1

Lecture 3

Mixed Logit Cinzia Cirillo

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Overview

1. Choice Probabilities 2. Random Coefficients 3. Error Components 4. Substitution Patterns 5. Approximation to Any RUM 6. Panel Data 7. Case Study

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1. Choice Probabilities

• Mixed logit (ML) is a highly flexible model that can estimate any RUM.

• ML overcomes the three major limitations of standard logit: – Random taste variation. – Unrestricted substitution patterns. – Correlation in unobserved factors over time.

• Unlike probit, ML is not restricted to a normal distribution. Like probit, ML have been known for many years but just recently increased in use with the advance of simulation.

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1. Choice Probabilities (cont.)

• ML models’ choice probabilities are expressed as follows:

𝑃𝑛𝑛 = �𝑒𝛽𝛽𝑛𝑛

∑ 𝑒𝛽𝛽𝑛𝑛𝑗∗ 𝑓 𝛽 𝑑𝛽

Where: – f(β) is a density function. – β is a set of coefficients. – Xni is the set of values of size N.

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2. Random Coefficients

• ML probability can be derived from utility-maximization behavior in many ways, random coefficients being the most popular one.

• Random coefficients represent the variation over people within a certain group (e.g., income level, drivers, bikers, elderly) in the value they put on a certain utility (i.e., cost).

𝑈𝑛𝑗 = 𝛽𝑛𝑥𝑛𝑗 + ε𝑛𝑗 – Where: Unj is the utility of person n for alternative j. εnj is a random term that is iid extreme value.

• The decision maker knows the value of his own βn and εnj for all j and chooses alternative i if and only if Uni > Unj V j ≠ i

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3. Error Components • ML can be used without random-coefficients interpretation by

representing error components that create correlations among the utilities for different alternatives:

𝑈𝑛𝑗 = α𝑛𝑥𝑛𝑗 + µ𝑛𝑧𝑛𝑗 + ε𝑛𝑗 – Where: xnj and znj are vectors of observed variables of alternative j. α is a vector of fixed coefficients. μ is a vector of random terms with zero mean. εnj is a random term that is iid extreme value.

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3. Error Components (cont.) • The terms in znj are random components that, along with εnj,

define the stochastic portion of utility. • The random part of the utility is ηnj = μnznj + εnj, which can be

correlated over alternatives depending on the specification of znj.

• For standard logit, znj is identically zero. Where as, if correlation exists (i.e. non-zero components), then: Cov (ηni, ηnj) = zniWznj, where W is the covariance of μn.

• Random coefficients and error components are formally equivalent. However, each one affects the researcher’s ML specification differently.

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4. Substitution Patterns • ML does not exhibit independence from IIA. • A ten percent reduction in one alternative does not imply a ten

percent reduction in each other alternatives. • The percentage change in the probability of one alternative

given a change in the mth attribute of another alternative is:

𝐸𝑛𝑛𝑋𝑛𝑛𝑚 = −1𝑃𝑛𝑛

�𝛽𝑚𝐿𝑛𝑛 𝛽 𝐿𝑛𝑗 𝛽 𝑓 𝛽 𝑑𝛽

– Where: βm is the mth element of β.

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5. Approximation to any RUM

• Any RUM can be approximated to any degree of accuracy through ML.

• Suppose the true model is Unj = αn*znj , where znj are variables related to alternative j and α follows any distribution f(α).

• The conditional probability is: 𝑞𝑛𝑛 𝛼 = 𝐼 𝛼𝑛′ 𝑧𝑛𝑛 > 𝛼𝑛′ 𝑧𝑛𝑗 ∀ 𝑗 ≠ 𝑖

– Where: I(.) is the 1-0 indicator of whether the event in parentheses occurs. • The unconditional probability is:

𝑄𝑛𝑛 = �𝐼 𝛼𝑛′ 𝑧𝑛𝑛 > 𝛼𝑛′ 𝑧𝑛𝑗 ∀ 𝑗 ≠ 𝑖 𝑓 𝛼 𝑑𝛼

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5. Approximation to any RUM (cont.)

• We can approximate the previous probabilities with a ML by scaling the utility by λ, so that

𝑈𝑛𝑗∗ =𝛼λ𝑧𝑛𝑗

• Then we add an iid extreme value εnj to obtain a ML. This does not change the model since it changes the utility of each alternative.

• The ML probability then is

𝑃𝑛𝑛 = �𝑒𝛼λ

′𝑧𝑛𝑛

∑ 𝑒𝛼λ

′𝑧𝑛𝑛

𝑗

𝑓 𝛼 𝑑𝛼

• As λ approaches zero, the ML probability Pni approaches the true probability Qni.

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6. Panel Data

• When using panel data, the integrand involves a product of logit formulas, one for each time period.

𝑃𝑛i = � L𝑛i 𝛽 𝑓 𝛽 𝑑𝛽

– Where:

L𝑛i 𝛽 = �𝑒𝛽′𝑛𝛽𝑛𝑛𝑖

∑ 𝑒𝛽′𝑛𝛽𝑛𝑛𝑖𝑗

𝑇

𝑡=1

• Lagged dependent variables can be added to ML without adjusting the probability formula or simulation method.

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6. Panel Data (cont.)

• So far, we have assumed that the βn is constant over choice situations for a given decision maker, which is only true if their tastes does not vary over a time period.

• The coefficients can then be specified to vary over time. One way to do this is to serially correlate each person’s tastes over choice situations:

𝑈𝑛𝑗𝑡 = 𝛽𝑛𝑡𝑥𝑛𝑗𝑡 + ε𝑛𝑗𝑡 𝛽𝑛𝑡 = 𝑏 + β𝑛𝑡∗

β𝑛𝑡∗ = 𝜌β𝑛𝑡−1∗ + 𝜇𝑛𝑡 – Where: b is fixed and μnt is iid over n and t.

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7. Case Study 1

• Consider a mixed logit of anglers’ choices of fishing sites. • Utility is Unjt = βnxnjt + εnjt , with coefficients βn varying over

anglers but not over trips for each angler. • The sample consists of 962 river trips taken in Montana by 258

anglers during the period of July 1992 through August 1993. A total of 59 possible river sites were defined.

• Simulation was performed using one thousand random draws for each sampled angler.

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7. Case Study 1 (cont.) • The following variables enter as elements of x for each site:

1. Fish stock, measured in units of 100 fish per 1000 feet of river. 2. A esthetics rating, measured on a scale of 0 to 3, with 3 being the highest. 3. Trip cost: cost of traveling from the angler’s home to the site, including the

variable cost of driving (gas, maintenance, tires, oil) and the value of time spent driving (with time valued at one-third the angler’s wage.)

4. Indicator that the Angler’s Guide to Montana lists the site as a major fishing site.

5. Number of campgrounds per U.S. Geological Survey (USGS) block in the site. 6. Number of state recreation access areas per USGS block in the site. 7. Number of restricted species at the site. 8. Log of the size of the site, in USGS blocks.

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16 TM Leuven, April 13 2005 16

Application 1: Mode choice model

• The data set (called Mobidrive) used in this research work was collected in 1999 in two cities of Germany: Karlsruhe and Halle.

• The Mobidrive study, whose main objective was to observe the variability and rhythms of daily life, involved 160 households and 360 individuals.

• Each individual was observed during six continuous weeks.

17 TM Leuven, April 13 2005 17

Activity patterns

• Worker’s daily activity chain » Morning commute » Midday tour » Evening commute » After tour

• Non worker’s daily activity chain » Before tour » Main tour (main activity) » After tour

18 TM Leuven, April 13 2005 18

Example of patterns and tours for two persons

Example of a worker Example of non-worker Trip chain Dep./Arr. Time Trip chain Dep./Arr. time 1 2 3 4

Home - Work Work-Shopping Shopping-Home

8:00am 8:20am 5:30pm 5:45pm 6:40pm 6:55pm

Home - Shopping Shopping - Home Home-Leisure Leisure-Home

11:10am 11:25am 12:45am 13:05pm 7:40pm 8:05pm 10:00pm 10:15pm

19 TM Leuven, April 13 2005 19

Mode choice model: Mobidrive data

Type of tour

Main mode Walking

Cycling

Vehicle driver

Vehicle Passenger

Public transport

Total All modes

% shares

Non-workers Morning tour Principal tour Evening tour Worker Morning tour Midday tour Work tour Evening tour All tour types (Share in %)

286 250

51

9 20

213 112 941

16.3%

328 203

31

10 53

474 144

1243

21.4%

638 541

89

31 33

561 181

2074

35.8%

182 264

25

1 3

76 170 721

12.4%

138 207

25

4 24

379 39

816

14.1%

1572 1465

221

55 133

1703 646

5795

27.1% 25.3%

3.8%

0.9% 2.3%

29.4% 11.2%

100.0%

20 TM Leuven, April 13 2005 20

Mode choice model: variables Level Household Individual Pattern LOS

Variables House hold location Age Marital status Professional Status Use of car Use of Public Transport Time budget [min/100] Sum of travel time [min] Tour Duration [min Number of stops Time [min] Cost [DM]

Categories Urban Suburban Age 18-25 Age 26-35 Age 51-65 Married with children Full Time worker Female and employed part-time Main car user Total annual mileage by car Number of season tickets 24 hours – time spent on previous activities (home stay included) and previous travel Sum of time spent traveling Sum of tour travel time and activity duration. Number of secondary activities observed within each tour

Including any parking fees

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Goodness of fit

n. of obs. L (0) L (C) L (β) K ρ2 adjusted

Multinomial Logit (MNL)

5795 - 8179.88 - 7503.82 - 6465.11

21 0.2070

MNL with interactions with socio-economic parameters

5795 - 8179.88 - 7503.82 - 6559.23

21 0.1955

Mixed Logit

5795 - 8179.88 - 7503.82 - 6446.88

26 0.2086

Mixed logit on panel data (day)

5795 - 8179.88 - 7503.82 - 6039.21

26 0.2585

22 22

VOT: Value Of Time study

Confidence interval (Armstrong et al., 2001)

( ) ( ) ( ) ( ) 2 2

c

2 2 c

2 2 t

2 c t

2

t

c

c

t 2 2

c

2 c t

t

c

c

t I , S

t t

t t t t t t t t t

t t t t t

t t

V −

− − − − ρ

θ θ

± − ρ −

θ θ

=

Variable Upper limit Lower limit VTTS point estimate correlation T-stat Number of observations

All purposes

14.47 8.60

11.06 0.093 - 4.76 5795

Work or education

13.57

4.50 8.23

0.061 - 4.65 1866

Shopping

757.09 13.02 26.99 0.077 - 6.03 1252

Leisure

80.61 7.07

15.30 0.09

- 3.24 1384

Other

50.74 9.09

17.94 0.057 - 6.78 1293

ρ

23 TM Leuven, April 13 2005 23

VOT by socio-economic characteristics Parameter ASC Car passenger ASC Public Transport ASC Walk ASC Bike Annual mileage by car Time Budget Tour Duration Sum of Travel Time Time Cost Interaction variables Time * urban household location Time * Age18-25 Time * Age26-50 Time * Age51-65 Time * Full-time worker Time * Female part-time Time * Married with children Time * Number of stops Time * Season Ticket Time * Main Car user

Alternative CP PT W B CD CD, CP PT B All All All All All All All All All All All All

β -1.379 -1.495 0.239

-0.491 0.046

-0.043 0.003

-0.005 -0.023 -0.113

0.018 0.021

-0.001 0.003

-0.001 -0.032 -0.027 0.015 0.002

-0.006

t-stat. -22.6 -11.6

1.7 -3.8 12.5 -2.9 17.2 -2.8 -9.4

-10.5

8.0 5.7

-0.1 1.3

-0.2 -8.4 -9.2 6.3 0.9

-2.5

VOT

12.04

2.35 0.82

12.48 10.35 12.31 29.19 26.52

4.19 10.79 14.98

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VOT per tour type VOT distribution for non-workers per tour type VOT distribution for workers per tour type

95th percentile VTTS [DM] 75th percentile VTTS [DM] Average VTTS [DM] Share of % negative VTTS

Morning pattern

32.2 18.8 13.5 6%

Principal pattern

55.25 28.4 15.4 25%

Evening pattern

14.6 8.2 5.4

14%

95th percentile VTTS [DM] 75th percentile VTTS [DM] Average VTTS [DM] Share of % negative VTTS

Morning pattern

n.a. n.a. n.a. n.a.

Commute pattern

15.3 9.3 7.6 0%

Evening pattern

64.7 35.0 21.4 19%

25 25

Travel time

0

0.2

0.4

0.6

0.8

1

-0.200 -0.150 -0.100 -0.050 0.000 0.050

Parameter value: travel time

Cumu

lative

prob

abilit

y

26 26

Travel cost

0

0.2

0.4

0.6

0.8

1

-0.200 -0.150 -0.100 -0.050 0.000 0.050 0.100 0.150 0.200

Parameter value: travel cost

Cumu

lative

prob

abilit

y

27 27

VOT

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30 35 40

Value of time (GM)

Cumu

lative

prob

abilit

y

28 28

Travel time: Before principal activity NW

0

0.2

0.4

0.6

0.8

1

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

Parameter value: travel time Before Principal Activity - non-workers

Cumu

lative

prob

abilit

y

29 29

Travel time: Principal pattern NW

0

0.2

0.4

0.6

0.8

1

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

Parameter value: travel time Principal pattern - non-workers

Cumu

lative

prob

abilit

y

30 30

Travel Time: Evening pattern NW

0

0.2

0.4

0.6

0.8

1

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

Parameter value: travel time Evening pattern - non-workers

Cumu

lative

prob

abilit

y

31 31

Travel time: Commute pattern W

0

0.2

0.4

0.6

0.8

1

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

Parameter value: travel time Commute pattern - workers

Cumu

lative

prob

abilit

y

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Heterogeneity across population ca be estimated with flexible forms of discrete choice models.

In numerous practical cases, parametric distributions are a priori specified and the parameters for these distributions are estimated.

This approach can however lead to many practical problems:

1. It is difficult to assess which is the more appropriate analytical distribution.

2. Unbounded distributions often produce values ranges with difficult behavioral interpretation.

3. Little is known about the tails and their effects on the mean of the estimates.

Flexible discrete choice models: accommodating heterogeneity across population.

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Alternative non-parametric approaches in the literature

Less restrictive non-parametric or semi-parametric approaches to the problem.

Bounded distributions: often obtained as simple transformations of normals (Train and Sonnier).

Mass point approach: Dong and Koppelman assume that distributions are represented by a finite number of points and use the Bayesian method to recover their mass.

Non-parametric: Fosgerau employs various non-parametric techniques to investigate the distribution

of the travel-time savings from a stated choice experiment. This method does not account for repeated observations and applies only to binomial choices.

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… more Global Sign conditions: Hensher resolves the problem of behaviorally incoherent sign changes by

imposing a global sign condition on the marginal disutility. He adopts a globally constrained Rayleigh distribution for total travel time.

Discrete Mixture of GEV: Hess, Bierlaire and Polak propose discrete mixture of GEV models over a finite set of distinctive support points. The major advantage of this approach is the lack of need for simulations.

Testing distributions: Recently, Fosgerau and Bierlaire have proposed a semi nonparametric (SNP) specification, based on Legendre polynomials, to test if a random parameter of a discrete choice model follows a given distribution

Willingness to pay space: Train and Weeks place distributional assumptions on the willingness to pay and derive the distribution of the coefficients.

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CONCLUSIONS!!!

“It is not possible to identify the distribution to use in all situations; the best distribution-fit is highly situation dependent.” Train and Weeks, 2005

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Mixed Logit model estimation Discrete set of alternatives available for individual Each alternative has some utility: Utility maximization principle: choose Probability choice of Aj by individual i: Heterogeneity in parameters inside the population: where random vector, vector of parameters

( )iAi :( )iAi ∈ ( ) ijijij VU εβ +=

( )iAAUUifA ninijj ∈∀≥

( ) ( ) ( ) ( )[ ]iAAVVPLP nininijijijij ∈∀+≥+== εβεββ( )θγββ ,=

( ) ( )[ ] ( ) ( )∫== γγθγθγθ γ dfLLEP ijijij ,,γ θ

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Drawing from random variable How to draw from univariate random (continuous) variable X? Fact: the cumulative distribution (CDF) FX (X) ~ U[0; 1]. This suggests to draw from an U[0; 1] and apply the inverse operation:

X = F -1X (U[0; 1]) If FX is unknown, construct F -1

X as a vector in some functional space (a linear combination of basis elements). Constraint: FX (and ) F -1X must be monotonically increasing.

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Multivariate distributions

Problem: the inverse CDF method cannot be applied directly to multivariate distributions. (In theory) decompose the CDF of the multivariate random variable as The inverse CDF method may then be applied sequentially. Independent distributions: => all random variables are considered independently; => we can then focus on unidimensional distributions.

( ) ( )1211211212112121 ...,|...|||)(),...,,(... ...−−

=dddddd XXXXXXXXXXXXxXxxxXXX PPPP

),...,,( 21 dXXX

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Univariate distributions

Assumption: each r.v. Xi has a bounded support.

One possibility to approximate

X : B-splines of degree p. Knots vector:

n basis functions, with B-spline curve:

where are the control points.

1−xF

{ }bbuuaaU pmp ,...,,,...,,,..., 11 −−+=

ipN 1−−= pUn

( ) ( )∑=

=n

ipii uNPuC

0,

iP

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can be chosen such that is monotonically increasing if

Example:

ipN

basis functions resulting splines

( )uC nPPP ....10 ≤≤

{ } 3,1,1,1,1,32,31,0,0,0,0 == pU

{ }15,3,5.0,5.0,3,15 −−−=P

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Log-likelihood maximization How to maximize the log-likelihood, under the monotonicity constraints? Use of trust-region methods. model (difficult) objective functions. . . . . . inside a trust region sufficient descent on the model contract/expand region

good theory + good practice

• The constraints can be easily dealt with projections.

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Congestion pricing There is a consensus among economists that congestion pricing represents the single most viable

and sustainable approach to reducing traffic congestion. Congestion pricing works by shifting purely discretionary rush hour highway travel to other

transportation modes or to off-peak periods, taking advantage of the fact that the majority of rush hour drivers on a typical urban highway are not commuters.

Although concerns are often expressed, surveys show that drivers support it because it offers them a reliable trip time. Transit and ridesharing advocates appreciate the ability of congestion pricing to generate both funding and incentives to make transit and ridesharing more attractive.

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Pricing strategies There are four main types of pricing strategies, each of which is discussed in more detail later in this

section: Variably priced lanes, involving variable tolls on separated lanes within a highway, such as

Express Toll Lanes or HOT Lanes, i.e. High Occupancy Toll lanes Variable tolls on entire roadways – both on toll roads and bridges, as well as on existing toll-free

facilities during rush hours Cordon charges – either variable or fixed charges to drive within or into a congested area within

a city Area-wide charges – per-mile charges on all roads within an area that may vary by level of

congestion

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Real life example - US Priced lanes on the State Route 91 in Orange County, California. HOT lanes on San Diego’s I-15. HOV converted to HOT on I-25 in Denver The State of Oregon is currently testing a pricing scheme involving per-mile charges, which it will

consider using as a replacement for fuel taxes in the future. HOT lanes on the Beltway DC ???

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Real life example - Europe In 2003, a cordon pricing scheme was introduced in central London. A similar scheme functioned in central Stockholm on a trial basis in 2006 from January through July. Area pricing has been introduced in Milan (Italy) in Fall 2007. The first results on congestion

reduction, increase of transit speed and more importantly decrease of pollutants in the atmosphere are very encouraging.

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Real case study: Estimating WTP on managed lanes

This dataset (called IRIS) is derived from a survey conducted in the region of Brussels (Belgium) in 2002.

The respondents are car users, intercepted during morning peak hours on the ring that gives access to the city from the suburban areas.

They were presented with up to three scenarios, each containing four choice options: 1. car, 2. car with delayed departure time, 3. public transport and 4. car on a High Occupancy Veh. lane (only prospective).

The original specification contained 18 exogenous variables, of which seven randomly distributed.

2602 observations belonging to 871 individuals are available

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Model Specification In total six model specifications have been estimated: 1. times normally distributed, cost constant (T N); 2. times and cost normally distributed (T-C N); 3. times log-normally distributed, cost constant (T L); 4. times and cost log-normally distributed (T-C L); 5. times B-Spline distributed, cost constant (T BS); 6. times and cost B-Spline distributed (T-C BS).

For the B-Spline coefficients, seven coefficients (P1,P2,...,P7) have been estimated, where P1 and P7

give the bounds of the distribution, and the knot vector is defined on the percentiles 0, 0.25, 0.5, 0.75, and 1.

48

Goodness of Fit

Distr. T N T-C N T L T-C L T BS T-C BS Final Log-Lik

-3.1460

-3.1399

-3.1604

-3.1511

-3.1453

- 3.1339

49

Congested Travel Time

50

Free-Flow Travel Time

51

Cost

52

Willingness to Pay

Distr. Quan. T N T-C N T L T-C L T BS T-C BS

Cong Time

25% 50% 75 % mean

4.75 15.31 25.89 15.32

3.36 4.52 10.49 12.96

6.41 12.64 24.88 20.90

16.10 37.61 87.77 13.17

7.20 11.05 26.12 7.90

2.42 4.36 16.68 4.74

Free Flow Time

25% 50% 75 % mean

2.56 13.72 24.91 13.74

2.10 6.24 11.35 11.61

5.33 10.84 22.02 18.80

14.03 32.65 75.74 12.35

0.89 13.53 22.18 12.64

0.70 3.07 14.35 7.47

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Willingness to pay (time and cost truncated at zero)

Distr. Quan. T N T-C N T BS T-C BS

Cong Time

25% 50% 75 % mean

10.25 18.54 27.98 19.93

4.65 5.59 12.54 7.94

7.84 15.03 28.47 18.35

4.56 6.95 17.01 7.83

Free Flow Time

25% 50% 75 % mean

9.64 17.97 27.68 19.58

4.40 4.87 12.62 7.92

9.55 17.11 25.67 18.59

3.96 5.60 15.23 7.65

54

Comments on WTP To summarize, we found that non-parametric B-Splines a) on both time and cost coefficients provide the best fit, b) reduce significantly the percentages of the population showing positive values for cost but leave

unchanged the proportion of positive time values, c) give VTTS ranges that do not suffer from fat tail effects. This suggests that the lognormal assumption, even if more coherent with the econometric theory, is not

reasonable here, and the non-parametric approach has the advantage over the normal to bound the distributions.

Computational time does increase with model flexibility; when three parameters are specified as non-parametric optimization time is about 74 minutes on a MacBook Pro, which is 6 times higher than the time required to three normal distributions instead.

55

Extensions Application to Finance The algorithm developed has been applied to a financial problem concerning central bank interventions

and dynamics in the foreign exchange market. The data used for our analysis have been collected from the Japanese Ministry of Finance’s website

(where they are publicly available) for the period April, 1991 to September, 2004. There are four possible outcomes of the central bank decision:

1. no intervention (W), 2. public intervention (Z), 3. secret intervention detected by the market (X), 4. secret intervention not detected by the market

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W (no intervention decision) deviation short deviation med deviation long misalignment statement interventiont? 1 RVt?1

Absolute level of short-term exchange rate deviation (%) Absolute level of medium-term exchange rate deviation (%) Absolute level of long-term exchange rate deviation (%) Absolute level of exchange rate misalignment (%) 1 if authorities made a statement expressing some discomfort with the exchange rate or confirming/discussing the intervention on the day of the operation 1 if there was an official intervention the day before Exchange rate realized volatility of preceding day, estimated at the end of the day

Z (public process) Leaning previous reported success inconsistence sum statement

1 if the intervention tries to reverse recent exchange rate trend 1 if the last detected intervention was a success 1 if the intervention direction is inconsistent with the reduction of the exchange Number of verbal interventions from the authorities signaling a discomfort with the exchange rate in the 5 days before the intervention

X (detection process) Amount coord success cluster

Amount invested in the daily intervention 1 if intervention is concerted 1 if the intervention moves the exchange rate in the desired direction 1 if there is at least one detected intervention over the last 5 preceding days

57

“Amount” distribution

58 58

Dynamic model of activity-type choice and

scheduling

59 59

Objectives • This paper presents a dynamic model for activity-type traveler choice and

scheduling estimated on a six-week travel diary. • The main focus of the study is the inclusion of past history of activity

involvement and its influence on current activity choice. • The econometric formulation adopted, explicitly accounts for both correlation

across alternatives and state dependency. • This is intended to be a first contribution to the evolution of demand model

into dynamic activity based framework.

60 60

State of art Activity-based models The need for more behaviorally sound framework has led to a new generation of transport model systems called activity-based models:

Bowman and Ben-Akiva, 2000;

Nagel and Rickert, 2001; Bhat and al., 2004; Arentze and Timmermans,

2004; Pendyala and al., 2004.

Day-to-day variability Jones and Clark study the policy

implication of variability analysis and encourage the collection of multi-day travel surveys.

Eric Pas distinguishes long-term patterns from daily behavior and finds that the latter is independent of individual characteristics.

Hanson and Huff, by using a specific measure of repetition and variability, conclude that one-week record of travel does not capture longer-term travel behavior.

Dynamic models Hirsh and al. (1986) estimate a

parametric model of dynamic decision-making process for weekly shopping activity behavior.

Mahmassani and Chang (1986) investigate the dynamics of departure time of urban commuters in a series of simulation experiments.

Dynamic models have been applied more extensively to explain car ownership behavior.

61 European Transport Conference Strasbourg 18-20 September 2006

61

Framework and Data

4952 activity episodes, 3212 daily schedules,

773 weekly schedules, 144 individual schedules.

Working day Non-working day Morning tour H Morning tour H

Morning commute H W Principal out-leg H P

Midday tour w Principal tour H

Evening Commute H W Principal ret-leg H P

Evening tour H Evening tour H

62 62

Mobidrive Number of activity episodes

Activity Scheduling

Shopping Leisure Personal business

Pick up / drop off

No extra activities

Home for

lunch

All tour types

Morning tour

41 14 41 5 101

Morning commute

44 24 65 26 159

Evening commute

207 103 105 40 455

Evening tour

143 376 90 54 663

Work as the only activity

547

547

W o r k i n g

d a y Midday

pattern 159 159

Morning tour

372 126 163 72 733

Principal tour

outbound leg

88

35

90

54

267

Principal tour

404 658 251 64 1377

Principal tour

return leg

73

59

40

14

186

Evening tour

97 93 62 53 305

N o n

w o r k i n g

d a y Total

1469

1488

907

382

4952

63 63

Model formulation: Mixed logit with repeated observations

• We develop this dynamic activity type choice model under the discrete choice analysis theory.

• The focus of the attention is on mixed logit models.

• The formulation adopted here is able to deal with correlation over alternatives in

the stochastic portion of utility, and to allow efficient estimation in presence of repeated choices by the same respondent.

64 64

Mixed logit with error components…

where:

and are vectors of observed variables relating to alternative j, is a vector of unobserved fixed coefficients, is a term of random terms with zero mean, is an unobserved random term i.i.d. extreme value distributed.

njtnjtnnjtnnjt zxU εµβ ++=

njtx njtz

nβ

nµ

njtε

65 65

…and repeated observations

Considering a sequence of alternatives, one for each time period , the probability that the person makes this sequence of choices is the product of logit formulas:

We consider here four groups k: • 1. an individual activity episode, • 2. the activity episodes of a person day, • 3. the activity episodes of a person week, • 4. the activity episodes of an individual.

( ) ∏∑=

+

+

=

T

1t jttnjznttnjxn

ttniznttnixnni

e

e,L µβ

µβµβ

66 66

The choice set structure Activity-type Scheduling Activity-type Scheduling

1. Just work 2.Home for lunch Shopping 3. Morning tour Shopping 19. Morning tour 4. Morning commute 20. Principal tour

outbound leg 5. Evening commute 21. Principal tour 6. Evening tour 22. Principal tour

return leg 23. Evening tour Leisure 7. Morning tour Leisure 24. Morning tour 8. Morning commute 25. Principal tour

outbound leg 9. Evening commute 26. Principal tour 10. Evening tour 27. Principal tour

return leg 28. Evening tour Personal Business

11. Morning tour

Personal Business

29. Morning tour

12. Morning commute 30. Principal tour outbound leg

13. Evening commute 31. Principal tour 14. Evening tour 32. Principal tour

return leg 33. Evening tour Pick up Drop off

15. Morning tour

Pick up Drop off

34. Morning tour

16. Morning commute 35. Principal tour outbound leg

17. Evening commute 36. Principal tour 18. Evening tour 37. Principal tour

return leg

W o r k i n g

d a y

N o n

w o r k i n g

d a y

38. Evening tour

67 67

List of independent variables - I

Level Variables Description / Categories Type

Day Activity duration (min)

It is referred to the activity to be undertaken and it is randomly drawn from the vector of activity (with the same purpose) durations, reported by the same individual, over the entire survey period.

continuous

Day Time budget (min)

It is calculated as 24 hours minus the time spent on previous activities (home stay included) and previous travel.

continuous

Day Available time before work (min)

It is the time available between the shops opening hour (8:00 am) and the arrival time to work.

continuous

Day Available time after work (min)

It is the time available after the departure from work and the shops closing hour (6:00 pm).

continuous

68 68

List of independent variables - II Level Variables Description / Categories Type

Week High week episode

It is a dummy variable, which is 1 if, for the week considered, the number of activity episodes with a specific purpose is greater than two.

dummy

Six-week Last time It is the number of days occurred between the day considered and the day when the same activity was undertaken.

discrete

Six-week Immobile days It is the number of days that the individual spent at home between the first day of the reported period and the day considered.

discrete

Level Of Service

Logsum It is the logsum of the mode choice model. continuous

Individual Socio-Economic Variables

Age Presence of children Professional Status

Age 6-18/ Age 26-35/ Age 51-65 Number of Children under 12 Number of working hours per week Full Time worker Female and employed part-time

dummy dummy continuous dummy dummy

69 69

Model Results I Logit Mixed – err comps Mixed - Day Mixed - Week Mixed - Individuals Variable Alt. Estimates t-stat Estimates t-stat Estimates t-stat Estimates t-stat Estimates t-stat Activity duration S -0.0007 1.04 -0.0007 0.97 -0.0007 1.03 -0.0007 1.03 -0.0007 0.78 L -0.0011 2.94 -0.0012 2.88 -0.0011 3.11 -0.0011 2.81 -0.0012 1.84 PB -0.0011 1.73 -0.0014 1.91 -0.0011 1.78 -0.0011 1.75 -0.0011 1.28 PD -0.0012 1.02 -0.0020 1.39 -0.0013 1.13 -0.0012 0.94 -0.0015 0.70 LH -0.0058 5.18 -0.0049 4.27 -0.0056 5.07 -0.0059 5.50 -0.0063 6.34 Time budget S -0.2830 14.00 -0.3969 11.40 -0.2959 14.40 -0.2857 14.50 -0.2930 16.48 L -0.6703 27.74 -0.8900 17.81 -0.6931 26.00 -0.6828 28.82 -0.7166 36.77 PB -0.3185 16.53 -0.4618 12.43 -0.3283 16.75 -0.3201 16.75 -0.3283 18.67 PD -0.4410 18.97 -0.7109 11.92 -0.4712 17.84 -0.4650 19.83 -0.4854 23.28 Available time before work S -0.0016 2.55 -0.0034 3.11 -0.0020 3.09 -0.0017 2.24 -0.0019 2.17 Available time post work S 0.0040 7.58 0.0073 6.58 0.0044 7.68 0.0041 8.06 0.0043 6.21 High week episode S 1.5222 18.94 2.5051 11.64 1.6169 17.08 1.5326 15.45 1.4756 12.07 L 0.4899 5.20 0.7163 5.40 0.5114 4.99 0.4720 4.67 0.4656 3.07 PB 1.1490 11.41 1.7147 8.36 1.1856 10.35 1.1556 9.14 1.1393 7.40 PD 2.0734 15.29 3.4511 10.13 2.1929 13.96 2.0999 11.48 1.8209 8.72 Last time S 0.5598 7.65 0.9304 6.33 0.6214 7.29 0.5647 7.45 0.5683 6.79 L 4.0883 23.70 5.9394 14.58 4.2972 21.67 4.2187 24.56 4.4828 23.82 PB 0.2022 9.51 0.3738 8.43 0.2220 9.54 0.2023 8.30 0.1978 7.32 PD 0.0020 5.47 0.0039 5.14 0.0025 6.38 0.0026 7.24 0.0030 7.75 Last time leisure W -0.2789 1.90 0.1271 0.72 -0.2345 1.59 -0.2791 1.98 -0.2766 2.05 Last time PB W -0.1242 2.36 -0.1064 1.75 -0.1234 2.24 -0.1252 2.20 -0.1346 2.43

70 70

Model Results II Logit Mixed-logit err comp Mixed logit - Day Mixed logit - Week Mixed logit -Ind Variables Alt. Estimates t-stat Estimates t-stat Estimates t-stat Estimates t-stat Estimates t-stat Immobile days S 0.0056 2.65 0.0085 3.24 0.0058 2.92 0.0058 2.73 0.0060 2.16 L 0.0056 2.80 0.0088 3.99 0.0058 3.26 0.0057 3.21 0.0061 2.77 PB 0.0059 2.94 0.0093 4.06 0.0062 3.36 0.0061 2.81 0.0063 1.98 PD 0.0056 2.75 0.0086 3.69 0.0058 3.15 0.0058 3.16 0.0062 2.94 Logsum All 0.1099 1.90 0.1738 1.92 0.1207 1.90 0.1460 2.38 0.2074 3.04 Age L -0.3866 1.77 -0.5175 1.71 -0.3920 1.35 -0.4364 1.46 -0.6407 0.91 PD 0.3956 2.97 0.6629 2.74 0.4111 2.70 0.4673 2.41 0.5624 1.76 LH -0.4198 2.51 -0.1579 0.91 -0.3929 2.42 -0.4054 2.33 -0.3111 2.16 Number of children under 12 L 0.1805 2.42 0.3369 3.28 0.1929 2.48 0.1814 2.27 0.2095 1.29 PD 0.4654 5.49 0.8233 5.13 0.5052 4.97 0.5302 4.57 0.7093 4.20 N of working hours per week W 0.0273 6.81 0.0336 6.72 0.0281 5.92 0.0272 7.04 0.0262 7.07 Full time worker S -0.7160 3.27 -1.4044 5.10 -0.8085 3.87 -0.7294 3.91 -0.5953 3.38 L -1.0030 4.55 -1.6624 6.62 -1.0914 5.29 -1.0212 5.41 -0.9009 4.29 PB -0.5727 2.61 -1.0877 4.08 -0.6433 3.13 -0.5896 3.27 -0.4504 2.91 PD -0.4398 1.83 -0.9277 2.87 -0.4908 2.05 -0.4840 2.02 -0.3161 0.91 W -1.7173 6.91 -2.5418 8.63 -1.8169 6.81 -1.7321 7.95 -1.6430 9.43 Female and part time S 0.5083 2.42 0.5245 1.82 0.5113 2.33 0.5108 2.26 0.5929 2.74 L 0.3888 1.79 0.4928 1.85 0.3848 1.70 0.3745 1.81 0.3512 1.27 PB 0.5978 2.86 0.7036 2.52 0.5969 2.72 0.5999 2.57 0.6916 3.44 PD 0.6523 2.43 0.7898 1.99 0.6956 2.36 0.6908 1.96 0.9444 1.27 Error components S - - 2.2158 8.48 0.6295 5.78 0.0146 0.02 0.1801 1.41 L - - 0.0118 0.01 0.1337 0.31 0.2444 2.10 0.5274 6.02 PB - - 2.0645 7.93 -0.4249 2.53 -0.0012 0.00 -0.0402 0.08 PD - - 2.5141 7.88 0.6895 3.66 0.5677 5.16 0.5688 2.69 Number of observations 4952 4952 4952 4952 4952 N. of obs with repetitions 4952 4952 3212 773 144 Log likelihood at zero -12268.27 -12268.27 -12268.27 -12268.27 -12268.27 Log likelihood final -9902.19 -9859.29 -9891.00 -9895.56 -9857.51 rho squared adjusted 0.18951979 0.19269057 0.19010586 0.18973417 0.19283566

71 71

Model fit (Final Log-likelihood / Rho-squared adjusted)

Model Degrees of freedom

Logit Mixed – err comps

Mixed - Day

Mixed - Week

Mixed - Individuals

Socio-economic variables

21

-11952.60

0.02434

-11845.09

0.03280

-11850.82

0.03231

-11685.33

0.04580

-11351.77

0.07299 Day behavioral variables

32 -11393.82

0.06899

convergence not

achieved

-11310.30 0.07548

-11162.79 0.08750

-10871.28 0.11126

Week behavioral variables

36 -10528.83

0.13917 -10503.48

0.14092 -10516.96

0.13982 -10525.89

0.13909 -10451.30

0.14517

Long-term behavioral variables

45

-9902.19

0.18952

-9859.29

0.19270

-9891.06

0.19010

-9895.56

0.18973

-9857.51

0.19283

72 72

Application: Week DAY

73 73

Application: Weekend DAY

Day 6 - Saturday

0100200300400500600700800

Sho

ppin

g W

D F

/R

Leis

ure

WD

F/R

PU

DO

WD

F/R

Per

sBus

WD

F/R

No

extr

a w

ork

act

F/R

Lunc

h H

ome

F/R

Tota

l WD

F/R

Sho

ppin

g N

onW

D F

/R

Leis

ure

Non

WD

F/R

PU

DO

Non

WD

F/R

Per

sBus

Non

WD

F/R

Tota

l Non

WD

F/R

ForecastReal

Day 7 - Sunday

0100200

300400500600

700800

Sho

ppin

g W

D F

/R

Leis

ure

WD

F/R

PU

DO

WD

F/R

Per

sBus

WD

F/R

No

extra

wor

k ac

t F/R

Lunc

h H

ome

F/R

Tota

l WD

F/R

Sho

ppin

g N

onW

D F

/R

Leis

ure

Non

WD

F/R

PU

DO

Non

WD

F/R

Per

sBus

Non

WD

F/R

Tota

l Non

WD

F/R

ForecastReal

74 74

Application: Week of survey

75 75

Conclusions

• The paper has implemented a model of activity type and timing choice. • The formulation includes variables, describing the dynamics of the day. • The socio-demographic variables, in other models dominant, loose their

prominence. • The personal preferences, here captured exclusively with the activity type specific

error components, are associated with leisure and picking up and dropping off people.