TREND ESTIMATION AT THE ENDS OF SERIES USING ADAPTIVE SEMI-PARAMETRIC LOCAL DYNAMIC MODELS.

Transcription

1 TREND ESTIMATION AT THE ENDS OF SERIES USING ADAPTIVE SEMI-PARAMETRIC LOCAL DYNAMIC MODELS. Alisair G. Gray and Peer J. Thomson Saisics New Zealand and Saisics Research Associaes Ld Alisair Gray, Analyical Suppor, Saisics New Zealand PO Box 9 Wellingon, New Zealand Ph: alisair_gray@sas.gov.nz Absrac End filers for non-seasonal ime series consruced using a minimum revisions crierion and opimal linear (biased) predicion wih respec o semi-parameric local dynamic models were firs inroduced by Gray and Thomson (1996), Research Repor CENSUS/SRD/RR-96/1, Saisical Research Division, Bureau of he Census, Washingon. There i was shown ha a suiable choice of a bias erm, relaed o X-11 s I / C raio can lead o minimum revisions ha are compeiive and someimes beer han hose achieved using ARIMA forecas exension. This paper exends he earlier work by addressing he quesion: have adapive BLIP filers he poenial o improve revisions a end of series? Keywords: Revisions; Linear Unbiased Predicion; Linear Biased Predicion. 1 INTRODUCTION Many seasonal adjusmen procedures decompose ime series ino rend, seasonal, irregular and oher componens using non-seasonal moving-average rend filers. End filers for non-seasonal ime series consruced using a minimum revisions crierion and opimal linear (biased) predicion wih respec o semi-parameric local dynamic models were firs inroduced in Gray and Thomson (1996). Two predicion mehods were considered. Bes Linear Unbiased Predicion (BLUP) and Bes Linear Biased Predicion, where he bias is ime invarian (BLIP). The BLIP end filers were shown o be a generalizaion of hose developed by Musgrave (1964) for he cenral X-11 Henderson filers and include he BLUP end filers as a special case. Secions and 3 summarize he relevan properies abou hese filers which are more fully discussed in Gray and Thomson (1999). The developmen of BLIP filers provides a unifying heory for many of he ideas which have moivaed rend filer design in Official Saisics organizaions in he las few decades. In Gray and Thomson (1996) some mehods were suggesed for esimaing he parameers in he local dynamic model and for esimaing he bias erm parameer used in he BLIP end filers. This paper considers more carefully he problem of esimaing he bias parameer as oulined in ha repor. Finally, and again building on he ideas presened in Gray and Thomson (1996), we show how he use of adapive bias erms in he end filers may lead o furher improvemens in rend esimaion. LOCAL DYNAMIC MODEL (LDM) The convenional paradigm for rend filer design is o consider a moving window of n = r + 1observaions wihin which an esimae of he rend is o be calculaed for he cenral ime poin. Wihin he finie window we choose o model he observaions as y = g + ε where he rend g is given by p g = β j= 0 and ε is a whie noise process wih variance σ. The zero mean sochasic process ξ is assumed o be correlaed, bu uncorrelaed wih ε, and ξ, ε are assumed o be no boh zero. In paricular we consider he siuaion where he β j and σ are parameers local o he window, bu p, n and he model for ξ / σ involve global parameers which are consan across windows. Thus, alhough he parameers involved wih he mean and variance of y vary across windows, he auocorrelaion srucure of y will be a funcion of ime invarian parameers in addiion o ime iself. j j + ξ 769

2 Loosely speaking, he finie polynomial is inended o capure deerminisic low order polynomial rend whereas ξ is inended o capure smooh deviaions from he polynomial rend. Noe ha i is he incorporaion of ξ which disinguishes his local model from he sandard siuaion where i is zero. Among he benefis of including ξ are lower values of p and improved performance a he ends of series. Because he window is no likely o be large he model will need o involve as few parameers as possible on he one hand, while allowing for a sufficienly flexible family of forms for g on he oher. Wih hese poins in mind we choose o model ξ as a (possibly inegraed) random walk wih iniial value zero. In paricular, if denoes he p p backwards difference operaor saisfying X = X X 1, we have in mind he siuaion where + 1 g = +1 ξ is a zero mean saionary process wihin he window. In keeping wih his raionale, we shall always assume ha he levels of inegraion of he random walk componens ha make up ξ do no exceed p + 1 This seems an appropriae and parsimonious model which should accoun for smooh deviaions from he deerminisic polynomial rend componen. I also provides a dynamic rend model for g which is essenially of he same form as ha used in he ARIMA srucural models ha have been successfully applied o economic and official daa. (See Bell 1993 and Kenny and Durbin 198 for example.) In he local linear case p = 1 a simple dynamic model for g is given by y g + ε = β + β + ξ + ε = 0 1 where β 0, β 1 are consans and ξ is a simple random walk saisfying ξ ξ + η wih ξ 0 = 0. Moreover ε and σ η = λσ respecively. = 1 η are muually uncorrelaed whie noise processes wih variances σ and 3 TREND FILTER DESIGN AT THE ENDS Now consider a finie window wihin which he observaions follow he local dynamic model given in Secion~. We consider he case where he rend g is o be esimaed by a given cenral moving-average rend filer r ĝ = w y. s s= r s In keeping wih sandard pracice, we assume ha he w are consrained by he requiremen ha ĝ is an unbiased esimaor of g. Noe ha his condiion is equivalen o he requiremen ha he moving average filer passes polynomials of degree p. A he ends of series he cenral moving average filer will involve fuure unknown observaions. How hese missing observaions should be reaed is open o quesion. A common and naural approach involves forecasing he missing values, eiher implicily or explicily, and hen applying he desired cenral filer. The forecasing mehods used range from simple exrapolaion o model based mehods, some based on he local rend model adoped, ohers based on global models for he series as a whole. The laer include he fiing of ARIMA models o produce forecass (see Dagum 1978 in paricular). Ye anoher way o handle he missing values in he window is o employ addiional crieria specific o he ends of he series. An imporan requiremen, especially among official saisicians, is o keep seasonal adjusmen revisions and herefore rend revisions o a minimum as more daa comes o hand. Thus, a he ends of series, a naural crierion o consider is where ha g ~ is a predicor of R q is minimized when R q = E{( ws y r s= r + s + s g ~ ) ĝ based on pas daa. In general, given a hisory of observaions g ~ = r w yˆ s s= r + s where y + s } y,, y, i is eviden ˆ denoes he bes predicor of y + s in he usual mean 1 T 770

3 squared error sense. Thus here is a close relaionship beween he minimum revisions approach and ha of forecasing he missing values in he window. The minimum revisions sraegy appears o have been originally proposed by Musgrave (1964) for he case where g ~ is resriced o be linear in he observaions wihin he window. (See he discussion in Dohery 1991). In Gray and Thomson (1999) we adoped he minimum revisions sraegy based on he moving window paradigm and he local dynamic rend model of Secion. A he ends of he series we chose o predic ĝ by a linear predicor of he form g ~ = q u yˆ s s= r where q = T wih 0 q < r, T denoes he ime poin of he las observaion and he us are dependen on q. We considered wo cases. The firs imposes he condiion ha g ~ be an unbiased predicor of ĝ. The second weakens his requiremen by considering biased predicors such as hose developed by Musgrave (1964) for X-11 (see in paricular Dohery 1991). 3.1 Unbiased predicors In Gray and Thomson (1999) we esablished he following resul. Theorem 1 T Le y follow he local dynamic model specified in Secion. Given δ = ( δ,, δ and observaions y r, y+ q + s r r ), 0 q < r, he bes linear unbiased predicor (BLUP) of δ s y + s is u u = ( u,, ) is given by E 1 1 r u q 1 T 1 T 1 1 T 1 1 T u = L1 ( I GL ( LGL ) L )δ T r s= r q s y + s s= r where Here G = E1 E1 C( C E1 C) C E1, where E1 is he covariance marix of ξ + ε in he window (which does no depend on he absolue of ime indexing he origin of he window) and C is he design marix of he polynomial regression in he model for g. r In paricular he BLUP of δ s y + s is given by δ s ŷ + s where y ˆ + s is he BLUP of y + s for q < s r and y + s oherwise s= r r s= r δ s by he weighs of he cenral moving-average rend filer means ha he weighs of he Replacing he arbirary unbiased predicor g ~ of ĝ are given by Theorem 1 and minimize predicors we refer o hese end filers as BLUP end filers. 3. Biased predicors R q. Because of heir dependence on BLUP In Gray and Thomson (1999) we also considered he siuaion where g ~ is a linear predicor, bu now is no longer required o be unbiased. There we required he bias o be ime invarian in he sense ha i does no depend on he absolue ime indexing he origin of he window, whaever he parameers of he local dynamic model adoped. This ime invariance requires he levels of inegraion of he random walk componens which make up ξ no o exceed p, raher han p + 1 so ha we are now using a resriced local dynamic model. Unlike he end filers based on BLUP predicors, he end filers we consruced no longer are independen of hese parameers. The requiremen ha he bias be ime invarian does, however, lead o relaively sraighforward procedures for esimaing he parameric quaniies involved. By virue of he local dynamic models adoped, he end filers derived generalize and exend he curren X-11 end filers which were developed by Musgrave (1964) and placed in a predicion conex by Dohery (1991). We esablished he following resul. 771

4 Theorem T Le y follow he local dynamic model specified in Secion. Given δ = ( δ,, δ and observaions y r, y+ q r r ), 0 q < r, he bes linear ime invarian predicor (BLIP) of δ s y + s is u T r s= r q s y + s s= r where u = ( u r,, u q ) is given by ~ T ~ 1 T u = L1 ( I GL ( LGL ) L )δ ~ ~ 1 1 Here 1 ~ 1 T ~ 1 T ~ ~ T G = E1 E1 C p 1( C p 1E1 C p 1) C p 1E1, where E 1 is E1 + β pc pc p and E 1 is as given in Theorem 1 and C is as given in Theorem 1 and is pariioned as C, c ]. r [ p 1 p In paricular he BLIP of δ s y + s is given by δ s ŷ + s where y ˆ + s is he BLIP of y + s for q < s r and y + s oherwise s= r r s= r Noe ha he BLIP predicor given by Theorem has he form of a shrinkage esimaor since i is exacly he same ~ as he BLUP predicor given by Theorem 1, bu wih E 1 replaced by E 1 and C replaced by C p 1. Indeed, when β p = 0, he BLIP predicor becomes he BLUP predicor for he reduced model where g is replaced by g = p 1 j= 0 j β + ξ if p > 0, and g = 0 if p = 0 j, bu ξ and ε remain he same. Replacing he arbirary δ s by he weighs of he cenral moving-average rend filer means ha he weighs of he ime invarian biased predicor g ~ of ĝ are given by Theorem and minimize R q subjec o he bias being ime invarian for he resriced local dynamic model. Because of heir dependence on BLIP predicors we refer o hese end filers as BLIP end filers. Now E 1 /σ does no depend on ~ σ and E 1 need only be known up o a consan of proporionaliy. Noe ha using Theorem on he Henderson cenral filer weighs yields he X-11 end filers derived by Musgrave (1964) and Dohery (1991) when ξ = 0, p = 1, and β p / σ = 4 /( π (3.5) ). Thus Theorem provides a generalizaion and exension of he curren X-11 end filers. Theorem 3 in Gray and Thomson (1999) provides an alernaive form of he BLIP end filers which explicily builds on he corresponding BLUP end filers. The resul also provides a means of exploring he effecs of misspecificaion of β /σ p. An immediae consequence of ha resul is ha end filers based on BLIP predicors will generally have smaller mean squared revisions han hose based on he corresponding BLUP predicors. Moreover BLIP end filers converge o heir corresponding BLUP end filers as β /σ p increases. The bes possible mean squared revisions are achieved when β p / σ = 0. Then he BLIP end filers become BLUP end filers for he local dynamic model wih order p 1, bu he same sochasic srucure. Boh he BLIP and BLUP end filers are dependen on he global parameers specified by p, n and he model for ξ /σ. Alhough hese global parameers are sufficien o deermine he BLUP end filers, he BLIP end filers furher require knowledge of β /σ p. The laer is a funcion of local parameers whose values will no normally be known in pracice and which will need o be esimaed from he daa. In his case i is possible ha a mis-specified value of /σ could resul in a BLIP end filer which has greaer mean squared revisions han is corresponding β p BLUP end filer. Analysis of he mis-specificaion error shows ha in pracice he choice of values /σ β p should, if anyhing, over-esimae BLIP end filer owards is BLUP counerpar. ˆ β p / σˆ for /σ β p hus conrolling he mis-specificaion error by shrinking he 77

5 3.3 Properies of he Filers In Gray and Thomson (1999) he properies of he BLUP and BLIP end filers were discussed. The following summarizes hose findings relevan o his paper. The mean squared revisions generally increase as λ and p increase. This is of marginal uiliy in pracice, since he local dynamic model chosen is deermined from he paricular ime series concerned. However i is possible ha a local linear model wih large λ migh describe a ime series as well as a local quadraic model wih small λ. I would appear ha, in such cases, he BLIP end filer for he quadraic model may have greaer capaciy o achieve lower revisions. ˆ For he local linear model, he BLIP end filers based on he X-11 value β p / σˆ = 4 /( π (3.5) ) appear o offer only marginal gains over he BLUP end filers. In he local quadraic case, hey are clearly oo conservaive and a lower value of ˆ β p / σˆ migh more profiably be considered. For boh models, he upper limi of ˆ β p / σˆ = 4 / π for hese paricular end filers leads o end filers wih unaccepably high revisions. This may explain, in par, he reason why forecas exension using ARIMA models has largely superseded he use of he X-11 end filers in pracice. The former yield (global) BLUP end filers wih properies ha one migh expec are close o he (local) BLUP end filers considered here. On he oher hand, for he local linear model, he I / C guidelines imply ha he X-11 end filers are inflexibly applied whenever β / σ 4 / π p. 4 ESTIMATING THE PARAMETERS OF THE LDM A naural bu complicaed approach o esimaing he parameers of he LDM would begin wih looking a he likelihood corresponding o he LDM. From he concenraed local likelihoods we would obain esimaes of λ. These esimaes are unsable across bandwidhs because he bandwidhs have few daa poins. So some mehod of averaging of hem would be required o produce he global esimae of λ. (Indeed a virue of making λ global raher han local is o avoid his insabiliy in local esimaes). In our formulaion of he LDM he likelihood spli ino non-overlapping bands has correlaion beween bands. This correlaion needs o be accouned for in esimaing λ hrough local likelihoods. We could use he ˆ β p and σˆ from he las bandwidh o esimae he parameers required for he BLIP filers. Apar from some iniial work, we have no pursued his approach furher. 4.1 Esimaing a global λ Anoher approach begins wih he reformulaion of he LDM by considering he appropriaely differenced series. To pu maers ino conex and for simpliciy we consider he simples LDM given a he end of Secion. Here he firs difference of his daa is given by y = β + η + ε 1 which is locally an MA(1) process wih non-zero mean β 1. More generally we could wrie his as y = m + su where he locaion and scale parameers m and s are evolving slowly over ime, bu u is a global MA(1) process wih ime invarian parameers wih lag one auocorrelaion 1 /( λ + ). This suggess recovering λ by fiing a global MA(1) model o he sandardized series ( y m )/ s. However, fiing an MA(1) his way needs o be handled wih care because of he filering. In Gray and Thomson (1996) we discussed esimaing m nonparamerically using a sandard low pass filer and esimaing s by smoohing y mˆ by he same filer. Given he problems wih he non-parameric esimaion mehods we now believe ha if one is o esimae λ hrough his reformulaion of he LDM, hen i will be necessary o fi a random walk ype model o m and a sochasic volailiy model o s. This would also provide a mehod of producing a local β /σ p. However, such a sraegy seems o a odds wih he semi-parameric approach we adoped in formulaing he LDM o sugges appropriae rend filers. Currenly we prefer he simpler and more direc mehods of esimaing λ adoped in Gray and Thomson (1999) which ake advanage of he BLUP predicors 773

6 Consider he family of BLUP end filers for a given cenral filer and given values of n and p. Noe ha since hese end filers are unbiased hey depend only on q and λ. Applying he end filers o he daa yields he revisions r q r ( q, λ) = ws y+ s us ( λ) y+ s r r where he w s are he given cenral filer weighs and he u s (λ) are he BLUP end filer weighs given by Theorem 1 where he arbirary δ s are replaced by he w s. An appropriae cos funcion such as P = r 1 q= 0 α qr ( q, λ) can now be consruced wih he given posiive weighs α q reflecing he relaive coss of he respecive revisions. Finally, if n is he lengh of he ime series, λ is deermined by minimising = wih respec o λ. In Gray and 1 Thomson (1999) we looked a resricing he cos funcion o he wors case so ha α 1 and α = 0 for q 0. Clearly deciding on an appropriae cos funcion is an imporan subask of esimaing λ using his approach. I is possible ha a differen choice of cos funcion may lead o a 'pah o revisions' which is less volaile han for example ARIMA exension filers, bu we have no explored his ye. /σ β p 4. Esimaing a local Consider again he reformulaion of he LDM in erms of an appropriaely differenced series. Under local Gaussian assumpions E y mˆ = σ ( / π )( λ + ) so given an esimae of λ an appropriaely scaled raio of mˆ and ŝ should provide a reasonable esimae of ˆ β p / σˆ and is evoluion. However, here may be a bias problem in esimaing he raio β /σ p even when he esimaes of β p and σ are no biased. We also suggesed anoher mehod of esimaing ˆ β p / σˆ via is relaionship o he I / C raio following Musgrave (1964) and Dohery (1991). Specifically for he local linear model, given an esimae of λ, and considering I C = E ε / E g o be he populaion parameer ha I / C is esimaing, under Gaussian assumpions / ~ ~ ~ β1 β1 λ 1 / ~ ~ exp β I C = Φ + Φ 1 1 πβ λ λ πβ λ ~ where β 1 = β1 / σ and Φ ( x) is he sandard normal cumulaive disribuion funcion. Adapive esimaes of numeraor and denominaor of he raio I / C can be obained non-paramerically using a sandard low pass filer as ~ before and hen solving for β 1 using he las equaion. One consequence of his idenificaion is ha as λ increases moderaely from zero he upper limi of he range of admissible values of I / C decreases rapidly. In looking a acual ime series, we have found series where he low pass filer esimae of he I / C raio is ouside he admissible range for he LDM of he series for many observaions in he daa window and occasionally for all observaions in he daa window. This may be a resul of mis-specifying eiher λ or he order of he LDM. However, his seems unlikely because for he I / C raios he required λ would ypically need o be zero and he alernaive mehods of esimaion of λ all give non-zero values. Furhermore graphical analysis suggess ha he order of he LDM is appropriae. We have no invesigaed his furher. Plos of esimaes of β 1 / σ for he Building Permi series (q.v. Gray and Thomson 1999 for deails) for boh hese mehods are given in Figure 1. Alhough he evoluion of β 1 / σ is broadly he same for he wo mehods, albei he second mehod providing a smooher graph, i is clear ha hey have quie differen levels, heir means, being 0.19 and 0.11 respecively. I is also clear ha using a global value for β /σ p can be a raher blun insrumen for improving revisions. I also seems unlikely from his graph ha using a bandwidh average say over a year for β /σ p would make many gains over a global average and his was he case wih he series we examined. n P 1 0 = q 774

7 Alhough boh hese mehods give a reasonable idea of he evoluion of β /σ p, hey do no provide esimaes a he ends. There is ypically a loss of eiher half a window or a full window a he ends of he daa. This is no problem if a global esimae is required, bu his is a problem if adapive esimaes are required since he esimaes would provide values a bes half a year behind he required value and so a 6-sep ahead forecas would be required. I is clear from Figure 1 ha wih real daa forecasing more han 1-sep ahead could produce worse revisions since wihin such a forecas horizon here is considerable variaion in he esimae of β 1 / σ. For he window of he series where we did have an esimae of β 1 / σ for he q = 0 filer we examined he effec on he revisions of using hese esimaes of β 1 / σ from he firs mehod and using 1-sep ahead forecass from exponenially smoohing hem. In boh cases he revisions were worse han using he global esimae of β 1 / σ. We believe ha his may be due o he bias problem wih esimaing he raio menioned before. So given hese wo problems wih hese non-parameric mehods we currenly prefer he more direc way, adoped in Gray and Thomson (1999), of esimaing he raio /σ, hough of as a single parameer. Consider he family of BLIP end filers which depend on he daa yields he revisions β p /σ β p as well as q and λ. Applying hese end filers o r q ~ r ( q, λ) = ws y+ s us ( ˆ β p / σˆ ) y+ s r r building permis nonparameric esimaion of bea/sigma using scaled m and s Time nonparameric esimaion of bea/sigma using I/C Time Figure 1: Two mehods of non-parameric esimaion of β 1 / σ. The horizonal lines are he global means. where he w s are he given cenral filer weighs and he u ˆ s ( β p / σˆ ) are he BLIP end filer weighs given by Theorem where he arbirary δ s are replaced by he w s and he value of λ is ha given by minimizing P. Finally β /σ p is deermined by minimising P ~ ( P wih r replaced by ~ r ) wih respec o β p /σ. Again we looked a he case wih α 0 = 1 and α q = 0 for q

8 In Gray and Thomson (1999) we esimaed λ as in Secion 4.1 and esimaed a global value of β /σ p in he sense ha we chose he β /σ which minimized n P ~ where n is he lengh of he ime series. I was shown ha p = 1 using BLIP end filers wih hese global esimaes gave revisions which were close o or beer han using X11 end filers wih ARIMA exension. To produce adapive esimaes of he value of ˆ β p / σˆ β /σ p, we ake λˆ as given in Secion 4.1 and for each ime poin deermine which minimizes P ~. This hopefully provides a ime series of opimal values for /σ β p up unil he las half window where predicion will be required. Of course we need o address he problem of only having esimaes up o he las half window of he filer, bu for he momen we consider ha his esimaion procedure provides us wih all he required esimaes. We inroduce some noaion o disinguish hree imporan cases of BLIP end filers. Le BLIP(inf) refer o he BLUP filer, which is he BLIP filer as β / σ p, BLIP(0) refer o he BLUP filer of reduced order, which is he BLIP filer when β p / σ = 0, and BLIP refer o he BLIP filer when β /σ p is a neiher boundary of is range. Theoreical consideraions and a small simulaion sudy using some simple LDM's wih varying values for he parameers λ and β /σ p suggess ha depending on he order of he model and he size of β /σ p he greaes revisions will mosly come from eiher he BLIP(inf) case or he BLIP(0) case bu in boh hese cases he revisions will mosly no be close o zero. However, he BLIP case does happen frequenly and hen if he exac value of β /σ p is known he revisions will be zero. Wih his in mind we produced he adapive values for β 1 / σ for he hree series in Gray and Thomson (1999). Plos of β 1 / σ for he Building Permi series are given in Figure. The graph of β 1 / σ has similariies wih and differences from hose in Figure 1. In paricular now for many ime periods β 1 / σ is 1 (which happened o be he maximum value used in he grid search, bu a β 1 / σ 1 is pracically he same as infiniy), indicaing ha a BLIP(inf) filer is appropriae. Around urning poins, ypically bu no always, β 1 / σ is zero, indicaing ha a BLIP(0) filer is appropriae. In Figure here seems o be more discriminaion han in say mehod of Figure 1. Finally here are periods when here are shor runs of β 1 / σ beween 0 and 1 where from he heoreical analysis in Gray and Thomson we would expec he revisions o be smalles. Similar graphs o Figure were observed wih he Merchandise Expor series and Permanen and Long Term Migraion series discussed in Gray and Thomson (1999). Figure 3 shows boxplos of revisions for he q = 0 case. where he Henderson 13 erm filer is used as he cenral filer. Using he adapive β 1 / σ insead of a global value improves he revisions considerably wheher looking a he medians or lower or upper quariles. When a BLIP filer is used he revisions are pracically zero. As expeced he BLIP(0) filer revisions are less han he BLIP(inf) revisions. Similar graphs o Figure 3 were observed wih he Merchandise Expor series and Permanen and Long Term Migraion series. For Merchandise Expors he BLIP(0) 776

9 building permis original series and rend from LDM wih p=1 lambda= Time bea /sigma Time Figure : Esimaed values of β 1 / σ ha minimize BLIP filer revisions. The horizonal line in he boom graph is he mean of he β 1 / σ and is almos exacly he same as he global esimae of β / σ building permis global adapive BLIP(0) BLIP BLIP(inf) Figure 3: Boxplos of revisions. The firs wo are for he global esimae of β 1 / σ and for he adapive esimae of β 1 / σ respecively; he nex hree are for he adapive esimae of β 1 / σ when respecively he BLIP(0) filer, BLIP filer or BLIP(inf) filer is chosen. The BLIP(0) filer is chosen 33% of he ime, he BLIP 31% of he ime and he BLIP(inf) 37% of he ime. These boxplos mixed in hese proporions will produce he adapive boxplo. 777

10 filer, BLIP filer, and BLIP(inf) filer are chosen 54% of he ime, 8% of he ime and 38% of he ime respecively. For Long Term Migraion he proporions were 38%,14% and 48% respecively. Figure 4 shows he Henderson 13 em cenral filer for he Building Permis series, and he q = 0 end filers. The end filer wih he adapive esimae of β 1 / σ performs similarly o he end filer wih he global esimae of β 1 / σ excep a urning poins where i racks more closely he cenral filer. So based on he limied cases we have examined, he quesion: have adapive BLIP filers he poenial o improve revisions a end of series? has an affirmaive answer. 4.3 Esimaing β /σ p a he ends The challenge here is o have a mehod of providing adapive esimaes for he curren ime period. which chooses fairly auomaically beween he BLIP(0), BLIP, and BLIP(inf) end filers. Recall ha pracically infiniy is reached raher quickly and any BLIP filer wih β p / σ 1 ypically has abou he same revisions as he BLIP(inf) filer. Wih ha in mind, i suggess ha we have a process wih hree saes: 0, a fixed β /σ p, and 1. Our curren idea is o model he β /σ p as a Markov chain, using he hisoric values o provide esimaes of he ransiion probabiliies, and o use as he fixed value of β /σ p he mean of he non-zero, non-infiniy values. This fixed value will ypically be much less han he global value of β /σ p, (e.g. for Building Permis he global esimae of β 1 / σ is.47 whereas he mean of he non-zero, non-infiniy values is.33). From he heoreical analysis in Gray and Thomson (1999) we know ha he greaes gains will come when β /σ p is small and if he esimae of β /σ p is greaer han he acual value. Thus by using his fixed value we would expec revisions somewhere beween hose for he global BLIP filer and he adapive BLIP filer. Furher research is required and we shall pursue hese ideas in a sudy we are currenly planning. building permis cenral global blip adapive blip Time Figure 4 : The Henderson 13 em cenral filer for he Building Permis series, and he q = 0 end filers using an LDM wih p = 1 and λ =. 046, and β 1 / σ esimaed globally wih a value of.47 and esimaed adapively respecively. 778

11 Bibliography Bell, W. (1993) Empirical comparisons of seasonal ARIMA and ARIMA componen (srucural) ime series models, Research Repor CENSUS/SRD/RR-93/10, Saisical Research Division, Bureau of he Census, Washingon, DC Dagum, E. B. (1978). "Modelling, forecasing and seasonally adjusing economic ime series wih he X-11 ARIMA mehod," The Saisician, 7, pp Dohery, M. J. (1991). Surrogae Henderson filers in X-11, Technical Repor, NZ Deparmen of Saisics, Wellingon, New Zealand Gray, A. G. and Thomson, P. J. (1996). Design of moving-average rend filers using fideliy, smoohness and minimum revisions crieria, Research Repor CENSUS/SRD/RR-96/1, Saisical Research Division, Bureau of he Census, Washingon, DC Gray, A. G. and Thomson, P. J. (1999). "On a family of moving-average rend filers for he ends of series," To appear in he Journal of Forecasing Kenny, P. B. and Durbin, J. (198) "Local rend esimaion and seasonal adjusmen of economic and social ime series," Journal of he Royal Saisical Sociey, Series A, 145, pp Musgrave, J. C. (1964). A se of end weighs o end all end weighs, Working paper, Bureau of he Census, Washingon, DC

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13 LOCALLY ADAPTIVE TREND-CYCLE ESTIMATION FOR X-11 Benoî Quenneville, Saisics Canada and Dominique Ladiray 1, INSEE Benoî Quenneville, Saisics Canada, R.H. Coas Building 3 rd floor, Oawa, Onario, K1A 0T6, Canada quenne@sacan.ca ABSTRACT The symmeric Henderson moving averages are used in X-11-based seasonal adjusmen procedures o calculae he rendcycle componen. In his paper, we invesigae how o exend hem o deal wih missing observaions a he end of he series. This is an imporan problem because he rend-cycle esimaes a he end of he series are used in he analysis of he curren sage of he business cycle while hey are subjec o revisions as more daa become available. The problems are hus o minimize he revisions in he rend-cycle esimaes, and o deec urning poins in hem. We compare wo approaches. The firs uses Musgrave asymmeric moving averages, which depend on a parameer. The second uses he symmeric Henderson moving average wih forecass for he missing observaions. We show ha, on average, he symmeric Henderson moving average wih forecass for he missing observaions minimizes he revisions, bu wih marginal gains over he Musgrave asymmeric moving averages wih is parameer esimaed from he daa. We also show ha he imely deecion of urning poins requires anoher sraegy, and we propose o produce wo ses of rend-cycle esimaes for he ime poins a he end of he series. The firs one by eiher using he symmeric Henderson moving average wih forecass for he missing observaions, or by using Musgrave asymmeric moving averages wih is parameer esimaed from he daa o minimize revisions. The second one by selecing a se of ailor-made Musgrave asymmeric moving averages designed o cach urning poins. Key Words: ARIMA models; Henderson moving averages; Local linear models; Musgrave asymmeric moving averages; Seasonal adjusmen; Turning poins. 1. INTRODUCTION In his paper we compare mehods for esimaing he rend-cycle componen for a seasonally adjused series. We invesigae how o exend he symmeric Henderson moving averages, used in X-11, o deal wih missing observaions a he end of he series. We sudy wo approaches o deal wih hese missing observaions. The firs uses Musgrave asymmeric moving averages. The second uses he symmeric Henderson moving average wih forecass for he missing observaions. In secion, we inroduce Musgrave asymmeric moving averages, and an alernaive, which is o use he symmeric Henderson moving average wih forecass for he missing observaions. We discuss rend-cycle esimaion a urning poins, and inroduce a se of ailor-made Musgrave asymmeric moving averages. Musgrave asymmeric moving averages depend on a parameer, and, in secion 3, we discuss various ways o esimae i. In secion 4, we presen a few ime series models used o forecas he missing observaions required for he symmeric Henderson moving average. In secion 5, he various rend-cycle esimaion mehods are evaluaed in erms of revisions and deecion of urning poins on wo specific examples. We give our conclusions in secion 6.. TREND-ESTIMATION FOR THE END POINTS.1 Musgrave Asymmeric Moving Averages The X-11 seasonal adjusmen mehod (Shiskin e al., 1967) and is subsequen generalizaions, such as X-11- ARIMA (Dagum, 1978, 1980) and X-1-ARIMA (Findley e al., 1998), use moving averages o esimae he rendcycle componen from a seasonally adjused series. For ime poins far enough from he beginning and end of he series, symmeric Henderson moving averages are used. Boh Dagum e al. (1996) and Findley e al. (1998) discuss how o calculae any Henderson moving average of an odd lengh, say N=n+1. X-11 selecs he lengh of he Henderson moving average using he so-called I/C-raio, he inverse of a signal o noise raio, which is he raio of 1 We wish o express our hanks o Professor Esela Bee Dagum, Mariea Morry, Norma Chhab, Guy Huo and Bernard Lefrançois for heir commens on earlier versions of his paper. 781

14 he average monh o monh absolue percen change (difference in X-11-addiive mode) in he irregular componen o ha of he rend-cycle componen. For he ime poins a he beginning and end of he series where he Henderson symmeric moving average canno be used, X-11 employs asymmeric moving averages derived from a known mehod due o Musgrave (1964), and repored in Findley e al. (1998). Musgrave's basic idea is o minimize he revisions beween he preliminary esimaes of he rend-cycle and is final esimae. The rend-cycle a he end of he series is assumed linear, say Y = α + β + ε wih ε ~ N (0, σ ). The only consrain imposed is ha he coefficiens sum o one. Equaion (B.3) from Findley e al. (1998) shows ha he = ( N 1) / depend on he raio n Musgrave asymmeric moving averages D = β / σ, bu are independen of he inercep parameer α. For he sake of clariy, he equaion o compue he coefficiens of a Musgrave asymmeric moving average is now given. Le ( w,..., w N ) he N coefficiens of he symmeric Henderson moving average, and le ( v,..., ) 1 be 1 v M be he M coefficiens of he asymmeric Musgrave moving average where n + 1 M N 1. Then he coefficiens v j, j = 1,..., M saisfying M j = 1v = 1 ha minimize are given by j v j = w j 1 + M N E w Y M v j j j j= 1 j= 1 Y j M + 1 j D N N w i + i i= M + 1 M 1 + M + 1 w. i ( M 1)( M + 1) i= M + 1 Findley e al. (1998) also repor ha he I/C-raio and he parameers β and R = I / C = σ /( π β ), and hence, D = 4 /( πr ). 1 D σ are relaed by he equaion X-11 asymmeric moving averages are obained wih pre-defined values for he I/C-raio, irrespecive of he rue heoreical values of he parameers β and σ for he ime series being seasonally adjused. In he X-11 mehod, he Musgrave asymmeric moving averages associaed wih he 9-erm Henderson use R = 1 ; hose associaed wih he 13-erm use R = 3. 5 ; and hose associaed wih he 3-erm use R = Musgrave asymmeric moving average dependence's on he parameer D can be relaxed by forcing he asymmeric moving averages o preserve a linear rend. Under Musgrave s assumpions, his ranslaes ino adding he consrain M = 1 jv = ( M +1) / on he coefficiens. This has also been sudied in a more general conex by Gray j j and Thomson (1996), and here, i corresponds o he case where D. In his case, he coefficiens v j, j = 1,..., M ake he form v j = w j 1 + M N 6 w + M ( j ( M + 1) ) ( M 1)( M + 1) M + 1 i wi. i i= M + 1 i= M + 1 In his paper, we propose o esimae he parameer D by a suiable mehod. We invesigae if his improve he revisions in he rend-cycle esimaes over keeping he parameer D fixed. We also invesigae if his can be an alernaive o using he symmeric Henderson moving average wih forecass for he missing observaions. Various ways o esimae he parameer D are discussed in secion 3.. Forecasing Missing Observaions The X-11-ARIMA mehod adds forecass for some of he missing observaions a he end of he original series; consequenly, asymmeric moving averages ha are closer o he cenral symmeric moving average are used, bu on a series exended wih forecass (Dagum, 1978). We quoe direcly from Kenny and Durbin (198), N 78

15 [ ] he use of a forecasing mehod is no more han a mahemaical device for consrucing an appropriae one-sided filer for use a he end of he series. The advanage of his filer compared wih he sandard X-11 filer is ha i can respond more flexibly o he behaviour of a paricular series. Here, insead of forecasing he original observaions before applying he X-11 mehod, as is done in X-11-ARIMA and X-1-ARIMA, we propose o forecas he componens a each sep of he X-11 mehod, i.e. whenever an asymmeric moving average would be used. Pierce (1980) showed ha predicing he missing observaions by heir condiional expecaion minimizes he revisions. Hence, heoreically, on average, minimum revision in he rendcycle esimaes will occur when he symmeric Henderson moving average is applied on he series exended wih forecass for he missing observaions. Consequenly, we expec Musgrave asymmeric moving averages wih he parameer D esimaed from he daa o give slighly larger revisions, bu no as large as when Musgrave asymmeric moving averages wih fixed parameer D are used. Our choice of forecasing models is furher discussed in secion 4..3 Trend-Cycle Esimaion a Turning Poins A urning poins, i is he imely deecion of he urning poin, in op of he minimizaion of he revision, ha is an imporan crierion. The inclusion of a consan in an ARIMA(p,1,q) model usually induces a slope in he forecass. On average, he slope esimae is on arge. A a urning poin, however, he rue local slope may flaen faser han is esimae, because he slope esimae depends oo much on all he previous daa poins, so ha he ARIMA forecass may miss he urning poins. Similar commens apply when a linear rend model is fied o he daa. To reac faser o he local changes in he values of he slope, we propose o locally esimae he parameers of he ARIMA model (mosly he consan erm) or he parameers β and σ of he Musgrave asymmeric moving averages. These local esimaion mehods are presened in he nex secions. Anoher rivial way of no missing a urning poin is o assume ha he rend is flaening. This ranslaes ino assuming β 0, and hence D 0, in Musgrave asymmeric moving averages. In his case, he coefficiens 1 N v j, j = 1,..., M ake he simple form v j = w j + wi. M i= M + 1 The examples will illusrae ha hese ailor-made Musgrave moving averages are BCS esimaors, and ha hey do exacly wha hey are supposed o do: well a urning poins, and quie bad on average. On he oher hand, assuming D, i.e. preserving linear rend, will obviously fail a urning poins. 3. ESTIMATION OF THE PARAMETER OF THE MUSGRAVE ASYMMETRIC MOVING AVERAGE This secion discusses various mehods o esimae eiher he parameers β and σ, or he parameer D, or he value of he I/C-raio. Any of hese can be used o compue Musgrave asymmeric moving averages by ransforming hem o he corresponding value of D. We consider he wo cases where he I/C-raio is fixed irrespecive of he ime series being seasonally adjused, and he case where he parameer D is allowed o change wih eiher he series and he number of observaions, or according o he local values of he parameers β and σ in he local window corresponding o he span of he symmeric Henderson moving average Fixed Value Mehods We consider he hree cases where 1) R is se o he value currenly used by X-11, i.e. 1.0, 3.5 and 4.5, which corresponds o he curren X-11's asymmeric moving averages (label is X11 or X in he sequel); ) where D which corresponds o he Gray-Thomson unbiased asymmeric moving averages for linear rend (label is GT or G in he sequel); and 3) where D 0 which corresponds o assuming β = 0 (label is Inf or In in he sequel). 783

16 3.. Adapive Mehods X-11 Esimae of he I/C-raio: (IoC or I) Whenever he rend-cycle is esimaed from a seasonally adjused series, X-11 esimaes he I/C-raio o selec he lengh of he symmeric Henderson moving average. The deails are provided in Ladiray and Quenneville (000). We use hese esimaes of he I/C-raio o generae he associaed se of Musgrave asymmeric moving averages. These esimaes of he I/C-raio may change whenever a new daa poin is added o he ime series. Minimizaion of Hisorical Revisions: (His or H) For each asymmeric Musgrave moving average, he parameer D akes he value ha has hisorically minimized he sum of squared revisions. The revisions are he differences beween he rend-cycle esimaes obained wih he given asymmeric moving average and he esimaes obained wih he symmeric Henderson moving average. The value of D varies wih each asymmeric moving average, and every ime a new daa poin is added o he ime series. Ordinary Leas-Squares Regression: (OLS or O) In his mehod, ordinary leas-squares (OLS) regressions are performed for he assumed linear model in he local window wih he available daa. The value of D varies wih each asymmeric moving average, and wihin each local window of M observaions. This is because he esimae of D for he Musgrave asymmeric moving average ( v1,..., vm ), M = n + 1,..., N 1 is based on he M available observaions in he local window. Weighed Leas-Squares Regression: (WLS or W) The fourh adapive esimaor of D is based on local 3 weighed leas-squares regression insead of OLS. The ricube weigh funcion, W ( z) ( 1 z ) 3 Y wihin he local window ( Y,..., Y0 ) is [ /( n +1) ] elsewhere, is used. The weigh for,..., n Y n j = for z < 1 and 0 W. 4. FORECASTING MODELS We consider various ARIMA models o forecas he missing observaions required for applying he symmeric Henderson moving average. According o Kenny and Durbin (198), he forecasing mehod is a choice of convenience. In addiion, Makridakis and Hibon (1997) concluded ha simple models could produce accurae possample forecass; however, Dagum e al. (1996) made i clear ha he ARIMA models used for he exrapolaions mus be consisen wih he implied models from X-11. These conclusions moivae our resricions o he ARIMA models presened below. These models will be evaluaed wih respec o he magniude of he revisions in he final rend-cycle esimaes. All he models are based on he firs differences of eiher he original or he log daa. This sresses he fac ha we model local slopes in he daa insead of heir local levels. An inercep is always included. I esimaes he average period o period increase (or percenage increase when log daa are used) in he ime series. Local ARIMA (1,1,0) Model: The firs model is a local AR(1) model on he firs differences of he original daa covered by he local window corresponding o he span of he asymmeric Musgrave moving average. The model is ( 1 B) Y = µ + φ( 1 B) Y 1 + e, = n,..., n j, j = n,..., 1. The daa are resriced o he previous n and fuure n j, j = n,...,1 observaions around he daa poin being filered. We use only hese M daa poins o obain a local esimae of he period o period increase. The mean µ is esimaed from he local sample mean of ( 1 B) Y. The auoregressive parameer φ is esimaed by heir firs order auocorrelaion. When he decomposiion model is muliplicaive, we use boh he unransformed (label is LARU or Lu) and he log (label is LARL or Ll) daa o compare he effec of esimaing/forecasing growh raes insead of simple period o period increase. Global ARIMA (1,1,0) Model: This model is he same as he local ARIMA(1,1,0), bu insead of using only he n previous daa poins, all he available ones are used. I will allow us o compare he local slope esimae wih he full-span slope esimae. As i is he case wih he local ARIMA(1,1,0) model, boh unransformed (label is GARU or Gu) and log (label is GARL or Gl) daa are used. Global ARIMA (0,1,1) Model: (GMA or Ma) This is he same as he Global ARIMA(1,1,0), bu i uses an MA(1) model insead of an AR(1) model. Here, he addiive decomposiion uses unransformed daa, and he muliplicaive 784

17 decomposiion uses log daa. This model can only be compared o he Global ARIMA(1,1,0) model and he nex model. Global ARIMA ( p, 1, q ) Model: (ARMA or Ge) The order p and q of he ARMA (p,q) model are seleced using he BIC. Boh p and q are resriced o be smaller han. The addiive decomposiion uses unransformed daa, and he muliplicaive decomposiion uses log daa. This model can only be compared o he Global ARIMA(1,1,0) model and he Global ARIMA(0,1,1) model. 5. ILLUSTRATED EXAMPLES In his secion, he various rend esimaion mehods are compared wih wo ime series. The series and he analysis ools are presened in he firs subsecion, while he main resuls are presened in he second one. 5.1 Series and Mehodology We consider wo ime series: he Toal Canadian Impors (Impors) and he Expors of Oher Animal Producs (Animexp) boh in curren dollars from January 1981 o December The seasonal adjusmen opions for hese wo series specify a muliplicaive decomposiion model wih he original series forecased wih an ARIMA model, and adjused for rading-day and Easer Holiday variaions. We used X-1-ARIMA o obain he X-11's D1, and D10 ables, from which we compued Table D11M=D1/D10. Table D1 conains he raw series adjused for Trading-day variaions, Easer Holiday, and exreme values. Table D10 conains he final seasonal facors. Table D11M is a modified seasonally adjused series used by X-11 o calculae he final rend-cycle via a symmeric Henderson moving average. Deails on X-11's ables are provided in Ladiray and Quenneville (000). The modified seasonally adjused series and heir rend-cycle values are superimposed for he Impors and Animexp series in Figure 1 and Figure respecively. For he Impors series, a 13-erm Henderson is used, and consequenly, he firs n = 6 and las n = 6 rend-cycle values are no available in Figure 1. For Animexp, a 3-erm Henderson is used, and he firs n = 11 and las n = 11 rend-cycle values are no available in Figure. The Impors series is characerized by a seady growh mainly caused by inflaion, which is ypical of many economic ime series expressed in curren dollars. The Animexp series is characerized by many urning poins, and is much noisier. The Absolue Percenage Errors (APE) are used o measure he size of he revisions. For a given asymmeric moving average, he APE is defined by he formula ( j 100 C ) C / C, where ( j C ) is he rend-cycle esimae for observaion Y using n j, j = n,..., 1 fuure observaions; and C is he final rend-cycle esimae. In he analysis o follow, he mean of he APE, he MAPE, will be used. The MAPE is sensiive o large revisions ha may occur from ime o ime, which official saisicians would like o avoid. Now, insead of abulaing he MAPE, heir ranks will be abulaed. For boh series and for each esimaion mehod, he APE's boxplos for he firs rend-cycle esimaes ( j = 6 for Impors, j = 11 for Animexp) and he penulimae rend-cycle esimaes ( j = 1) are also graphed. These MAPE's ranks and he associaed boxplos provide a full picure for he APE's disribuions. 5. Main Resuls The heory holds rue Table 1 and Figures 3 and 4 show ha he heory holds rue for he Impors series. On average he smalles MAPE occur when he symmeric Henderson moving average is used wih forecass for he missing observaions. This is followed by mehods where he I/C-raio is esimaed, and followed by mehods where he I/C-raio is fixed over he full range of he series. When compared wih GT moving averages, which impose he preservaion of a linear rend, we see ha X11's MAPE can be considered as being equal or smaller. This resul was heoreically expeced because he X11 and GT moving averages are derived he same way, excep for he addiional consrain. Mehod His, designed o give he smalles roo mean squared error (RMSE), yields MAPE ha are smaller han hose obained wih X11, GT, and Inf, where he I/C-raio is fixed over he full range of he series. Mehod His also gives MAPE ha are smaller han hose wih OLS and IoC and abou he same size as hose wih WLS. The able and figures confirm ha here are no major differences beween he ARIMA models, wheher or no he log ransform 785

18 was used, and wheher or no he esimaion of he parameers was done wih local daa alone. Finally, mehod Inf gives much larger APE han any oher mehods. This was o be expeced because his esimaion mehod assumes a locally fla rend, which is rarely he case for he Impors series. Tha mehod, however, is no o be alogeher discarded. Bu he differences are marginal The boxplos in Figures 3 and 4 show ha, excep for mehod Inf, he revisions are all abou he same size; around 0.75% for he firs rend-cycle esimaes, and around 0.04% for he penulimae rend-cycle esimaes. I does no really maer which mehod is used for he Impors series. Wih more noise hings sar o change When he series is more perurbed, some differences sar o show up: For he Impors series, we have also considered Table B6 insead of Table D11M. Table B6 is he firs seasonally adjused series produced by he X-11 mehod on which he Henderson moving average will be applied for he firs ime. Compared o Table D11M, Table B6 conains all he exreme observaions; so, i can be considered as a more erraic series han Table D11M. Table gives he MAPE for he rend esimaion mehods using Table B6 insead of Table D11M. Here, he final rendcycle is obained by applying he 13-erm Henderson o he B6-series. Clearly, local ARIMA models perform poorly a he firs esimaion of he rend-cycle esimaes j=6,5,4. Table also gives a case where Musgrave asymmeric moving averages wih he parameers locally esimaed by WLS can be preferred over forecasing he missing observaions wih local models, a leas for j=6,5,4. This resul is also confirmed wih series Animexp in Figures 5 and 6, and Table 3 where now j=11,...,1, and where again local ARIMA models poorly perform a he firs few esimaions of he rend-cycle esimaes j=11,10,9,8,7. Excluding local ARIMA models, Table 3 shows ha he mehods ranking for he Animexp series is similar o ha of he Impors series. Change he problem, change he mehod For he Impor series, consider he wo urning poins ha occur a he ime poins =4 in December 198 and a =17 in April Growh raes in he final rend-cycle in November and December 198 are 0.058% and 0.603%. Tha is, he rend-cycle flaens in November 198, and urns up in December 198. In March 1995, he rend-cycle flaens wih a growh rae of % and i urns down in April 1995 wih a growh rae of %. Growh raes were calculaed, as hey would have been made available o he analys in a real life applicaion. Mehods Inf, GARU, GARL, GMA, and ARMA did cach he sign reversal in he growh rae in December 8. All he ohers had o wai unil January 83. Tha experimen was repeaed for he April 1995 urning poin. Mehod Inf deeced he sign reversal in he growh rae in March 1995; mehods IoC, WLS, LARL, LARU, GARU, GARL, GMA, and ARMA deeced i in April 95; and finally mehods X11, GT, His, and OLS had o wai unil May 95. For he Animexp series, he ime poins = 8, 45, 55, 87, 98, 116, 13, 154, 161, where he final esimaion in he rend-cycle changes direcion, are considered. Table 4 indicaes how many monhs i ook before he rend-cycle esimaes show a sign reversal in he growh rae, he las column indicaes he average ime lag in monhs over he nine daa poins. Clearly, mehod Inf deecs he sign reversal before he oher mehods. I even has a negaive average ime lag: hese ailor-made asymmeric moving averages deeced some of he urning poins before he ime poin daed by he symmeric Henderson moving average. This is cerainly a desirable feaure for an esimaor designed o cach urning poins. Mehod WLS ranks second as far as he average ime lag is concerned. This illusraes he adapabiliy of he esimaes of he parameers β and σ o heir local behavior. Mehods based on forecasing he missing observaions wih global ARIMA models deec he urning poins faser han mehods ha use local ARIMA models. Mehods IoC, His and X11 rank nex, and before mehod GT ha ranks las, as expeced. Table 5 averages he APE a he nine urning poins in consideraion, and gives heir ranks. The sriking feaure of his able is he superior performance of mehod Inf, especially for he firs five preliminary esimaes of he rendcycle. On he oher hand, GT moving averages perform poorly because hey are designed o preserve curren linear rend in he daa. Mehods based on forecasing he missing daa wih a global ARIMA model on he firs differences have a rend slope esimae given by ( logy logy1 )/( 1). As soon as Y sars o eiher increase or decrease, so does he slope in he forecass, and his migh explain why hese mehods perform relaively well a 786