The utility of journeys, from Dupuit’s constant-time bridge crossing hops to commutes of chosen duration and reliability in the Paris region Marc Gaudry Département de sciences économiques et Agora Jules Dupuit (AJD), www.e-ajd.net ou http://e-ajd.mkm.de/ Université de Montréal, Montréal, Canada,
[email protected]
This first version of this paper, originally entitled “The demand for journey duration and reliability: Paris Region work trips by mode and sex, 2010-2011” (Gaudry, 2016a), was presented in Paris by the author on March 11, 2016, at the Séminaire sur les méthodes de prise en compte de la fiabilité des temps de transport dans les évaluations of the Direction générale des infrastructures, des transports et de la mer (DGITM). It was also kindly presented by Alain Bonnafous on July 12, 2016, at the Special Dupuit Session of the 14th World Conference on Transport Research (WCTR) in Shanghaï. It is based on research work carried out for, and financially supported by, the Société du Grand Paris (SGP) in charge of the planning and construction of the Grand Paris Express (GPE) automatic metro, a mass transit extension program consisting primarily in 200 km of new lines and 68 stations eventually doubling the length of the current metro line network. The background research (Gaudry, 2015) primarily summarized here was supported in Paris by the DGITM and could not have been carried out without strong data support from the Régie Autonome des Transports Parisiens (RATP) and without the close collaboration with Vincent Leblond on the data supplied, on their significance and modeling (data models), and on the relevance of the adopted multi-moment methodology (models of data). Computational assistance was also received from Cong-Liem Tran and Kim Tran (Oikometra, Montreal), Benjamin Cuillier (Stratec, Bruxelles), Ralf Klar (MKmetric, Karlsruhe), and Charles Varrod (RATP, Paris). Taken into account were significant comments or information supplied by Sophie Dantan, Lasse Fridstrøm, Nathalie Picard, Jean-Claude Prager, Alain Sauvant and, repeatedly, by Émile Quinet. The author, grateful for this embarras de richesse, is responsible for all remaining errors. Profoundly revised for Transport Policy where it is forthcoming in 2018, this final version summarizes original version and complementary document source contents.
Université de Montréal Agora Jules Dupuit - Publication AJD-166 28th July, 2017 (with minor corrections on 21th December 2017) 1
Abstract We seek first to establish the existence of a safety margin built into any transport journey of duration T. This duration, assumed to combine an endured network service time component S and a precautionary margin I constructed by the traveler, is explainable by that same service S and other factors X. Should the adjunction hypothesis T f(I,S) be false, duration T is equivalent to the service provided by the network S and the elasticity of duration demand with respect to this endured service equals one, the econometric explanation being found wanting and of little interest beyond this result. Should it be true, and T include more than S due to incorporation of an endogenous safety margin I, the same elasticity differs from one and the legitimized econometric explanation becomes more interesting. In that case, an elasticity smaller than one denotes a palliative role for the safety margin. If and when the adjunction hypothesis holds, it is interesting and useful to extract from explained duration demand estimates the marginal rates of substitution among its moments of various orders, determinate up to the fourth, understood without formal proof to match Allais’ famous rates of substitution among all moments of a random prospect relevant to its utility with respect to the first. The proposed unit elasticity tests, aimed both at establishing the existence of the precautionary time margin and at extracting relevant reliability valuations from built-up durations that incorporate it, are performed on home-based work trips by private car driven alone (PC) and public mass transit (PT) longer that 3 km of Great Circle Distance and shorter that 180 minutes, starting or ending on a work day between 7h30 and 9h30 in Île-de-France, as reported in a survey of 18 000 households in 2010. We strongly reject, by commuter mode and sex, the unitary value of the elasticity of duration T to endured network service S and show that empirical valuations of the reliability of trips differ significantly across the commuter groups analyzed (men and women; car drive alone and transit user). For all of these 4 groups, the asymmetry of trip durations is much better modeled by Box-Cox transformations than by other forms (linear, log-log) of the regression variables. And, for all such best fit models, elasticities of duration moments with respect to endured network service, estimated with very great precision, are smaller than one because margins and service are substitutes. Exogenous service level changes induce offsetting changes in the time profiles of trip demand assumed at the transport planning outset, thereby closing the transport demand model. Results imply that putting new roads into service or even automating existing metro lines will contract peak time demand profiles and produce demand super-peaks of strongly increased multiple-moment (reliability) utility in spite of misleading fourth moment deteriorations in road speed or transit congestion comfort at the peaks. At given user prices, the utility of new road investments or transit line modernizations cannot then be calculated merely with flow and mean peak speed variations because all four first moments matter to utility, some in fact more affected (negatively or positively and proportionately speaking) by service improvements than the first. Only the fourth moment implicitly appears in simplistic analyses claiming for instance that, if congestion conditions that road projects were originally designed to improve rapidly come back, even visibly higher flows are somehow pointless in view of the return of speed levels at best identical to those prevailing ex ante the newly deplorable superpeaks. Our approach is based on the assumption that preferences for distributions are moment dependent, i.e. characterized solely by their empirical moments, and that all time moments of the travel “lottery” matter to utility, notably the fourth. Transport project evaluation should not be concerned merely with Dupuit-like utility of new trips or ton-km, but also with to Allais-inspired reliability matters, including fast fill-ups of new highways and super-peaks on new or newly automated transit lines. ______________________________________________________________________________ ________________________________________________________________________________________________________________________________________________________________________
Key words: Jules Dupuit; Maurice Allais; trip demand; trip duration demand; trip time reliability demand; home-based commuting; Île-de-France; Enquête Globale Transport 2010; Moment Dependent Utility (MDU); preferences for distributions characterized solely by their empirical moments; safety time margin; public mass transit; private car; Box-Cox regression; elasticities and marginal rates of substitution among moments of a random dependent variable.
Journal of Economic Literature (JEL) classification: D-12, R-41, C-41, C-51. 2
Table of contents 1. Enriching Dupuit’s famous 1844 bridge crossing formulation.............................................................. 4 2. BCT , correlation, meaning and the fitting of skewed outcomes ................................................. 12 2.1. Beta-Lambda BCT analysis, statistical correlation, and the signs of variables .............. 12 2.2. The behavioral meaning of forms ...................................................................................... 12 2.3. Regression explanations of asymmetric outcomes ........................................................... 13 3. Paris region work trips by transit and car: data, model, and results .................................................... 17 3.1. Data: the asymmetry of observed Paris area trip durations .............................................. 17 3.2. Model: summary of economic and econometric specifications ....................................... 17 3.3. Existence and structure of the safety margin..................................................................... 18 3.4. Results: unit elasticity tests at optimal form, moment demand elasticities and MRS .... 18 4. Potential application to rail line automation ......................................................................................... 21 5. Conclusion and explicit policy implications for project evaluation .................................................... 22 6. Acknowledgements................................................................................................................................. 23 7. References ............................................................................................................................................... 23 List of figures Figure 1. Modernization of line Z of the Paris metro: the first 4 moments of in-vehicle time (RATP) . 5 Figure 2. Signs of k coefficients and TBC values in univariate and multivariate regression ............. 13 Figure 3. Standard and Box-Cox cumulative logistic distributions ......................................................... 14 Figure 4. Asymmetric behavior of some probability weighting functions used in CPT ........................ 14 Figure 5. Distribution of Paris area morning peak home-based work trip durations (2010) ................. 17 List of tables Table 1. Definitions of the first and of the next three central moments of a random variable y ............. 5 Table 2. Presumed direction of preference (effects on utility) of increases in moments of y .................. 7 Table 3. Risk-aversion, risk-seeking, and expected signs of the MRS among the first three moments.. 7 Table 4. Four notions of the elasticity of y relative to a variable X k ................................................... 9 Table 5. Derivatives of moments of y explained by regression with Gauss-Laplace errors .................. 16 Table 6. Safety margin existence test based on a unitary elasticity of duration w.r.t. endured service 18 Table 7. Work trip durability: functional forms and the four Case 3 optima, each with two BCT ....... 19 Table 8. Three own moment elasticities of car and transit durations w.r.t. to service modifications ... 19 Table 9. Marginal rates and elasticities of substitution among the first three moments of duration ..... 21
3
1. Enriching Dupuit’s famous 1844 bridge crossing formulation The measurement of economic utility and the theory of value certainly owe much to Jules Dupuit 1. In contrast with predecessors, such as Cournot (1838) who had formulated market demand (and supply) functions, he was the first to formulate a demand curve derived for a given utility level (Dupuit, 1844; and, to a lesser degree2, 1849), a point conceded by even the most demanding retrospective analyses (e.g. Ekelund & Hébert, 1999; Bonnafous & Baumstark, 2010). Walras, Marshall and others later refined and generalized his approach but some, like Marx, ignored it to their detriment 3 because Dupuit is at least as relevant to the human sciences as Darwin, who published on 24th November 1859 The Origin of Species read by Marx before publishing Capital in 1867. Dupuit’s seminal formulation does not confuse value with numéraire (be they pieces of apple pie or units of labor) and never implicitly reduces the dynamic demand-supply interplay and equilibria of proper estimable schedules to mere fixed coefficients. It influenced many, directly and indirectly, including Hotelling 4, and was completed by the development of econometric identification methodology (Working, 1927) to unravel interactions among postulated schedules ‒ including notably transport demand ones, our concern here. Indeed, our purpose is to: (i) briefly summarize some key transport demand developments since Dupuit as background steps to the particular empirical enrichments we proposed in a set of source papers on the demand for metropolitan Paris journeys to work (Gaudry, 2015, 2016a & b, 2016c); (ii) give an idea of how our proposed Moment Dependent Utility (MDU) approach can also be applied to the interpretation of random utility model (RUM) Logit choice model streams with Box-Cox specifications in transport and with Cumulative Prospect Theory (CPT) specifications elsewhere; (iii) justify in depth use of Box-Cox models adopted here; (iv) recall only the key points and results of our application to the Greater Paris region, referring the reader for details to the source documents. A. The 1844 starting point: constant and certain endured transport service time. Dupuit’s non stochastic (errorless) trip demand function for crossings of a given bridge n may fairly be written as: (1)
Demn q Pn ; X s ; A d n ,
Qn
where: (i) price Pn denotes his key variable, progressively manipulated to trace the demand curve; (ii) road users may well have different socio-economic characteristics Xs, s=1, ..., S; (iii) the vector A ( A1 ,..., Aa ) denotes trip generation activities for persons and goods; (iv) endured trip duration Durn d n is implicitly certain and invariant across users or demand levels ‒ a dubious axiom if any. By contrast with his assumption of a unique price, his hypothesized uniqueness and certainty of travel time d n , per force vague and obscure because trips of certain durations do not exist, helps to focus the mind on the drawing of the demand schedule driven by manipulations of the price. Similar vagueness plagues SP surveys to-day if respondents must choose between “certain” trip times (not defined further) and “uncertain” ones (however defined): such choices are not proper because “certain” trip times ‒ in contrast with uncertain ones ‒ have no empirical reference and exhibit a null probability of occurrence. Below, we dodge these traps of trip time certainty notions by arguing that consumers: (i) perceive the first four moments of actual trip time distributions, as defined in Table 1 and illustrated in Figure 1 with real subway data; (ii) compare them; (iii) offset their uncertainty by precautionary time constructs made up of the same moments of time. For simplicity, without loss of generality, we focus whence on extant, or real, uncertain travel time, and discuss cost, uncertain or not, only in passing. 1
This paragraph is taken from the author’s opening statement, as Scientific Chairman, of the 6th International Association for Travel Behavior (IATB) Conference in Quebec City, 22-24 May 1991 (Gaudry, 2016d). It prompted the creation of the Jules Dupuit prize awarded since 1992 by the World Conference on Transport Research (WCTR). 2 Only the first paper is usually translated in English and found in various Readings in Welfare Economics. 3 None of Dupuit’s works were part of Karl Marx’s own personal library of more than 800 books or annotated documents; and he apparently never read the Annales des Ponts et Chaussées sitting in the British Museum Reading Room, reputedly favoring desk 07. Other London authors, such as Alfred Marshall, read them and explicitly refer to Jules Dupuit. 4 In June 1970, wishing to read the original 1844 paper in the Annales at the Princeton University Engineering Library, the present author noted only one previous reader listed on the library card, namely Harold Hotelling, in October 1929.
4
In our view, service as such does not exist over, above and independently from its moments, notably those of in-vehicle time shown in Figure 1 (wait time is not included) where, as one might recall, skewness is the obvious visual asymmetry (here to the right) of all 8 graphs and kurtosis their excess peakedness. Note in Figure 1.A the difference between peak and off-peak service in each direction before automatization and compare it with the same metro service after modernization, shown in 1.B. Changes in the four moments brought about by modernization occur in the same direction as changes between peak and off-peak slices. Notably, automatization has turned the metro into a metronome with an extraordinary increase in the fourth moment, but has increased skewness. Does it all matter? Table 1. Definitions of the first and of the next three central moments of a random variable y 1st
3rd
Mean
e E ( yg )
Skewness
E ( yg E ( yg ) 3
2nd
3
4th
Standard error
E ( yg E ( yg )
Kurtosis (Fisher)*
4 E ( yg E ( yg ) 2 E ( yg E ( yg )
2
1
2
e 3 44 3 2
*Fisher’s definition, with the shift of -3, implies a kurtosis of zero for the Normal distribution, in contrast with Pearson’s for which that distribution has a kurtosis of 3: it therefore denotes peakedness in excess of that the Normal distribution.
Figure 1. Modernization of line Z of the Paris metro: the first 4 moments of in-vehicle time (RATP) A. Before modernization B. After modernization
Analysis shows that both changes (from Peak to Off-peak before modernization, and from any time period and direction before modernization to any afterwards) have the following properties: (a) higher speed lowers mean invehicle time and its standard error; (b) its skewness increases, but less in the first set of changes (Peak/Off-peak) that in the second set; (c) but percentage increases in skewness are small relative to those of kurtosis, which can be huge.
B. Endured transport service defined by the first two moments of travel time. Basic demand theory only uses cost and time features of modes or paths in their utility functions, and for scores of years only their mean values mattered. Neglecting cost, the seminal RUM Abraham-McFadden discrete choice approach (Abraham, 1961; CRA, 1972) and independent Logit practice (Warner, 1962) were concerned only with mean time, as were early aggregate Logit formulations explicitly linked to utility (Setec et al., 1959; Rassam et al., 1970, 1971) and their successors. The 2nd moment appeared recently as component of perceived endured transport service S np , as in Senna (1994): (2-A)
Demn q Pn ; Snp ; X s ; A ,
Qn
characterized by vertical concatenation of two moments (mean e and standard error ) of travel time: (2-B)
Snp r1te( n ) r2 t ( n ) , 5
r1 and r2 denoting weights and bars indicating that, for the user, moments are given and endured ‒ not chosen. As in the joint demand formulations of Demsetz (1970) and Samuelson (1969), the components of duration are concatenated in (2-B) by a vertical summation over time moment demand schedules. Moments of time replace these authors’ jointly supplied goods: for the former, hides and meat provided in fixed ratios by steers; for the latter (see also Samuelson, 1954, 1955), individuals’ schedules of willingness to pay for public goods. Horizontal summation again yields market demands. C. Recognizing the lurking third moment of endured travel time: Allais’ famous r(y) MRS. Trade-offs between the first two moments are long known in finance (Tobin, 1957/1958; 1965) and estimation of marginal rates of substitution between them, as joint determinants of the demand for asset y, MRS X k , X y X k y X X X k , is well accepted. But higher moments are ignored because it is logically impossible to calculate the partial derivative of a utility function with respect to, say, the third central moment without affecting lower order ones (Brockett & Garven, 1998). Transport authors therefore state dogmatically that only mean and variance determine utility but, in empirical tests, rapidly abandon that untenable assumption, perhaps by replacing mean and variance determinants by the median and high percentile points Pyz of the time distribution. Such slippages from clean analytical moments to merely descriptive Pyz (all un-optimized multiples of 5!) abound, as in Logit models of choices between free and tolled California State Road 91 lane options. With SP data (400 answers) on lane choices, Lam & Small (2001) finally retain the median (P50) and the 90 th (P90) percentiles of the time distribution; but with RP data on the very same SR91 case, Small et al. (2005) end up using P50 and P80-P50. These manually chosen percentile point replacements “yield a better fit” than the mean and standard error pair; and the estimated signs of, say P50 and P90, coefficients match in practice, implying that latent first and third moment effects dominate those of the variance. But, of course, such descriptive ad hoc P80 or P90 terms, silent about the form of the distribution, are indicators that only “work” because actual SR91 lane durations have positively skewed distributions: understanding casual Pyz results (why not Pyz+x?) always requires appeal to a missing key latent third moment with the same sign effect on utility as the first, obviously lurking despite the Brocket-Garven point that moment ordering is a necessary but not sufficient condition for an ordering of utility (stochastic dominance). To recognize this role of asymmetry , assume that preferences are for distributions characterized solely by their empirical moments. Moment Dependent Utility (MDU) makes trips depend on all relevant moments of endured service and allows us to posit an empirical extension of Senna (op. cit.): (2-C)
Snp s(r1te( n ) r2 t ( n ) r3 t ( n) ) ,
with valuation weights r (r1 , r2 , r3 ) . If people are known to correctly identify, even on the basis of few observations, at least distributions as varied of those by Gauss, Poisson and Erlang, as well as various power functions (Griffiths & Tenenbaum, 2006), they will a fortiori identify sequences of moments, which does not require to identify their distributions, perhaps not unique (Gut, 2002). As service takers, travelers face or suffer exogenously determined components of S np . On this extant role of the third moment, one may then quote Charcot 5: “Theory is good, but it does not prevent from existing”.
To empirically dodge the logical impossibilities of utility function derivatives, we therefore assume without proof that our exploitation of empirically and behaviorally determined MRS is consistent in particular with Allais’ (1987) famous r(y) marginal rates of substitution approach, the intuitive description of which he provided as follows. For an individual of wealth C, the utility of the monetary value V of a random prospect is assumed to be: u(C, V) = û + R( 2, ..., P), where û denotes the mathematical expectation of the ui (the cardinal utilities corresponding to the various gains gi) and the p the moments of order p of these utilities ui), and where the ratio r= R/û: «can be considered as an 5
Declared by Jean-Martin Charcot in his lectures between October 13, 1885 and February 28, 1886, dates of Sigmund Freud’s stay in Paris and attendance of Charcot’s lectures. Freud quoted this statement by Charcot, made in response to a student claiming that a particular theory ruled out the possibility of one of Charcot’s claims, numerous times and notably in his eulogy of Charcot in 1893 (Freud, 1984, p. 23).
6
index of the propensity for risk: for r = 0, the behaviour is Bernoullian [only the first moment matters], for r positive, there is a propensity for risk and for r negative there is a propensity for security». In fact, on this point: «[this] adds to the Bernoullian formulation a specific term R characterising the propensity to risk which takes account of the distribution as a whole...» (op. cit.). This taking into account of the distribution as a whole means for instance that, if one makes a financial investment, one will have an interest at least in: (i) its average yield or return e( y ) ; (ii) the standard error of this return ( y) ; (iii) the asymmetry of the return ( y ) , or relative chances of yield increases (upside risk) or decreases (downside risk); (iv) higher moments like kurtosis. D. Endured moments and the direction of preference. Whether the prospect is a «good» like financial return or a «bad» like transport time or accident, naturally reverses the sign of its marginal effect on utility but not the sign of the MRS between any two such effects because we adopt the view that signs of U (moment i) alternate (Scott & Horvath, 1980), as summarized in Table 2 where signs of the first two moments are considered as obvious and those of higher orders as reasonable. Table 2. Presumed direction of preference (effects on utility) of increases in moments of y Moment 1st 2nd 3rd Prospect: good + + Prospect: bad + -
As a consequence, for an investment, increased yield is pleasant (+) but not increased variability (-), whereas more asymmetry (less negative on the left or more positive on the right, depending on its nature) is also pleasant (+). Analogously, and mutatis mutandis, the occurrence of longer transport times or more road crashes is unpleasant (-) and one wishes that the standard variability of this outcome increased (+), but not its asymmetry (by becoming less negative or more positive) because higher asymmetry (in absolute value) of a bad is unpleasant (-). But, on this, do riscophobes and riscophiles differ? The first line of Table 2 has assumed the preferences of a risk-averse investor, which imply the signs of the MRS ratios of matrix [A] in Table 3 ‒ matrix [B] are a risk-seeker’s. Table 3. Risk-aversion, risk-seeking, and expected signs of the MRS among the first three moments Two sets of marginal rates of substitution among moments: -[ (moment i)/ (moment j)]
[A]. Riscophobe
i\j e
e 1
+ 1
1
[B]. Riscophile
e 1
1
+ 1
assumed MRS signs derived MRS signs
Both sets are mathematically coherent and admissible (cf. Tran et al., 2008, Table 6, p. 36). Risk seeking and risk aversion are defined in Table 3 by assuming the existence of certain signs for MRS between the first and other moments. The MRS between the second and third moments are derived from the first set, which explains their common negative value in figures [A] and [B]. A trade-off equal to 1 or 0 (arising from horizontal indifference curves in Tobin’s (1965) risk neutral formulation) are ignored in these matrices, which are based on the existence of the usual preference maps… E. Endured exogenous, chosen precautionary, and total endogenous compunction travel time. Some transport researchers have long complained, without obtaining much subsequent satisfaction, that demand (or mode choice, in most cases) levels were explained solely in terms of modal (price and) time service levels and without any role attributed to precautionary travel time choice, called «idle time» (D. Starkie, 1971) or «safety margin» (T.E. Knight, 1974) in English and «temps de précaution» in French, as if such time margins somehow cancelled out across modes or did not really matter. According to such critiques, all transport demand or mode choice models which do not include a safety margin term are misspecified and fail to yield unambiguous or unbiased coefficients. In contrast to the dearth of safety margin variables, we propose here an application of F.H. Knight’s (1921) famous distinction between risk and uncertainty. He reserved the term risk for strictly calculable probabilities from functionally specified distributions (probabilities of non-failure) and the 7
word uncertainty to globally or, in his terminology, “directly perceived” probabilities (of unpredictability). Consequently, here we call endured expected perceived transport service S np “risk”, and we call the possibility that it could be something else, i.e. its unreliability I n* , “uncertainty”. This ˆ ˆ ˆ latter punction is an unobservable chosen precaution margin I n* i(rt 1 e ( n ) r2t ( n ) r3t ( n ) ) where the endogeneity (designated by hats) of the palliative time vector moment components tˆmom i is of the essence.
F.H. Knight’s distinction is useful, if not essential to the issue at hand: the intuitive notion of precautionary or safety travel margin can hardly be better justified than as an Inescapable uncertainty compensation, concerning provided Expected transport service. In practice, authors such as Starkie (op. cit.) and T.E. Knight (op. cit) add to this pure unreliability adjustment a so-called «schedule delay» adjustment. It consists in an adaptation of one’s schedule, arising from the need to adjust it to the fact that at most one person in a queue can arrive at work on time and that all others are either early or late, and may be called strict schedule delay. In transit systems, a further adjustment may be required to compensate for increasing discrepancies between planned vehicle schedules and the scheduled start of work occurring when the frequency of transit service diminishes. Whether schedule delay is conceived as a reaction to expected service characteristics (e.g. infrequent service) or as a reaction to pure uncertainty, and whether it is conceptually translated into a preferred arrival time or not, does not make much difference here. Any reaction not arising solely from pure uncertainty of service has own endogenous moments that cannot be distinguished from those of uncertainty-based safety margins. There is only one identifiable «reaction» to supplied transport service, the time margin reaction: should preferred arrival time or any schedule delay exist, their moments are fused with the identifiable moments of duration, and notably reflected in its asymmetry. Compunction time Tnc* I n* Snp is then the sum of unobserved punction I n* and observed punction S np . With the first three moments of time as arguments of both margin and service components, we have: ˆ* ˆ* ˆ* r1eˆ(Tˆnc* ) r2ˆ (Tˆnc* ) r3ˆ(Tˆnc* ) (rt (3-A) 1 e ( n r2t ( n ) r3t ( n ) ) (r1te( n ) r2 t ( n ) r3 t ( n ) ) . This compunction time sum, driven by the chosen time margin component with endogenous levels both of moments and of their flexible proportions, is a multiple joint choice of ( dˆe ( n ) = tˆe ( n ) te ( n ) ), ( dˆ ( n ) = tˆ ( n ) t ( n ) ) and ( dˆ ( n ) = tˆ ( n ) t ( n ) ). It is not observable as such, contrary to its random outcome ‒ Duration (Durn), equal to the difference between rendezvous and departure times ( H rdv H dep ) ‒ which contains an outcome error term ( H rdv H dep Tnc uT c* ) that generates an observable Tnc . The n
identification and estimation of unobserved compunction time Tnc* I n* Snp Tnc uT c by the random n
outcome term from explained observed journey duration Dur between departure time Hdep and rendezvous time Hrdv leads to: (3-B)
Durn H rdv H dep Tnc uT c* Tnc d Pn , (r1te ( n ) r2 t ( n ) r3t ( n ) ), X s un,T c . n n
Dn
F. Demand for transport and demand for trip duration on a unique link. The endogenous time margin choice component of observed trip duration and reliability construct (3-A) then requires enrichment of the original single-level (Q n) process (1) into a two-level (Q n)-(D n) system of estimable demand equations for journeys Q and for their duration and reliability D. For a unique origindestination (OD) market, it may be written in a simpler form, with u H a duration observation error: (4-A) (4-B)
Demn q Pn ; Tnc* ; X s ; A uQn , with Tnc* I n* Snp Tnc uT c , and n p ( H rdv H dep )n d Pn ; Sn ; X s un , with un uH uT c , n
Qn Dn
p n
where, for this simple case: (i) Pn : price of unique OD transport mode n; (ii) S : service risk, or expected value of the unique OD endured transport service Snp s(r1te( n ) r2 t ( n ) r3 t ( n ) ) ; (iii) I n* : * ˆ ˆ ˆ uncertainty of S np , or its unobservable precaution margin I n i(rt 1 e ( n ) r2t ( n ) r3t ( n ) ) chosen by
8
commuters as offset to the global unpredictability of S np ; (iv) X s : workers’ socioeconomic characteristics, such as sex, age, position in family structure, etc.; (v) A : labor force qualities and employment conditions ( A1 ,..., Af ,..., AF ) at the origin or destination of the hypothetical unique transport service link served.
Even if the demand for trips is not for a unique OD service and link but is given by a mode choice model with modal utility functions V n containing Tnc and the functions are jointly estimated with equality constraints on the r (r1 , r2 , r3 ) in each, the MRS calculated for the duration equation (4-B) cannot be interpreted solely as mere distribution fitting parameters. The endogeneity of r1eˆ(Tˆnc* ) r2ˆ (Tˆnc* ) r3ˆ(Tˆnc* ) implies MRS that must reflect Allais’ relative moment utility valuations. If transport demand in (4-A) is aggregate in nature and trip duration data are averaged across users, the system (4-A)-(4-B) is simultaneous and cannot be recursive because supplied service S np in (4-B) then really should depend also on would-be demand. However, in cases such as Paris area work trip data, durations are individual and endured service S np is then strictly exogenous: it does not introduce a simultaneity bias if and when the duration equation is estimated conditionally upon chosen modes. Both PT (public transit) and PC (private car) observations are independent over time: the sample from 18 000 households required months of surveying, carried out over 2010 and 2011 (OMNIL, 2012). If such a bias still arises through contemporaneous correlation of the two errors, we cannot easily correct for it because we are only going to estimate the duration equation (4-B). G. Statistics of interest. We have an interest in a number of statistics derived from estimates of (4-B): i) elasticity of demand of one or many moments of duration. For any explanatory variable, we are interested in its effect on duration moments measured either absolutely: dˆ X ; dˆ X ; dˆ X ; (4-C) e( n)
k
( n)
k
( n)
k
or in elasticity terms, the most relevant notions of which are recalled in Table 4, where the arc elasticity approximation expressions for dummy variables are drawn from Dagenais et al. (1987): Table 4. Four notions of the elasticity of y relative to a variable X k Name of elasticity concept
A. Point elasticity for ordinary continuous variable Xk
y X k X k y
B. Arc elasticity approximation* for dummy variable Xk
X k Xk
1.
Sample elasticity of y with respect to Xk
2.
Elasticity of e(y) with respect to Xk
e ( y, X k )
e( y ) X k X k e( y )
X k ( y, X ) ( y, X k ) Xk
3.
Elasticity of (y) with respect to Xk
( y, X k )
( y ) X k X k ( y )
X k ( y, X ) ( y, X k ) Xk
4.
Elasticity of (y) with respect to Xk
( y, X k )
( y) X k X k ( y )
( y, X k ) ( y, X k )
s ( y, X k )
s ( y, X k ) s ( y, X k ) e
k
k
e
X k Xk
* In Column B, X k denotes the sample mean of the dummy variable and X k its sample mean recalculated only on its positive values (usually all equal to 1, but any arbitrary constant will do).
Elasticities of moments found in Column A of Table 4 are not all new. Some are rarely used despite Goldberger’s (1968) known proof that elasticities of the sample, the expected value and the median of y are identical in a logarithmic model, yielding for all of them and for any observations t: (4-D)
s yt X kt e ( yt ) X kt m m( yt ) X kt ( yt , X kt ) ( yt , X kt ) ( yt , X kt ) k . X kt yt X kt ( yt ) X kt m( yt )
In the case of explanatory variable S np , or of its components, the effects are particularly interesting (a) because values different from unity, i.e. (Tnc , Snp ) 1 , demonstrate that duration Tnc I n* Snp 9
cannot be reduced to expected service but includes a precautionary margin ‒ otherwise the effects would be strictly proportional, i.e. (Tnc , Snp ) 1 , and the regression of y on S np would not be legitimate, in effect regressing S np on itself (and other X k ) ‒; (b) because the strict notions together account for the impact of transport service on the time profile of demand for duration. This feedback closes traffic model (4-A); ii) marginal rates of substitution between explanatory variables. With respect to relative effects between explanatory variables, it is always possible to compute the so-called casual rates of substitution MRS X k , X y X k y X X X k but the quantitative model also allows for said strict MRS between any X k and X (components of S np included) with respect to moments of y:
MRS X k , X
(4-E)
E ( y ) X k ( y ) X k ( y ) X k y X k X E ( y) X ( y) X ( y) X y X X k
which have the surprising property that the ratio X X k , for any pair of variables, is equal6 for all moments of y and for the simple “sample” construct y X k y X that altogether ignores the error term distribution. This equality property, valid unless variables X and X k are also used to explain heteroskedasticity of residual un in (4-B), forms below the basis of a statement concerning the robustness of our results to potential collinearity. iii) marginal rates of substitution among moments of duration. After verification that (Tnc , Snp ) 1 , one might be interested in extracting the MRS m ,ms among the moments of duration Tnc , namely, with m1 e( y) , m2 ( y) and m3 ( y) : m m X k MRS m ,ms , ( s 1,3 and k=1,...K) . (4-F) ms ms X k Relative valuation of moments: from marginal rates to elasticities of substitution. More explicitly, with three moments considered, there will be only three MRS among them (their inverses being counted as identical) generated by the following development of all too compact formula (4-F): (5.0)
(moment i( y)) X k (moment j( y)) X k (moment i( y))
(moment j ( y)) , X k ,
which indeed holds for all X k even if, in (4-E), [(mom. i( y)) / X k ] [(mom. i( y)) / X ] . They are:
e( yt ) e( yt ) X kt e( yt ) X t ; ( yt ) ( yt ) X kt ( yt ) X t
me , m MRS me ,m (5.1*)
e( yt ) e( yt ) X kt e( yt ) X t ; ( yt ) ( yt ) X kt ( yt ) X t
(5.2*)
( yt ) ( yt ) X kt ( yt ) X t m m m ,m m ,m m ; (5.3*) MRS , ( yt ) ( yt ) X kt ( yt ) X t m m m
(5.1)
MRS me ,m
(5.2)
MRS
me , m
(5.3)
MRS
m , m
me , m
MRS
me , m
m me m , me m me m me m , me m me
where (5.1) is without units, but not (5.2) and (5.3). Legibility is reestablished by defining corresponding elasticities of substitution m ,ms , for ( s 1,3), namely (5.1*)-(5.2*)-(5.3*), all pure numbers equal to the MRS multiplied by matching inverses of moments, evaluated below at sample means of variables. Substitution effects as trades. Formally, substitution involves the sacrifice of a unit of X1, traded against a unit of X2 while maintaining the value of their combination constant: it is the slope of an isoquant, defined as minus the ratio of partial derivatives of the output or utility quantity held constant with respect to X1 and X2 (Allen, 1968, p. 42). We have neglected the negative sign in the above six 6
As documented in the program manual (Tran et al. 2008, Section 2.5), this surprise can be partly understood by noticing that any effect on a given moment mechanically implies structured effects on higher moments because, once e( y ) X is known, it is easy to deduce ( y ) X and ( y ) X with Jacobians of transformations from one moment to the next. k
k
k
10
expressions in order to keep the natural signs of the partial derivatives and ease the understanding, in Tables 2 and 3, of the impacts of partial derivatives of moments on utility. Later, tables of results take signs into account. H. Justification of regression method adapted to skewed outcomes. A quantification method is never neutral, so we need to explain how all these derivatives and elasticities of moments, as well as their trade-off ratios, are extracted from notoriously asymmetric random road accident or trip duration outcome data. Our extraction is effected within the forthcoming Box-Cox model (6-A)-(6-F) by computing derivatives of desired moments of the dependent variable y in a purely mechanical extension of popular practices typically limited to a first moment focus. But this model adoption requires a justification, provided presently and in part graphically, that matters for scientific purposes, such as the establishment of statistical correlation and the fitting of skewed outcomes essential to our problem, but also for policy reasons because incorrect statistical models will yield incorrect predictions and policy implications. Below, in our models of chosen trip duration levels (taking after road accident applications of the same MDU approach previously), and also in the Logit models to be referred to (with utility functions containing duration moments), the common Box-Cox transformation (BCT) defined in (6-B) is always used. Tukey’s shift parameter k specified in the original Box & Cox (1964) article, but typically neglected since by most authors, in not introduced or considered for reasons too long to discuss here.
11
2. BCT
,
correlation, meaning and the fitting of skewed outcomes
The Box-Cox transformation of variables effects a local approximation of curvature, in contrast with the Fourier transformation (Gallant, 1980) that should eventually displace it. To-day, there are very strong reasons to use BCT, rather than say simple power transformations producing derivatives of a sign that depends on the very sign of the power parameter itself, discontinuous at 0, and failing to preserve the order of the data (Johnston, 1984, p. 63). Still, the BCT is not yet dominant in practice despite 40 years of demonstrations, since Gaudry & Wills (1978), that untested fixed a priori forms yield dubious results in models of levels and Logit models. Many fixed form analysts still maximize expected sign findings (rather than likelihood values), minimize computation cost or mix up form and randomness of coefficient dimensions in said mixed Logit models (cf. Orro et al., 2005, 2010). Urban transport demand level analysis is still plagued by models of untested fixed a priori forms, for instance logarithmic in Batley et al. (2011) for Greater London rail and Duranton & Turner (2011) for US city roads7. In Logit model practice, despite the fact that early practitioners (Warner, op. cit.; Setec, op. cit.) carefully compared linear and logarithmic forms and found that the latter yielded better fits, many assume linearity, since Domencich & McFadden (1975), without tests. Need one worry?
2.1. Beta-Lambda BCT analysis, statistical correlation, and the signs of variables Firstly, the transport literature is full of examples of regressor signs that change with form, especially after ad hoc attempts to maximize expected sign findings, and is relatively empty of proper BCT analyses, for fear of the results. Yet the due existence, i.e. the size and sign, of statistical correlation depends on the form of variables, as shown in Figure 2 where monthly data (181 observations) on aggregate Montreal PT trips by schoolchildren are regressed on average transit system travel time TT, real average weekly earnings of households RAWEM, or both. The coefficients shown (from Gaudry, 2016e) are not obtained by matrix inversion but by calculating slopes b1 and b2 of the least squares regression plane relating observations on a vector y to observations on a constant X0 of arbitrary size and on variables X1 and X2, as in textbooks writing y and the latter pair of vector values in terms of their standard errors and of pairwise (“linear” or “simple Pearsonian”) correlations among them based on raw unranked data (e.g. Johnston, 1984, p. 81). Note that, except for the first, all graphs of Figure 2 have slopes that change in sign for some ranges of BCT values. David Hume’s “constant conjunction” is therefore always implicitly conditional on form, as made clear by this Beta-Lambda BCT analysis.
2.2. The behavioral meaning of forms Secondly, nobody can be indifferent to the interpretation of forms. In classic models of levels, it matters whether partial derivatives with respect to any regressor have additively independent or multiplicatively interacting effects, and whether these vary with the level of regressand and regressor variables, or not. The frequent childish fixation on the constant values of the elasticities of presumed Log-Log models is unscientific, as we will point to below with duration (and road accident) models. This critique also holds for the Trans-Log model because Box-Cox forms that nest the Trans-Log can be shown to dominate it infinitely, statistically speaking, at least in some transport (infrastructure cost) applications (Gaudry & Quinet, 2016) where interaction among variables is definitely not of Log-Log form but can be demonstrated to be of a much more complex type better captured by flexible BCT. In Logit models, linearity of, say, time (or cost) hypothesizes a constancy of marginal utility that is often just plain silly because no extra unit of money or time should without test be assumed to mean 7
Their demand function has no gasoline price. The unpublished research report version of the paper (Duranton & Turner, 2009, Footnote 10) states that the variable was removed because it was “imprecisely estimated and sensitive to the exact specification”. An optimal form approach, combined with a rough estimation of the autocorrelation parameter, instead of untested first differences in logs (which exclude null values of explanatory variables, contrary to advanced BCT practice), might have yielded more interesting results. Indeed, there exists a considerable literature on the significant impact in the US of gasoline price on mileage and, by consequence, even on road fatalities (cf. Grabowski & Morrisey, 2004, for references). One can even detect the specific impact of changes in the nominal state gasoline tax (Grabowski & Morrisey, 2006) due to the frequency of such changes: “Over the period of study (1982-2000), there were 253 instances where states changed their nominal state gasoline tax and every state changed its tax at least once except Georgia”.
12
the same for trips of 10 minutes and of 10 hours, a matter which piecemeal segmentation of variables by ranges may not handle well, if at all. In a survey of more than 50 Logit models using BCT on travel time and cost, it was found, except for vacation trips8, that the value of time always increased with distance, as occurs if TIME COST 0 , and that the BCT were systematically above unity ( TIME 1 ) in urban and below unity ( TIME 1 ) in intercity passenger and freight markets, arguably the first structural behavioral difference ever found between urban and intercity markets (Gaudry, 2010). Figure 2. Signs of k coefficients and TBC values in univariate and multivariate regression UNIVARIATE correlation, or slope ryx between transformed y and X1 or transformed y and X2
(2 y 1.65), (2 X1 2)
1.
(2 y 1.65), (2 X 2)
2.
2
'beta2.txt'
'beta3.txt' Beta 3
Beta 2
8e-009 7e-009 6e-009 5e-009 4e-009 3e-009 2e-009 1e-009 0 -1e-009
2.5e-009 2e-009 1.5e-009 1e-009 5e-010 0 2 1.5
-2
1
-1.95
0.5 -2
-1.95
-1.9
0 -1.9
-1.85
-0.5 -1.85
Lambda(XCSCNFA)
-1.8
Lambda(XCSCNFA)
-1 -1.75
-1.5 -1.7
-1.8 -1.75 -1.7
Lambda(TT)
-2 -1.65
-1.65 -2
-1.5
-0.5
-1
1
0.5
0
1.5
2
Lambda(RAWEM)
MULTIVARIATE correlation, or slope ryx between transformed y and one of the two transformed X1 and X2
3.
(6.5 y 1), (2 X1 1.5) X 0.5 2
4.
(6.5 y 1), (1.5 X 1.5) X1 0.5 2
'beta2.txt'
'beta1.txt' Beta 1
Beta 2
6.0e-008 5.0e-008 4.0e-008 3.0e-008 2.0e-008 1.0e-008 0.0e+000 -1.0e-008 -2.0e-008
5.0e-008 4.0e-008 3.0e-008 2.0e-008 1.0e-008 0.0e+000 -1.0e-008 -2.0e-008
-6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -0.5 -2.5 -1.0 -2.0 -1.5 -1.5 -1.0 -2.0 Lambda(XCSCNFA)
1.0
0.5
0.0
Lambda(TT)
(1 y 0.4), (5 X1 6) X 5.6
5.
-6.5 -6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -1.5 Lambda(XCSCNFA)
1.5
2
6.
1.5 1.0 0.5 0.0 -0.5 -1.0 Lambda(RAWEM)
(1 y 0.4), (6 X 3.5) X1 5.6 2
'beta2.txt' Beta 2
'beta3.txt' Beta 3
200 0 -200 -400 -600 -800 -1000 -1200 -1400
2e-011 1.5e-011 1e-011 5e-012 0 -5e-012 -3.5
6 -4
5.8 -1
5.6 -0.9
-0.8
-1
5.4 -0.7
-0.6
-0.5
5.2 -0.45
-4.5 -0.9
-0.8
-5 -0.7
Lambda(TT)
Lambda(XCSCNFA)
-0.6
-0.5
-5.5 -0.4-6
Lambda(RAWEM)
Lambda(XCSCNFA)
2.3. Regression explanations of asymmetric outcomes Thirdly, as some outcomes, like trip durations, are always and everywhere asymmetrically distributed, should one be expected to properly explain them with symmetric outcome models like Ordinary Least 8
On this point, see also Mandel et al. (2017) in the context of a Europe-wide model with 3 trip purposes.
13
Squares (OLS) for classical regression applications and linear Logit for logistic regression ones? The two SR91 papers referred to above make non linear adjustments by multiplying time by distance and squared distance. This is not surprising in view of the strong and fluctuating asymmetry of P80-P50 travel time differences between the ordinary and the tolled lanes during the day (Small et al., op. cit., Figure 1). Much of the art of regression involves the creation of asymmetry to match that of the explained outcome, be it a Logit road lane, path or mode choice probability, or a trip demand level. Box-Cox (BC) Logit models, for instance, do comply with Allais’ first idea, expressed above, of the relevance of all moments of a distribution by indeed modifying all moments of the logistic density function, and notably its asymmetry, as compared with its linear starting point, as obvious in Figure 3. Figure 3. Standard and Box-Cox cumulative logistic distributions Cumulative logistic distributions of p1 with linear predictor ( 1 ), predictor transformed by 1 and by 1 ( i , TIME )
If X1 = speed S and predictor gi i ,TIME Si
, two
[sequences of signs] of ( pX1 pX1, nonlinear pX1, linear ) are possible, depending on the optimal
i BCT value:
pn 0 for low values of X1; under-prediction pn 0 in the middle range; over-prediction pn 0 for high values of X1]; -if X 1 : [under-prediction pn 0 for low values of X1; over-prediction pn 0 in the middle range; under-prediction pn 0 for high values of X1 ]. -if X 1 : [over-prediction
(7)
But compliance with a second idea (Allais, 1953), of responses everywhere higher at low probabilities and lower at high probabilities, rather requires Cumulative Prospect Theory (CPT) Logit models with Vi representative utilities based on monotonic probability weighting wi and value v( xi ) functions. The former embody the attitude towards risk and the latter the attitude towards outcomes, their co-monotonic combination defining a matrix of relevant h(Ti ) i ,TIME wi v(Si ) predictor possibilities, as in Stott (2006), all fulfilling that second idea. For instance, Figure 4 presents, for a given v( xi ) , Prelec’s (1998) graph of cumulative logistic distributions associated to estimated specifications of wi : all have an inflexion point at p1=0,37 but estimation generates an asymmetry that is specific to each. Figure 4. Asymmetric behavior of some probability weighting functions used in CPT Simulated empirical functions Estimated empirical functions Tversky & Kahneman (1992), data on gains
Tversky & Kahneman (1992), data on losses
Tversky & Fox (1994)
Wu & Gonzalez (1996)
Prelec (1998)
14
These BC-Logit and CPT-Logit model classes are both behaviorally reinterpretable as MDU models because agents’ decisions dictate moments of endogenous chosen outcomes in ways that resemble endogenous trip duration moment construction above. Best fits arise from the very asymmetry of fitted outcome choice probabilities as if this asymmetry generated utility directly, and not through assumed indirect Vi utility function indices presumed to do so for each outcome (cf. Gaudry, 2016c). A Rosett-Nelson (1975) model with Gauss-distributed9 N ( w2 , I ) errors wt , collapsing to the BoxCox (1964) in the absence of limit observations on y, can yield asymmetric fitted durations E ( yt ) : ( y )
yt
(6-A) with
0 k 1 k X kt( k ) wt , k K
yt 0, X kt 0 , t=1,…,T,
10
X vt
(6-B) and
E ( yt )
(6-C)
wt ( )
( v )
( X vt ) v 1 v ln ( X vt )
(w)dw
wt ( )
wt ( )
yt ( w)dw
,
0,
,
0,
wt ( )
( w)dw, ( and 0) ,
( yt ) Var ( yt ) E yt E ( yt ) E ( yt2 ) E ( yt ) , 2
(6-D)
( yt )
(6-E)
e3
3
2
E ( yt3 ) E ( yt ) 3E ( yt2 ) 2 E ( yt )
2
3
, with e E( y ) ; t
where, for powers r = 1,2,3, one has: (6-F)
E ( ytr ) if
0 and v
0
when
=
wt ( )
ytr ( w)dw r
wt ( )
(w)dw
does not exist
= ytr ( w)dw
= ytr ( w)dw
y 0
y 0
y 0
wt
because, for (6-C), it has been assumed that observed yt is, as in a two-limit Tobit model, censured both upwards and downwards: ≤ yt ≤ where and denote the strictly positive limits assumed identical for all observations11. Note that the measure of 1st moment fit calculated for (6-A) with a normally distributed error is then indeed (6-C) and not the following unrolled version of (6-A): (6-G)
( ˆ ) yˆt 1 ˆy ˆ0 k ˆk X kt k
1
ˆy
.
Measure (6-C) is best, even in the Case 4 logarithmic case ( y 0 ) of Table 4, when it collapses to: (6-H)
E ( yt ) kˆ exp
ˆk X kt , ( h )
k
the particulars of which require an unbiased estimate of k, the sample mean of the log-normal random variable exp(wt) for the sample in question. But estimates of k drawn from the Maximum Likelihood adjustments raise further issues that make calculating the full previous expression (6-C) preferable. For linear and non linear cases to be reasonably nested in (6-A), it is assumed that E ( yt ) in (6-C) is large enough relative to w (Davidson & MacKinnon, 1985, p. 501), a fair hypothesis with our data. 9
In Box-Cox regression, the residual cannot be strictly normally distributed because y must be strictly positive. Should one then specify a weighted Likelihood function to take into account the probability to be below the lower limit (and symmetrically for the upper limit)? No: macroeconomic models explaining national GNP components C, I or G should not be rejected on the grounds that their errors cannot be strictly normally distributed. Consequently, our algorithm is constructed in the following way: (i) one first assumes normality and the a priori absence of limit observations, in which case the Likelihood function of the two-limit Tobit specified by Rosett & Nelson (1975) reduces to that of Box & Cox (1964) themselves; (ii) one then verifies ex post the reasonableness of the latter assumption by calculating an index of the probability (see Appendix 1, Part III, line 3 of Gaudry, 2016a) of each fitted value to be at the limits and (defined by the user) as recommended by Olsen (1978) on the lines of remarks by Draper & Cox (1969) on this point. 10 It is possible, in certain circumstances, to apply a BCT to an Xk containing zeroes, as we do in the source papers. 11 Tobin’s original formulation (Tobin, 1958) had different limits across observations. It does not require that the sample actually contain limit observations: our Likelihood function formulation assumes none are present.
15
This model makes it possible to estimate the derivatives of the first three moments of yt with respect to explanatory variable X kt , as outlined in summary manner in Table 4, and its impact on MRS among moments and their elasticities. We are not here talking about casual (“sample”) measures of the impact of variables, such as yt X kt k X ktk 1 , but of strict impact measures such as (3-C) and (5) that take into account the fact that the dependent variable is random because error term wt exists: proper analytical measures, pertaining to moments of y, not to sampled y, are found in source papers. Case 1 is known: we have [ yt ] with Gaussian errors where y is explained linearly [implicitly with ( y 1 )] by OLS and changes of Xk has no effect on ( yt ) and ( yt ) , as indicated by the null values. It may be forgotten that, in Case 2 with ( yt 0 ) as required by the BCT on y, ( yt ) is sensitive to changes in Xk (symmetry of Gaussian errors remaining) despite the linearity of y. Table 5. Derivatives of moments of y explained by regression with Gauss-Laplace errors Case
Domain of y
Box-Cox power of y
1
yt
(OLS)12
2 3 4 5
yt 0
DXe kt
E ( yt ) X kt
DX kt
( yt ) X kt
DX kt
( yt ) X kt
0
0
0
y 0 y 0
y 0
A
B
C
y 1
Partial derivative of a moment of y
= the partial derivative in question with respect to Xkt in columns A, B or C is not zero. More generally, when, as in Cases 3, 4, and 5, the non linearity of y produces a variable with an asymmetric distribution, all moments of y react to changes in Xk , a known property of models that are «non linear in y» (there are of course many others than the Box-Cox model), which we use to extract the freely determined MRS among its moments: here ( E ( y)) ( 2 ( y)) 1 , but the Poisson equality restriction13 is here testable, as demonstrated with road accident count data by Fridstrøm (1999, 2000). Analytical derivatives of the 3 moments of yt with respect to Xkt are written out in the source papers. The idea that preferences for distributions characterized solely by their moments provide an alternative to the mean-variance approach, and the need to extend a Box-Cox algorithm then limited to calculations of the 1st moment to the extraction of MRS among 3 moments of y, were prompted by the serendipitous reading of Allais (1987) in 1988. It led to a congruent research proposal in 1989 and to a report of results to the funding agency (FCAR, 1991) on a 2-moment application to road accident frequencies by severity category in Quebec, with results clearly inconsistent with Poisson restrictions (Gaudry, 1997, 2000). Similar 3-moment applications to Quebec and West Germany yielded tables of MRS of the same type as in Table 9 below (Blum & Gaudry, 2000) showing almost identical MRS among moments in the two regions and leading to implied RP values of a statistical life derived from such trade-offs among accidents frequencies of different severity and cost categories (Gaudry, 2006). Moments of road accident frequencies by severity category can be said to be endogenously chosen and concatenated, just like those of trip durations. There is indeed no need to limit oneself to the first moment of road speed, as Ashenfelter & Greenstone (2004) do: yet they simultaneously recognize that speed variance also matters, as Lave (1985) had noted, but do not dare consider it as also properly endogenous, as if drivers had blinkers with MDU preferences characterized only by picking the mean. 12
The application of OLS in Case 1 involves no transformation of y. If it is transformed, only the intercept is rescaled. A Normal law has 2 parameters (mean and variance) but the Poisson law has only one due to the equality constraint (mean of y) = (variance of y). The general heteroskedasticity formulation that allows Fridstrøm to impose this restriction on the former model is that of Gaudry & Dagenais (1979). 13
16
3. Paris region work trips by transit and car: data, model, and results 3.1. Data: the asymmetry of observed Greater Paris region trip durations The 3836 Paris region trip durations from the EGT 2010 Île-de-France survey, shown in Figure 5, have distributions14 more skewed for car (PC) and women that for transit (PT) and men. For observed Tnc in (4-B), consider that arrival time at work (arr) approximates unknown work starting time (rdv):
eˆ(Tnc ); ˆ (Tnc ); ˆ(Tnc ) . eˆ(Tnc ); ˆ (Tnc ); ˆ(Tnc ) Hrdv Hdep H arr Hdep
(7-A)
Figure 5. Distribution of Paris area morning peak home-based work trip durations (2010) Base 2
A. Men (979 obs.), Mass Transit (PT)
Base 2
350
350
300
300
250
250
200
200
150
150
100
100
50
50
B. Men (851 obs.), Private car (PC) alone
0
0 15
e 61 minute Base 3 s 350
30
45
60
75
90
105
120
135
150
165
15
180
24 minutes
1.00 A. Women (1180 obs.), Mass Transit (PT)
30
e 42 minutes Base 3
45
60
75
90
105
120
135
150
165
180
24 minutes
1.50 B. Women (826 obs.), Private car (PC) alone
350
300
300
250
250
200
200
150
150
100
100
50
50 0
0 15
30
45
60
75
90
105
120
135
150
165
180
15
30
45
60
75
90
105
120
135
150
0.75 e 35 e 60 24 minutes 19 minutes minute minutes s3.2. Model: summary of economic and econometric specifications
165
180
1.40
To model durations of home-based work trips starting or ending between 7h30 and 9h30, and longer 15 than 3km as defined by Great Circle Distance in the RATP network of 2294 zones, we write: (7-B)
( S ) (Tnc )( T ) =f [(Day of week, Direction, Parking type); S PT or PC ; ( Age
( Age )
, Income
( Income )
, etc.)],
where 4 continuous variables are applied BCT and all other variables (18 for PT and 26 for PC) but one are dummies. Service variable SPT is the antilog of the logsum of the RATP Logit itinerary path choice model, which does not yet include a transit path price but uses Box-Cox transformed in-vehicle time (Leblond & Langlois, 2013) weighted by tested comfort functions based on passenger densities, and rail transit line type dummies (Prat & Leblond, 2014). SPC comes from a Wardrop (1952) user equilibrium yielding a unique time value by OD pair ‒ but not the paths used. It does not yet come from an enriched procedure allowing for path identification (Bar-Gera, 1999, 2006; Bar-Gera & Boyce, 1999) and consequently making the derived construction of road itinerary logsums possible. For each of these four heterogeneous markets (Men-PT, Men-PC, Women-PT, Women-PC), four distinct BCT specification sets were tested: the linear, corresponding to Case 2 in Table 4; the logarithmic, corresponding to Case 4; free BCT on Duration Tnc and Service S np , with further relaxation of linear constraints on Age and household Income ‒ which added nothing, making Case 3 the optimal specification with 2 BCT. Hence, Table 7 only presents 3 sets of estimates per market. 14
By definition, individual PC car trips are strictly drive-alone. Note that asymmetry of total (for all purposes) trip durations is found for all modes in all EGT surveys: PT, PC, Two-wheel, Other motorized, On foot (DREIF, 2004, p. 31). 15 Censuring short trips, to limit the effect of rounded stated departure and arrival times, makes women’s PT trip durations close to men’s (Fig. 5), whereas their uncensored values (for both modes, cf. Picard et al., 2013, Table 1) are much shorter.
17
3.3. Existence and structure of the safety margin We are first interested in the elasticity of duration (cf. Table 4) with respect to endured network service because this elasticity is the foundation of the model, an argument summarized in Table 6. Table 6. Safety margin existence test based on a unitary elasticity of duration w.r.t. endured service Given (Tnc , Snp ) ˆ , an estimate of the elasticity of duration w.r.t. endured network service derived from a ( )
Box-Cox model with regressand (Tnc ) y and regressors ( Snp )( x ) and other variables X, (n = PC or PT); if ˆ 1 , duration and service are identical ( Tnc Snp ) : no safety margin exists. The model is not legitimate; if ˆ 1, the safety margin legitimately exists ( Tnc f ( I nc , Snp ) , but its (internal) structure is unknown; and ˆy 0
the additional assumption of a multiplicative margin I n* Snp allows for the identification of its elasticity w.r.t. service, ( I n* , Snp ) ˆ 1 , but no known author has ever considered anything but an additive safety margin;
and ˆ 1
one understands that I n* compensates for the uncertainty of S np ;
one may calculate the MRS among moments of (external) duration for any (ˆ , ˆy , ˆx ) set.
to which one might add the following comments: i) Non unitary elasticity of duration w.r.t. endured service as a test of the existence of the safety margin. The econometric model is legitimate if (Tnc , Snp ) 1 : otherwise, it amounts to a regression of service on itself (and other variables X), which is of little interest beyond that very point. Were the units of Tnc and S np identical, one could imagine a test of the value of the regression coefficient analogous to the unit root tests of time series. But as soon as units differ and both variables are subjected to BCT, a test of the unitary value of the elasticity is much preferable, even if it remains intuitive. ii) The internal structure of compunction time Tnc conditional to a non-unitary elasticity. Once it is established that (Tnc , Snp ) 1 , can anything be said about the internal structure of Tnc ? If ˆy 0
and one further assumes that I n* Snp , it is possible to calculate ( I n* , Snp ) from (Tnc , Snp ) 1 ˆ because the elasticity of a product is the sum of the elasticities of its components. More generally, ˆy 0 and one can say nothing about the internal structure of Tnc . But the proof of existence with
ˆ 1 does not require to take a position on whether the safety margin is multiplicative, additive (as all think and build into models) or something else such as (Tnc )
( y )
1 ( I n* )( I ) 2 ( Snc )( S )
( y )
.
iii) The palliative interpretation of I n* when ˆ 1 . One expects ˆ 1 because I n* is understood to compensate for the uncertainty of S np : improvement of service reduces the safety margin, and the reverse occurs when service worsens. In the particular case of ˆ 0 , the offsetting (negative) role y
of I directly follows from ( I , S ) = ˆ 1 when ˆ 1 . * n
* n
p n
iv) Robustness of derived MRS. Explicitation of the moment structure of Tnc is always feasible, no matter the exact values of the set of estimated parameters ( ˆ , ˆ , ˆ ). On this point, Equation y
x
(4-E) supports the belief, extensively tested in a source paper (Gaudry, 2015), that MRS estimates are for all practical purposes insensitive to the level of potential multi-collinearity present.
3.4. Results: unit elasticity tests at optimal form, moment demand elasticities and MRS A. Neither linear, nor logarithmic but Box-Cox optimal forms with 0,11 y 0, 42 . Maximum likelihood estimates reported in Table 7 were verified to ensure that they corresponded to global maxima, a numerical necessity because it has long been known in practice, and more recently explored formally (Kouider & Chen, 1995), that the Box-Cox Log-Likelihood function need not be 18
unimodal. Here, logarithmic Case 4 dominates linear Case 2 almost infinitely but itself performs much less well than Case 3 in three of the markets and somewhat less well in the last (Men-PC). All BCT sets of estimates significantly differ from one another, as documented in detail in the source papers. Table 7. Work trip durability: functional forms and the four Case 3 optima, each with two BCT Public Transit (PT)
Table 5 reference: Base 2 (Men) Variant run number Number of observations Number of k coefficients on Duration (y) on PT service on PC service
BCT
Log-Likelihood Base 3 (Women) Variant run number Number of observations Number of k coefficients on Duration (y) on PT service on PC service
BCT
Log-Likelihood
Private Car drive alone (PC)
Case 2
Case 4
Case 3
Case 2
Case 4
Case 3
15 979 22
16 979 22
17 979 22
20 851 30
21 851 30
22 851 30
1.00 1.00 1.00 -4412
0.00 0.00 0.00 -4039
0.19 0.05 -----4014
1.00 1.00 1.00 -3495
0.00 0.00 0.00 -3305
0.11 ----0.21 -3300
29 1180 22
30 1180 22
31 1180 22
33 826 30
34 826 30
35 826 30
1.00 1.00 1.00 -5326
0.00 0.00 0.00 -5027
0.42 0.10 -----4909
1.00 1.00 1.00 -3142
0.00 0.00 0.00 -3023
0.24 ----0.29 -3008
In particular: (i) women’s BCT values are roughly twice the size of men’s and PT values twice PC values; (ii) the optimal form of the denominator of the public transit path choice model used as measure of public transit service (the anti-logsum) is not logarithmic despite estimated values of 0,05 and 0,10 that are numerically close to 0: the logsum is here rejected in favor of a non-zero Box-Cox transformed simple sum as the adequate perceived summary measure of available transit path choices. B. Moment demand elasticities w.r.t. regressors, in particular endured transport service. Source papers report on coefficient t-statistics and on elasticities of moment demands w.r.t. all regressors, mostly of limited interest here (Rounding of declared trip duration time, Day of week, Direction towards or away from Paris intra muros, PT ticket rights, Parking type, Age, Income, etc.). Table 8 reports specifically on elasticities w.r.t. endured transport Service, decisive for traffic model closure. Table 8. Three own moment elasticities of car and transit durations w.r.t. to service modifications A. Public Transit (PT) B. Private Car drive alone (PC) Xmk mk mkX X
Xp Xmk mk Xp X Xp1 k mk
Xp Xp1
Xmk k
0,016
f
)
f
n
r
f
)
)
n
r
n
f
f
u
r
n
r
n
u
r
D
u
u
D
D
(
(
D
(
E
u
1
2
)
k
X
,
f
n
r
u
(
Variant run 31
(e( Durnf ), X k )
=
0,740
19,00
( ( Dur ), X k )
=
0,560
1,40
( ( Dur ), X k )
=
-0,180
f n
COLUMN
1
f
f
n
)
)
)
n
r
f
f
r
u
n
r
n
n
f
u
D
(
D
f n
D
-10,85 -2,00
D
Women
x
y
)
k
X
,
f
-0,091
(
0,048
n
=
35,00
-0,110 -0,067
r
( ( Dur ), X k )
(
0,727 = 0,134 -0,521 = -0,096
u
1,50
E
=
0,220
=
D
(
0,660
COLUMN
D
( ( Durnf ), X k )
0,75
=
1,193
0,750
=
f n
)
k
( ( Durnf ), X k )
24,00
=
=
( ( Dur ), X k )
(e( Dur ), X k ) f n
60,00
k
k
24,00
f n
Variant run 35 X
,
n
f
Women
x
(e( Dur ), X k )
42,00
r
=
-0,065
-10,85 -2,00
X X k y Variant run 17
f n
r
( ( Dur ), X k )
=
-0,081
Men
y OBSERVED
D
1,00
f n
r
-10,85=-0,22* 36,6* 0,014 -2,00=-0,22* 12,2* 0,75
u
( ( Dur ), X k )
0,879 = 0,162 0,705 = 0,130 -0,174 = -0,032
SAMPLE ELASTICITY of (y, Xk) =
k
u
24,00
f n
=
X mk
X X mk pm
u
(e( Dur ), X k )
u
x
)
k
X
,
n
f
r
u
D
(
61,00
D
y
Variant run 22
f n
(
U
Men
U U y
ELASTICITY w.r.t. IN-VEHICLE TIME BY CAR (MEAN OF 32,5 MIN)
D
y OBSERVED
=
k Xmk km
SAMPLE ELASTICITY of (y, Xk) =
Xmk mk mkX X
ELASTICITY w.r.t. IN-VEHICLE TIME OF A PT LINE AT 36,6 OR 12,2 MIN (MEAN: 24,4 MIN)
2
How then do commuters modify their chosen precautionary time in reaction to improved transit and road services? Rather, are their multidimensional [ dˆe ( n ) ; dˆ ( n ) ; dˆ ( n ) ] trip profile durations adjusted by responses equal and of opposite signs to that of better services? Do commuters just sleep longer?
19
Casual elasticities of duration with respect to service. The top part of Table 8 contains sample elasticities and the bottom part strict elasticities, all formally defined in Table 4. For the private car where service is simply Average in-vehicle time, the sample elasticity formula is straightforwardly k X kk y y , but for public transit service it is a product of k U U y y , the elasticity with respect to the anti-Logsum variable U (the sum of the terms found in the denominator of the Logit model mk determining the path shares pm), by k X mk pm , the elasticity of U with respect to increased In-vehicle time on path m. In part A of Table 8, two reference path times are supplied (36,6 and 12,2 minutes) and, to compute U, we used the fact that the coefficient of in-vehicle time in the RATP 6-path model is (-0,22) and we took in-vehicle time at 24,4 minutes16 and out-of-vehicle time at 27,5 minutes. To every network service action its multi-moment trip duration profile reaction. The bottom part of Table 8 lists, for the optimal Case 3 variants of Table 7, strict elasticity estimates of the three moments. And we note immediately that the elasticity of skewness is always of a sign opposite to that of the mean and of the standard error. So, clearly, commuters do not just sleep longer when service improves. But more can be said on Table 8 results: i) signs of moment duration elasticities with respect to Service are in fact the same. Improved X 0; dˆ X 0; dˆ X 0 ] of the same sign on both Service X has effects [ dˆ k
e( n)
(n)
k
k
(n)
k
k
modes: e.g. lower in-vehicle time X lowers mean trip duration in Columns A.2 and B.2; ii) sizes of moment duration elasticities are of the same order of magnitude and all ˆ 1 . All elasticities, precisely estimated because the t-statistics of underlying service variable regression coefficients (not shown in Table 8) are equal to 30, are smaller than 1: it is obvious in part B. In mk part A, we used two reference values to compute k X mk pm , which generates two values in Column A.2. In the upper ranges of values, elasticities of the first two moments are of the same order of magnitude in the two modes but skewness is more sensitive to improved (or to worsened) service for transit trip durations than for car trip durations, especially for women. All elasticities are smaller than unity, as required by Table 6 tests; iii) superpeak or fast fill-up quandaries and infrastructure capacity. The weakness of the elasticities of skewness relative to those of the first two moments implies that, for both modes, improved Service X k (requiring the minus sign of X k ) concentrates the time profiles of duration demand: (7-C)
(e( Durn ), X k ) ( ( Durn ), X k ) ( ( Durn ), X k ) SUPERPEAK of kurtosis ,
but this relative super-peak effect on kurtosis, derived from Table 8, is larger for car services than for transit services (even if women exert some particular moderating influence through skewness). No wonder new urban highways are said to fill up fast; and improved transit services are also said to fill up rapidly. This increased 4th moment of the time profile of demand, or peakedness effect, occurring for a given number of trips, might pose capacity problems, notably for transit planning; iv) closure of the transport demand model. Of course, the demand model should be closed in the sense that the time profiles of duration demand assumed to kick-off the demand process should be modified and lead to new modal demands and new equilibria, but that is another matter. C. The MRS m ,ms among moments of duration defined in Equation (5). Two matters of robustness of Table 9 contents require comment. First, note that, because the sets of BCT values by sex and mode framed in two columns of Table 7 differ significantly among themselves, the derived rates and elasticites of substitution listed in Table 9, along with other statistics about the four Case 3 models, must also differ significantly among themselves. Second, the source papers report Belsley-Kuh-Welsh (Belsley et al., 1980) condition indices17 for these four cases and point to the existence of very small effects of multicollinearity on estimated regression coefficients, if any, but find no effect whatsoever on the MRS m ,ms . These key remarks made, consider the following comments on the results listed in Table 9: (i) the higher the moment, the less well it is fitted, if the fit is measured by the simplistic Fitted value evaluated at the means of all variables. Clearly, reasonable indices of fit have yet to be developed for 16
In-vehicle time is not provided in EGT 2010 and must be obtained from transit assignments. Interpreted on the lines of Erkel-Rousse (1995) who showed that Belsley-Kuh-Welsh condition indices had arbitrary levels that depended on units of measurement of the variables. There yet exists no procedure to adapt such units in order to minimize the value of indices and no demonstration that this would be helpful to the treatment of multicollinearity. 17
20
all moments in order to extend to higher moments than the first appropriate measures for non linear models in the spirit of the long-used Pseudo-(L)-R2 (Aigner, 1971, p. 85-90); (ii) in all four models, behavior is riscophobic, in the sense of matrix [A] of Table 3, and elasticities of substitution closely follow the marginal rates of substitution; (iii) reliability is not valued in the same way across the modes: independently from their sex, car commuters value standard error less, and asymmetry more, than transit commuters; (iv) reliability is not valued in the same way across the sexes: independently from the mode used, women value standard error more, and asymmetry less, than men. The existence of strongly different MRS m ,ms between men and women shows that these reflect constructed durations rather than the service structure of transit services common to both sexes. In their paper on lane choice mentioned above, Palma and Picard (2005) find fluctuating sign differences between modes and between men and women, depending on specification. And, on this last point, Lam & Small (2001) find that women are more averse to risk than men. It appears that explaining choices by some distance between mean or median time and a high percentile measure of the travel time distribution (Pxy), or some risky alternative (lottery) involving time reliability, rather captures 1st and 3rd moment effects and in fact says little about the 2nd moment for which the trade-offs with the 1st are opposite in sign and relative size (by sex and mode) in Table 9. Non Allaisian literature fails to distinguish analytically between 2 nd and 3rd moments that each specifically matter. Table 9. Marginal rates and elasticities of substitution among the first three moments of duration Model Public Transit (PT) Private Car alone (PC) Base 2. Men
Variant run 17
Variant run 22
Marginal rates of substitution [Eq. 5.1; 5.2; 5.3)]
e e +1
+ 4.5 +1
e 330 + 1 73 +1
+ 3.5 + 1.0
400 110 +1
Elasticities of substitution [Eq. 5.1*; 5.2*; 5.3*)]
e +1
+ 1.2 +1
5.2 4.2 +1
+ 1.1 + 1.0
8.2 7.2 +1
Moments
i\j
Sample moment value Fitted value at the means Mean of fitted values Base 3. Women Moments i\j
Marginal rates of substitution [Eq. 5.1; 5.2; 5.3)] Elasticities of substitution [Eq. 5.1*; 5.2*; 5.3*)]
61 42 61
e e +1 e +1
Sample moment value Fitted value at the means Mean of fitted values
60 42 60
24 1.00 13 0.68 18 0.71 Variant run 31
+1
42 42 42
24 1.50 14 0.88 19 0.98 Variant run 35
+ 5.4 +1
e 180 + 1 33 +1
+ 4.5 + 1.0
220 49 +1
+ 1.7 +1
2.4 1.4 +1
+1
+ 1.3 +1
4.1 3.1 +1
24 13 17
0.75 0.56 0.08
35 36 35
19 10 16
1.40 0.67 1.30
4. Potential application to line automation: a metro as metronome The approach developed here could be useful to the evaluation of the automation of urban rail lines the effects of which on the first four moments of in-vehicle time are obvious in Figure 1, the metro being transformed into a metronome. Little is published about the impact of metro line automation proper, or about its valuation. The first automated line, the Victoria Line in London, was automated twice: at birth in 1968 and in 2013. Much is known about the impact of its opening on (average) time savings (Foster & Beesley, 1963) and on their valuation in terms of the hourly wage rate (Beesley, 1965 or 1970) but we could find no discussion of the impact of its automation as such, even when the recent second modernization increased train frequency per hour from 27 to 33 trains. 21
One way to study this question is to distinguish between service before (or in the absence of) automation and service after automation or general modernization (ceteris paribus, with the same supply of train-hours). Given that service does not exist as such over, above and independently from its moments, what then happens to the moments of in-vehicle time and wait time with automation? An important question is that of the impact of the dramatic increases in the kurtosis of in-vehicle time brought about by automation: with a positive effect on utility (Table 2 signs constituting an alternating series) which could more than outweigh the negative effect of increased skewness, as it apparently did on automated metro line M14 in Paris and could do with the Grand Paris Express extensions of 200 km of new automated lines (and 68 stations) with a commercial speed of 65 km/h. But one should conceptually distinguish the effects of in-vehicle from those of wait-time based on headways. In a simplified system of uniform commercial speed (which includes the stopping times) among all stations of a line or closed circuit, wait-time is inversely proportional to train-hours of service VH and to commercial speed. With higher commercial speeds made possible by automation, and if supplied train-hours VH do not change, the first two moments and of in-vehicle time TT and of wait-time TA will vary proportionately and in the same direction ( TA kVH TT and TA kVH TT ), but Fisher’s dimensionless moments of skewness and kurtosis ( and ) of wait-time will vary identically with the moments of in-vehicle time18 affected by the line automation.
5. Conclusion and explicit policy implications for project evaluation We seek first to establish the existence of a safety margin built into any transport journey of duration T. This duration, assumed to combine an endured network service time component S and a precautionary margin I constructed by the traveler, is explainable by that same service S and other factors X. Should the adjunction hypothesis T f(I,S) be false, duration T is equivalent to the service provided by the network S and the elasticity of duration with respect to this endured service equals one, the econometric explanation being found wanting and of little interest beyond this result. Should it be true, and T include more than S due to incorporation of an endogenous safety margin I, the same elasticity differs from one and the legitimized econometric explanation becomes more interesting. In that case, an elasticity smaller than one denotes a palliative role for the safety margin. If and when the adjunction hypothesis holds, it is interesting and useful to extract from explained duration the marginal rates of substitution among its moments of various orders (here no greater than the fourth), understood without formal proof to match Allais’ famous rates of substitution among all moments of a random prospect with respect to the first, all assumed relevant to its utility. The proposed unit elasticity tests, aimed both at establishing the existence of the precautionary time margin and at extracting relevant reliability valuations from built-up durations that incorporate it, are performed on home-based work trips by private car driven alone (PC) and public mass transit (PT) longer that 3 km of Great Circle Distance and shorter that 180 minutes, starting or ending on a work day between 7h30 and 9h30 in Île-de-France, as reported in a survey of 18 000 households in 2010. We strongly reject, by commuter mode and sex, the unitary value of the elasticity of duration T to endured network service S and show that empirical valuations of the reliability of trips differ significantly across the commuter groups analyzed (men and women; car drive alone and transit use). For all of these 4 groups, the asymmetry of trip durations is much better modeled by Box-Cox transformations than by other forms (linear, log-log) of the regression variables. And, for all such best fit models, elasticities of duration moments with respect to endured network service, estimated with very great precision, are smaller than one because margins and service are substitutes. Exogenous service level changes induce offsetting changes in the time profiles of trip demand assumed at the transport planning outset, thereby closing the model. Results imply that putting new roads into service or even automating existing metro lines will contract peak time demand profiles and produce demand super-peaks of strongly increased multiple-moment (reliability) utility in spite of misleading first moment deteriorations in road speed or transit congestion comfort measured at the peaks. 18
To see this, recall that kVH is equivalent to a change in the scale or units of measurement s, for which we have E(sX)=sE(X) and Var(sX)=s2 Var(X), but for which the other two moments are invariant: (sX)= (X) and (sX)=(X).
22
As a consequence, the utility of new road investments or transit line modernizations cannot be calculated, at given user prices, merely with flow and mean peak speed variations because all four first moments matter to utility, some in fact more affected (negatively or positively and proportionately speaking) by service improvements than the first. Only the fourth moment implicitly appears in simplistic analyses claiming for instance (e.g. Dutzig et al., 2017) that, if congestion conditions that road projects were originally designed to improve rapidly come back, even visibly higher flows are somehow pointless in view of the return of speed levels at best identical to those prevailing ex ante the newly deplorable superpeaks. Our approach is based on the assumption that preferences for distributions are moment dependent, i.e. characterized solely by their empirical moments, and that all moments of the travel time lottery matter to utility, including the fourth. Transport project evaluation cannot then be concerned merely with Dupuit-like utility of new trips or tonne-km, but also with to Allais-inspired reliability matters, including fast fill-ups of new highways and super-peaks on new or newly automated transit lines. This revealed preference (RP) approach to trip time profile demand and model closure obviates the need to explain flexible departure time by unobserved variables, such as preferred arrival time implausibly considered as exogenous (Vickrey, 1969). Also, we need estimate neither an arbitrary number of quintile linear regression slices (Koenker & Hallock, 2001) nor would-be symmetric twomoment time distributions on ever asymmetric itinerary times with safety margins (Fosgereau, 2015) assumed proportional to the standard deviation of travel time [sic]. The demonstration that time durations are proper endogenous constructs explained by a new and additional duration demand equation has consequences for the pre-existing traditional equations of the system, those of modal demand for trips proper (or for the choice of mode Trip duration constructs containing precautionary time differ deeply from traditional simplistic time (and cost) explanatory variables specified for endured services without extant offsetting precautions against their uncertainty.
6. Acknowledgements This first version of this paper, originally entitled “The demand for journey duration and reliability: Paris Region work trips by mode and sex, 2010-2011” (Gaudry, 2016a), was presented in Paris by the author on March 11, 2016, at the Séminaire sur les méthodes de prise en compte de la fiabilité des temps de transport dans les évaluations of the Direction générale des infrastructures, des transports et de la mer (DGITM). It was also kindly presented by Alain Bonnafous on July 12, 2016, at the Special Dupuit Session of the 14 th World Conference on Transport Research (WCTR) in Shanghaï. It is based on research work carried out for, and financially supported by, the Société du Grand Paris (SGP) in charge of the planning and construction of the Grand Paris Express (GPE) automatic metro, a mass transit extension program consisting primarily in 200 km of new lines and 68 stations eventually doubling the length of the current metro line network. The background research (Gaudry, 2015) primarily summarized here was supported in Paris by the DGITM and could not have been carried out without strong data support from the Régie Autonome des Transports Parisiens (RATP) and without the close collaboration with Vincent Leblond on the data supplied, on their significance and modeling (data models), and on the relevance of the adopted multi-moment methodology (models of data). Computational assistance was also received from Cong-Liem Tran and Kim Tran (Oikometra, Montreal), Benjamin Cuillier (Stratec, Bruxelles), Ralf Klar (MKmetric, Karlsruhe), and Charles Varrod (RATP, Paris). Taken into account were significant comments or information supplied by Sophie Dantan, Lasse Fridstrøm, Nathalie Picard, Jean-Claude Prager, Alain Sauvant and, repeatedly, by Émile Quinet. The author, grateful for this embarras de richesse, is responsible for all remaining errors. Profoundly revised for Transport Policy, this final version summarizes original version and complementary document source contents.
7. References Abraham, C. (1961). La répartition du trafic entre itinéraires concurrents: réflexions sur le co mportement des usagers, application au calcul des péages. Revue générale des routes et aérodromes 357, pages 5760, 65-72, 75-76. Aigner, D.J. (1971). Basic Econometrics. Prentice-Hall, Englewood Cliffs, N. J.
23
Allais, M. (1953). Le Comportement de l’Homme Rationel Devant le Risque, Critique des Postulats et Axiomes de l’École Americaine. Econometrica 21, 503–546. Allais, M. (1987). Allais Paradox. In Atwell, J., Milgate, M., Newman, P. (eds), The New Palgrave Dictionary of Economics 1, 80-82, The Macmillan Press, London. Allen, R.G.D. (1968). Macro-Economic Theory: A Mathematical Treatment. Macmillan, 420 p. Ashenfelter, O., Greenstone, M. (2004). Using Mandated Speed Limits to Measure the Value of a Statistical Life, Journal of Political Economy 112, 226-267. Bar-Gera, H. (1999). Origin-based algorithms for transportation network modeling. University of Illinois at Chicago, 197 pages. Bar-Gera, H. (2006). Primal Method for Determining the Most Likely Route Flows in Large Road Networks. Transportation Science 40, 3, 269-286, August. Bar-Gera, H., Boyce, D. (1999). Route flow entropy maximization in origin-based traffic assignment. In Ceder, A. (ed.) Transportation and Traffic Theory. Proceedings of the 14th International Symposium Transportation Traffic Theory. Elsevier Science, Oxford, UK, 397–415. Batley, R., Dargay, J., Wardman, M. (2011). The impact of lateness and reliability on passenger rail demand. Transportation Research E 47, 61-72. Beesley, M.E. (1965). The Value of Time Spent in Travelling: Some New Evidence. Economica 32, 174185. Belsley, D., Kuh, E., Welsh, R.E. (1980). Regression Diagnostics. John Wiley, New York. Beesley, M.E. (1970). The Value of Time Spent in Travelling: Some New Evidence. Ch. 11 in Quandt, R.E. (ed). The Demand for Travel: Theory and Measurement, pages 221-234. Heath Lexington Books, Lexington Mass.. Blum, U., Gaudry, M. (2000). The SNUS-2.5 Model for Germany. Ch. 3 in Gaudry, M., Lassarre, S. (eds), Structural Road Accident Models: The International DRAG Family, Pergamon, Elsevier Science, Oxford, 67-96. Bonnafous, A., Baumstark, L. (2010). Calcul économique et politique de transport: de Jules Dupuit à Maurice Allais. Ch. 23 in Diemer, A., Lallement, J., Munier, B. (eds). Maurice Allais et la science économique. Paris, Clément Juglar. Box, G.E.P., Cox, D.R. (1964). An Analysis of Transformations. Journal of the Royal Statistical Society, Series B 26, 211-243. Brockett, P.L., Garven, J.R. (1998). A Reexamination of the Relationship between Preferences and Moment Orderings by Rational Risk Averse Investors, Geneva Papers on Risk and Insurance Theory 23, 2, 127-137. Clark, H.M. (2017). Who Rides Public Transportation. American Public Transportation Association (APTA), 86 p., January. Cournot, A.-A. (1838). Recherches sur les Principes Mathématiques de la Théorie des Richesses. CRA (1972). A disaggregated behavioural model of urban travel demand. Report No. CRA-156-2 prepared under contract No. FH-11-7566 “Development of a behavioural modal split model” for the Federal Highway Administration. Charles River Associates Incorporated, Cambridge, Massachusetts, March. Dagenais, M.G., Gaudry, M.J.I., Liem, T.C. (1987). Urban Travel Demand: the Impact of Box-Cox Transformations with Nonspherical Residual Errors. Transportation Research B 21, 6, 443-477. Davidson R., MacKinnon, J.G. (1985). Testing linear and loglinear regressions against Box-Cox alternatives. The Canadian Journal of Economics / Revue canadienne d'Economique 18, 3, 499-517. August. Demsetz, H. (1970). The private production of public goods. Journal of Law and Economics 13, 2, 293306. October. Domencich, T.A., McFadden. D. (1975). Urban Travel Demand: A Behavioral Analysis, North Holland, Amsterdam. Draper, N.R., Cox, D.R. (1969). On distributions and their transformations to normality. Journal of the Royal Statistical Society, Series B 31, 472-476. DREIF (2004). Les déplacements de Franciliens en 2001-2002. Enquête Globale Transport. La Documentation Française, 43 p. Dupuit, J. (1844). De la mesure de l'utilité des travaux publics. Annales des Ponts et Chaussées, 2° série, Mémoires et Documents, n°116, t. VIII, pp. 332-375.
24
Dupuit, J. (1849). De l'influence des péages sur l'utilité des voies de communication, Annales des Ponts et Chaussées, 2° série, p. 170-248. Duranton, G., Turner, M.A. (2009). The Fundamental Law of Road Congestion: Evidence from US Cities. Working paper 15376, National Bureau of Economic Research, 47 p., September. Duranton, G., Turner, M.A. (2011). The Fundamental Law of Road Congestion: Evidence from US Cities, American Economic Review 101, 2616-2652, October. Dutzig, T., Bradford, A., Weissman, G., Frontier Group, Aragon, L., U.S. PIRG Education Fund (2017). Highway Goondoggles 3. Big Projects, Bigger Price Tags, Limited Benefits. U.S. PIRG Education Fund and Frontier Group, 47 p., April. Ekelund, R.B., Hébert, R.F. (1999). Secret Origins of Modern Microeconomics: Dupuit and the Engineers. 468 p., The University of Chicago Press. Erkel-Rousse, H. (1995). Détection de la multicolinéarité dans un modèle linéaire ordinaire: Quelques éléments pour un usage averti des indicateurs de Belsley, Kuh et Welsch. Revue de Statistique Appliquée 43, 4, 19–42. FCAR (1991). Rapport d’activité sur les 10 mois d’opération du Programme multidimensionnel en sécurité routière, Projet 5.2. Estimations dans le cadre du modèle DRAG, Tâche 1. Action concertée sur la sécurité routière FCAR-67. Fonds pour la formation des chercheurs et l’aide à la recherche. Fosgereau, M. (2015). The valuation of travel time variability. Draft Discussion Paper 2015-xx, International Transport Forum, OECD, 22 p., 15th September. Foster, C.D., Beesley, M.E. (1963). Estimating the Social Benefit of Constructing an Underground Railway in London. Journal of the Royal Statistical Society, Series A, 126, 46-79. Freud, S. (1984). Freud présenté par lui-même [1925-1935], Gallimard, Paris, 1984. Fridstrøm, L. (1999). Econometric Models of Road Use, Accidents, and Road Investment Decisions. Report 457 292, Institute of Transport Economics, Norwegian Centre for Transport Research, TØI, Oslo. www.e-ajd.net. Fridstrøm, L. (2000). The IRPOSKML Procedure of Estimation. Ch. 13 in Gaudry, M., Lassarre, S. (eds). Structural Road Accident Models: The International DRAG Family, Pergamon, Elsevier Science, Oxford, 325-334. Gallant, A.R. (1981). On the Bias in Flexible Forms and an Essentially Unbiased Form: the Fourier Flexible Form. Journal of Econometrics 15, 211-245. Gaudry, M. (1997, 2000). Some Perspectives on the DRAG Approach and Family of National Road Safety Models. Ch. 12 in Holst, H.v., Nygren, Å., Andersson, Å.E. (eds). Transportation, Traffic Safety and Health ‒ Prevention and Health. Third International Conference, Washington, D.C., U.S.A, 1997. pp. 123-168, Springer Verlag, Heidelberg, 198 p., 2000. Gaudry, M. (2006). Life, Limb and Bumper Trade-Offs Calculable from Road Accident Models: a Note on a Multi-moment Portfolio Analysis and Life Asset Pricing Model (LAPM). Transport Reviews 26, 4, 501-520. Gaudry, M. (2010). Quebec-Windsor Corridor High Speed Rail Market Forecast Profiles in Context: Level-of-Service Response Curvature Sensitivity and Attitude to Risk or to Distance in Forty Logit Core Model Applications of the Law of Demand. 100 pages, Publication AJD-127, Agora Jules Dupuit, Université de Montréal (www.e-ajd.net) and Working Paper DEST No 12, Institut National de Recherche sur les Transports et leur Sécurité, Université Paris-Est. ResearchGate. Gaudry, M. (2015). La demande de durée et de fiabilité des trajets de transport: tests d’une méthode à trois moments sur les durées domicile-travail constatées en Île-de-France. Rapport d’expertise déposé en version finale par la Mission d’appui scientifique à la Société du Grand Paris pour l’utilisation des modèles de transport (MAPA 2013-32), Étape 2, Oikometra (Montréal) et Stratec (Bruxelles), 134 p., le 28 décembre 2015. Publication AJD-153, Agora Jules Dupuit, Université de Montréal. www.researchgate.net/profile/Marc_Gaudry. Gaudry, M. (2016a). The demand for journey duration and reliability: Paris Region work trips by mode and sex, 2010-2011. Publication AJD-154, Agora Jules Dupuit, Université de Montréal, Version 2 of 11th January, 35 p.. www.researchgate.net/profile/Marc_Gaudry. Gaudry, M. (2016b). Appendix 1 of: Gaudry, M. (2016). The demand for journey duration and reliability: Paris Region work trips by mode and sex, 2010-2011. Publication AJD 154, Agora Jules Dupuit, Université de Montréal. 19 p., January. www.researchgate.net/profile/Marc_Gaudry Gaudry, M. (2016c). Préférences MDU pour des distributions caractérisables par leurs seuls moments: applications aux transports et réinterprétation des fonctions de pondération des probabilités et de
25
valeur en théorie cumulative des perspectives. Publication AJD-156, Agora Jules Dupuit, Université de Montréal, 25 p., 8 mars. www.researchgate.net/profile/Marc_Gaudry. Gaudry, M. (2016d). A minor acknowledgement of a great man, Jules Dupuit. Publication AJD 162, Agora Jules Dupuit, Université de Montréal, 1 p.. www.researchgate.net/profile/Marc_Gaudry. Gaudry, M. (2016e). Méthodes Box-Cox, algorithmes de TRIO et demande de transport: trois consignes occamiennes pour faire rendre sens aux coefficients de régression de modèles simples et logistiques discrets ou agrégés. Publication AJD-150F, Agora Jules Dupuit, Université de Montréal, 93 p., mars. ResearchGate. Gaudry, M., Dagenais, M.G. (1979). Heteroscedasticity and the Use of Box-Cox Transformations. Economics Letters 2, 3, 225-229. Gaudry, M., Quinet, É. (2016). Box-Cox transformations of terms nesting the Trans-Log: the example of rail infrastructure maintenance cost. Publication AJD-155, Agora Jules Dupuit, Université de Montréal and PSE Working Papers n°2016-01, Paris-Jourdan Sciences Économiques, 20 p. January. www.researchgate.net/profile/Marc_Gaudry. Goldberger, A.S. (1968). The interpretation and estimation of Cobb-Douglas functions. Econometrica 35, 464-472. Grabowski, D.C., Morrisey, M.A. (2006). Do higher gasoline taxes save lives? Economics Letters 90, pp. 51–55. Griffiths, T., Tenenbaum, J. (2006). Optimal predictions in everyday cognition. Psychological Science 17, 9, 767-773, September. Gut, A. (2002). On the moment problem. Bernouilli 8, 3, 407-421. Johnston, J. (1984). Econometric Methods, Third Edition, McGraw-Hill Book Company. Knight, F.H. (1921). Risk, Uncertainty and Profit, Houghton, Mifflin. Knight, T.E. (1974). An approach to the evaluation of changes in travel unreliability: a “safety margin” hypothesis. Transportation 3, 4, 393-408. Koenker, R., Hallock, K.F. (2001). Quantile regression. Journal of Economic Perspectives 15, 4, 143156, Fall. Kouider, E., Chen, H. (1995). Concavity of Box-Cox log-likelihood function. Statistics and Probability Letters 25, 171-175. Lam, T.C., Small, K.A. (2001). The value of time and reliability: measurement from a value pricing experiment, Transportation Research Part E 37, 231-251. Lave, C.A. (1985). Speeding, Coordination and the 55 MPH limit. The American Economic Review 75, 5, 1159-1164, December. Leblond, V., Langlois, C. (2013). Modelling detailed passengers’ behaviours in a public transportation network. DIT-IDD-PRI, Régie Autonome des Transports Parisiens (RATP), 12 p., presented at the European Transport Conference, Frankfurt, September 30-October 2, 2013. Mandel, B., Gaudry, M., Ungemach, D. (2017). Europe-wide aviation connectivity measures and the PATH theorem. Publication AJD-161, Agora Jules Dupuit, Université de Montréal, 17 p., March. ResearchGate. Olsen, R.J. (1978). The method of Box and Cox, a pitfall. Institution for Social and Policy Studies, Yale University, Working Paper No 804. OMNIL (2012). Enquête Globale Transport 2010. Observatoire de la mobilité en Île-de-France (OMNIL), 20 p., 1er septembre. Orro, A., Novales, M., Benitez, F.G. (2005). Nonlinearity and Taste Heterogeneity Influence on Discrete Choice Model Forecasts. 18 p., Association for European Transport and contributors. Orro, A., Novales, M., Benitez, F.G. (2010). Box-Cox mixed Logit model for travel behaviour analysis. Paper presented at the 8th International Conference of Numerical Analysis and Applied Mathematics, Rhodes, 4 pages, September. Palma, A. de, Picard, N. (2005). Route choice decisions under uncertainty. Transportation Research A 39, 295-324. Picard, N., Palma, A. de, Dantan, S. (2013). Intra-household discrete choice models of mode choice and residential location. International Journal of Transport Economics 40, 3, 419-445. Prat, C., Leblond, V. (2014). Passengers’ density impact on route choice in urban public transport. DITIDD-PRI, Régie Autonome des Transports Parisiens (RATP), 21 p., presented at the European Transport Conference, Frankfurt, September 29-October 1, 2014. Prelec, D. (1998). The Probability Weighting Function. Econometrica 66, 3, 497-527, May.
26
Rassam, P.R., Ellis, R.H., Bennett. J.C. (1970, 1971). The N-Dimensional Logit Model : Development and Application. Peat Marwick and Mitchell & Co, Washington, D.C., 42p., 1970. Also in Highway Research Record 369, 135-147, 1971. Rosett, R.N., Nelson, F.D. (1975). Estimation of the two-limit Probit regression model. Econometrica 43, 141-146. Samuelson, P.A. (1954). The Pure Theory of Public Expenditure. The Review of Economics and Statistics 36, 4, 387-389. October. Samuelson, P.A. (1955). Diagrammatic Exposition of a Theory of Public Expenditure. The Review of Economics and Statistics 37, 4, 350-356. November. Samuelson, P.A. (1969). Contrast Between Welfare conditions for Joint Supply and for Public Goods. The Review of Economics and Statistics 51, 1, 26-30. February. Santos, A., McGuckin, N., Guckin and Hikari Yukiko Nakamoto, H.K., Gray, D., Liss, S (2011). Summary of Travel Trends: 2009 National Household Travel Survey. Federal Highway Administration, US Department of Transportation, 82 p. Scott, R.C., Horvath. P.A. (1980). On the Direction of Preference for Moments of Higher Order than the Variance, Journal of Finance 35, 4, 915-919. Senna, L.A.D.S. (1994). The influence of travel time variability on the value of time. Transportation 21, 203-228. Setec (1959). Tunnel sous la Manche – Étude de Trafic et de Rentabilité: Partie B – Annexe. The Economist Intelligence Unit Ltd (Londres), De Leuw, Cather and C o (Chicago) et Société d’Études Techniques et Économiques (Paris), pages iii-ix (pages 196-202, Volume 3), novembre. http://temis.documentation.developpement-durable.gouv.fr/document.xsp?id=Temis0005494&qid=sdx_q0&n=1&q=. Small, K.A., Winston, C., Yan, J. (2005). Uncovering the distribution of motorists’ preferences for travel time and reliability. Econometrica 73, 4, 1367–1382. Starkie, D.N.M. (1971). Modal Split and the Value of Time - a Note on Idle Time. Journal of Transport Economics and Policy 5, 2, 216-220. Stott, H.P. (2006). Cumulative prospect theory’s functional menagerie. Journal of Risk Uncertainty 32, 101–130. Tobin, J. (1957/1958). Liquidity preference as behaviour towards risk. The Review of Economic Studies 25, 65-86. Tobin, J. (1958). Estimation of Relationships for Limited Dependent Variables. Econometrica 25, 24-36. Tobin, J. (1965). The Theory of Portfolio Selection. Ch. 1 in Hahn, F., Brechling, F. (eds), The Theory of Interest Rates, 3-51. Macmillan, London. Tran, C.-L., Gaudry, M., Dagenais, M. (2008). LEVEL: The L-1.6 estimation procedures for BC-GAUHESEQ (Box-Cox Generalized AUtoregressive HEteroskedastic Single EQuation) regression and multi-moment analysis. Publication AJD-105, Agora Jules Dupuit, Université de Montréal, 59 p., April. www.e-ajd.net or ResearchGate. Tversky, A., Fox, C.R. (1994). Weighting Risk and Uncertainty. Psychological Review 102, 269-283. Tversky, A., Kahneman, D. (1992). Cumulative Prospect Theory: An analysis of Decision Under Uncertainty. Journal of Risk and Uncertainty 5, 297-323. Warner, S.L. (1972). Stochastic Choice of Mode in Urban Travel: A Study in Binary Choice, Northwestern University Press, Evanston, Ill., 1962. Vickrey, W.S. (1969). Congestion theory and transport investment. American Economic Review 59, 251261. Wardrop, J.G. (1952). Some theoretical aspects of road traffic research. Proceedings, Institute of Civil Engineers, PART II, Vol. 1, pp. 325-378. Working, E.J. (1927). What do statistical “demand curves” show? Quarterly Journal of Economics, 41, 212-35. Wu, G., Gonzalez, R. (1996). Curvature of the Probability Weighting Function. Management Science 42, 1676-1690.
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