf for a specified value, y. Alhough his is a cdf esimaor, i can be used o esimae quaniles by ieraively evaluaing he righ hand side of he expression for differen values of y unil he desired value for he cdf esimaor, ˆ ( y) F T, is achieved o a required degree of olerance. Fˆ T ( y) T λ T = 1 = T = 1 I ( y < y) λ T (3.5) Esimaing he quanile using expression (3.5) is exacly equivalen o he VaR mehod of Boudoukh e al. (1998), described in Secion 2. Their mehod, herefore, is equivalen o EWQR wih a consan erm and no regressors. In Secion 2, we also briefly described he Gorr and Hsu (1985) adapive quanile mehod. Expression (2.2) of heir mehod is similar o he cdf exponenial smoohing expression (3.4). 3.2. Applying EWQR o he univariae ime series conex For he univariae forecasing of he quaniles of a ime series, he use of EWQR wih a consan and no regressors is one possibiliy. As wih simple exponenial smoohing, he muliperiod forecass would be idenical o he one sep-ahead predicion. In Taylor s (2006) applicaion of he approach o financial reurns, he only regressor considered in EWQR was a dummy variable, which was used o capure he asymmeric leverage effec in he volailiy. The applicaion of he mehod o sales daa moivaes consideraion of oher regressors. For a rending ime series, a rend erm can be included in he EWQR. Using EWLS o fi models ha are funcions of ime is ermed general exponenial smoohing (GES) (see 6
Gardner, 1985). The inclusion of funcions of ime in he EWQR is, herefore, he exension of GES o quanile forecasing. The use of a linear forecas funcion in exponenial smoohing mehods has been criicised because i ends o overshoo he daa beyond he shor-erm. Gardner and McKenzie (1985) address his problem by including an exra parameer in Hol s mehod o dampen he projeced rend. Empirical sudies show ha his ends o lead o improvemens in accuracy (e.g. Makridakis and Hibon, 2000). The EWQR minimisaion wih a damped linear rend is shown in expression (3.6). The values of he damping parameer, φ, and weighing parameer, λ, mus be specified before he minimisaion is performed. We discuss opimisaion of hese parameers in Secions 4 and 5. min a, b ( θ) T 1 T 1 T i T T i θ y ( a b φ ) + λ ( 1 θ ) y ( a b φ ) i= y Q ( θ) i= T λ y Q < (3.6) If he daa is seasonal, i could be deseasonalized prior o he EWQR. The resuling quanile forecass would hen need o be reseasonalized. A more direc approach o seasonaliy, which has been used in GES, is o include dummy variables or sinusoidal erms. Noe ha, if he seasonaliy is muliplicaive, hese erms mus be muliplied by he rend. 3.3. Using EWQR for robus poin forecasing In Secion 1, we explained ha he sales series ha we consider in his paper are characerised by high volailiy and skewness, and ha hese feaures may be varying over ime. Furhermore, Figure 1 shows ha he series occasionally have large values. These feaures moivae consideraion of poin forecasing mehods ha are robus o non-gaussian disribuions and oulying observaions. Dunsmuir e al. (1996) presen expression (3.5) in a sudy ha inroduces he idea of an exponenially weighed moving median (EWMM), which hey propose as a robus poin forecasing alernaive o he sandard exponenially weighed moving average. They use he expression o deermine he cdf esimaor, ˆ ( y) F T, for a range of values of y. The EWMM 7
esimae is hen se as he smalles y value for which ˆ ( y) F T 0.5. We would sugges ha a simpler way o derive he EWMM esimae is by running he EWQR for θ = 0.5, wih a consan and no regressors. Using θ = 0.5 is equivalen o weighed leas absolue deviaions regression, which was proposed by Cipra (1992) as a robus form of exponenial smoohing. Koenker and Basse (1978) describe how esimaors of differen quaniles can be combined o form a robus esimaor of he cenral locaion of a disribuion. I, herefore, seems naural o combine he EWQR quanile esimaors in a similar way o provide robus poin forecass. Koenker and Basse propose he use of Tukey s (1970) rimean and he Gaswirh (1966) esimaor. Judge e al. (1988) sugges an alernaive five-quanile esimaor. We presen hese esimaors in he following expressions: Trimean: y ˆ = 0.25Qˆ (0.25) + 0.5Qˆ (0.5) + 0.25Qˆ (0.75) Gaswirh : y = 0.3Qˆ ( 1 ) + 0.4Qˆ (0.5) + 0.3Qˆ ( 2 ) ˆ 3 3 Five-quanile: y ˆ = 0.05Qˆ (0.1) + 0.25Qˆ (0.25) + 0.4Qˆ (0.5) + 0.25Qˆ (0.75) + 0.05Qˆ (0.9) A Winsorised esimaor involves rimming a daase of is upper and lower 100α % of values (Tukey, 1962). In he ime series conex, all in-sample observaions below heir corresponding in-sample fied EWQR α quanile would be se equal o his value, and all insample observaions above heir corresponding in-sample fied EWQR (1- α) quanile would be se equal o his value. A sandard poin forecasing mehod, such as simple exponenial smoohing, would hen be applied o he resuling series, wih forecass produced from he simple exponenial smoohing mehod in he usual way. As each new observaion becomes available, he EWQR esimaed α and (1-α) quaniles would be updaed, he rimming would be performed for he new observaion, and hen he poin forecas updaed for he nex period. We are no aware of he previous use of any of he esimaors described in his secion for univariae forecasing, apar from he EWMM. 8
4. Empirical comparison of poin forecasing mehods 4.1. Descripion of he sudy In his secion, we compare he accuracy of esablished poin forecasing mehods wih he EWQR-based robus poin forecasing mehods discussed in he previous secion. The sudy used ime series of daily observaions of sales of differen iems from an oule of a large UK supermarke chain. As greaer ineres and benefi lies wih improved forecasing for higher volume series, we considered only hose series wih median daily sales greaer or equal o 5. We fel ha none of he forecasing mehods ha we wished o compare, including he company s own mehod, would be suiable for series ha conained a large number of successive days wih zero sales. Therefore, we did no include series ha consised of long periods of zeros or series ha conained only inermien observaions. This resuled in 256 series, which varied in lengh from 72 o 1436 observaions wih a median of 764 observaions. Figure 2 presens he hisogram for he lenghs of he series. ---------- Figures 2 o 5 ---------- In Figures 3 o 5, we presen summary informaion for he deseasonalised series using he same measures and hisograms considered by Fildes e al. (1998) for differen deseasonalised daa. To deseasonalise he series, we used muliplicaive seasonal decomposiion, which is based on raio-o-moving averages and was used in he M3- Compeiion (Makridakis and Hibon, 2000). The hisogram in Figure 3 shows ha he correlaion beween he deseasonalised series and a deerminisic linear rend is high for few of he series. The measure summarised in Figure 4 is he adjused R 2 for he regression of deseasonalised sales on a deerminisic linear rend and auoregressive erms of orders one, wo and hree. This is used by Fildes e al. as an approximae measure for he sysemaic variaion which can be forecas using simpler mehods. The resuls in Figure 4 are consisen wih our earlier suggesion ha he series are characerised by high volailiy. Figure 5 summarises he proporion of ouliers in he series, where an oulier is defined, as in 9
Fildes e al. They define an oulier o be any observaion for which he corresponding observaion in he differenced series is ouside he iner-quarile range of he differenced series by a disance ha is a leas 1.5 imes he widh of he iner-quarile range. As we explained in Secion 1, he series were characerised by inra-week seasonaliy and high volailiy, wih few displaying rend. Inra-year seasonaliy is also apparen in some of he series bu, wih a mos four years of observaions for any series, we followed he pracice of he company and did no ry o incorporae i in any mehod. The company s oules are open seven days a week, bu no bank holidays. We replaced bank holidays by he average of he corresponding day of he week from he week before and he week afer he day in quesion. Using a similar approach, we removed he Chrismas period from he daa because he mehods considered in his sudy are no designed for such unusual periods. This is consisen wih he approach aken by he company, as hey do no use heir exponenial smoohing approach for he Chrismas period. We esimaed mehod parameers using he firs 80% of observaions in each series. We used he final 20% for pos-sample forecas evaluaion. For many of he producs in our daase, a forecas lead ime of 5 o 7 days was of mos imporance o he company. However, his varied across he range of producs, and so, in consulaion wih he company, we considered forecas horizons of one o 14 days in our sudy. For each series, we rolled he forecas origin forward hrough he pos-sample evaluaion period o produce a collecion of forecass from each mehod for each horizon. For he one-day horizon, his resuled in each mehod being evaluaed on a oal of 42,633 pos-sample sales observaions. We employed he saisical programming language Gauss for all compuaional work in his paper. 4.2. Poin forecasing mehods based on smoohing he level We included in he sudy eigh mehods ha involve exponenial smoohing of he level of each series. These mehods are lised below. We derived parameer values for all 10
mehods, excep P2 and P7, by he common procedure of minimising he sum of squared insample one-sep-ahead forecas errors, wih parameers consrained o lie beween zero and one. We used simple averages of he firs few observaions o calculae iniial values for he smoohed componens (see Williams and Miller, 1999). Using muliplicaive seasonal decomposiion, we deseasonalised he daa prior o using Mehods P1 o P5, as hese mehods are appropriae for series wih no seasonaliy. The resulan forecass were reseasonalised before comparison wih he raw sales daa. We also implemened random walk forecass, bu he resuls were very poor due o he highly volaile naure of he series. Mehod P1. Simple exponenial smoohing. Mehod P2. Simple exponenial smoohing using leas absolue deviaions (LAD). Gardner (1999) proposes ha he smoohing parameer is opimised by minimising he sum of he absolue value of he one-sep-ahead forecas errors. This has he appeal of robusness, and so is a useful benchmark for he robus EWQR-based mehods, which we presen in Secion 4.3. Mehod P3. Hol s exponenial smoohing. Mehod P4. Damped Hol s exponenial smoohing. Mehod P5. Brown s double exponenial smoohing. Mehod P6. Hol-Winers exponenial smoohing for muliplicaive seasonaliy. Mehod P7. Company s mehod. This involves smoohing he oal weekly sales, W, and he spli, L, of he weekly sales across he days of he week. The parameers are se a he same subjecively chosen values for all series; α = 0.7 and γ = 0.1. W 6 = α y i + ( 1 α ) W 1 i= 0 L = γ y 7 i= 1 y i + ( 1 γ ) L 7 The forecass are given by: y ˆ ( m) = W L + m 7 for m = 1 o 7 y ˆ ( m) W L for m = 8 o 14 = + m 14 11
Mehod P8. Opimised company s mehod. We opimised values for he parameers α and γ. 4.3. Robus poin forecasing mehods based on EWQR We implemened he following robus mehods, which were described in Secion 3.5: Mehod P9. Median. This is equivalen o he EWMM approach of Dunsmuir e al. (1996). Mehod P10. Trimean. Mehod P11. Gaswirh. Mehod P12. Five-quanile. Mehod P13. Winsorised wih α = 0.05. For Mehods P13 o P15, simple exponenial smoohing was applied o he Winsorised series. For simpliciy, he values of he rimming percenage, α, were chosen arbirarily. An alernaive is o opimise α based on in-sample fi. Mehod P14. Winsorised wih α = 0.10. Mehod P15. Winsorised wih α = 0.25. For all seven mehods, we esimaed quaniles using EWQR wih a consan and no regressors. As his approach clearly canno capure seasonaliy, we deseasonalised he daa prior o using EWQR. Apar from he appeal of simpliciy, we oped for he use of only a consan erm in he EWQR because, as we show in Secion 5, i led o beer quanile forecass, for our daa, han hose produced by EWQR wih regressors. For he series wih fewer han 364 observaions (one year of daa) in he esimaion sample, we performed EWQR wih he whole esimaion sample. For he series wih a leas 364 observaions, we used a moving window of jus he mos recen 364 observaions. This saved compuaional running ime, wihou noiceably affecing quanile forecas accuracy. In Secion 3.1, we commened ha he mehod of Boudoukh e al. (1998) is he same as EWQR wih jus a consan erm. They also used a moving window of he mos recen year of observaions, which for heir finance applicaion amouned o 250 rading days. 12
Opimisaion of he weighing parameer λ, in he EWQR minimisaion of expression (3.1), proceeded by he use of a rolling window of 364 observaions o esimae one sepahead quanile forecass for each of he remaining observaions in he esimaion sample. The value of λ deemed o be opimal was he value ha produced one-sep-ahead quanile forecass leading o he minimum value of he summaion in he sandard quanile regression minimisaion (QR Sum), which is given in expression (4.1). We compued his summaion over a grid of values for λ beween 0.7 and 1, wih a sep size of 0.005. We performed he opimisaion separaely for each of he differen values of θ (i.e. for each differen quanile). We did no perform he opimisaion for series for which here were fewer han 548 (=364+182) observaions in he esimaion sample. We fel ha 182 observaions (six monhs of daa) was a reasonable minimum from which o evaluae he in-sample quanile forecass. QR Sum = θ Q θ ) + ( 1 θ ) y ( y Q ( θ ) (4.1) y Q ( θ) y < Q The resuling opimised values for λ varied quie subsanially across he series. In view of his insabiliy, and given ha we did no have opimised values of λ for he shorer series, we eleced o use for all series, for esimaion of he θ quanile, he median of he opimised values of λ corresponding o he same θ quanile. A similar approach o opimising exponenial smoohing parameers is considered by Fildes e al. (1998), who find ha using a commonly occurring value for all series can be preferable o using he value opimised for each series. The median λ values for a range of values of θ are presened in Figure 6. Larger values of λ imply ha he older observaions in he moving window of 364 are given a larger weighing. Giving a sizeable weigh o all 364 observaions would seem o be more imporan for he ail quaniles because hese quaniles require more observaions for heir esimaion. I is, herefore, inuiive ha in Figure 6 he values of λ are greaer for he ail quaniles. ---------- Figure 6 ---------- ( θ) 13
4.4. Poin forecasing resuls We evaluaed pos-sample forecasing performance a each forecas horizon for each series using he mean absolue error (MAE) and he roo mean squared error (RMSE). Since he order of magniude of he forecas errors varied across he series, i was no appropriae o average he error measures across he 256 series. Percenage error measures can be averaged, bu using a percenage error measure is no recommended when he value of he series can be close o zero, which was ofen he case for our sales daa. In order o summarise performance across all he series, we calculaed a measure of average performance relaive o simple exponenial smoohing. For he MAE resuls, he calculaion proceeded by compuing, for each series and each forecas horizon, he raio of he MAE for ha mehod o he MAE for simple exponenial smoohing, Mehod P1. We hen calculaed he weighed geomeric mean of his measure across he 256 series, where he weighing was proporional o he number of possample observaions in each series. Expression (4.2) presens he measure: N i 256 256 MAE i N j 1 100 1 (4.2) P j= 1 i= 1 MAEi P1 where MAE i and MAE i are he MAE for he mehod being considered and for Mehod P1, respecively; and N i is he number of pos-sample observaions for series i. Lower values of he measure are beer, wih negaive values indicaing ha he mehod ouperforms Mehod P1. For conciseness, Table 1 repors he average of he accuracy measure for pairs of forecas horizons. The values in bold highligh he bes performing mehod a each horizon. The final column of he able presens he average of he accuracy measure across he 14 horizons. We also calculaed he relaive measure using RMSE insead of MAE in expression (4.2), bu we do no repor he resuls in deail here because hey were similar o hose in Table 1. ---------- Table 1 ---------- 14
Le us firs consider he resuls in Table 1 for he sandard poin forecasing mehods, Mehods P1 o P6. Perhaps of greaes surprise is he poor performance of he muliplicaive Hol-Winers mehod. This is likely o be due o he unsable raios ha resul wihin he mehod when he ime series akes values close o or equal o zero. The superior resuls for Mehods P1 o P5 indicae ha muliplicaive seasonal decomposiion is a more robus approach o handling he seasonaliy. We eleced no o implemen he addiive version of Hol-Winers because our belief was ha he seasonaliy was no addiive. This view is suppored by he choice of muliplicaive seasonal modelling wihin he company s mehod. Comparing he resuls for Mehods P1 and P2 indicaes ha here is no gain in using LAD o opimise he simple exponenial parameer. The fac ha simple exponenial smoohing was more accurae han Hol s, damped Hol s and Brown s suppors our earlier commen ha few of he sales series conain rend. Comparing he resuls for Mehods P7 and P8 shows ha opimising he parameers in he company s mehod provides a clear improvemen over he company s subjecive choice of parameers. This finding is consisen wih he lieraure on exponenial smoohing mehods (e.g. Fildes e al. 1998). The company s mehod wih opimised parameers, Mehod P8, performs well for he early lead imes, bu he posiive value in he final column of he able shows ha, overall, i is ouperformed by simple exponenial smoohing, Mehod P1. Turning o he mehods based on EWQR, Mehods P9 o P15, we see from he final column in Table 1 ha overall hey all ouperformed all he level smoohing mehods, Mehods P1 o P8. The bes resuls were achieved using he Winsorised approach wih rimming α se a 25%, which implies rimming of all observaions ouside he inerquarile range. Table 2 aims o provide a lile more insigh ino he resuls. I shows he percenage of he 256 series for which he MAE for his mehod is lower han he MAE for he opimised company s mehod. The Winsorised mehod can be seen o dominae beyond he early horizons. ---------- Tables 2 and 3 ---------- 15
In Secion 4.1, we commened ha he 256 series vary in lengh quie considerably. In order o esablish wheher he relaive performances of he mehods depends on he lengh of he series, we calculaed he relaive percenage measure separaely for he following four caegories of series lenghs: fewer han one year of observaions, beween one and wo years of observaions, beween wo and hree years, and more han hree years. Table 3 provides he resuls for he opimised company s mehod and he EWQR-based Winsorised approach wih rimming α se a 25%. The only noiceable difference in he rankings of he wo mehods across he four caegories is for one and wo day-ahead predicion, where he EWQR-based mehod seems o be slighly preferable for he shorer series and a lile worse for he longer series. 5. Empirical comparison of quanile forecasing mehods 5.1. Descripion of he sudy In his secion, we compare mehods for forecasing he following four quaniles: 0.025, 0.25, 0.75 and 0.975. As in Secion 4, we compare pos-sample forecass for lead imes from one o 14 days for he final 20% of observaions in each of he 256 sales series. We deseasonalised he daa prior o forecasing wih all he mehods, excep Mehod Q7. 5.2. Quanile forecasing mehods based on a poin forecas Of he sandard poin forecasing mehods in Secion 4, he mehod ha overall was he mos accurae was simple exponenial smoohing. In view of his, we consruced quanile forecass as he simple exponenial smoohing poin forecas plus he quanile of he forecas error disribuion esimaed using he following wo differen approaches: Mehod Q1. Empirical. For each lead ime, we used he disribuion of he forecas errors, for ha lead ime, in a moving window of he mos recen 364 periods. Mehod Q2. Theoreical variance. We used a Gaussian assumpion wih he simple exponenial smoohing muli sep-ahead heoreical error variance formula: 16
var where ˆ ( k) is he k sep-ahead forecas; y k 2 ( α ) 2 ( y yˆ ( k) ) = σ 1+ ( k 1) k e 2 σ e is he one sep-ahead variance calculaed from a moving window of he mos recen 364 days; and α is he smoohing parameer. 5.3. Adapive quanile forecasing mehods We implemened he adapive CAViaR mehod and he mehod of Gorr and Hsu (1985), which we presened in Secion 2. For boh mehods, muliperiod forecass are idenical o he one day-ahead predicion. We performed parameer opimisaion for each mehod. Mehod Q3. Adapive CAViaR. Mehod Q4. Gorr and Hsu. 5.4. Quanile forecasing mehods based on EWQR We considered he following hree differen versions of he EWQR approach: Mehod Q5. EWQR wih a consan and no regressors. Mehod Q6. EWQR wih a consan and a linear rend erm. Mehod Q7. EWQR wih a consan and he sinusoidal seasonal erms in expression (5.1). The b i are consan parameers, and d() is a repeaing sep funcion ha numbers he days of he week from 1 o 7. Preliminary analysis for our daa indicaed ha wo sine and wo cosine erms was preferable o one or hree. b d ( ) d ( ) d ( ) d ( ) ( 2π ) b cos( 2π ) + b sin( 4π ) + b cos( π ) 1 sin 7 2 7 3 7 4 4 + (5.1) We also considered he damped rend formulaion of expression (3.6). However, he median value of he opimised damping parameer was one, which implied no damping. We suspec ha his is due o here being rend in few of he series. We implemened he hree EWQR mehods in he same way as described for our poin forecasing sudy of Secion 4. The median opimised values of λ for he four quaniles 7 17
are presened for he hree mehods in Table 4. I is ineresing o see ha he values of λ are higher for he EWQR implemenaions ha included regressors. The implicaion of his is ha giving a sizeable weigh o all 364 observaions is more imporan when regressors are included. This is inuiive because i indicaes ha fiing regressors requires more informaion han simply fiing a consan erm. Noe ha, alhough a λ value of 0.995 may seem high, i does resul in noiceable discouning over a reasonable size window. For example, over a four-week period, he weigh reduces o 0.995 28 =0.869, and over a one-year period, he weigh reduces o 0.995 364 =0.161. ---------- Table 4 ---------- 5.5. Quanile forecasing resuls The uncondiional coverage of a θ quanile esimaor is he percenage of observaions falling below he esimaor. Ideally, he percenage should be θ. To summarise uncondiional coverage across he four quaniles, we calculaed chi-squared goodness of fi saisics for each mehod, a each lead ime, for each of he 256 series. We calculaed he saisic for he oal number of pos-sample observaions falling wihin he following five caegories: below he 0.025 quanile esimaor, beween each successive pair of quanile esimaors, and above he 0.975 quanile esimaor. Assuming independence of he resuling chi-squared saisics for he differen series, we hen summed hese saisics for all he 256 series. The only wo of he resuling summed chi-squared saisics ha were no significan, a he 5% level, were hose for one day-ahead predicion from he wo adapive quanile modelling mehods, Mehods Q3 and Q4. The chi-squared saisics, from differen lead imes for he same mehod, are no independen. Therefore, saisical esing would be invalid if hese saisics were summed or averaged. Neverheless, for conciseness, Table 5 repors he average of he chi-squared saisics for pairs of forecas horizons, and he average across he 14 horizons. Lower values of he measure are beer. 18
---------- Table 5 ---------- Table 5 shows ha he bes performing mehod a all lead imes is he adapive CAViaR mehod. The fac ha i ouperforms he Gorr and Hsu mehod suggess ha he smoohing in expression (2.2) of he Gorr and Hsu mehod is no beneficial. The use of EWQR wih a consan bu no regressors, Mehod Q5, also performed well. By conras, he resuls are very poor for he EWQR mehod wih rend or seasonal erms. As we have already discussed, few of he supermarke series conain rend, so we should no have expeced benefi in including he rend erm. The poor resuls in Secion 4 for he Hol-Winers mehod indicaed ha modelling seasonaliy was difficul for his daa, and ha seasonal decomposiion prior o modelling was preferable. The same seems o be he case wih regard o he use of he EWQR mehod. Tesing for uncondiional coverage is insufficien, as i does no assess he dynamic properies of each quanile esimaor (Chrisoffersen, 1998). The essence of ess of condiional coverage is o examine wheher here is auocorrelaion in he hi variable, defined as Hi = I ( y Q ( θ )) θ ˆ. Unforunaely, when evaluaing ail quaniles, he hi variable can someimes conain no variaion, in which case he es canno be performed. For our 256 supermarke series, his was frequenly he case for one or more of he seven mehods. A formal esing of condiional coverage was, herefore, no possible. Insead, we evaluaed he dynamic properies of he mehods using he QR Sum, which is presened in expression (4.1). This measure can be viewed as he equivalen of he MAE and RMSE for evaluaing quanile forecas accuracy. I is one of he measures repored by Engle and Manganelli (2004) in heir VaR sudy. We calculaed he pos-sample QR Sum for each mehod applied o each series a each forecas horizon. Replacing he MAE erms in expression (4.2) by he QR Sum, we hen compued he relaive percenage measure, wih he QR Sum for he Empirical mehod, Mehod Q1, in he denominaor of he expression. Lower 19
values of he measure are beer, wih negaive values indicaing ha he mehod ouperformed Mehod Q1. Using he relaive QR Sum measure, overall, he relaive performance of he mehods was similar o ha in Table 5 for he chi-squared saisic. Tables 6 and 7 repor he resuls for he 0.025 and 0.975 quaniles. I seems sensible o focus on he hree mehods ha performed reasonably in Table 5: adapive CAViaR, Gorr and Hsu and EWQR wih a consan and no regressors. The resuls in Tables 6 and 7 show ha, of hese hree mehods, he EWQR mehod has he bes dynamic properies for boh quaniles a all lead imes. ---------- Tables 6 and 7 ---------- 6. Summary and concluding commens The daily sales series considered in his paper are characerised by high volailiy and skewness, which are boh ime-varying. In addiion, he series have occasional oulying observaions. These feaures moivae consideraion of poin and inerval forecasing mehods ha are robus wih regard o disribuional assumpions. As predicions are required for many differen iems, he mehod mus be suied o implemenaion wihin an auomaed procedure. Our proposal is o generae poin and inerval forecass from quanile forecass generaed using EWQR. We have shown ha his can be viewed as simple exponenial smoohing of he cdf. The EWQR framework has several benefis. Firs, i enables efficien soluion hrough he use of linear programming. Second, i allows regressors, such as a rend and seasonal erms, o be included in he quanile model. Third, he regression framework provides scope for saisical inference. For he sales series, he quanile forecass from EWQR wih only a consan erm compared favourably wih a variey of oher mehods. Anoher mehod ha performed well was he adapive CAViaR mehod, which has previously only been applied o financial daa. 20
Including rend or seasonal erms in he EWQR led o poor resuls, which moivaes furher experimenaion wih rending and seasonal series. Poor accuracy resuled from quanile forecass based on a Gaussian disribuion cenred a he simple exponenial smoohing poin forecas wih variance calculaed using he heoreical error variance formula. This suggess ha he saisical models underlying he sandard exponenial smoohing mehods are no well suied o his daa. Indeed, poin forecass from hese sandard mehods, as well as he company s own approach, were ouperformed by robus poin forecasing mehods based on he EWQR quanile forecass. Acknowledgemens We are graeful o he members of he sales forecasing group a he collaboraing company, and in paricular Jon Tasker. We also acknowledge he very helpful commens of Jan De Gooijer and Everee Gardner on an earlier version of his paper. We are also graeful for he helpful commens of wo anonymous referees. References Boudoukh, J., Richardson, M., Whielaw, R.F., 1998. The bes of boh worlds, Risk 11 May 64-67. Chafield, C., 1993. Calculaing inerval forecass, Journal of Business and Economic Saisics 11 121-135. Chrisoffersen, P.F., 1998. Evaluaing inerval forecass, Inernaional Economic Review 39 841-862. Cipra, T., 1992. Robus exponenial smoohing, Journal of Forecasing 11 57-69. Dunsmuir, W.T.M., Sco, D.J., Qiu, W., 1996. The disribuion of he weighed moving median of a sequence of iid observaions, Communicaions in Saisics: Simulaion and Compuaion 25 1015-1029. 21
Engle, R.F., Manganelli, S., 2004. CAViaR: Condiional auoregressive value a risk by regression quaniles, Journal of Business and Economic Saisics 22 367-381. Fildes, R., Hibon, M., Makridakis, S., Meade, N., 1998. Generalising abou univariae forecasing mehods: Furher empirical evidence, Inernaional Journal of Forecasing 14 339-358. Gardner, E.S., Jr., 1985. Exponenial smoohing: he sae of he ar, Journal of Forecasing 4 1-28. Gardner, E.S., Jr., 1988. A simple mehod of compuing predicion inervals for ime-series forecass, Managemen Science 34 541-546. Gardner, E.S., Jr., 1999. Noe: Rule-based forecasing vs. damped rend exponenial smoohing, Managemen Science 45 1169-1176. Gardner, E.S., Jr., McKenzie, E., 1985. Forecasing rends in ime series, Managemen Science 31 1237-1246. Gaswirh, J.L., 1966. On robus procedures, Journal of he American Saisical Associaion 61 929-948. Gorr, W.L., Hsu, C., 1985. An adapive filering procedure for esimaing regression quaniles, Managemen Science 31 1019-1029. Harvey, A.C., 1990. Forecasing, Srucural Time Series Models and he Kalman Filer, Cambridge Universiy Press, New York. Hyndman, R.J., Koehler, A.B., Ord, J.K., Snyder, R.D., 2005. Predicion inervals for exponenial smoohing using wo new classes of sae space models, Journal of Forecasing 24 17-37. Judge, G.G., Hill, R.C., Griffihs, W.E., Lükepohl, H., Lee, T.-C., 1988. Inroducion o he Theory and Pracice of Economerics, Wiley, New York. Koenker, R.W., Basse, G.W., 1978. Regression quaniles, Economerica 46 33-50. 22
Makridakis, S., Hibon, M., 2000. The M3-Compeiion: resuls, conclusions and implicaions, Inernaional Journal of Forecasing 16 451-476. Taylor, J.W., 2006. Esimaing value a risk using exponenially weighed quanile regression, Working Paper, Universiy of Oxford. Taylor, J.W., Bunn, D.W., 1999. A quanile regression approach o generaing predicion inervals, Managemen Science 45 225-237. Tukey, J.W., 1962. The fuure of daa analysis, Annals of Mahemaical Saisics 33 1-67. Tukey, J.W., 1970. Explanaory Daa Analysis, Addison-Wesley, Reading, MA. Williams, D.W., Miller, D., 1999. Level-adjused exponenial smoohing for modeling planned disconinuiies, Inernaional Journal of Forecasing 15 273-289. 23
Figure 1 Daily supermarke sales of a single iem. sales 40 35 30 25 20 15 10 5 0 0 182 364 546 728 910 1092 days Figure 2 Hisogram for number of observaions in he daily supermarke sales series. Frequency (%) 40 35 30 25 20 15 10 5 0 72 o 200 201 o 400 401 o 600 601 o 800 801 o 1000 1001 o 1200 1201 o 1400 1401 o 1436 Number of observaions 24
Figure 3 Hisogram for srengh of deerminisic linear rend in he daily supermarke sales series. Frequency (%) 16 14 12 10 8 6 4 2 0-0.8 o -0.7-0.7 o -0.6-0.6 o -0.5-0.5 o -0.4-0.4 o -0.3-0.3 o -0.2-0.2 o -0.1-0.1 o 0 0 o 0.1 0.1 o 0.2 0.2 o 0.3 0.3 o 0.4 0.4 o 0.5 0.5 o 0.6 0.6 o 0.7 Correlaion wih deerminisic linear rend Figure 4 Hisogram for variaion explained by deerminisic linear rend and AR erms in he daily supermarke sales series. Frequency (%) 30 25 20 15 10 5 0-5 o 0 0 o 5 5 o 10 10 o 15 15 o 20 20 o 25 25 o 30 30 o 35 35 o 40 40 o 45 45 o 50 50 o 55 55 o 60 60 o 65 65 o 70 70 o 75 Adjused R 2 (%) Figure 5 Hisogram for proporion of oulying observaions in he daily supermarke sales series. Frequency (%) 35 30 25 20 15 10 5 0 0 o 1 1 o 2 2 o 3 3 o 4 4 o 5 5 o 6 6 o 7 7 o 8 8 o 9 9 o 10 10 o 11 11 o 12 12 o 13 Proporion of oulying observaions (%) 25
Figure 6 Median of he opimised values of he weighing parameer, λ, for he exponenially weighed θ quanile regression of expression (3.1). Median λ 1.00 0.98 0.96 0.94 0.92 0.90 0.0 0.2 0.4 0.6 0.8 1.0 θ Table 1 Evaluaion of poin forecasing mehods using he relaive MAE measure of expression (4.2), calculaed for all 256 series. Accuracy measured relaive o Mehod P1. Lower values are beer. Forecas Horizon 1-2 3-4 5-6 7-8 9-10 11-12 13-14 All Level Smoohing Mehods P1. Simple 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 P2. Simple using LAD 0.7 0.6 0.7 0.6 0.5 0.5 0.6 0.6 P3. Hol s 0.8 1.1 1.2 1.7 2.5 2.9 3.2 1.9 P4. Damped Hol s 0.7 0.7 0.7 0.7 0.7 0.7 0.6 0.7 P5. Brown s 1.4 2.2 2.9 4.1 5.5 6.6 7.7 4.4 P6. Hol-Winers 4.7 4.7 5.3 6.4 7.0 7.4 8.1 6.2 P7. Company -0.7-1.6-0.6 1.8 3.3 4.0 5.4 1.7 P8. Opimised Company -1.9-2.4-1.3 0.7 1.6 2.4 3.6 0.4 EWQR Mehods P9. Median -1.0-1.5-1.1-1.0-1.8-1.8-1.4-1.4 P10. Trimean -1.2-1.8-1.3-1.4-2.1-2.1-1.8-1.7 P11. Gaswirh -1.3-1.8-1.4-1.4-2.1-2.1-1.8-1.7 P12. Five Quanile -1.0-1.5-1.1-1.2-2.0-2.0-1.7-1.5 P13. Winsorised α = 0.05-0.9-1.1-1.0-1.0-1.2-1.1-1.1-1.0 P14. Winsorised α = 0.10-1.2-1.5-1.3-1.3-1.6-1.5-1.4-1.4 P15. Winsorised α = 0.25-1.6-2.0-1.6-1.7-2.3-2.1-1.9-1.9 26
Table 2 Percenage of he 256 series for which he MAE for he EWQR-based Winsorised Mehod P15 wih α=0.25 is lower han he MAE for he opimised company s Mehod P8. Bold indicaes significanly greaer han 50% a he 5% significance level. Forecas Horizon 1 2 3 4 5 6 7 8 9 10 11 12 13 14 % 50 49 49 54 54 55 60 74 75 75 73 76 76 78 Table 3 Evaluaion of poin forecasing Mehods P8 and P15 using he relaive MAE measure of expression (4.2), calculaed for series of differen lenghs. Accuracy measured relaive o Mehod P1. Lower values are beer. Forecas Horizon 1-2 3-4 5-6 7-8 9-10 11-12 13-14 All 83 series wih observaions 364 P8. Opimised Company -2.5-2.7-2.4 0.6 2.5 3.2 4.5 0.5 P15. EWQR Winsorised α = 0.25-2.6-2.1-2.1-2.2-2.6-2.3-1.7-2.2 39 series wih 365 observaions 728 P8. Opimised Company -1.2-1.3-0.7 1.5 2.2 2.6 3.4 0.9 P15. EWQR Winsorised α = 0.25-1.6-1.8-1.5-1.4-2.0-2.0-2.0-1.7 28 series wih 729 observaions 1092 P8. Opimised Company -1.6-2.0-1.4 0.4 1.8 2.4 3.8 0.5 P15. EWQR Winsorised α = 0.25-0.9-1.0-0.9-0.9-1.0-0.9-0.9-0.9 106 series wih 1093 observaions 1436 P8. Opimised Company -2.0-2.6-1.2 0.6 1.5 2.3 3.6 0.3 P15. EWQR Winsorised α = 0.25-1.6-2.2-1.7-1.8-2.5-2.3-2.1-2.1 All 256 series P8. Opimised Company -1.9-2.4-1.3 0.7 1.6 2.4 3.6 0.4 P15. EWQR Winsorised α = 0.25-1.6-2.0-1.6-1.7-2.3-2.1-1.9-1.9 27
Table 4 Median of he opimised values of he weighing parameer, λ, for hree differen versions of he exponenially weighed θ quanile regression of expression (3.1). θ 0.025 0.25 0.75 0.975 Q5. EWQR no regressors 0.990 0.950 0.925 0.9725 Q6. EWQR rend 0.995 0.985 0.970 0.995 Q7. EWQR seasonal 0.995 0.990 0.980 0.995 Table 5 Evaluaion of quanile forecasing mehods using uncondiional coverage chisquare saisic summarising performance across all four quaniles. Lower values are beer. Forecas horizon 1-2 3-4 5-6 7-8 9-10 11-12 13-14 All Mehods Based on a Poin Forecas Q1. Empirical 2459 2648 2734 2952 3267 3489 3797 3049 Q2. Theoreical variance 4142 4360 4540 4666 4902 5128 5420 4737 Adapive Mehods Q3. Adapive CAViaR 1018 1371 1587 1822 2073 2413 2744 1861 Q4. Gorr and Hsu 1193 1537 1712 1956 2163 2474 2804 1977 EWQR Mehods Q5. EWQR no regressors 1528 1708 1907 2156 2505 2812 3136 2250 Q6. EWQR rend 3609 4249 4902 5720 6690 7572 8471 5888 Q7. EWQR seasonal 3145 3124 3222 3492 3820 3890 3938 3519 28
Table 6 Evaluaion of 0.025 quanile forecass using relaive QR Sum measure. Accuracy measured relaive o Mehod Q1. Lower values are beer. Forecas horizon 1-2 3-4 5-6 7-8 9-10 11-12 13-14 All Mehods Based on a Poin Forecas Q1. Empirical 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Q2. Theoreical variance 2.7 2.0 1.4 0.9 0.0-0.1 0.0 1.0 Adapive Mehods Q3. Adapive CAViaR -20.1-22.0-23.5-24.8-26.6-27.4-28.0-24.6 Q4. Gorr and Hsu -19.6-21.5-23.0-24.4-26.1-26.9-27.4-24.1 EWQR Mehods Q5. EWQR no regressors -21.1-23.0-24.5-25.8-27.6-28.4-28.9-25.6 Q6. EWQR rend -19.4-20.9-22.1-23.3-24.7-25.3-25.7-23.0 Q7. EWQR seasonal -18.4-20.6-22.3-23.6-25.2-26.1-26.7-23.3 Table 7 Evaluaion of 0.975 quanile forecass using relaive QR Sum measure. Accuracy measured relaive o Mehod Q1. Lower values are beer. Forecas horizon 1-2 3-4 5-6 7-8 9-10 11-12 13-14 All Mehods Based on a Poin Forecas Q1. Empirical 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 Q2. Theoreical variance -0.1-0.1-0.3-0.4-0.5-0.8-1.1-0.5 Adapive Mehods Q3. Adapive CAViaR 10.6 9.6 9.8 9.8 9.5 9.2 9.6 9.7 Q4. Gorr and Hsu 11.0 10.1 10.3 10.3 9.9 9.6 9.9 10.2 EWQR Mehods Q5. EWQR no regressors 5.0 4.7 5.3 5.6 5.6 5.7 6.1 5.4 Q6. EWQR rend 13.0 13.2 14.6 15.3 15.3 15.8 16.6 14.8 Q7. EWQR seasonal 11.5 8.9 8.3 8.2 7.1 5.4 4.5 7.7 29