Coping with Insufficient Data: The Case of Household Automobile Holding Modeling by Ryuichi Kitamura and Toshiyuki Yamamoto

Transcription

1 Coig with Isufficiet Data: he Case of ousehold utomobile oldig odelig by Ryuichi Kitamura ad oshiyuki Yamamoto It is ofte the case that tyically available data do ot cotai all the variables that are desired for the aalysis of the behavior of iterest. I case of the aalysis of household automobile holdig behavior, for examle, iformatio o the cost of holdig a automobile is rarely available i data, but has to be geerated by the aalyst based o a set of assumtios. I fact, uless the data have bee collected secifically to aalyze household automobile holdig behavior, iformatio that is eeded to determie the cost of automobile holdig ad oeratio e.g., make ad model of the automobile, acquired ew or used, urchase rice, fuel cosumtio rate, or isurace costs is tyically uavailable. ossible aroach whe data are isufficiet is to develo a theoretical model which, based o exteral riciles, embodies relatioshis amog observed variables. I this study, a utilitaria model of household automobile holdig is develoed based o the assumtio that a household holds a otimum umber of automobiles at the time of observatio. uique feature of the model is the adotio of the otio of base auto owershi cost. his is the miimum exediture that is required er uit time to hold a automobile, ad each household is assumed to sed a oegative amout of moey i additio to the base cost to hold a better automobile that offers more ameities. With the utilitaria assumtio that the household otimizes its vehicle holdig ad use, the model exresses the utility of automobile holdig i terms of icome ad household size, without requirig variables that ca hardly be measured, e.g., uit cost of auto ad trasit travel. Let = umber of automobiles = umber of adult household members Y = household icome = mobility by automobile er adult household member (erso-km) = mobility by ublic trasit er adult household member (erso-km) = auto ameities exediture er automobile X = exediture er adult household member for other goods = auto variable cost (er erso-km) = trasit variable cost (er erso-km) q = uit auto ameity cost C % = base auto cost er uit time = rice of other goods ( Z ) = miimum mobility er household (erso-km) ( Z ) = maximum mobility er automobile (km) Z = vector of household attributes automobile ameities exediture refers to the amout set over the miimum amout to hold a higher quality automobile or a automobile with more otios.

2 ssumig that utility is roduced by travelig to egage i activities, by cosumig auto ameities, ad by cosumig other goods, let the utility of a household, give the umber of automobiles it ows ( > 0), be ( ) η U = U(,,, X ) = X α β γ δ η is a modifier that reresets the effect of auto availability o the utility of auto mobility. Now, takig the logarithm of Eq. (1), let the household s automobile holdig behavior be deicted as ax l U(,,, X ) = α l + αη l( ) + β l + γ l + δ l X Subject to + + ( q+ C% ) + X = Y ( + ) ( Z) ( Z) (2) the first coditio reresets icome costrait, the secod coditio idicates that the miimum mobility requiremets be met, ad the third coditio reresets the ceilig o the use of household automobiles i terms of total vehicle kilometers. he Lagragea is give as L = α l + αη l( ) + β l + γ l + δ l X λ( + + ( q+ C% ) + X Y) (3) μ( ( Z) ( + )) ρ( ( Z)) I this aer, oly ubouded solutio ( μ = ρ ) is examied. Normalizig the utility fuctio by lettig δ = 1, the first-order coditios for the otimum are, by differetiatig the Lagragea with resect to,, ad X, α β γ 1 =, =, =, X =. (4) λ λ λ q λ Substitutig these ito the icome costrait, α + β + γ + 1 K1 λ = = Y C% Y C% K1 = α + β + γ + 1. he otimum solutio is Y C α Y C β Y C γ Y C = %, = %, = %, X = % K1 K1 q K1 K1 he idirect utility fuctio is obtaied as 1 (1) (5) (6)

3 % U = K1 l( Y C) αl + αη(l l ) βl γ l q l + αlα + βl β + γ lγ K l K 1 1 Rearragig this, U = K1 l( Y C% ) γ l + αηl ( α + β)l αηl l αl βl γ l q l + αlα + βl β + γ lγ K1l K1 = K l( Y C% ) ( γ αη)l ( α + β + αη + 1)l + K 1 2 K = α l βl γ l q l + αlα + βl β + γ lγ K l K. (9) Note that there is o elemet i K 2 that is associated with the attributes of the household. (7) (8) I case the household does ot hold a automobile ( ), let the household behavior be formulated as ax U(, X) = β l + δ l X Subject to (10) + X = Y ( Z) he idirect utility fuctio is obtaied as U = β(l β + ly l l( β + 1)) + (ly l l( β + 1)) = ( β + 1)l Y ( β + 1) l βl l + βl β ( β + 1)l( β + 1) = ( β + 1)l Y ( β + 1)l + K 3 (11) K3 = βl l + βl β ( β + 1)l( β + 1) (12) I sum, the idirect utility of automobile holdig is give as ( β + 1)l Y ( β + 1)l + K3 if U = ( α + β + γ + 1)l( Y C% ) ( γ αη)l (13) ( α + β + αη + 1)l + K2 if Lettig C = K3 K2 ad itroducig radom error terms, the ε i( ), redefie the idirect utility as ( β + 1)l Y ( β + 1)l + C + εi(0) if Ui ( ) = ( α + β + γ + 1)l( Y C% ) ( γ αη)l (14) ( α + β + αη + 1)l + εi( ) if i refers to the household. With the assumtio that the ε i( ) are i.i.d. Gumbel, the arameters ca be estimated by formulatig a multiomial logit model of automobile owershi with the idirect utility fuctio as defied i Eq. (13).

4 For this urose, it may be more coveiet to rewrite (13) i terms of relative idirect utility, by addig ( α + β + αη + 1)l to the right-had side: ( 1)l (1 )l (0) if 0 β + Y + α + η + C + εi = Ui ( ) = ( α + β + γ + 1)l( Y C% ) ( γ αη)l + εi( ) if (14 ) Note that the differeces i utility across differet values of Eq. (14) ad Eq. (14 ). remai uchaged betwee Eq. (14) may aear strage at first because the relative utility of owig o automobile icreases with the umber of adult household members. Note, however, that whe household icome is give, the owig a automobile may become less advatageous as the umber of adult members icreases because icome er adult member decreases with more adult members. O the other had, the relative utility of owig more automobiles icreases whe ( γ αη) is egative, or, whe αη > γ. ssumig that the maximum umber of automobiles a household may hold is robability that household i will hold automobiles ca be exressed as { β } ex ( + 1)lY + C K i( ) = ex {( α + β + γ + 1)l( Y C% ) ( γ αη)l α(1 + η)l} K (15), max { β 1)l } K = ex ( + Y + C max = 1 max { α β γ Y C γ αη α η } + ex ( )l( % ) ( )l (1 + )l Ukow arameters are α, β, γ, η ad C. he sum, ( α + β + γ + 1), ca be estimated as the coefficiet of l( Y C% ), α(1 + η) as the coefficiet of l, ad ( γ αη) as the coefficiet of l. he variable, C %, is also ukow. It is roosed that alterative values be ostulated for C % whe estimatig the model, ad the oe that offers the largest likelihood value be used as the value of C %. hus, the miimum cost of holdig a automobile is estimated through the exercise of estimatig the model of this study. Imortatly, the formulatio does ot require that,, the ad be kow. Sice these variables are ot i tyically available data sets ad sice it is ot easy to emirically determie the values of these variables, use of the theoretical formulatio show here aids i the quatificatio of the idirect utility fuctio of household automobile owershi. Yet, there are at least two major issues that remai. First, boudary solutios must be icororated ito the estimatio rocess. Secod, it is logical to assume that the arameters α, (16)

5 β ad γ, ad ossibly η, vary across households deedig o their attributes. Oce we allow them to vary, however, the costat term, C, of Eq. (14 ) varies from household to household. Estimatig the arameters, the, requires that, ad be kow.