Long-Term Demand Prediction using Long-Run Equilibrium Relationship of Intrinsic Time-Scale Decomposition Components

Transcription

1 Proceedings of he 2012 Indusrial and Sysems Engineering Research Conference G. Lim and J.W. Herrmann, eds. Long-Term Demand Predicion using Long-Run Equilibrium Relaionship of Inrinsic Time-Scale Decomposiion Componens Akkarapol Sa-ngasoongsong School of Indusrial Engineering and Managemen Oklahoma Sae Universiy Sillwaer, OK 74078, USA Absrac Long-erm demand predicion is crucial for indusries wih long produc developmen cycles. Especially in case of auomobile producs, where ypical concep-o-release imes are long (12-60 monhs), he process of demand evoluion follows a nonlinear, nonsaionary dynamics hindering accurae predicion of he fuures of auomobile demand. This paper presens a new predicion approach based on Inrinsic Time-Scale Decomposiion (ITD) o forecas auomobile demand over exended ime-horizons. The ITD algorihm decomposes a ime series signal ino a sum of componens called proper roaion componens and monoonic rend. A key advanage of his approach is ha his mehod can be used o idenify long-run equilibrium relaionship of auomobile demand and ITD componens of economic indicaor. Once he relaionship is idenified, i akes advanage of he coinegraion propery of a Vecor Error Correcion Model (VECM) o forecas auomobile demand. Hence, long-erm impac of economic indicaor componens on auomobile demand can be quanified. The empirical resuls sugges ha his ITD-based predicion approach can significanly improve predicion accuracy in erms of RMSE (23%) and MAPE (26%) for long-erm predicion of auomobile demand (12-monh ahead predicion), compared o classical and advanced ime series echniques. Keywords Long-Term Demand Predicion, Inrinsic Time-Scale Decomposiion, Vecor Error Correcion Model 1. Inroducion Demand predicion is essenial par of any business aciviy. For indusries wih long produc developmen cycles, long-erm demand predicion serves as an inpu o many business decisions which usually affec profiabiliy of he organizaion. As in he case of auomobile producs, where ypical concep-o-release imes are long (12-60 monhs), reliable long-erm demand predicion makes an imporan conribuion o successful service, revenue, producion and invenory planning. Auomobile demand predicion has received significan aenion in he lieraure. Many heoreical models for auomobile demand predicion have been proposed [1-5]. Mos of hem are economeric approaches imposing a cerain srucure of economic heory on he daa. Only recenly, a few effors have been made o address he auomobile demand predicion problem using ime series and daa-driven approach [6-8]. However, none of hem have addressed an auomobile demand predicion for long-erm horizon. In economic area, some recen developmens on ime series echniques have been specifically designed o quanify relaionships among endogenous and exogenous variables. These echniques include Vecor auoregressive (VAR) and Vecor error correcion Model (VECM) [9, 10]. Especially in he case of nonsaionary variables, VECM has been broadly recognized as a powerful heory-driven model ha can be used o describe he long-run dynamic behavior of mulivariae ime series. However, he choice of endogenous and exogenous variables in VECM is problemaic for auomobile demand predicion. Demand is known o be influenced by many exogenous facors, such as adverising, sales promoions, reail price and echnological sophisicaion [11]. In auomobile marke, adverising and sales promoions end o have subsanial effecs, however, hese effecs on demand are rarely persisen [12]. To selec variables for auomobile demand predicion, some economic indicaors, such as Consumer Price Index (CPI), unemploymen rae, ec., have been suggesed o have persisen effecs on auomobile demand, bu he resuls of empirical model show ha long-

2 run equilibrium relaionship among hese economic indicaors and auomobile demand does no exis. Considering a nonlinear behavior of hese economic indicaors, one would expec ha a saisical hypohesis es of long-run equilibrium using linear assumpion may perform poorly due o model misspecificaion. This paper presens a new predicion approach based on Inrinsic Time-Scale Decomposiion (ITD) echnique. The ITD mehod is a recenly developed nonparameric decomposiion echnique for signals ha are nonlinear and/or nonsaionary in naure. ITD decomposes a signal ino a sum of componens called proper roaion componens and monoonic rend. As advanages of nonparameric mehod, ITD requires very limied assumpions abou he daa. This advanage makes ITD suiable for nonlinear daa from unknown underlying processes. ITD also provides unbiased decomposing componens, compared o parameric algorihm due o parameer esimaion. The key aspec of his new predicion approach is o develop a mehodology o idenify a causal and long-run equilibrium relaionship from variables of ineres and ITD componens of relaed indicaors, and use his idenified relaionship for predicion. This new predicion approach can be applied o any sysem of nonlinear and/or nonsaionary variables. In his paper, he propery of ITD leads o an applicaion of deermining long-run equilibrium relaionship of auomobile demand and ITD componens of nonlinear economic indicaor. Our invesigaion indicaes ha he long-run equilibrium of auomobile demand and ITD componens of unemploymen rae does exis, and he ITDbased predicion can significanly improve predicion accuracy in erms of RMSE (23%) and MAPE (26%) for longerm predicion of auomobile demand (12-monh ahead predicion), compared o classical and advanced ime series echniques. The organizaion of his paper is as follows: Secion 2 presen ITD algorihm in deails. VECM and relaed saisical hypohesis ess are presened in Secion 3. Secion 4 is a mehodology secion of ITD-based predicion approach. The implemenaion deails and empirical resuls are given in Secion 5. Conclusion and suggesed fuure work are presened in he las secion of his paper. 2. Inrinsic Time-Scale Decomposiion Given a nonlinear ime series signal X, where ( 1,2,..., T), a general represenaion of a dynamical sysem of X can be represened using Volerra expansion [13] as X i i ij i j ijk i j k ijkl i j k l i0 i0 j0 i0 j0 k 0 i0 j0 k 0 l0... (1) where is deerminisic componen, ~ IID and i, ij, ijk, ijkl,... is a se of coefficiens. Eq. (1) is nonlinear if i has nonzero coefficiens ij, ijk, ijkl,... on he higher-order erms. The Volerra expansion in Eq. (1) can be ransformed ino an auoregressive represenaion [14] as X f ( X 1, X2,...) (2) where f ( X 1, X2,...) is some nonlinear funcion of he pas values of X. The difficuly in esimaing Eq. (2) is ha he funcional form f ( ) is unknown when working wih real world daa. There are number of procedures in he lieraure ha have been developed o deermine he form of nonlineariy. However, no se of ess can acually discern he correc form of nonlineariy. Raher, he ess can only sugges he form of he nonlineariy. Inrinsic Time-Scale Decomposiion (ITD) is a recenly developed nonparameric algorihm aiming o enhance an analysis of nonlinear and nonsaionary signals [15]. The advanage of ITD is ha i does no require parameric funcional form of he original series. The ITD mehod represens a nonlinear, nonsaionary signal as a summaion of componens called proper roaion componens and monoonic rend. ITD decomposes he signal ino a sequence of proper roaions of successively decreasing insananeous frequency a each subsequen level of he decomposiion as X HX LX HX ( H L) LX ( H(1 L) L ) X ( H L L ) X (3) 2 a 1 k k 0 where H is defined as a proper-roaion-exracing operaor and L is defined as baseline-exracing operaor. HL k X is he (k+1) s level proper roaion and L a X is he monoonic rend. The following procedure of ITD is used o exrac proper roaion componens from a ime series X : (1) Idenify he locaion of all local exrema of X denoed as τ k where k {1,2, } (2) Assume ha X is available for [0, τ k+2 ], define a baseline signal L using a piece-wise linear baselineexracing operaor, L, beween successive exrema as follows Lk 1 Lk LX L Lk ( X X k ), X k 1 X ( k, k 1] (4) k a

3 where X k and L k denoe X(τ k ) and L(τ k ) respecively, and k 1 k Lk 1 [ X k ( )( X k 2 X k )] (1 ) X k 1, k 2 k 0 1 (5) (3) The firs proper roaion can be exraced from X using a proper-roaion-exracing operaor, H, as HX ( 1 L) X H X L (6) (4) The process (1) o (3) will perform ieraively, using he baseline signals as inpu, unil he monoonic baseline signal is obained. The following properies provide some key conceps of ITD: Remark 1: The decomposiion is nonlinear in he sense ha he decomposiion of wo signals need no produce componens equal o he sum of he componens obained from decomposing each signal individually, i.e. S k k k X W HL S HL X HL W Remark 2: The edge effec of ITD process is confined o he inerval [ 0, 2] a he beginning and [ k, k 2] a he ending of each proper roaion and monoonic rend. 3. Vecor Error Correcion Model Vecor Error Correcion Model (VECM) [16] is a dynamical nonsaionary muli-facor model wih long-run relaionship characerisics among variables called coinegraing relaionships. Le y ( y 1,..., y k )', = 1,2, denoe a k-dimensional ime series vecor of random variables of ineres. The vecor error correcion model wih he coinegraion rank r ( k), denoed as VECM(p), can be wrien as y 1 p 1 i1 i y y (7) where Δ is differencing operaor, such ha y y y1 ; ', where and are k r marices; k k 1 i is a marix. The coinegraing vecor,, is someimes called he long-run parameers, and is adjusmen coefficien. In he case of coinegraion wih exogenous variables, he VECM wih exogenous variables, VECMX(p, s), can be wrien as follows: y 1 p1 s i y 1 i1 i0 i zi y (8) Theoreically, VECM is applicable if and only if y is coinegraing, as defined in he following definiion: Definiion 1: The componens of he vecor y y,..., y )' are said o be coinegraed of order d, b, denoed by ( 1 k y ~ CI(d, b) if (I) All componens of y are inegraed of order d, I(d), and (II) There exiss a vecor ( 1, 2,..., n) such ha he linear combinaion y 1 y1 2y2... n yn is inegraed of order (d - b), I(d - b), where b > 0. In order o idenify he coinegraing vecor from a sysem of variables, y, using Definiion 1, some saisical hypohesis ess are required. The following subsecions provide he deails of saisical hypohesis ess involving in coinegraing vecor idenificaion. 3.1 Uni Roo Tes By definiion, coinegraion necessiaes ha all variables be inegraed of he same order. The Augmened Dickey- Fuller (ADF) es [17, 18] is seleced o prees each variable o deermine is order of inegraion. There are hree main versions of he es equaion as shown in Eq. (10)-(12). y y 1 1y 1... p1y p1 (9) y y 1 1y 1... p1y p1 (10) y y 1 1y 1... p1y p1 (11) The null hypohesis is 0 agains he alernaive hypohesis of 0. Eq. (9) is he es equaion of a uni roo wih zero mean. Eq. (10) is he es equaion of a uni roo wih drif and Eq. (11) is he es equaion of a uni roo es wih drif and deerminisic ime rend. The appropriae lag lengh (p) can be seleced using he general-ospecific mehodology [19] or an informaion crierion such as he Akaike Informaion Crierion (AIC) or Bayesian Informaion Crierion (BIC). Dickey and Fuller [20] have shown ha he es saisic of Dickey-Fuller es does no

4 follow a -disribuion, so ha he usual percenile of -disribuion canno be used as comparison numbers for he calculaed. In heir Mone Carlo sudy, hey found ha he criical values for 0 depend on he form of he regression and on sample size. The saisics called, and are he appropriae saisics o use for Eq. (9), (10), and (11), respecively., 3.2 Weak Exogeneiy Tes In a coinegraed sysem, a variable is weakly exogenous if no useful informaion is los when his variable is condiioned wihou specifying is generaing process. In he oher words, if a variable does no respond o he discrepancy from he long-run equilibrium relaionship, i is weakly exogenous. The weak exogeneiy es is used o idenify he weak exogeneiy effec of each variable o he ohers. To es which variables should be reaed as endogenous, and which ones as exogenous in he equaion, he k-vecor of I(d) random variables y is iniially ' ' 1 2 ' ' ( 1, 2 pariioned ino he k 1 -vecor y 1 and he k 2 -vecor y 2, where y ( y, y ) and k = k 1 + k 2. From he VECM model in Eq. (7), he parameers can similarly be decomposed as ), ( 1, 2), i ( 1i, 2i), and he variance-covariance marix as (12) The condiional model for y 1 given y 2 is y and he marginal model for y 2 is p1 ' 1 y2 ( 2 i1 1 2 ) y1 ( 1i 2i) y i ( 1 2 ) ( 1 ) (13) p 1 ' y2 2 y 1 2iy i 2 2 i1 ' where The null hypohesis of he es of weak exogeneiy is 0 2 disribuion. (14) 2 ' ', and he weak exogeneiy es saisic follows 3.3 Coinegraion Tes For a es of coinegraion, he Johansen s reduced rank mehodology [21, 22] is employed. Le y y,..., y )', = 1,2,,k denoe a k-dimensional ime series vecor. As in ADF es, y 1 p 1 i1 y can be wrien as i ( 1 k y y (15) The key of he rank mehodology is rank of marix π. The rank of π is equal o he number of independen coinegraing vecors which can be obained by checking he significance of he characerisic roo of π. The es for he number of characerisic roos can be conduced using wo es saisics, he race and maximum eigenvalue saisics: n ir1 i ( r) T ln(1 ) (16) race max ( r, r 1) T ln(1 r1) (17) where λ i is he esimaed values of he characerisic roos (also called eigenvalues) obained from he esimaed π marix, and T is he number of usable observaions. The null hypohesis is λ i = 0 for i = r+1,, k. The asympoic disribuion of he race and maximum eigenvalue saisic is given by r(a) and max(a) where A is he marix defined as 1 ~ 1 1 ' ~ ~ ' ~ ' A ( dw ) W WW dr 0 0 W( dw ) (18) 0 where r(a) is he race of marix A and max(a) is he maximum eigenvalue of a marix A. W is he k r dimensional Brownian moion, and W ~ is he derended Brownian moion. Tables of he criical values of hese saisics can be found in [23]. 1 i

5 4. ITD-based Predicion Mehodology As presened in he inroducion secion, he ITD-based predicion mehod akes an advanage of coinegraion propery of VECM by idenifying long-run equilibrium relaionship of auomobile demand and ITD componen of economic indicaor. The ITD-based predicion mehodology is consising of he following hree seps. The firs sep of he predicion mehodology is an ITD sep. In his sep, economic indicaor will be decomposed ino a finie number of proper roaion componens and monoonic rend using ITD mehod. The second sep of mehodology is variable selecion sep. This sep is used o selec endogenous and exogenous variables for VECM. In he second sep, nonsaionary condiion and inegraion order for each componen will be idenified using uni roo es. Since auomobile demand is nonsaionary, saionary ITD componens will be reaed as exogenous variables. Oher exogenous variables will be idenified using weakly exogenous es. Long-run equilibrium relaionship of endogenous variables, including auomobile demand and seleced ITD componens, will be idenified using coinegraion es. The hird sep of predicion mehodology is a predicion sep. The VECM model of seleced endogenous and exogenous variables in he second sep will be parameerized. Ou-of-sample forecasing will be made for auomobile demand. The comparison of ou-of-sample forecasing wih oher classical and advanced ime series echniques will be done in his sep. The overall framework of ITD-based predicion mehodology is shown in Figure 1. ITD Sep Variable Selecion Sep Uni Roo Tes Economic Indicaor Inrinsic Time-Scale Decomposiion (ITD) Auomobile demand and ITD componens Coinegraion Tes Weak Exogeneiy Tes Ou-of-Sample Forecasing Comparison Vecor Error Correcion Model (VECM) Seleced Endogenous and Exogenous Variables Predicion Sep Figure 1: ITD-based Predicion Mehodology Framework 5. Implemenaion Deails and Empirical Resuls In his paper, he number of monhly reail sales (Moor vehicle and pars dealers) in U.S. during a period of January December 2010 is used o represen an aggregae auomobile demand. An unemploymen rae during he same period of ime is seleced as economic indicaor o help predic demand. In order o predic demand for long erm, ITD componens of unemploymen rae are hypohesized o have long-run equilibrium relaionship wih auomobile demand. This paper will invesigae and es hypoheses of long-run relaionship of auomobile demand and ITD componens of unemploymen rae. Table 1 provides he deails of demand and unemploymen rae. For simpliciy, logarihm ransformaion will be used wih boh variables in subsequen analysis. The ime series in original scale of boh variables are shown in Figure 2. (a) Figure 2: (a) Auomobile Demand and (b) Unemploymen rae (b)

6 From lieraure [24], many research papers have indicaed an exisence of nonlinear behavior in he U.S. unemploymen rae. To presen he effec of nonlinear behavior in he unemploymen rae on he long-run equilibrium relaionship wih auomobile demand, he resul of coinegraion es beween unemploymen rae and auomobile demand is shown in Table 2. Table 1: Summary of variables Variables Source Publishing Inerval Descripion Auomobile Demand BLS Monhly A monhly reail sales of moor vehicles and par accessories Unemploymen rae BLS Monhly A naional unemploymen rae (16 years or over) Bureau of Labor Saisics (Seasonally adjused daa) Table 2: Coinegraion Rank Tess (Auomobile Demand and Unemploymen Rae) Trace Tes Maximum Eigenvalue Tes H 0 : Rank = r H 1 : Rank > r Trace Saisic 5% 10% Max Saisic 5% 10% Table 2 repors he es saisics and he corresponding asympoic criical values of boh Trace and Maximum Eigenvalue ess a he 5% and 10% significance levels. The empirical resuls of coinegraion ess show ha he null hypohesis of no coinegraing vecor canno be rejeced for boh ess a 5% and 10% significance levels. The es resuls indicae ha here is saisically no long-run equilibrium relaionship beween unemploymen rae and auomobile demand. Considering coinegraing vecor requiremen, VECM canno be direcly applied o hese variables. As discussed in he inroducion secion, he coinegraion ess using linear assumpion may perform poorly in he case of nonlinear variables. The nex sep in his secion is o apply he ITD-based predicion mehodology presened in Secion 4 o invesigae and es he long-run equilibrium relaionship beween auomobile demand and ITD componens of unemploymen rae. The implemenaion deails are as follows: 5.1 Inrinsic Time-Scale Decomposiion Sep The firs sep of predicion mehodology is o decompose unemploymen rae using ITD mehod described in Secion 2. The ITD algorihm decomposes unemploymen rae ino four proper roaion componens and monoonic rend as shown in Figure 3. The procedure of ITD algorihm allows a perfec reconsrucion of original unemploymen rae as presened in Eq. (3). Figure 3: ITD componens of Unemploymen Rae 5.2 Variable Selecion Sep In his second sep, endogenous and exogenous variables for VECM are seleced from auomobile demand and ITD componens of unemploymen rae obained from he firs sep. Saionariy characerisic of auomobile demand and

7 each ITD componens of unemploymen rae is idenified using uni roo ess. Exogenous variables are seleced, based on resuls from weak exogeneiy ess. In order o conduc he Augmened Dickey-Fuller uni roo es, i is imporan o use he correc number of lags in he equaion. The appropriae lag lengh is seleced, based on he general-o-specific mehodology. The procedure begins wih esimaing an auoregressive model wih relaively long lag lengh. The idea is o pare down he model by he usual -es and/or F-ess. If he -saisic on he las lag (p) is insignifican a some specified significance level, reesimae he model wih a lag lengh of p-1, hen repea he process unil he las lag is significanly differen from zero. The ADF es resuls wih opimal lag lengh seleced using a general-o-specific mehodology are shown in Table 3. Table 3: ADF Uni Roo Tes of Original and Firs Differenced Variables Variables Opimal Lag Uni Roo wih Drif and Uni Roo wih Drif Lengh Deerminisic Time Trend Original Variables p τ μ Pr < τ μ τ τ Pr < τ τ Auomobile Demand (AD) Proper roaion 1 (H1) Proper roaion 2 (H2) Proper roaion 3 (H3) Proper roaion 4 (H4) Monoonic Trend (R) Firs Differenced Variables p τ μ Pr < τ μ τ τ Pr < τ τ ΔAD ΔH ΔH ΔH ΔR From he ADF uni roo es resuls, only proper roaion 2 (H2) is saisically saionary a 1% significance level. The null hypohesis of uni roo canno be rejeced for auomobile demand and all oher ITD componens of unemploymen rae a 1% significance level. Applying firs difference o all variables, excep H2, he ADF es saisics rejec he null hypohesis of uni roo for all differenced variables a 5% significance level. Based on he resuls of ADF ess, all variables, excluding H2, are reaed as nonsaionary I(1) series. To selec exogenous variables for VECM, he resuls of weak exogeneiy ess on I(1) series as shown in Table 4. Table 4: Weak Exogeneiy Tess (Seleced Variables) Variables Degree of Freedom χ 2 (4) Pr > χ 2 AD H <.0001 H H < R The weak exogeneiy es begins wih he vecor of five variables (AD, H2, H3, H4 and R) in Table 3. For a sysem of five variables, here can exis a mos four coinegraing vecor. The resuls in Table 4 show ha he null hypohesis of weak exogeneiy canno be rejeced for monoonic rend (R). Table 5: Weak Exogeneiy Tess (Re-esimae) Variables Degree of Freedom χ 2 (4) Pr > χ 2 AD H H H Hall e al. [25] sugges re-esimaing he model, using only rejecing weakly exogenous variables as endogenous variables, o avoid sensiiviy on he model specificaion. Table 5 shows he resuls of re-esimaing weak exogeneiy ess of four endogenous variables, excluding monoonic rend. The es saisics coninue o rejec he null

8 hypohesis of weak exogeneiy a 5% significance level for all four variables. Based on resuls of uni roo and weak exogeneiy ess, auomobile demand (AD), and hree ITD componens of unemploymen rae (H2, H3 and H4) are seleced as endogenous variables. For exogenous variables, due o possible colineariy issue beween auomobile demand (AD) and monoonic rend (R), only proper roaion 1 (H1) is seleced as exogenous variable. To invesigae an exisence of long-run equilibrium relaionship among endogenous variables, Table 6 repors he es saisics and corresponding asympoic criical values a 5% and 10% significance level of coinegraion rank ess. The es resuls show ha he null hypohesis of one coinegraing vecor canno be rejeced a boh 5% and 10% significance level. There is saisically long-run equilibrium relaionship or coinegraing vecor among four endogenous variables (AD, H2, H3 and H4) as shown in Eq. (19). AD H 2 H2 H3 H3 H 4 H4 (19) The underlying process of hese variables are random in he shor erm bu end o move ogeher in he long erm horizon. Since all four variables are I(1) series, he coinegraing vecor in Eq. (19) is saionary process (I(d-b) = I(0) where d = b = 1). Long-Run parameer (β) esimaes of he coinegraing vecor are presened in Table 7, where he parameer esimae of auomobile demand is normalized (β=1). Table 6: Coinegraion Rank Tess (Auomobile Demand and seleced ITD componens of unemploymen rae) Trace Tes Maximum Eigenvalue Tes H 0 : Rank = r H 1 : Rank > r Trace Saisic 5% 10% Max Saisic 5% 10% Table 7: Long-Run Parameer Bea Esimaes of Coinegraing Vecor (AD, H2, H3 and H4) Variable AD H2 H3 H4 Bea β Β H2 Β H3 Β H4 Parameer Esimae Predicion Sep From resuls of seps 1 and 2, i shows ha VECM is applicable for an auomobile demand and ITD componens of unemploymen rae since coinegraing vecor of hese variables can be idenified. The ITD-based predicion using VECMX wih 4 endogenous (AD, H2, H3 and H4) and 1 exogenous (H1) variable is esimaed using Eq. (9). To evaluae he forecasing performance of he ITD-based predicion using VECMX, he ou-of-sample forecasing of his model is compared wih hose from hree rival models. The firs model is an auoregressive inegraed moving average (ARIMA) model. The ARIMA model is he mos general class of univariae ime series models which can be applied o nonsaionary variable. The second model is an auoregressive inegraed moving average wih exogenous variables (ARIMAX) model. The ARIMAX model of auomobile demand is seleced as a baseline model since i allows o use unemploymen rae as exogenous variable in he model, however i canno handle feedback from one variable o he oher. The hird model is vecor auoregressive (VAR) model. The VAR model can handle feedback from one variable o he oher, bu does no impose coinegraing resricions as in he VECMX. I reas auomobile demand and unemploymen rae symmerically. Two model selecion crierions are seleced for ou-ofsample comparison. They are roo mean square error (RMSE) and mean absolue percenage error (MAPE). The ou-of-sample forecasing iniially idenified each model specificaion over he sample period of January 1992 o May 2007, hen each model is used o generae forecass of four-, eigh- and welve-sep ahead predicions. The sample is hen rolled forward for one monh, and anoher se of four- o welve-sep ahead predicions is generaed. As a resul, 24 four-, eigh- and welve-sep ahead predicions are obained. Table 8 provides RMSE and MAPE values of four-, eigh- and welve-sep ahead predicions of auomobile demand (AD) for each model. Figure 4(a) and 4(b) show he ou-of-sample forecasing comparison of four models using RMSE and MAPE. Table 8: Ou-of-Sample Forecasing Comparison Model 4-sep ahead predicion 8-sep ahead predicion 12-sep ahead predicion RMSE MAPE RMSE MAPE RMSE MAPE ARIMA ARIMAX VAR ITD-based Predicion

9 Comparing four models in Table 8, ITD-based predicion model is he bes model in erms of RMSE and MAPE. I can significanly improve a predicion accuracy of auomobile demand for all four-, eigh-, and welve-sep ahead predicions. For 4-sep ahead predicion, ITD-based predicion model can reduce RMSE and MAPE by 25% and 30% respecively, compared o he second bes model (VAR model). For 8-sep ahead predicion, ITD-based predicion model can reduce RMSE and MAPE by 24% and 25% respecively, compared o VAR model. In he case of 12-sep ahead predicion, ITD can reduce RMSE and MAPE by 23% and 26% respecively, compared o VAR model. (a) Figure 4: (a) Ou-of-Sample Forecasing Comparison using RMSE and (b) Ou-of-Sample Forecasing Comparison using MAPE 6. Conclusion and Suggesed Fuure Work In his paper, a new predicion approach based on Inrinsic Time-Scale Decomposiion (ITD) is proposed for longerm predicion applicaion of nonlinear and/or nonsaionary variables. ITD is a recenly developed nonparameric algorihm aiming o enhance an analysis of nonlinear and nonsaionary signals. The ITD mehod represens a nonlinear and/or nonsaionary signal as a summaion of componens called proper roaion and monoonic rend. The advanage of ITD-based predicion approach is ha, from sysem of variables wih no long-run equilibrium relaionship or coinegraing vecor, his approach can be used o idenify coinegraing vecor of one variable and ITD componens of oher variables. In his sudy, he ITD-based predicion approach is applied o a long-erm predicion of auomobile demand wih he aid of economic indicaor. Considering economic indicaor as a combinaion of muliple componens wih differen characerisics resuling from facors underlying each componen, he propery of ITD leads o a predicion applicaion using a long-run equilibrium relaionship or coinegraing vecor of auomobile demand and ITD componens of economic indicaor. The empirical resuls show ha auomobile demand has a long-run equilibrium relaionship wih some of ITD componens of unemploymen rae, and ITD-based predicion approach can significanly improve a predicion accuracy for long-erm predicion of auomobile demand. To improve a predicion accuracy using ITD algorihm furher, one may consider he edge effec of ITD as discussed in Secion 2 of his paper as suggesed fuure work. The edge effec of ITD on he mos recen daa is confined o he inerval of he las wo exremas. This is because he mos recen daa poin of he signal is reaed as an exremum in which i may no necessarily be rue when more daa become available. In a predicion applicaion, he edge effec of ITD has a significan effec on a predicion accuracy. One of possible alernaives o solve his issue is o consruc he decomposiion beyond he las available daa poin. However, his exension of he daa is a risky procedure because he edge effec will depend on an accuracy of he exended daa. Currenly, no simple approach or general heory exiss o solve he edge effec issue of ITD. However, wih propery of ITD ha a proper roaion is monoonic beween wo consecuive local exremas, ITD has an advanage o help in he analysis. Tha is he whole daa span needed no o be exended. The only hings needed are he value and locaion of he nex wo exremas. Considering he advanage of ITD, his ask of improving a predicion accuracy by solving he edge effec issue is sill challenging. Acknowledgemens This research is suppored by Naional Science Foundaion (CMMI and CMMI ). (b)

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