A NOVEL MODEL UPDATING METHOD: UPDATING FUNCTION MODEL WITH GROSS DOMESTIC PRODUCT PER CAPITA

Transcription

1 A NOVEL MODEL UPDATING METHOD: UPDATING FUNCTION MODEL WITH GROSS DOMESTIC PRODUCT PER CAPITA Nobuhiro Graduae School of Business Adminisraion, Kobe Universiy, Japan -1 Rokkodai-cho, Nada-ku, Kobe -01 Japan Phone & fax: sanko@kobe-u.ac.jp words + ables + 1 figure = words Submied: 1 Augus 01 1

2 ABSTRACT When daa are available from wo poins in ime: older daa wih a larger number of observaions and more recen daa wih a smaller number of observaions, hen model updaing is uilised o use he meris of he boh daases. However, he auhor s previous sudy quesioned he meris of convenional model updaing echniques: ransfer scaling, join conex esimaion, Bayesian updaing, and combined ransfer esimaion. Alhough hese model updaing mehods uilise daases from wo poins in ime, models using only he more recen daa ofen produced saisically significanly beer forecass han he models updaed. The presen sudy proposes a novel updaing mehod (called updaing funcion model ), where parameers are assumed o follow funcions of gross domesic produc per capia. The mehod was originally proposed by he auhor, bu he presen sudy aims o demonsrae ha i is a novel updaing mehod. While convenional model updaing applies o a case where he number of observaions from he more recen ime poin is smaller han ha from he older ime poin, he presen sudy also considered a case where he number of observaions from he more recen ime poin is larger han ha from he older ime poin. In boh of he wo cases, he presen sudy demonsraed ha he updaing funcion model ofen produced saisically significanly beer forecass han models using only he more recen daa. The sudy concluded ha he updaing funcion model is a useful model updaing echnique and exended he applicabiliy of he model updaing o he case where he number of observaions from he more recen ime poin is larger han ha from he older ime poin.

3 INTRODUCTION Forecass using disaggregae ravel demand models ofen are based on daa from he mos recen ime poin, even when cross-secional daa is available from muliple ime poins. However, his is no a good use of he daa. When daa are available from wo poins in ime: older daa wih a larger number of observaions and more recen daa wih a smaller number of observaions, hen model updaing is uilised o use he meris of he boh daases. To dae, four updaing mehods have been proposed: ransfer scaling, join conex esimaion, Bayesian updaing, and combined ransfer esimaion. However, advanages of hese updaing mehods have been quesioned. Some papers argue ha when several hundreds of observaions are available from he more recen ime poin, model updaing mehods, which uilise boh ha several hundreds of observaions and he larger number of observaions from he older ime poin, conribue lile o improve forecasing performance (1, ). The auhor s previous sudy (), which uilises he daases used in he presen sudy, compared he forecasing performance by he four updaing mehods and ha by models wih only he more recen daa. Wih a use of boosrapping echnique, he auhor found ha he models using only he more recen daa ofen produced saisically significanly beer forecass han he updaing mehods, bu ha he updaing mehods never produced saisically significanly beer forecass han he models using only he more recen daa. This means ha he convenional model updaing mehods conribue lile o improve forecasing performance from a saisical poin of view. The auhor () proposed a mehod ha joinly uilises cross-secional daa from muliple ime poins and demonsraed ha he proposed mehod produced beer forecass han a model uilising daa from only he mos recen ime poin. The auhor formulaed ha parameers are funcions of ime (year), meaning ha he parameer values vary over ime and he fuure parameer values can be forecas. The auhor () also examined anoher model formulaion, where he parameers are assumed o follow funcions of GDP (gross domesic produc) per capia, and he funcions of GDP per capia produced beer forecass han he funcions of ime. However, hese sudies uilised large numbers of observaions form each ime poin, and he auhor was no ineresed in comparing he proposed mehod wih convenional model updaing mehods. Also, he resuls were no saisically esed. This sudy aims o demonsrae wheher he auhor s proposed mehod has appropriae characerisics as a model updaing mehod. A research quesion of he presen sudy is summarised below. Noe ha he auhor s proposed mehod uilising daa from muliple ime poins (wo ime poins in he presen sudy) is ermed updaing funcion model, since he mehod assumes a funcion which updaes parameers. The model uilising only he more recen daa is ermed more recen daa model. Research Quesion: Suppose ha daa from wo poins in ime is colleced: n 1 and n observaions from older (year y 1) and more recen (year y ) ime poins, respecively. In which combinaions of ime poins of daa colleced and he numbers of observaions from wo poins in ime, he updaing funcion models (using boh n 1 and n observaions from y 1 and y, respecively) produce saisically significanly beer forecass han he more recen daa models (using n observaions from y ). Noe ha boh cases of n 1 n and n 1 < n are considered. I is worh saing he meaning of he above wo inequaliies. While he n 1 n means a case where convenional model updaing mehods have been applied, he n 1 < n means a case where he convenional updaing mehods have no been applied bu he updaing funcion mehods are applied in he presen sudy. (Noe ha convenional model updaing is an approach o updae models esimaed wih he older daa wih a larger number of observaions by using small number of observaions colleced from he more recen ime poin.) The auhor invesigaed his issue in he conex of journey-o-work mode choice behaviours by uilising household ravel survey daa colleced in Nagoya, Japan, in,,, and 001, where he firs hree ime poins were used for model esimaion and he las ime poin was used for validaion. Two ime poins (one for he older ime poin and he oher for he more recen ime poin) are chosen from he,, and, and differen numbers of observaions (ranging from 0 o 000) are

4 randomly seleced from he chosen wo poins in ime. The forecasing performance for 001 is invesigaed. In order o obain insighs wih saisical meaning, he boosrap echnique is employed. All of he household ravel survey daa used for his sudy was implemened by he same governmenal bodies, and he survey was conduced in a similar manner in each year. Therefore, i is reasonable o assume ha he daa is similar in qualiy across he years. Hence, he daases used in he presen sudy are appropriae for analysing he wo ime poins, where he daa are colleced, and he numbers of observaions colleced from he wo poins in ime. Since his is he firs aemp o saisically es he advanages of he auhor s proposed mehod, he analysis is kep as simple as possible and uilises daases available o he auhor. Time poins when he daa are colleced and he numbers of observaions are he wo dimensions of ineres in he presen sudy. Oher dimensions which migh affec emporal ransferabiliy bu are no considered in he presen sudy include: (i) he underlying heory of ravel behaviour (e.g., uiliy maximisaion vs. lexicographic; rip-based vs. our-based), (ii) he mahemaical model srucure (e.g., logi vs. nesed logi), and (iii) he empirical specificaion (e.g., choice of explanaory variables, linear vs. non-linear formulaion of explanaory variables, consideraion of heerogeneiy). (Sikder () proposed his classificaion. He also inroduced (iv) model parameer esimaes (e.g., ransferabiliy of coefficiens of explanaory variables and oher parameers such as elasiciies and value of ime measures). The wo dimensions of ineres of he presen sudy relae o he (iv). They firs impac he model parameer esimaes and hen he forecasing performance.) This sudy assumes uiliy maximisaion uilising linear-in-parameers mulinomial logi models and uses a single model specificaion hroughou he paper. The daa used for his sudy comes from he 001 period. The 001 daa is 1 years old, bu his is less of a concern. This paper is organised as follows. The Lieraure review secion presens papers which compared convenional model updaing mehods and models using only he more recen daa. Papers on updaing funcion models also are explained. The Daa secion describes he daases. The Mehodology secion describes a mulinomial logi model, he more recen daa model and updaing funcion model, he boosrapping procedure, and hypohesis esing. The Resuls and discussion secion repors he esimaed parameers and he forecasing performance of he models using saisical ess and discusses in which case he proposed updaing funcion models ouperformed he more recen daa model. The Conclusions secion presens he concluding remarks. LITERATURE REVIEW Convenional Model Updaing Mehods in Comparison wih More Recen Daa Model There exis limied sudies comparing model updaing mehods for improving emporal ransferabiliy wih he more recen daa model, especially focusing on he wo dimensions of ineres in he presen sudy. However, he following sudies have been repored. Badoe and Miller (1) uilised daa from Torono, Canada colleced in 1 and 1 and evaluaed ransferabiliy of morning peak work rips mode choice models. They esimaed models uilising large number of observaions from 1, which is fixed, and small number of observaions from 1, which is varied. They also esimaed models uilising only small number of observaions from 1, which is again varied. Four updaed models (ransfer scaling, join conex esimaion, Bayesian updaing, and combined ransfer esimaion) and models wih small number of 1 daa are applied o forecas behaviours in 1, and he forecasing performance was compared. Noe ha 1 is used boh for model esimaion and validaion, since hey have daases from only wo poins in ime. They found ha when small number of observaions, a leas 00 00, is available from 1, he above four updaing mehods resuled in lile or no conribuion o improving forecasing performance of models wih ha small number of observaions. Similar analysis o he above (1) was conduced by Karasmaa and Pursula () uilising daa from Helsinki, Finland in and 1, esimaing mode and desinaion choice models of home-based work rip, and resuling in similar conclusions. The presen auhor believes ha he above wo sudies have he following limiaions: (a) he

5 resuls are no saisically esed, and (b) daa is available from wo poins in ime and he daa from he second ime poin is uilised boh for model esimaion and evaluaion. () uilised he same daases used in he presen sudy and saisically esed he forecasing performance of convenional updaing mehods and he more recen daa models. Wih a use of boosrapping echnique, hree combinaions of ime poins (y 1 and y ) and combinaions of he number of observaions (n 1 and n ) (ranging from 0 o 000) are invesigaed. The auhor found ha he more recen daa models ofen produced saisically significanly beer forecass han he updaing mehods. However, he updaing mehods never produced saisically significanly beer forecass han he more recen models in any combinaions of y 1 and y, and n 1 and n. Alhough he updaing mehods someimes produced beer forecass han he more recen daa model wihou saisical significance, resuls wihou saisical significance are weak o suppor he usefulness of he convenional updaing mehods. Updaing Funcion Models Only he mos recen daa has been used o develop models for forecasing, even when cross-secional daa is available for muliple ime poins. () examined he possibiliy of improving forecasing performance by using he mos recen daa ogeher wih older daa. He quesioned one of he assumpions made by previous sudies: parameers are fixed beween ime poins. He assumed ha he parameers o be funcions of ime (year), which allows he parameers o change over ime and allows fuure parameer values o be prediced. More specifically, he parameers consis of a par ha is independen of ime and a par ha is dependen on ime, which is expressed as a funcion of ime. He uilised he same daases used in he presen sudy, and applied his mehod o commuing mode choice behaviours. He esimaed models uilising daa from,, and joinly and a model uilising daa only from. Forecasing performances for behaviours in 001 were compared, and he proposed model (uilising daa from hree poins in ime joinly) provided beer forecass han a model using only he mos recen daa from. Alhough he funcions of ime ascribed he parameer changes o he rends of he imes, here migh be oher facors ha affec parameer changes. One of he facors ha he auhor has examined was GDP per capia. () uilised he same daases used in he above sudy () and found ha he funcions of GDP per capia produced beer forecasing performance han he funcions of ime. In he above wo sudies, all of mode choice parameers were influenced by he funcions of ime or GDP per capia, bu differen parameers migh be influenced by differen facors. (), which uilised he same daases used in he above sudies, invesigaed which parameers of he mode choice models were which funcions of which variables. models were esimaed by assuming ha he differen parameers of mode choice models are funcions of differen variables, i.e., ime (in linear form) and GDP per capia (in linear, square, and square roo forms). The resuls sugges ha he funcions of ime were oo rained on he esimaion daases and produced poor forecass. Oher han he funcion of ime, few differences can be found regarding he choice of funcional forms. A significan difference beween he funcions of ime and funcions of GDP per capia is ha he former ascribes he parameer changes o he rends of he imes wihou any economic reasons while he laer ascribes i o economic facors. The sudy also considered funcional forms of: female social paricipaion (in linear form) and Nagoya Ciy s subway lengh (in linear form). However, hese are considered for only wo of eigh parameers esimaed and he resuls differ lile from he funcion of GDP per capia (in linear form). Therefore, he presen auhor believes ha he funcion of GDP per capia (in linear form) is he bes choice of funcional forms. However, his sudy assumed ha here are a lo of numbers of observaions (and equal numbers of observaions) from all ime poins, and he sudy did no ineresed in a proposal of new updaing mehod. Also, he resuls were no esed saisically. DATA The repeaed cross-secional daa used in his sudy came from household ravel surveys in,,, and 001 in he Nagoya meropolian area of Japan. This sudy uilises daa colleced in,, and for modelling and he daa colleced in 001 solely for validaion purposes. The household

6 ravel survey has been implemened in a similar manner by he same governmenal bodies over he years. The modelled rips were journeys o work (commues). Three alernaive ransporaion modes were considered: rail, bus, and car. The daases are fully described in (), bu wo poins mus be resaed. Firs, his sudy does no consider ravel cos, since mos companies provide allowances for employees o purchase commuing passes or fuel. (Rules for allowances differ among companies, and some companies se an upper limi, which mos employees do no exceed.) Second, he shares of ravel modes have changed subsanially beween and, bu since hen have changed less. Afer cleaning he daa for esimaion purposes, he shares of ravel modes among rail, bus, and car for commuing purposes in,,, and 001 were %, %, %, and %, respecively, for rail, 1%, %, %, and %, respecively, for bus, and 1%, %, %, and %, respecively, for car. The GDP per capia of Japan in consan 00 JPY was 0.1 in, 0. in, 0. in, and 0. in 001 (). METHODOLOGY Since his is he firs sudy o saisically examine he advanages of he updaing funcion model, simple mulinomial logi models were employed. However, he mehodology is applicable o oher model formulaions. This secion presens mulinomial logi models, followed by he more recen daa models and updaing funcion models, he boosrapping procedure, and hypohesis esing uilising he boosrap. Noe ha he boosrapping and hypohesis esing are uilised in a similar manner in he auhor s recen works (, ). In he following explanaion, 1 and represen he older and more recen ime poins, respecively. Mulinomial Logi Models Random uiliy models are assumed and oal uiliy is decomposed ino a deerminisic componen and an error componen. Under he assumpions of linear-in-parameers mulinomial logi models, he deerminisic componen of individual p s uiliy for alernaive i a ( = 1 or in he following explanaions), V, is expressed as Eq. (1). where ip V ip + = α β x (1) α i denoes an alernaive-specific consan for alernaive i a ; i k ik ikp x ikp denoes he k-h explanaory variable for individual p for alernaive i a, and β ik denoes is relaed parameer. However, he scale parameer, which is fixed o a uniy value, is no explicily menioned in Eq. (1), since he scale parameer and α and β canno be separaely idenified. Assuming independen and idenical Gumbel disribuions for he error componens, mulinomial logi models are derived, where he probabiliy of individual p s choosing alernaive i among alernaive j s in his/her choice se a, P, is expressed as: L : ip P ip = j exp exp ( Vip ) ( V ) jp The log-likelihood funcion, L, is defined by he sum of log-likelihood from each ime poin of, = L = p j jp ( P ) L y ln () jp ()

7 where y jp denoes an indicaor ha akes a value of one if individual p chose alernaive j a and zero oherwise. More Recen Daa Model and Updaing Funcion Model wih GDP per Capia More Recen Daa Model Models are esimaed as shown in he Mulinomial logi models secion by uilising daa from =. The log-likelihood funcion, which is maximised for esimaion, as shown in Eq. (), is expressed as L = L. A forecasing performance is evaluaed by a log-likelihood on 001 daa (L 001 ), which is calculaed by uilising Eqs. (1) (), where α ˆ and β ˆ are from he esimaed models bu x and y are from he 001 daase. Noe ha a ha (^) indicaes an esimae. Updaing Funcion Model wih GDP per Capia In Eq. (1), he following formulaion applies o α i and β ik. α + i = αi α digdp (a) ik = βik βdik gdp (b) β + where, α i and β ik denoe parameers independen of ime ( base parameers ) and α di and β dik denoe hisorically changing facors for corresponding parameers ( hisorically changing parameers ). The α di and β dik express pars changing according o he GDP per capia. The gdp denoes GDP per capia (consan 00 price) a (unis in million JPY). The presen sudy names Eq. () as an updaing funcion, since i updaes parameers for each ime poin of. Models are formulaed by applying = 1 and o Eqs. (1), (), and (). The log-likelihood funcion, which is maximised for esimaion, as shown in Eq. (), is expressed as L = L 1 + L. A forecasing performance is evaluaed by a log-likelihood on 001 daa (L 001 ), which is calculaed by uilising Eqs. (1) (), where αˆ, βˆ, αˆ d, and βˆ d are from he esimaed models bu x and y are from he 001 daase, and he GDP per capia is from 001 (consan 00 price). Noe ha his sudy assumes ha he parameers are expressed in linear form based on findings explained in he Lieraure review secion. The updaing funcion model is idenical o esimaing he main effecs and ineracions beween explanaory variables and he GDP per capia. The locaion parameers and scale parameers of he Gumbel disribuions migh differ beween ime poins; he auhor assumes ha alernaive-specific consans wih funcional forms accoun for he differences in he locaion parameers and ha he scale parameers are consan over ime. While convenional models assume ha neiher he scale parameers nor he parameers of explanaory variables change over ime (see Fox and Hess () for a review), he updaing funcion model allows he parameers relaed o explanaory variables o change, which is more flexible. Boosrap Boosrapping, a mehod proposed by Efron and Tibshirani (), is applied o his sudy in he following way. Firs, 000 commuing rips were randomly seleced from each ime poin of,, and. In he following analysis, a smaller number of observaions was chosen from hese 000 observaions. The same number of observaions was chosen from each ime poin o avoid any impac on forecasing performance ha migh occur should differen numbers of observaions from each ime poin

8 be used. 000 commuing rips also were seleced randomly from he 001 daase ha was used for evaluaing forecasing performance. Three noaions y, n, and b are defined below. y denoes he year when he daa was colleced (,, and ). n denoes he number of observaions. The auhor examined 1 values for n (0, 00, 00, 00, 00, 00, 00, 00, 00, 00, 000, and 000). b denoes a boosrap repeiion. Boosrapping was repeaed 00 imes (b = 1,,..., 00). From each y, n observaions were randomly drawn 00 imes, wih replacemen from 000 commuing rips already seleced from each year. (Noe ha for each of he b-h draw from he same y, large n observaions conain all he records included in he small n observaions.) In oal, y s 1 n s 00 b s = 00 daases were generaed. This sudy is ineresed in a case where daases are colleced from wo poins in ime. Therefore, furher noaions are inroduced. The older and more recen ime poins are expressed as y 1 and y, respecively; he numbers of observaions from y 1 and y are denoed as n 1 and n, respecively. Realisable combinaions of y 1, y, n 1, and n are 1 =, which is a muliplicaion of hree combinaions relaing o y 1 and y ((y 1, y ) = (, ), (, ), and (, )) by 1 1 = 1 combinaions relaing o n 1 and n. A working procedure is as follows. For he more recen daa models, only combinaions of y and n are examined. Therefore, 1 00 = 00 models are esimaed and applied o forecasing behaviours for 001. For he updaing funcion models, combinaions of y 1, y, n 1, and n are examined. Therefore, 1 00 = 00 models are esimaed and applied o forecasing behaviours for 001. A forecasing performance is evaluaed by a log-likelihood on he 001 daase defined in he More recen daa model and updaing funcion model wih GDP per capia secion. The log-likelihood values by uilising he b-h repeiion for n 1 and n samples from y 1 and y, respecively are expressed as L1 (, y,, n, b) and L (y 1, y, n 1, n, b) for he more recen daa model and updaing funcion model, respecively. Hypohesis Tesing This secion proposes ess o compare he forecasing performance of updaing funcion models and ha of more recen daa models. Suppose ha here are wo combinaions of y and n: y 1 and n 1 and y and n. The following variable x b is defined for b = 1,,, 00. Noe ha he same b is used for boh L1 (, y,, n, b) and L (y 1, y, n 1, n, b). x b = L (y 1, y, n 1, n, b) L1 (, y,, n, b) () Noe ha x b is defined only when boh L s are calculaed. The calculaion of x b s is unsuccessful for some b s, which is likely o happen when n 1 and/or n are small. If he updaing funcion models produce beer forecass, hen x b is more posiive. Null and alernaive hypoheses, represened as H 0 and H 1, respecively, are defined below. H 0: x b = 0 H 1: x b 0 The saisic, z, is defined as Eq. (). where, x s b z = () b ( x ) b x and ( ) s denoe mean and sandard deviaion of x b, respecively. x b

9 If x b is assumed o follow a normal disribuion, hen he null hypoheses are rejeced a he five percen level of significance when z 1.. RESULTS AND DISCUSSIONS This secion presens esimaes uilising all 000 commuing rips chosen from each year, followed by he resuls of hypohesis esing. Esimaes The following dummy variables are defined: male (1 for male, 0 for female), 0 years old or older (1 if 0 years old or older, 0 if younger), years old or older (1 if years old or older, 0 if younger), and Nagoya (1 if origin and/or desinaion of he rip are in Nagoya Ciy, 0 if no). Descripive saisics for he variables included in he mode choice models are fully inerpreed in (). Table 1 reproduced he journey-o-work mulinomial logi mode choice model esimaes using daa from each ime poin independenly (). More recen daa models required for he presen sudy correspond o esimaes for he and daases. However, a model using he 001 daa also was esimaed and presened as a reference. (Noe ha he 001 daa is used solely for validaion.) The auhor examined numerous combinaions of variables and repored he bes resuls. The auhor did no include car ownership as an explanaory variable, since i is highly relaed o mode choice (or car choice) and he wo are regarded as being endogenous o some exen. The model specificaion presened here is used hroughou he presen paper. Noe ha ravel cos is no included in he models for he reason menioned in he Daa secion. Models are fully inerpreed in (), Table produced he journey-o-work mulinomial logi mode choice model esimaes by he updaing funcion models. The esimaes of he base parameers and hisorically changing parameers are shown a he op and boom of he able, respecively. The base parameers mus be inerpreed carefully, however, since hey express parameers where he GDP per capia = 0, which is highly unlikely. For he comparison o be fair, he parameers for,, and mus be calculaed using he esimaes in Table and Eq. (). (For example, he ravel ime parameer for in / model is (= GDP per capia in ) = -1.. Readers migh refer o () for more deailed inerpreaion.) The choice of explanaory variables is he same as ha for he more recen daa models shown in Table 1, and he model specificaion presened here is used hroughou he presen paper. Forecasing performances of models shown in Tables 1 and for he 001 daase are compared (see he rows labelled Log-likelihood on 001 daa ). The / model produced he bes forecasing performance, followed by he /,, /, and. This means ha if he more recen daa comes from, addiional use of daa from older ime poin in updaing funcion models conribues o improve he forecasing performance. The same applies o a case where he more recen daa comes from.

10 TABLE 1 Esimaes of more recen daa models 001 a Variables Es. -sa. Es. -sa. Es. -sa. Consan (B) Consan (C) Travel ime [hr] Male dummy (R) Male dummy (C) years old or older dummy (C) years old or older dummy (B) Nagoya dummy (C) N (randomly drawn) L(β) L(0) Adj rho-squared Log-likelihood on 001 daa No applicable Noe: (R), (B), and (C) noaions refer o alernaive-specific variables for rail, bus, and car, respecively. Variables wihou noaions are generic. a 001 is he arge year of forecas, and a model from 001 is no required bu is presened for a comparison purpose. TABLE Esimaes of updaing funcion models / / / Variables Es. -sa. Es. -sa. Es. -sa. Base parameers (α i, β ik) Consan (B) Consan (C) Travel ime [hr] Male dummy (R) Male dummy (C) years old or older dummy (C) years old or older dummy (B) Nagoya dummy (C) Hisorically changing parameers (α di, β dik) Consan (B) Consan (C) Travel ime [hr] Male dummy (R) Male dummy (C) years old or older dummy (C) years old or older dummy (B) Nagoya dummy (C) N (randomly drawn) L(β) L(0) Adj rho-squared Log-likelihood on 001 daa Noe: (R), (B), and (C) noaions refer o alernaive-specific variables for rail, bus, and car, respecively. Variables wihou noaions are generic.

11 Hypohesis Tesing Figure 1 shows he resuls of ess o deermine in which case he updaing funcion models produce beer forecass han he more recen daa models. Also esed were combinaions of older and more recen ime poins: and in panel (a), and in panel (b), and and in panel (c). All es resuls are based on x b, which is calculaed when forecasing performances from boh mehods are available. Kolmogorov Smirnov ess (no presened in his paper) did no rejec he hypohesis ha he x b is normally disribued in all combinaions of y 1 and y when he sample size is 000 for boh he older and more recen ime poins. Quanile Quanile plos (no presened in his paper) also suggesed ha he x b is normally disribued. This jusifies he auhor s proposed ess, which assume ha x b is normally disribued. In each panel, he horizonal and verical axes represen he number of observaions from he older and more recen ime poins, respecively. The cells are shaded in black and dark grey if z 1. and 1. > z > 0.0, respecively, indicaing ha he updaing funcion models produced saisically significanly beer forecass a five percen level of significance and produced beer forecass wihou five percen level of significance, respecively. Diagonal cells from he lower lef o he upper righ of he panels (dashed lines are drawn o faciliae reader undersanding) indicae ha he numbers of observaions from wo poins in ime are he same (n 1 = n ). Cells below he diagonal cells (lower righ of he panel) represen areas where he sample size from he more recen ime poin is smaller han ha from he older ime poin (n 1 > n ), which was ineress of convenional model updaing. On he oher hand, cells above he diagonal cells (upper lef of he panel) represen areas where he sample size from he more recen ime poin is greaer han ha from he older ime poin (n 1 < n ). The n 1 < n was no ineres of he convenional model updaing, bu he presen sudy is ineresed in his area since he updaing funcion models migh produce beer forecass. Firs, a case when he numbers of observaions from wo poins in ime are he same is examined (n 1 = n ). When n 1 = n = 000 or 000, he updaing funcion models produce saisically significanly beer forecass. When n 1 > n (pars below he diagonal cells), he updaing funcion models someimes produced saisically significanly beer forecass. Alhough his area was ineress of convenional model updaing echniques, () demonsraed ha convenional four model updaing mehods never produced saisically significanly beer forecass han he more recen daa model. Therefore, he presen auhor proposes wih confidence he updaing funcion models as a novel model updaing mehod. In addiion, n 1 < n (pars above he diagonal cells) was no ineress of convenional model updaing mehods. However, he updaing funcion models someimes produced saisically significanly beer forecass in panels (b) and (c). Therefore, he updaing funcion models have anoher novely, which exended he possibiliy of model updaing. On he oher hand, z -1. has never been observed, which means ha he more recen daa models never produced saisically significanly beer forecass han he updaing funcion models. Figure 1 panels (b) and (c), where he more recen daa comes from, are compared. The wo ime poins span 0 and years in panels (b) and (c), respecively. The panel (b) has more cells shaded in black and dark grey, implying ha a use of daa from wider range of ime poins conribue o improve he forecasing performance. One of he disadvanages of he updaing funcion model is ha he fuure GDP per capia mus be forecas. Therefore, a sensiiviy analysis wih respec o he fuure GDP per capia is required. (1) conduced a sensiiviy analysis wih respec o he fuure GDP per capia, when 000 observaions are uilised from,, and. A similar approach is applicable o he presen sudy.

12 n from n from n 1 from (a) y 1= and y = n from n 1 from (b) y 1 = and y = n 1 from (c) y 1 = and y = Noe: Cells are filled in black and dark grey for z 1. and 1. > z > 0.0, respecively. In panel (c), saisical es was no performed for (n 1, n ) = (0, 0) due o he smaller number of boosrap repeiions for successfully calculaing x b. The cells, where he dashed lines are passing hrough, mean n 1 = n. FIGURE 1 Saisical ess.

13 CONCLUSIONS This sudy proposed an updaing funcion model as a novel model updaing echnique. The mehod was originally proposed by he auhor o uilise cross-secional daa from muliple ime poins o produce beer forecas han a model uilising daa from only he mos recen ime poin. The updaing funcion model expresses parameers as funcional form, and he presen sudy expresses hem as funcions of GDP per capia (in linear form). The auhor examined a case where daa comes from wo ime poins: y 1 and y. This sudy examined hree combinaions of y 1 and y, and 1 combinaions of n 1 and n (ranging from 0 o 000), which represen he numbers of observaion from y 1 and y, respecively. The auhor uilised repeaed cross-secional daa colleced in Nagoya, Japan, and commuing mode choice behaviours are analysed. The main findings are lised below. When he number of observaions from he more recen ime poin is equal o or smaller han ha from he older ime poin, he updaing funcion models someimes produced saisically significanly beer forecass han he more recen daa models. Convenional four updaing mehods of ransfer scaling, Bayesian updaing, join conex esimaion, and combined ransfer esimaion never produced saisically significanly beer forecass han he more recen daa models (). Therefore, he auhor concludes wih confidence ha he updaing funcion model is a novel model updaing mehod. When he number of observaions from he more recen ime poin is larger han ha from he older ime poin, which was no ineres of convenional model updaing, he updaing funcion models someimes produced saisically significanly beer forecass han he more recen daa models. This means ha he updaing funcion models exended he possibiliy of model updaing. In any combinaions of he ime poins when he daa was colleced and he numbers of observaions, more recen daa model never produced saisically significanly beer forecass han he updaing funcion models. This sudy examines a single case, bu wih hree combinaions of older and recen ime poin, bu his is a good sar o invesigae his issue. Since his is he firs aemp o analyse his wih saisical ess, he auhor adops simple model srucure of mulinomial logi models. This sudy uilised he daases in hand, so he mos recen daa (from 001) is already 1 years old. This was less of a concern. The use of more recen daases wih more recen rends of ravel behaviours, such as peak car, also is a opic for fuure research. A sensiiviy analysis wih respec o he fuure GDP per capia also is a fuure research opic. Budge consrains for ranspor survey in recen years need analyses like he presen sudy o deermine efficien survey inerval and number of observaions. ACKNOWLEDGEMENT This work was suppored by JSPS KAKENHI Gran Numbers 0 and 1K01. The auhor acknowledges he use of daa provided by he Chubu Regional Bureau, Japan s Minisry of Land, Infrasrucure, Transpor and Tourism, and he NUTREND (Nagoya Universiy TRanspor and ENvironmen Dynamics) Research Group. REFERENCES 1. Badoe, D. A., and E. J. Miller. Comparison of Alernaive Mehods for Updaing Disaggregae Logi Mode Choice Models. In Transporaion Research Record: Journal of he Transporaion Research Board, No. 1, Transporaion Research Board of he Naional Academics, Washingon D.C., 1, pp Karasmaa, N., and M. Pursula. Empirical Sudies of Transferabiliy of Helsinki Meropolian Area Travel Forecasing Models. In Transporaion Research Record: Journal of he Transporaion Research Board, No., Transporaion Research Board of he Naional Academics, Washingon D.C., 1, pp..., N. Crieria for Selecing Model Updaing Mehods for Beer Temporal Transferabiliy. Compendium of Papers of he h Annual Meeing of he Transporaion Research Board,

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