Time Scale Evaluation of Economic Forecasts

Transcription

1 CENTRAL BANK OF CYPRUS EUROSYSTEM WORKING PAPER SERIES Tme Scale Evaluaon of Economc Forecass Anons Mchs February 2014 Worng Paper

2 Cenral Ban of Cyprus Worng Papers presen wor n progress by cenral ban saff and ousde conrbuors. They are nended o smulae dscusson and crcal commen. The opnons expressed n he papers do no necessarly reflec he vews of he Cenral Ban of Cyprus or he Eurosysem Address 80 Kennedy Avenue CY-1076 Ncosa, Cyprus Posal Address P. O. Box CY-1395 Ncosa, Cyprus E-mal publcaons@cenralban.gov.cy Webse hp:// Fax Papers n he Worng Paper Seres may be downloaded from: hp:// Cenral Ban of Cyprus, Reproducon s permed provded ha he source s acnowledged. 2

3 Tme Scale Evaluaon of Economc Forecass Anons Mchs 1 Absrac A maxmal overlap dscree wavele ransform s used o oban me scale decomposons of economc forecass and her errors. The generaed me scale componens can be used n loss measures and ess for comparng forecas accuracy o evaluae wheher he forecass accuraely capure he cyclcal feaures of he daa. Keywords: forecas accuracy; loss measures; me scales; waveles JEL classfcaon: C53; E37 1 The opnons expressed n hs paper are hose of he auhor and do no necessarly reflec he vews of he Cenral Ban of Cyprus or he Eurosysem. Correspondence: Anons A. Mchs, Sascs Deparmen, Cenral Ban of Cyprus, 80 Kennedy Avenue, P.O.Box 25529, CY-1395, Ncosa, Cyprus. Emal: AnonsMchs@cenralban.gov.cy 3

4 1. Inroducon Waveles consue a relavely new bu powerful ool for he analyss of me seres ha can be used n many economc applcaons. Yogo (2008) used waveles o decompose he US GDP no dfferen me scales (or cycles), whch perms he denfcaon of he busness cycle componen n he daa. The me scale componens generaed by waveles can also be used o examne relaonshps beween economc varables across frequences. For example, Gallega e al. (2011) nvesgaed he relaonshp beween wage nflaon and unemploymen n he US across frequences and over me. Waveles have also been used n economerc esmaon and esng. Jensen (2000) and Glles e al. (2009) examned he use of waveles n he esmaon of models for long memory processes, and Mchs and Sapanas (2007) consruced wavele nsrumens o mprove he effcency of GMM esmaors. Addonal applcaons of waveles nclude esng for seral correlaon of unnown form n panel daa regresson models as suggesed by Hong and Kao (2004) and he un roo ess proposed by Fan and Gencay (2010). In hs sudy, he maxmal overlap dscree wavele ransform (MODWT) s used o decompose economc forecass and her assocaed forecas errors no dfferen me scales. The generaed me scale componens capure dfferen cyclcal feaures of he daa, whch perms an evaluaon of forecas accuracy across he cycle. Because he me scale componens have equal lengh wh he acual me seres, hey can be ncorporaed n sandard loss measures (e.g., he mean squared error) and ess for comparng forecas accuracy (e.g., he es proposed by Debold and Maranno, 1995). The wavele me scale decomposons are brefly explaned n Secon 2. Secon 3 demonsraes how he generaed me scale componens can be used o evaluae forecass over dfferen cycles, and Secon 4 provdes an emprcal sudy usng four forecass for shppng volume daa. 2. Tme scale decomposons of economc me seres Waveles consue famles of bass funcons defned whn he se of square negrable funcons L 2 ( R ). Tme seres or funcons can be represened by a sequence of proecons ono 4

5 a bass of faher ( ) and moher ( ) waveles ha are defned as follows (see Gallega e. al., 2011): / 2 2, 2 2 and / 2 2, 2. 2 A un decrease n he value of proporonal o 1,2,3,...,J expands he range of he moher wavele 2, whch reduces s wdh and doubles s frequency. In conras, he faher wavele s no affeced by changes n, bu a un ncrease n shfs he locaon of boh he faher and moher waveles. For a me seres,, wh lengh be expressed as J N 2, he wavele mul-resoluon approxmaon can s ) d ( ) d ( )... d ( ). J, J, ( J, J, J 1, J 1, 1, 1, The faher and moher wavele coeffcens are, respecvely: sj, J, ( ) d and d,, ( ) d. The faher wavele coeffcens capure he smooh, low-frequency rend behavour n he daa, and he moher wavele coeffcens capure all hgh-frequency, shor-erm devaons from he rend. The wavele mul-resoluon approxmaon can also be wren as S J D J D J 1... D 1 wh me scale componens S J sj, J, ( ) and D d,, ( ). Therefore, he wavele mul-resoluon analyss decomposes he daa no J me scales. Tme scale componens wh hgher values of capure long-erm cycles n he daa, and he 5

6 me scale componen, S J, assocaed wh he faher waveles, capures he smooh rend behavour n he daa. In conras, me scales wh small values of capure he hgh-frequency, shor-erm cyclcal movemens n he daa. For a me seres wh lengh wees, he wavele mul-resoluon analyss provdes a decomposon no nne me scales. The elemens of he frs me scale componen ( D 1) capure frequency varaon over duraons of 2 o 4 wees. The elemens of he second componen ( D 2 ) capure frequency varaon over duraons of 4 o 8 wees, and accordngly, he hrd componen ( D 3 ) s assocaed wh frequency varaon over duraons of 8 o 16 wees. Ths s he case up o level 9. In hs sudy, a mul-resoluon analyss based on he MODWT s used. The MODWT does no provde an exacly orhogonal decomposon of he me seres, bu s more effcen han he basc dscree wavele ransform. I also generaes me scale componens of equal lengh wh he acual me seres ha can be used n loss measures and ess of forecas accuracy. In pracce, he MODWT s compued wh a pyramd algorhm ha eravely flers he me seres wh a hgh- and a low-pass fler o produce he vecors of wavele coeffcens (see Percval and Walden, 2000, pp. 174). 3. Tme scale evaluaon of economc forecass Several approaches have been proposed n he leraure for evaluang he accuracy of economc forecass. These range from smple loss measures such as he mean squared error (MSE) and he mean absolue error (MAE) o sascal ess for comparng he accuracy of wo (see, e.g., Debold and Marano, 1995) or more forecass (see, e.g., Hansen, 2005). Snce he me scale componens generaed by he MODWT have equal lengh wh he acual me seres daa, hey can be used n loss measures and ess of forecas accuracy o evaluae wheher he forecass accuraely capure he cyclcal feaures of he daa. To demonsrae hs, le be he h -perods ahead forecas errors assocaed wh, h h h mehod a me. Raher han evaluang he forecass, h, a he observed samplng rae of 6

7 he daa, he daa are frs decomposed no dfferen me scales usng he MODWT. In a second sep, separae forecas errors are formed for each me scale as follows:., h D, h D, h The me scale componen,,, refers o he acual daa ( h ), and he me scale D h componen, D h,, refers o he forecass generaed wh mehod a me ( h ). By dervng he forecass errors assocaed wh each me scale, s possble o draw conclusons concernng he forecasng accuracy of he mehod across he cycle. For example, s useful o now wheher a mehod s more accurae n forecasng shor-erm han long-erm changes n an economc varable. Usng he above me scale defnon for he errors, s possble o calculae he MSE and he MAE for each me scale as follows: 1 T 2 1 T MSE ( 1, ) h and MAE T 1, h. T The me scale errors can also be used n he conex of he Debold-Marano es o compare he accuracy of wo compeng mehods n forecasng specfc cycles n he daa. Usng a squared loss funcon L( 2, h ) (, h ), he loss dfferenal beween forecasng mehods a b a and b a me scale s defned as d L ) L( ). The h -perods ahead, (, h, h forecass are assumed o be repeaedly compued a me perods,..., T. For a covarance 0 saonary seres, he Debold-Marano es sasc ( S ) follows he sandard normal dsrbuon under he null hypohess of zero expeced loss dfferenal (equal forecas accuracy) d S ~ N(0,1). Vˆ T d 7

8 The sample mean loss dfferenal a me scale s d T 0 M d, / T. Vˆ d ( ) s M d an esmaor of he varance of he mean loss dfferenal usng he sample auocovarance, d, a dsplacemens and where 1/ 3 M T (see Debold, 2004, p.300). 4. An emprcal sudy In hs secon, he proposed mehods are appled o he Alanc Eas rade lane (cargo) shppng volume forecass descrbed n Debold (2004, p ). In addon o he acual cargo volume daa, hs daase consss of wee ahead volume forecass generaed wh wo dfferen mehods: () based on a quanave model (Quan.) and () based on a udgmenal mehod (Judgme.). Debold also suggesed he use of a regresson combnaon mehod (Regress.) usng a model wh MA(1) errors. For he purposes of hs sudy, a fourh mehod was also consdered ha conssed of smple averages (Aver.) of he quanave and udgmenal forecass. Because he me seres nclude 499 observaons, he daa were padded wh he las value of he seres o ncrease her lengh o 512 (a power of 2), as suggesed by Gencay e al. (2002, p. 144). The MODWT was performed wh Wavehresh sofware usng he Daubeches leas asymmerc famly of waveles wh a fler lengh of 8, whch provded good resoluon for he daa. The accuracy of each mehod a each me scale was evaluaed usng he MSE and MAE loss measures descrbed n Secon 3. The resuls are ncluded n Table 1. The shaded areas n he able ndcae he mos accurae mehods by me scale. Boh loss measures ndcae he same mehods. For example, he resuls n me scale TS9 ha capures he low-frequency, long-erm cyclcal movemens n he daa ndcae ha he quanave model provded he mos accurae forecass. For me scales TS1 and TS2, whch are assocaed wh hgh frequency, shor-erm cyclcal movemens n he daa wh lenghs of 2 o 4 and 4 o 8 monhs, respecvely, he regresson combnaon mehod provded he bes resuls. For me scales TS3 (8 o 16 wees) and TS4 (16 o 32 wees), he smple averagng mehod was he mos accurae. 8

9 Table 1 Loss measures by me scale Tme-scale Measure Quan. Judgme. Aver. Regres. TS9 MSE TS8 MSE TS7 MSE TS6 MSE TS5 MSE TS4 MSE TS3 MSE TS2 MSE TS1 MSE TS9 MAE TS8 MAE TS7 MAE TS6 MAE TS5 MAE TS4 MAE TS3 MAE TS2 MAE TS1 MAE The bes mehods denfed by me scale n Table 1 were also compared agans he oher mehods usng he Debold-Marano es. The generaed p-values are ncluded n Table 2. Wh he excepon of me scale TS5, he null hypohess of zero expeced loss dfferenal was reeced n all me scales, confrmng he superory of he mehods ncluded n column 2. In me scale TS5 he quanave model was found o be equal o he averagng mehod. Consequenly, f he man purpose of he forecasng exercse s o predc shor-erm movemens n cargo shppng volume, hen he regresson combnaon mehod should be used. In a dfferen case, f he goal s o predc he long-erm cyclcal movemens n he seres, hen he quanave model should be used. 9

10 Table 2 Debold-Marano es p-values by me scale Tme scale Bes mehod Quan. Judgme. Aver. Regres. TS9 Quan TS8 Quan TS7 Quan TS6 Quan TS5 Quan * TS4 Aver TS3 Aver TS2 Regres TS1 Regres *Equal predcve accuracy. References Debold, F.X., Elemens of forecasng, 3 rd ed. Thomson, Souh-Wesern, Oho. Debold, F.X., Marano, R.S, Comparng Predcve Accuracy. Journal of Busness and Economc Sascs 13(3), Fan, Y. and Gencay, R., Un roo ess wh waveles. Economerc Theory, 26(5), Gallega, M., Gallega, M., Ramsey, J.B., Semmler, W., The US Phlps curve across frequences and over me. Oxford Bullen of Economcs and Sascs 73 (4), Gencay, R., Selcu, F., Whcher B., An Inroducon o Waveles and Oher Flerng Mehods n Fnance and Economcs. Academc Press, New Yor. Glles, F., Moulnes, E., Roueff, F., Taqqu, M.S., Esmaors of long-memory: Fourer versus waveles. Journal of Economercs 151(2), Hansen, P., A es for superor predcve ably. Journal of Busness and Economc Sascs 23(4), Hong, Y., Kao, C., Wavele-based esng for seral correlaon of unnown form n panel Models. Economerca 72(5), Jensen, M.J., An alernave maxmum lelhood esmaor of long-memory processes usng compacly suppored waveles. Journal of Economc Dynamcs and Conrol 24 (3),

11 Mchs, A.A., Sapanas, T., Wavele nsrumens for effcency gans n generalzed mehod of momens models. Sudes n Nonlnear Dynamcs and Economercs 11(4). Percval, D.B., Walden, A.T., Wavele mehods for me seres analyss. Cambrdge Unversy Press, Cambrdge. Yogo, M., Measurng busness cycles: A wavele analyss of economc me seres. Economcs Leers 100(2),