EXPONOMIAL MODEL FOR MULTIPLE DISCRETE-CONTINUOUS …

The Pennsylvania State University

The Graduate School

EXPONOMIAL MODEL FOR MULTIPLE DISCRETE-CONTINUOUS CHOICES:

ANALYSIS OF ACTIVITY TIME-USE PATTERNS IN DUAL EARNER HOUSEHOLDS

A Thesis in

Civil Engineering

by

Renato Guadamuz-Flores

© 2020 Renato Guadamuz-Flores

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

May 2020

ii

The thesis of Renato Guadamuz-Flores was reviewed and approved* by the following:

Rajesh Paleti

Assistant Professor of Civil and Environmental Engineering

Thesis Adviser

Vikash Varun Gayah

Associate Professor of Civil and Environmental Engineering

Sukran Ilgin Guler

Assistant Professor of Civil and Environmental Engineering

Shelley Marie Stoffels

Professor of Civil and Environmental Engineering

Chair of the Graduate Program

iii

ABSTRACT

In single choice modeling, methods like the popular multinomial logit (MNL) are focused

on estimating the probability of each alternative to be chosen given specific conditions. This can

be very limiting for scenarios where the decision makers consume more than one alternative.

Multiple discrete-continuous (MDC) models address this issue by accounting for the allocation of

a constrained budget (e.g., money or time) across a set of available alternatives, rather than a binary

consumption or not.

The standard approach for MDC models is the Multiple Discrete-Continuous Extreme

Value (MDCEV) choice model and is based on a Gumbel distribution for the stochastic component.

From a behavioral perspective of the consumers, the positive skewness of a Gumbel distribution

does not accurately describe the expected nor observed rational consumption of goods. A

negatively skewed distribution for the stochasticity terms describes better the perceived value from

the decision makers for the available alternatives.

In the single choice framework, the Exponomial choice has been proved to offer better

behavioral and data fitness properties compared to a regular MNL. This work presents the

development and properties of the Multiple Discrete-Continuous Exponomial Choice (MDCEC)

that holds an elegant closed form for the likelihood function that, unlike the MDCEV, offers

easiness of implementation for heteroscedasticity across alternatives.

The ability from MDCEC to retrieve the true value of the parameters is demonstrated using

simulated data under a variety of conditions and later, the MDCEC is compared to the MDCEV in

an empirical case of activity time use for activity-based travel demand applications, where the

MDCEC approach provides a significant better fit to the empirical data.

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TABLE OF CONTENTS

LIST OF TABLES ................................................................................................................... v

ACKNOWLEDGEMENTS ..................................................................................................... vi

Chapter 1 Introduction ............................................................................................................ 1

Chapter 2 Literature Review ................................................................................................... 5

Activity based travel demand ........................................................................................... 5 Multiple discrete-continuous modeling ............................................................................ 7

Chapter 3 Methodology .......................................................................................................... 11

Chapter 4 Simulation analysis ................................................................................................ 19

Chapter 5 Empirical application ............................................................................................. 22

Chapter 6 Concluding remarks ............................................................................................... 32

References ................................................................................................................................ 34

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LIST OF TABLES

Table 5-1. Summary statistics of time allocation for each type of activity. ............................ 23

Table 5-2. Summary statistics of explanatory variables for MDC modeling. ........................ 24

Table 5-3. Results from the MDCEC homoscedastic model. ................................................. 27

Table 5-4. Results from the MDCEV homoscedastic model. ................................................. 28

Table 5-5. Results from the MDCEC heteroscedastic model. ................................................ 30

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ACKNOWLEDGEMENTS

I would like to convey my deepest and most sincere gratitude to my advisor and committee

chair, Prof. Rajesh Paleti, whose inspiration, curiosity, enthusiasm, and knowledge were

undoubtedly essential to the success of this project and for endorsing me on other academic goals.

Without his guidance and extensive support, I could have not completed this project.

I would like to thank the rest of the committee for their insightful comments. To Prof.

Vikash Gayah, whose passion for teaching and sharing knowledge inspired me many times to give

an extra effort and whom I consider a true professional role model, as well as Prof. Ilgin Guler,

from whom I learned a lot through her lectures that served me directly into this research and

outstanding help during the final stages of this study.

My genuine thanks to my classmates, laboratory colleagues and all my friends, that

together, we encouraged each other to strive for the best, helped me expand my professional and

personal horizons, stood up during difficult times and extended me their friendship with

unforgettable memories and experiences.

Finally, but being one of the most important pillars in my life, I thank my family for their

endless support and love and with whom I can count under any conditions.

Chapter 1

Introduction

Discrete choice modeling serves as a powerful tool to analyze events where the decision

makers are offered a specific set of alternatives for consumption. In single choice, the alternatives

are considered as perfect substitutes, because they offer a very similar or identical use, for instance,

from a set of soft drinks, all of them satisfy the consumer needs to hydrate, refresh, and enjoy a

tasty drink according to the unique preferences of each individual, but no drink has significant

differences compared to the rest.

Recently, the literature has presented more interest into discrete analyses beyond traditional

multiple-choice models, such as multinomial logit (MNL) or multinomial probit. One particular

area of development arises when the decision makers can spend from a continuous budget (e.g.,

time, money, or mileage) into a set of imperfect substitutes, for instance, a regular soft drink, an

energy drink, orange juice, and water, in the context of soft drinks. These are imperfect substitutes

because each product satisfies different needs of the consumer, unlike the example where all the

alternatives were soft drinks. The consumption of these products is typically constrained to a

budget, in this case how much money to allocate on each type of drink; this is known in the literature

as multiple discrete-continuous (MDC) models.

Because of the generality of MDC modeling, it has multiple applications: from marketing,

production and consumption of goods and services (like the drinks example above), to allocation

of investments and diversification of a portfolio in specific financial instruments (e.g., stocks,

bonds, commodities, real estate, etc.).

In the transportation field, MDC models can be applied to estimate the household usage

from a set of available vehicles (e.g., sedan, SUV, convertible, truck, motorcycle), where each

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vehicle serves better for different trip purposes. More recently, it can be used to analyze the

ownership and usage of electric, hybrid or other alternative-fuel vehicles compared to more

standard types of vehicles in the household, and in the future, usage of autonomous vehicles in the

household. The usage can be estimated as the time (from all the time spent traveling or commuting)

or mileage allocated into each type of vehicle. On a broader sense of commuting, it can also be

applied to estimate the allocation of weekly or monthly travel into multiple modes (e.g., private

vehicle, bus, train, walking, cycling, taxi, ride-hailing, etc.).

From a business perspective, the vehicle fleet from a delivery company, transit agency,

airlines, cruise ships, cargo ships, freight or passenger trains, or in general, any business that

operates with a fleet can be modeled for different types of ground vehicles, aircrafts or ships and

the miles or time that is allocated for each type of vehicle. For instance, a transit agency might use

small of buses for long distance routes with low ridership and articulated buses with more capacity

for urban routes with high rush hour demand. A delivery company might use big trucks for long

distance between cities and small vehicles for final delivery.

From a city planning perspective, the types of land use and how much to build for each

land use can also be modeled for land development decisions, whether for development companies

or in a macro level of consumption of the city land resources.

For activity demand, MDC models can be used to estimate how much time do individuals

spend in a set of discrete activities (e.g., staying at home, working, shopping, visiting, eating out)

in a regular day. In long-distance leisure travel, it can be used to analyze where the individuals go,

how many days, or how much money to spend in different activities. This application in activity

demand is particularly relevant for its applications for activity-based travel demand to generate

more accurate and realistic predictions for travel demand based on activities and time use.

Traditionally, travel demand has been modeled using the “four-step travel model” (trip

generation, trip distribution, mode choice, and route assignment), but uses aggregated data that

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does not reflect the behavioral nature of the decision makers (Bradley, M., Bowman, J., & Lawton,

1999) and therefore, more detailed methods are needed to address these limitations from the

traditional models (Ben-Akiva & Bowman, 1998; Bhat & Koppelman, 1999; Pinjari & Bhat, 2011).

Instead of modeling the travel demand itself and given that the travel demand is a consequence of

a broader activity demand (Ben-Akiva & Bowman, 1998; Bowman, J. L., & Ben-Akiva, 2000;

Jones, P., Koppelman F., 1990), an alternative approach focuses of activity-based travel demand,

in which the time and features of the activities from each individual are estimated and later

incorporated into the travel demand modeling (Castiglione, Bradley, & Gliebe, 2015).

When modeling the activity demand, it is assumed that the individuals can choose from a

set of discrete activities and they allocate as much time as they want on each alternative, constrained

to the available total time for all activities.

Most of the MDC development has focused around models based on Gumbel distributed

errors, i.e., extreme value type I distribution, which yields the name MDCEV and can be seen as a

generalization of the MNL for MDC modeling. Although the MDCEV performs well, it ignores

some important behavioral aspects of the decision makers regarding the amount of consumption,

given the willingness to pay or perceived value of each alternative. The exponomial choice

approach for MDC (MDCEC) considers these behavioral attributes by using negative exponential

distributed error terms. We present an empirical application of the proposed MDCEC model for

time allocation of individuals in the context of activity-based travel demand. The proposed

MDCEC model is also compared to the standard MDCEV approach, resulting in more adequate

behavioral properties and better data fit to the empirical data.

The objectives of this study include to 1) present in detail the fundamentals and

development of the MDCEC model, 2) implement MDCEC for convenient use under different

scenarios, 3) evaluate the appropriateness of the model to retrieve the true value of the parameters

using synthetic datasets, and 4) compare the performance of MDCEC and MDCEV under identical

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homoscedastic conditions with an empirical application to predict the time allocation in different

activities.

The rest of the thesis is organized as follows: Chapter 2 presents a summary of the previous

studies both in travel demand and in discrete-continuous modeling. Chapter 3 describes the

methodological and theoretical development of the proposed MDCEC. Synthetic datasets are used

in Chapter 4 to demonstrate the appropriateness of the MDCEC to retrieve the true parameter values

under different conditions. In Chapter 5, we compare the proposed model to the standard state-of-

the-art MDCEV using empirical data for time-use in a context of activity-based travel demand. The

concluding remarks presented in Chapter 6 summarize the results and delineate the future work

needed.

Chapter 2

Literature Review

In light of the interests of this study, the consulted literature can be separated into two

categories: 1) the methods related to travel demand as a consequence of activity demand, and 2)

the current state-of-the-art modeling for discrete-continuous data. Both cases are explained in detail

in this chapter.

Activity based travel demand

Travel demand estimation and forecast are essential for public agencies that develop

policies in this field, therefore, it is pertinent to model travel demand in the most realistic manner

as far as the complexity and cost of the models remain reasonable. The classic approach for travel

demand has been the four-step model (Bhat & Koppelman, 1999; Castiglione et al., 2015), which

is based on individual trips from one origin to one destination and does not depict an accurate

representation of the decision-making process of the individuals when traveling to multiple

destinations (Bradley, M., Bowman, J., & Lawton, 1999; Castiglione et al., 2015).

Trip-based models ignore the time of the day of trips because it is not captured by the

aggregated models or is only included in a limited way through time-of-day factors (Bhat &

Koppelman, 1999). Additionally, the dependence between home-based and non-home-based trips

is ignored since both models are fitted separately when the analysis unit is the trip, and in a multi-

stop tour (multiple single trips chained one after another), the chosen mode depends on the

characteristics of all the single trips, but this is ignored when the analysis is trip-based, as the trips

are treated as independent among them (Bhat & Koppelman, 1999).

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In the last decades, trip-based approaches have evolved into more realistic and appropriate

activity-based models (since travel demand derives from a broader activity demand), where time

allocation for different activities is estimated in a behavior-oriented manner for the each individual

to later aggregate the flows generated by these tours (Ben-Akiva & Bowman, 1998; Bhat &

Koppelman, 1999; Bowman, J. L., & Ben-Akiva, 2000; Davidson et al., 2007; Jones, P.,

Koppelman F., 1990; Pinjari & Bhat, 2011).

The activity-based travel demand focuses on the utility maximization of the individuals

when choosing the time to allocate for each activity, where and when to make such activities, how

many individual tours or multi-tours to endeavor, time and distance of each tour, mode choice and

other features of the travels (Ben-Akiva & Bowman, 1998). There are three main common

components of the activity-based travel demand analyses (Davidson et al., 2007):

• Activity-based platform: general framework of daily activities by household and

its members, using time as unit of analysis for activity behavior (Bhat &

Koppelman, 1999).

• Tour-based travel demand: instead of focusing on the trip as the elemental unit for

travel.

• Microsimulation: focused on the probability of discretely choosing from fully-

disaggregate activities and travel choices.

The methods developed for the activity-based platform include several variations of the

multiple discrete-continuous choice models (MDC) that account for the discreteness of the possible

activities from which to choose and also for the continuous amount of time can be allocated to each

activity under a constrained budget of available time (Bhat, 2005).

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Multiple discrete-continuous modeling

MDC are applicable not only to time use for travel demand but are general to discrete-

continuous problems where multiple alternatives area available to the decision makers. The

framework has been developed heavily in the transportation area but has seen applications in other

fields.

Most of the effort and a high load of the model design has been established by Chandra

Bhat and de transportation engineering department at the University of Texas at Austin. Bhat

developed (Bhat, 2005) and later clarified (Bhat, 2008) the general framework for the MDC models,

based on previous definitions of the MDC models (Kim, Allenby, Rossi, Greg, & Peter, 2002),

where the utility structure for the likelihood function and the role of parameters (translation and

satiation) are explained in detail. This approach of the MDC modeling includes the implementation

of the error terms under a Gumbel distribution, i.e., Generalized Extreme Value Type I distribution

(which leads to the model name MDCEV). From this definition, the MDCEV has been widely used

as the standard for MDC models.

The MDCEV assumes the willingness to pay for a product (or more generally, the

perceived value of an alternative) to follow a Gumbel distribution, which is positively skewed. This

is reasonable when consumers have very limited information about the real value of a product (e.g.

fine wine, art, mansions, jewelry, etc.) (Alptekinoglu & Semple, 2016). However, if the consumers

are well informed about products and their real value (or face value), they are reluctant to overpay

and even unwilling to spend from their budget in specific alternatives at all if the cost is considered

as excessive or unjust, making the utility function to decrease quickly above a latent threshold price

(Paleti, 2020). One can even expect a longer left tail (i.e., negatively skewed distribution) because

more consumers would be more willing to underpay that overpay, since consumers can lower their

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perceived value for various idiosyncratic reasons that make the alternatives less than ideal

(Alptekinoglu & Semple, 2016).

The choice probabilities can be described as an exponomial (a linear combination of

exponential terms) (Duffin, 1961). This was briefly introduced into a single choice modeling

framework as the negative exponential distribution (NED) by Daganzo (Daganzo, 1979) and

greatly expanded by Alptekinoglu and Semple (Alptekinoglu & Semple, 2016) under the name of

exponomial choice (EC). The NED or EC is negatively skewed and has an upper bound on the

perceived attractiveness for any alternative (Daganzo, 1979), which makes the modeling more

realistic to the expected expenditure in each alternative and reflects behavioral sensitivities (Paleti,

2019). Also, the concave log-likelihood function of the EC allows fast and efficient computational

performance for its optimization.

The MDCEV is a generalization of the widely used MNL for the MDC choice analysis and

collapses exactly to the MNL when only one alternative is chosen (Bhat, 2008). In a single choice

framework, NED models have been applied to different contexts, including fiducial limits (Grubbs,

1971; Pierce, 1973), risk of heart disease (Lee, Fry, & Hamling, 2012), and traffic modeling

(Rahmani, Afzali-Kusha, & Pedram, 2009). Also, literature has demonstrated that NED has a better

data fit than MNL for different transportation modal choices (Currim, 1982) and transit alternatives

(Aouad, Feldman, & Segev, 2018). Moreover, Berbeglia et al. (Berbeglia, Garassino, & Vulcano,

2018) empirically found out that EC models perform better than MNL, regardless of the amount of

historical data available, data structure (random vs. price-based), number of alternatives or

consistency of underlying preferences. Although the EC model takes slightly more time to compute

than an MNL, it takes on average the same order-of-magnitude of time (Berbeglia et al., 2018),

which is also relatively irrelevant in modern times, given the great advances in computational

power.

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EC models do not have the limitation of independence of irrelevant alternatives (IIA) from

the MNL (Alptekinoglu & Semple, 2016; Paleti, 2019), that although can be relaxed for the MNL,

may affect other assumptions (Fosgerau & Bierlaire, 2009; Li, 2011). Additionally, the EC

heteroscedastic extension has a closed-form choice probability that is simpler than the

heteroscedastic extreme value choice (Alptekinoglu & Semple, 2018; Bhat, 1995; Paleti, 2020).

The MDC models are mathematically general and can be applied to many different fields,

for instance, in the transportation area, specific types of vehicle ownership and usage (Ahn, Jeong,

& Kim, 2008; Bhat & Sen, 2006), annual mileage of households (Jäggi, Weis, & Axhausen, 2013),

long-distance travel demand (van Nostrand, Sivaraman, & Pinjari, 2013), and applications to time-

use and activity-based travel patterns (Bernardo, Paleti, Hoklas, & Bhat, 2015; Bhat et al., 2013;

Castro, Bhat, Pendyala, & Jara-Díaz, 2012; Paleti & Vukovic, 2017; Spissu, Pinjari, Bhat,

Pendyala, & Axhausen, 2009; Srinivasan & Bhat, 2006).

Apart from the applications of the MDCEV, few substantial theoretical improvements have

been incorporated, namely, the expansion to panel data (Paleti & Vukovic, 2017; Spissu et al.,

2009), Generalized Extreme Value distribution for the error terms (Pinjari, 2011), multiple

constraints (Castro et al., 2012) and a more flexible approach for the discrete components (Bhat,

2018).. To the best of our knowledge, the only improvement for the underlying behavioral

distribution for the stochastic components of MDC models was developed by Bhat et al. (Bhat, C.

R., Dubey, S. K., Alam, M. J. B., & Khushefati, 2015) using normally distributed errors, which

corresponds to an extension of the probit model for single choice, however, no models have been

proposed to include a negatively skewed distribution.

The EC presents better theoretical properties for choice modeling under most realistic

circumstances and certainly for activity-based travel demand, including some evidence of

providing a better fit to empirical data for single choice modeling. Therefore, the EC generalized

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to MDC choice analyses presents a tremendous opportunity to improve the estimations of activity-

based travel demand modeling.

Chapter 3

Methodology

The total utility function is assumed to be an additively separable direct utility function

specified as the sum of sub-utility functions of each alternative. This implies that none of the

alternatives are a priori inferior and that the marginal utilities are independent to the consumption

level of the other alternatives (Bhat, 2008). For ease of presentation, we momentarily assume

absence of outside goods, i.e., zero consumption levels are allowed for all alternatives (Bhat, 2008).

The total utility function is given by:

𝑈(𝑥𝑘) = ∑𝛾𝑘

𝛼𝑘

𝐴

𝑘=1

× 𝜓𝑘 × [(𝑥𝑘

𝛾𝑘+ 1)

𝛼𝑘

− 1]

𝜓𝑘 = 𝑒𝑥𝑝(𝛽′𝑧𝑘 − 𝜀𝑘)

Equation 3-1

where 𝜀𝑘 is an exponential random variable with scale parameter 𝜆𝑘 that captures unobserved

factors that influence the marginal utility at zero consumption of alternative k (out of 𝐴 total

alternatives) and is independent across choice alternatives, 𝜓𝑘 is the marginal utility at zero

consumption and 𝑒𝑥𝑝(𝛽′𝑧𝑘) is the ideal (or maximum) marginal utility for alternative k from

predictors 𝒛𝑘. Unlike the MDCEV (Bhat, 2008), the stochastic term has a negative sign to account

for a NED and represents the heterogeneity across decision makers (Alptekinoglu & Semple, 2016).

The parameters 𝛾𝑘 and 𝛼𝑘 control the satiation by translating and exponentiating the

consumption quantity, respectively. Since both parameters control the satiation, only one of those

is estimated while the other must be fixed (Bhat, 2008).

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The observed vector of consumptions (𝒙) is assumed to be an outcome of maximizing 𝑈(𝒙)

subject to a budget constraint ∑ 𝑥𝑘𝐴𝑘=1 = 𝐵.

Maximizing the total utility through a constrained optimization leads to the following

Lagrangian function:

ℒ = ∑𝛾𝑘

𝛼𝑘

𝐴

𝑘=1

× 𝑒𝑥𝑝(𝜷′𝒛𝑘 − 𝜀𝑘) × [(𝑥𝑘

𝛾𝑘+ 1)

𝛼𝑘

− 1] − 𝜇 × (∑ 𝑥𝑘

𝐴

𝑘=1

− 𝐵)

where 𝜇 > 0 is the Lagrangian multiplier for the budget constraint.

The observed vector of consumptions (𝒙) must satisfy the Karush-Kuhn-Tucker (KKT)

conditions of optimality given by:

𝜕𝑈(𝑥)

𝜕𝑥𝑘− 𝜆 = 0, 𝑖𝑓 𝑥𝑘 > 0 , ∀ 𝑘

𝜕𝑈(𝑥)

𝜕𝑥𝑘− 𝜆 < 0, 𝑖𝑓 𝑥𝑘 = 0 , ∀ 𝑘

which leads to:

𝑒𝑥𝑝(𝜷′𝒛𝑘 − 𝜀𝑘) × (𝑥𝑘

𝛾𝑘+ 1)

𝛼𝑘−1

− 𝜇 = 0 , ∀ 𝑥𝑘 > 0

𝑒𝑥𝑝(𝜷′𝒛𝑘 − 𝜀𝑘) × (𝑥𝑘

𝛾𝑘+ 1)

𝛼𝑘−1

− 𝜇 < 0 , ∀ 𝑥𝑘 = 0

Bringing 𝜇 to the right-hand side of the equations and then taking logarithm leads to:

𝜷′𝒛𝑘 − 𝜀𝑘 + (𝛼𝑘 − 1) × 𝑙𝑛 (𝑥𝑘

𝛾𝑘+ 1) = 𝑙𝑛(𝜇) , ∀ 𝑥𝑘 > 0

𝜷′𝒛𝑘 − 𝜀𝑘 + (𝛼𝑘 − 1) × 𝑙𝑛 (𝑥𝑘

𝛾𝑘+ 1) < 𝑙𝑛(𝜇) , ∀ 𝑥𝑘 = 0

If 𝑉𝑘 is defined as 𝑉𝑘 = 𝜷′𝒛𝑘 + (𝛼𝑘 − 1) × 𝑙𝑛 (𝑥𝑘

𝛾𝑘+ 1), then the optimality conditions can

be written as:

𝑉𝑘 − 𝜀𝑘 = 𝑙𝑛(𝜇) , ∀ 𝑥𝑘 > 0

𝑉𝑘 − 𝜀𝑘 < 𝑙𝑛(𝜇) , ∀ 𝑥𝑘 = 0

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Let 𝑽𝑐 denote a 𝑀 × 1 vector of 𝑉𝑘 entries for the 𝑀 chosen alternatives (i.e., 𝑥𝑘 > 0, ∀ 𝑘 ∈

𝑽𝑐). Pick the alternative with the least 𝑉𝑘 value in 𝑽𝑐 as the first alternative, i.e., 𝑉1𝑐 = 𝑚𝑖𝑛{𝑽𝑐}.

Also, let 𝑽𝑛𝑐 denote an (𝐴 − 𝑀) × 1 vector of 𝑉𝑘’s of the non-chosen alternatives (i.e., 𝑥𝑘 =

0, ∀ 𝑘 ∈ 𝑽𝑛𝑐) sorted in the increasing order (i.e., 𝑉1𝑛𝑐 < 𝑉2

𝑛𝑐 < ⋯ . 𝑉𝐴−𝑀𝑛𝑐 ). The optimality

conditions can be re-written as:

𝜀𝑘 = 𝑉𝑘 − 𝑉1𝑐 + 𝜀1 , ∀ 𝑥𝑘 > 0

𝜀𝑘 > 𝑉𝑘 − 𝑉1𝑐 + 𝜀1 , ∀ 𝑥𝑘 = 0

The probability that the individual chooses the first 𝑀 alternatives is given by:

𝑃(𝑥1, 𝑥2, … 𝑥𝑀 , 0,0, … ,0)

= |𝐽| ∫ [∏ �̅�(𝑉𝑟𝑛𝑐 − 𝑉1

𝑐 + 𝜀1)

𝐴−𝑀

𝑟=1

] × [∏ 𝑓(𝑉𝑟𝑐 − 𝑉1

𝑐 + 𝜀1)

𝑀

𝑟=2

] × 𝑓(𝜀1) × 𝑑𝜀1

∞

𝜀1=0

where 𝐽 is the Jacobian whose elements are given by:

𝐽𝑟,𝑠 =𝜕[𝑉1

𝑐 − 𝑉𝑟+1𝑐 + 𝜀1]

𝜕𝑥𝑠+1

|𝐽| = (∏1 − 𝛼𝑘

𝑥𝑘 + 𝛾𝑘

𝑀

𝑘=1

) (∑𝑥𝑘 + 𝛾𝑘

1 − 𝛼𝑘

𝑀

𝑘=1

)

Equation 3-2

where 𝑟, 𝑠 ∈ [1, 𝑀 − 1]. This differs slightly from the definition of the determinant of the Jacobian

from (Bhat, 2008), since this assumes a unitary price for any good, provided that every unit of time

allocated in any activity has the same cost across alternatives, but can be easily expanded to include

different prices for each alternative as presented in (Bhat, 2008). Also, the definition the probability

density function (𝑓) and the cumulative distribution function (�̅�) correspond to:

𝑓(𝑉𝑟𝑐 − 𝑉1

𝑐 + 𝜀1) = 𝜆𝑟𝑐 × 𝑒−𝜆𝑟

𝑐×(𝑉𝑟𝑐−𝑉1

𝑐+𝜀1).1

1 𝑓(𝑉𝑟

𝑐 − 𝑉1𝑐 + 𝜀1) is never zero because 𝑉𝑟

𝑐 − 𝑉1𝑐 is always greater than 0 given that 𝑉1

𝑐 = 𝑚𝑖𝑛{𝑽𝑐}.

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�̅�(𝑉𝑟𝑛𝑐 − 𝑉1

𝑐 + 𝜀1) = 𝑃(𝜀𝑘 > 𝑉𝑘 − 𝑉1𝑐 + 𝜀1) = {

1 𝑖𝑓 𝑉𝑟𝑛𝑐 − 𝑉1

𝑐 + 𝜀1 < 0

𝑒−𝜆𝑟𝑛𝑐×(𝑉𝑟

𝑛𝑐−𝑉1𝑐+𝜀1) 𝑖𝑓 𝑉𝑟

𝑛𝑐 − 𝑉1𝑐 + 𝜀1 ≥ 0

The vector 𝑽𝑛𝑐 can be split into two parts such that the first 𝑅 entries are lower than 𝑉1𝑐

and the remaining (𝐴 − 𝑀 − 𝑅) entries are greater than 𝑉1𝑐 as follows:

−∞ ≤ 𝑉1𝑛𝑐 ≤ 𝑉2

𝑛𝑐 ≤ ⋯ 𝑉𝑅𝑛𝑐 ≤ 𝑉1

𝑐 ≤ 𝑉𝑅+1𝑛𝑐 ≤ ⋯ 𝑉𝐴−𝑀

𝑛𝑐

Subtracting each element of the above inequality from 𝑉1𝑐 implies the following:

∞ ≥ 𝑉1𝑐 − 𝑉1

𝑛𝑐 ≥ 𝑉1𝑐 − 𝑉2

𝑛𝑐 ≥ ⋯ ≥ 𝑉1𝑐 − 𝑉𝑅

𝑛𝑐 ≥ 0 ≥ 𝑉1𝑐 − 𝑉𝑅+1

𝑛𝑐 ≥ ⋯ ≥ 𝑉1𝑐 − 𝑉𝐴−𝑀

𝑛𝑐

Therefore, the likelihood function can be written as the sum of (𝑅 + 1) integrals:

𝑃(𝑥1, 𝑥2, … 𝑥𝑀 , 0,0, . .0) =

|𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟

𝑛𝑐−𝑉1𝑐+𝜀1)𝐴−𝑀

𝑟=𝑅+1 −∑ 𝜆𝑟𝑐×(𝑉𝑟

𝑐−𝑉1𝑐+𝜀1)𝑀

𝑟=1 𝑑𝜀1

𝑉1𝑐−𝑉𝑅

𝑛𝑐

𝜀1=0

+

|𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟


𝑟=𝑅 −∑ 𝜆𝑟𝑐×(𝑉𝑟


𝑟=1 𝑑𝜀1

𝑉1𝑐−𝑉𝑅−1

𝑛𝑐

𝜀1=𝑉1𝑐−𝑉𝑅

𝑛𝑐

+ ⋯ +

|𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟


𝑟=2 −∑ 𝜆𝑟𝑐×(𝑉𝑟


𝑟=1 𝑑𝜀1

𝑉1𝑐−𝑉1

𝑛𝑐

𝜀1=𝑉1𝑐−𝑉2

𝑛𝑐

+

|𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟




𝑟=1 𝑑𝜀1

∞


𝑛𝑐

Then, the probability of the observed consumption vector (𝑥𝑘) can be written as the sum

of (𝑅 + 1) integrals as follows:

𝑃(𝑥1, 𝑥2, … 𝑥𝑀 , 0,0, . .0) =

|𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟




𝑟=1 𝑑𝜀1

𝑉1𝑐−𝑉𝑅

𝑛𝑐

𝜀1=0

+

15

|𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟




𝑟=1 𝑑𝜀1

𝑉1𝑐−𝑉𝑅−1

𝑛𝑐


𝑛𝑐

+ ⋯ +

|𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟




𝑟=1 𝑑𝜀1

𝑉1𝑐−𝑉1

𝑛𝑐


𝑛𝑐

+

|𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟




𝑟=1 𝑑𝜀1

∞


𝑛𝑐

Define 𝐺(𝑠) as:

𝐺(𝑠) =𝑒− ∑ 𝜆𝑟

𝑛𝑐×(𝑉𝑟𝑛𝑐−𝑉𝑅−𝑠+1

𝑛𝑐 )𝐴−𝑀𝑟=𝑅−𝑠+2 −∑ 𝜆𝑟

𝑐×(𝑉𝑟𝑐−𝑉𝑅−𝑠+1

𝑛𝑐 )𝑀𝑟=1

∑ 𝜆𝑟𝑛𝑐𝐴−𝑀

𝑟=𝑅−𝑠+2 + ∑ 𝜆𝑟𝑐𝑀

𝑟=1

For instance,

𝐺(1) =𝑒− ∑ 𝜆𝑟

𝑛𝑐×(𝑉𝑟𝑛𝑐−𝑉𝑅

𝑛𝑐)𝐴−𝑀𝑟=𝑅+1 −∑ 𝜆𝑟

𝑐×(𝑉𝑟𝑐−𝑉𝑅

𝑛𝑐)𝑀𝑟=1


𝑟=𝑅+1 +∑ 𝜆𝑟𝑐𝑀

𝑟=1

𝐺(2) =𝑒− ∑ 𝜆𝑟

𝑛𝑐×(𝑉𝑟𝑛𝑐−𝑉𝑅−1

𝑛𝑐 )𝐴−𝑀𝑟=𝑅 −∑ 𝜆𝑟

𝑐×(𝑉𝑟𝑐−𝑉𝑅−1



𝑟=𝑅 +∑ 𝜆𝑟𝑐𝑀

𝑟=1

𝐺(3) =𝑒− ∑ 𝜆𝑟


𝑛𝑐 )𝐴−𝑀𝑟=𝑅−1 −∑ 𝜆𝑟




𝑟=𝑅−1 +∑ 𝜆𝑟𝑐𝑀

𝑟=1

𝐺(𝑅) =𝑒− ∑ 𝜆𝑟

𝑛𝑐×(𝑉𝑟𝑛𝑐−𝑉1

𝑛𝑐)𝐴−𝑀𝑟=2 −∑ 𝜆𝑟


𝑐 )𝑀𝑟=1


𝑟=2 +∑ 𝜆𝑟𝑐𝑀

𝑟=1

First integral:

|𝐽| × [ ∏ 𝜆𝑟𝑐𝑀

𝑟=1 ] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟




𝑟=1 𝑑𝜀1𝑉1

𝑐−𝑉𝑅𝑛𝑐

𝜀1=0

= −𝑒− ∑ 𝜆𝑟

𝑛𝑐×(𝑉𝑟𝑛𝑐−𝑉𝑅

𝑛𝑐)𝐴−𝑀𝑟=𝑅+1 −∑ 𝜆𝑟

𝑐×(𝑉𝑟𝑐−𝑉𝑅

𝑛𝑐)𝑀𝑟=1



𝑟=1+

𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟

𝑛𝑐−𝑉1𝑐 )𝐴−𝑀


𝑐−𝑉1𝑐 )𝑀

𝑟=1



𝑟=1

= −𝐺(1) +𝑒− ∑ 𝜆𝑟


𝑐 )𝐴−𝑀𝑟=𝑅+1 −∑ 𝜆𝑟


𝑐)𝑀𝑟=1



𝑟=1

16

Second integral:

|𝐽| × [ ∏ 𝜆𝑟𝑐𝑀

𝑟=1 ] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟





𝑐−𝑉𝑅−1𝑛𝑐


𝑛𝑐

= −𝑒− ∑ 𝜆𝑟







𝑟=1+


𝑛𝑐−𝑉𝑅𝑛𝑐)𝐴−𝑀


𝑐−𝑉𝑅𝑛𝑐)𝑀

𝑟=1



𝑟=1

= −𝑒− ∑ 𝜆𝑟







𝑟=1+


𝑛𝑐−𝑉𝑅𝑛𝑐)𝐴−𝑀


𝑐−𝑉𝑅𝑛𝑐)𝑀

𝑟=1



𝑟=1

= −𝐺(2) +∑ 𝜆𝑟

𝑛𝑐𝐴−𝑀𝑟=𝑅+1 +∑ 𝜆𝑟

𝑐𝑀𝑟=1



𝑟=1× 𝐺(1)

Third integral:

|𝐽| × [ ∏ 𝜆𝑟𝑐𝑀

𝑟=1 ] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟


𝑟=𝑅−1 −∑ 𝜆𝑟𝑐×(𝑉𝑟



𝑐−𝑉𝑅−2𝑛𝑐

𝜀1=𝑉1𝑐−𝑉𝑅−1

𝑛𝑐

= −𝑒− ∑ 𝜆𝑟


𝑛𝑐 )𝐴−𝑀𝑟=𝑅−1 −∑ 𝜆𝑟





𝑟=1+


𝑛𝑐−𝑉𝑅−1𝑛𝑐 )𝐴−𝑀

𝑟=𝑅−1 −∑ 𝜆𝑟𝑐×(𝑉𝑟

𝑐−𝑉𝑅−1𝑛𝑐 )𝑀

𝑟=1



𝑟=1

= −𝐺(3) +∑ 𝜆𝑟

𝑛𝑐𝐴−𝑀𝑟=𝑅 +∑ 𝜆𝑟

𝑐𝑀𝑟=1



𝑟=1× 𝐺(2)

(𝑅 + 1)𝑡ℎ integral:

|𝐽| × [ ∏ 𝜆𝑟𝑐𝑀

𝑟=1 ] × ∫ 𝑒− ∑ 𝜆𝑟𝑛𝑐×(𝑉𝑟




𝑟=1 𝑑𝜀1∞


𝑛𝑐

=𝑒− ∑ 𝜆𝑟


𝑛𝑐)𝐴−𝑀𝑟=1 −∑ 𝜆𝑟


𝑛𝑐)𝑀𝑟=1



𝑟=1=



𝑟=1



𝑟=1× 𝐺(𝑅)

The likelihood function for the MDCEC under no outside good conditions can be written

in its closed form as follows:

17

𝑃(𝑥1, 𝑥2, … , 𝑥𝑀 , 0,0, … ,0)

= |𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

]

× {𝑒− ∑ 𝜆𝑟


𝑐)𝐴−𝑀𝑟=𝑅+1 −∑ 𝜆𝑟


𝑐)𝑀𝑟=1


𝑟=𝑅+1 + ∑ 𝜆𝑟𝑐𝑀

𝑟=1

− ∑ 𝐺(𝑠)

𝑅

𝑠=1

+ ∑ (∑ 𝜆𝑟

𝑛𝑐𝐴−𝑀𝑟=𝑅−𝑠+2 + ∑ 𝜆𝑟

𝑐𝑀𝑟=1


𝑟=𝑅−𝑠+1 + ∑ 𝜆𝑟𝑐𝑀

𝑟=1

× 𝐺(𝑠))

𝑅

𝑠=1

}

Equation 3-3

In the special case when 𝑅 = 0, the likelihood function simplifies even further to:

𝑃(𝑥1, 𝑥2, … , 𝑥𝑀 , 0,0, … ,0) = |𝐽| × [∏ 𝜆𝑟𝑐

𝑀

𝑟=1

] × {𝑒− ∑ 𝜆𝑟


𝑐)𝐴−𝑀𝑟=𝑅+1 −∑ 𝜆𝑟


𝑐)𝑀𝑟=1


𝑟=𝑅+1 + ∑ 𝜆𝑟𝑐𝑀

𝑟=1

}

In some cases, there is an external alternative that is always consumed, this is known in the literature

as an outside good (Kim et al., 2002). In the activity-time use context, this can be illustrated with

activities performed at home (e.g., sleeping, daily personal care, cooking at home, etc.). When

outside goods are considered, the outside goods are labeled as the first goods and usually one of

those is considered as the base alternative with a unitary price. Then, the total utility function is

modified into (Bhat, 2008):

𝑈(𝑥) = ∑𝛾𝑘

𝛼𝑘exp(𝜷′𝒛𝑘 − 𝜀𝑘) [𝑥𝑘

𝛼𝑘 − 1]

𝐴𝑂𝐺

𝑘=1

+ ∑𝛾𝑘

𝛼𝑘exp(𝜷′𝒛𝑘 − 𝜀𝑘) [(

𝑥𝑘

𝛾𝑘+ 1)

𝛼𝑘

− 1]

𝐴

𝑘=𝐴𝑂𝐺+1

where 𝐴𝑂𝐺 is the number of alternatives considered as outside goods and the rest of the formulation

described above stands. The determinant of the Jacobian from Equation 3-2 still holds, as well as

the closed form of the likelihood function from Equation 3-3.

All conditions presented above account for heteroscedastic scale parameters for the

stochastic component (𝜀𝑘) across alternatives (i.e., 𝜆𝑖 ≠ 𝜆𝑗 for some 𝑖 ≠ 𝑗 ∈ {1, 𝐴}), this implies

that the distribution of the error terms for the alternatives may differ for at least one pair. However,

there may also be the case in which all the error terms are independent and identically distributed

18

(i.i.d.) (i.e., 𝜆𝑖 = 𝜆𝑗 ∀ 𝑖 ≠ 𝑗 ∈ {1, 𝐴}), which is known in the literature as the homoscedastic case.

For both cases, the likelihood function from Equation 3-3 is valid and the homoscedastic likelihood

function collapses to a much simpler form where all the 𝜆 values are the same.

Chapter 4

Simulation analysis

In order to evaluate the ability of the proposed model to properly estimate the true values

of the parameters, different synthetic datasets were generated and the estimates from the MDCEC

model were compared to the known true values of the parameters that generated the synthetic data.

The parameters of the model were estimated using the maximum likelihood inference method under

the likelihood specification from Equation 3-3. The details on how the synthetic data was generated

and the ability of the MDCEC model to estimate these values are presented in detail in this section.

Four different cases were estimated, specifically, homoscedastic scale parameters without

outside good, heteroscedastic scale parameters without outside good, homoscedastic scale

parameters with outside good, heteroscedastic scale parameters with outside good. For each case,

100 synthetic datasets were generated including 5,000 decision makers (𝑄 = 5,000) and three

alternatives (𝐴 = 3) each.

The partial (𝑢𝑘) and total (𝑈) utility equations for the data generation for the

heteroscedastic case without outside good, based on Equation 3-1 are constructed as follows:

𝑢𝑥(𝑥𝑘) =𝛾𝑘

𝛼𝑘× 𝑒𝑥𝑝(𝛽′𝑧𝑘 − 𝜀𝑘) × [(

𝑥𝑘

𝛾𝑘+ 1)

𝛼𝑘

− 1]

𝑢1(𝑥1) = 𝑒1 × 𝑒𝑥𝑝(1.0 ∗ 𝑧1 + 0.5 ∗ 𝑧2 − 𝜀1) × [𝑥1

𝑒1]

𝑢2(𝑥2) = 𝑒2 × 𝑒𝑥𝑝(1.0 ∗ 𝑧4 + 0.5 ∗ 𝑧5 – 3.0 ∗ 𝑧3 + 0.5 − 𝜀2) × [𝑥2

𝑒2]

𝑢3(𝑥3) = 𝑒3 × 𝑒𝑥𝑝(1.0 ∗ 𝑧6 + 0.5 ∗ 𝑧7 – 3.0 ∗ 𝑧3 – 1.0 − 𝜀3) × [𝑥3

𝑒3]

𝑈(𝒙) = 𝑢1(𝑥1) + 𝑢2(𝑥2) + 𝑢3(𝑥3)

where the 𝑧 variables are drawn from a specific random normal distribution that does not change

across datasets and serve as predictors or deterministic generators of the data. The stochastic term

is defined as 𝜀𝑘~ 𝐸𝑥𝑝(𝜆𝑘) with 𝜆 = {1,1,1} in the homoscedastic case and 𝜆 = {1,2,3} in the

20

heteroscedastic case, and varies across alternatives and datasets for both cases. The stochasticity

varies for the 𝑧 and 𝜀𝑘 for each decision maker.

The consumption vector, 𝑥𝑘 = {0,0,0} in the initial stage and is increased and assigned to

the alternative with the highest utility in that step; the process is repeated continuously until the

available budget (𝐵 = 100) is depleted. Both satiation parameters from Equation 3-1 cannot be

estimated simultaneously, therefore, alpha is set to 1 to estimate the gamma values, which is known

in the literature as a 𝛾𝑘-profile (Bhat, 2008).

The mean (�̅�) and standard deviation (𝜎) from each of the parameters being estimated were

calculated for each dataset and summarized across the 100 datasets for each case. To evaluate the

performance of the model under each scenario, the following metrics were calculated to compare

the model estimation to the known true value of the parameter (𝛽) that generated the data: 1) the

absolute percent biases (APB), calculated as |�̅�−𝛽

𝛽| and reported as a percentage indicating the

extent of bias, 2) the mean APB across all parameters, which is equivalent to the mean absolute

percent error (MAPE), and 3) the t-statistic calculated as �̅�−𝛽

𝜎 to determine if the bias is statistically

significant (abosulte values less than 1.96 denote a 95% confidence interval of no difference)

(Paleti, 2020).

The results from the synthetic data estimation are presented in Table 4-1 for the four

analyzed cases. The maximum likelihood inference approach successfully retrieves all parameters

from the models, as suggested by the low bias percentages from the ABP and the t-statistic less

than 1.96, showing no statistical difference between the estimations and the true value of the

parameters.

Different values of the same class of parameters (i.e., different 𝛾 and 𝜆 values) were

implemented to demonstrate that the implemented MDCEC model performs well and also retrieves

the correct parameter values under a true heteroscedastic process.

21

Table 4-1. Estimation results from the MDCEC on synthetic data.

Parameter

to estimate

True

value

Homoscedastic Heteroscedastic

Mean

estimate APB t-stat Mean

estimate APB t-stat

No

Outs

ide G

ood

𝛽1 1 1.000 0.01% 0.006 1.000 0.01% -0.004

𝛽2 0.5 0.499 0.23% 0.106 0.500 0.00% -0.001

𝛽3 -3 -2.999 0.05% -0.039 -3.001 0.03% 0.011

𝐴𝑆𝐶2 0.5 0.493 1.47% 0.135 0.500 0.04% 0.002

𝐴𝑆𝐶3 -1 -1.005 0.51% 0.116 -1.001 0.09% 0.013

ln (𝛾1) 1

ln (𝛾2) 2 2.006 0.32% -0.122 2.000 0.01% 0.002

ln (𝛾3) 3 3.005 0.17% -0.122 3.001 0.02% -0.008

ln (𝜆1) 1

ln (𝜆2) 2 2.003 0.16% -0.020

ln (𝜆3) 3 3.001 0.05% -0.005

MAPE 0.42% 0.05%

Log-

likelihood -24830.4 -15119.7

With

Ou

tsid

e G

oo

d

𝛽1 1 1.001 0.13% -0.133 1.000 0.02% -0.012

𝛽2 0.5 0.499 0.15% 0.076 0.500 0.02% -0.008

𝛽3 -3 -2.997 0.09% -0.190 -3.000 0.01% -0.016

𝐴𝑆𝐶2 0.5 0.501 0.27% -0.025 0.500 0.03% 0.001

𝐴𝑆𝐶3 -1 -0.998 0.22% -0.070 -0.999 0.08% -0.018

ln (𝛾1) -100

ln (𝛾2) 1 1.000 0.04% 0.007 1.000 0.04% -0.004

ln (𝛾3) 2 1.998 0.09% 0.042 1.999 0.05% 0.014

ln (𝜆1) 1

ln (𝜆2) 2 2.003 0.17% -0.019

ln (𝜆3) 3 3.000 0.01% 0.001

MAPE 0.14% 0.06%

Log-

likelihood -28315.9 -14306.5

Chapter 5

Empirical application

The developed MDCEC model was additionally used to estimate the time allocation in

different out-of-home activities in dual-earner households which are considered to have time

poverty, i.e., very limited available time to spend in leisure, sport, relaxation, and social activities

(Bernardo et al., 2015; Paleti & Vukovic, 2017). The data used is based on the study by Paleti and

Vukovic (Paleti & Vukovic, 2017) regarding activity-time use and telecommuting patterns.

The cases were filtered to only include households where at least one of the spouses had

the option to engage in telecommuting activities and those in which both workers had fixed working

places. Households surveyed during weekends and with missing or incomplete information were

filtered out from the sample (Paleti & Vukovic, 2017). After applying these filters, the sample is

composed of 4,222 workers from 2,111 households with detailed records for travel diary and

activity purpose that were aggregated in eight categories as: home, work, escorting children,

shopping, maintenance, eating out, visiting, and discretionary.

As expected, workers spend a lot of time at home, especially when they have the option to

telecommute. Table 5-1 shows that over 94% of the time reported in the activities is spent at home

or working, which reinforces the concept of time poverty, where they only have 78 minutes on

average every day to allocate in out-of-home activities. The rest of activities aside from being at

home or working, do not have much absolute difference on average, but in comparative difference,

discretionary activities take around twice the time as shopping or eating out or even four times the

average time spent escorting children. More importantly, these differences are estimated for each

individual using MDC models and can be later aggregated for travel demand purposes.

23

Table 5-1. Summary statistics of time allocation for each type of activity.

Activity Mean (min) Std Dev. (min) Percentage of use

Home 906.52 227.68 67.2%

Work 365.40 228.36 27.1%

Escorting children 6.28 29.64 0.5%

Shopping 13.20 36.17 1.0%

Maintenance 12.46 40.64 0.9%

Eating out 13.12 34.20 1.0%

Visiting 7.41 40.25 0.5%

Discretionary 25.58 65.21 1.9%

The time allocations from Table 5-1 were estimated using the predictors shown in Table 5-

2, which explain mostly person attributes and some household social demographics (Paleti &

Vukovic, 2017). The data used is balanced for sex and contains mainly highly educated individuals,

with high income, working full time, and with flexibility to work. Other important factors included

in the analyses are the immigration status, vehicle sufficiency (defined as the ratio of number of

vehicles by driver-aged adults in the household) and commuting variables, like commuting distance

and frequency of telecommuting.

24

Table 5-2. Summary statistics of explanatory variables for MDC modeling.

Explanatory variable Level

0: base 1 2 Sex [base: female, male] 0.50 0.50

Disability status [base: no, yes] 0.98 0.02

Job category [base: other jobs, sales and services] 0.87 0.13

Has work flexibility [base: no, yes] 0.32 0.68

Vehicle sufficiency [base: equal, lower, higher] 0.64 0.04 0.32 Income [base: low, mid, high] 0.02 0.15 0.84 Has Bachelor’s degree or higher [base: no, yes] 0.31 0.69

Immigration status [base: immigrant, US-born citizen] 0.11 0.89

Works full time [base: no, yes] 0.15 0.85

Safety concerns are a big issue [base: no, yes] 0.95 0.05

Explanatory variable Mean Std. Dev. Min Max

Age (years/100) 0.46 0.10 0.16 0.86

Commute distance (mi/100) 0.16 0.20 0 5.62

Telecommuting frequency (days in a month) 1.77 3.74 0 31

Number of children aged 0 to 5 0.05 0.21 0 2



Because all individuals spend time in activities at home, at-home-activities can be

considered as an outside good from the out-of-home activities, which also improves the accuracy

of the estimations. Additionally, given the complex definition of MDCEV under heteroscedastic

conditions, MDCEC and MDCEV models are compared under homoscedastic assumptions and

considering at-home activities as an outside good. Because of space limitations, the results from

this comparison are presented in Table 5-3 for the MDCEC and Table 5-4 for the MDCEV

separately.

The log-likelihood is used as a measure of goodness of fit of the model to the empirical

data; both models have the 56 estimated parameters (seven constants, 42 explanatory variables, and

seven translation parameters (𝛾𝑘)), although both models have the same number of parameters, the

Bayes Information Criterion (BIC) is used to compare the models instead of the log-likelihood to

extend the comparison to other models using the following definition:

25

𝐵𝐼𝐶 = −2 ln(𝐿𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑) + 𝑘 ∙ 𝑙𝑛(𝑛)

where 𝐿 is the likelihood, 𝑘 is the number of free parameters being estimated and 𝑛 is the number

of observations in the sample. Since BIC penalizes for the number of parameters, the lower value

is preferred; a difference of over 10 units in the BIC can be considered as very strong or decisive

(Fabozzi, F. J., Focardi, S. M., Rachev, S. T., & Arshanapalli, 2014; Kass & Raftery, 1995). The

BIC for the MDCEC homoscedastic is 133,233 and the BIC for the MDCEV is 134,274, a

difference of over 1,000 units, meaning that the MDCEC model fits the empirical data significantly

better than the MDCEV. This finding supports the expected better performance of the MDCEC,

since it accounts better for the behavioral choices of individuals.

The magnitude and sign of the coefficients obtained for the MDCEV is similar to those

found in (Paleti & Vukovic, 2017) using the same data and a very similar model definition

regarding the selection of predictor variables. Only the estimates that are statistically significant to

a 95% confidence level are presented in the MDCEV framework; the estimates that are not

presented for some activities refer to not statistically significant estimates. The selection of

explanatory variables included in the MDCEC approach is not necessarily optimal and is used to

compare both models under identical specifications.

The mean estimated values from the MDCEC tend to be closer to zero, but this does not

necessarily affect the significance of the explanatory variables, because the standard errors are also

reduced. This could indicate that the estimates are less biased (because of the better behavioral

properties of EC) and more accurate (because of lower standard errors) to the real parameter value.

However, this cannot be properly compared for the empirical case because the selection of

predictors and model definition were developed to be optimal for the MDCEV homoscedastic only,

therefore, the model structure used to compare both models could include insignificant predictor

variables and omit others that may have been significant under an optimal MDCEC approach. For

instance, the job description (sales and services compared to other job categories) is significant for

26

MDCEV but not significant for MDCEC, still kept in both models. Overall, the results suggest that

the estimations from the MDCEC are more accurate than those obtained from an MDCEV

approach.

27

Table 5-3. Results from the MDCEC homoscedastic model.

Explanatory variable MDCEC homoscedastic (base: Home)

W E S M EO V D

Constant -6.07 -6.60 -6.86 -7.12 -7.03 -7.33 -7.17

Sex

Male [base: female] 0.05 -0.08 -0.13 -0.03 0.05 0.08

Age (years/100) -0.03 -1.47 -0.02 0.22

Disability status

Yes [base: no] -0.13 -0.28 0.19 -0.23

Has Bachelor’s degree or higher

Yes [base: no] -0.05 0.08 0.04 0.07 0.09

Safety concerns are a big issue

Yes [base: no] 0.07

Immigration status

US-born citizen [base: immigrant] 0.07

Number of children aged 0 to 5 -0.03 0.07 Number of children aged 6 to 10 0.16 0.04

Number of children aged 11 to 15 0.14 0.04

Vehicle sufficiency

Less cars than drivers [base: equal] -0.10 Vehicle sufficiency

More cars than drivers [base:

equal] 0.05

Medium income [base: low] 0.18

High income [base: low] 0.17

Works full time [base: does not] 0.29 0.02 0.04

Has work flexibility [base: does not] 0.02 0.06

Commute distance (mi/100) 0.21 -0.17 Job category

Sales and services [base: other

jobs] 0.02

Telecommuting frequency -0.02 -0.01 -0.01 Translation parameter ln(𝛾𝑘) 5.66 4.23 4.94 5.32 5.19 7.19 6.10

Log-likelihood -66382.6

Note: W=Working, E=Escorting children, S=Shopping, M=Maintenance, EO=Eating Out,

V=Visiting, D=Discretionary.

28

Table 5-4. Results from the MDCEV homoscedastic model.

Explanatory variable MDCEV homoscedastic (base: Home)

W E S M EO V D

Constant -6.28 -7.83 -7.63 -8.93 -8.20 -9.43 -8.43

Sex

Male [base: female] 0.13 -0.49 -0.38 -0.20 0.15 0.19

Age (years/100) -0.78 -3.71 0.58 1.78

Disability status

Yes [base: no] -0.62 -1.18 0.66 -0.89


Yes [base: no] -0.10 0.37 0.15 0.19 0.43


Yes [base: no] 0.39

Immigration status

US-born citizen [base: immigrant] 0.22



Vehicle sufficiency

Less cars than drivers [base: equal] -0.35

Vehicle sufficiency

More cars than drivers [base:

equal] 0.17



Works full time [base: does not] 0.82 -0.21 -0.21


Commute distance (mi/100) -0.39 -0.49

Job category

Sales and services [base: other

jobs] 0.26


Log-likelihood -66903.5



A better fit to the data can be achieved by relaxing the assumption of homoscedastic

stochastic terms across alternatives, i.e., including heteroscedastic scale parameters. This can be

achieved relatively easy for the MDCEC, keeping its closed form presented in Equation 3-3 with

29

minimal changes, unlike the MDCEV, that needs major modifications to account for

heteroscedastic stochasticity. Because of this and the scope of the research, the heteroscedastic

approach is only presented for the MDCEC in Table 5-5, still considering in-home activities as an

outside good. The BIC of the heteroscedastic models is 130,649, which is 2,584 lower than the

homoscedastic MDCEC model and 3,625 units lower than the homoscedastic MDCEV.

Consequently, the heteroscedastic approach provides a significantly better fit to the empirical data

than any of the homoscedastic choice models and stands out the importance of a heteroscedastic

apporach that is easy to estimate.

The coefficients vary in different ways compared to the homoscedastic approach, males

and disability status mean estimates become more negative (except for eating out for males),

meaning that workers with these conditions are less likely to engage in outside activities compared

to females and people without disabilities. Also, safety concerns and the number of children at

home have more impact in the heteroscadastic case to allocate more time in activities outside of

home.

Vehicle sufficiency maintains its trend, as more cars are available, more time is allocated

in activities outside of home and it is more noticeable for household with more available vehicles

when compared to the homoscedastic case. The household income shows the same effect, however,

medium income participate more in escort activities for their children, which is also more common

with a hgher number of children aged 6 to 10 present at home.

As expected, when working full time, more time is allocated into work activities, however,

the effect is better captured in the heteroscedastic case by a higher mean estimate and better

accuracy (t-statistic of 29.2 for heteroscedastic and 16.1 for homoscedastic model). Also, if work

flexiblity is available, more time is allocated in maintenance outside of home and eating out.

30

Table 5-5. Results from the MDCEC heteroscedastic model.

Explanatory variable MDCEC heteroscedastic (base: Home)

W E S M EO V D

Constant -6.06 -6.33 -5.88 -4.37 -5.21 -6.09 -4.68

Sex

Male [base: female] 0.05 -0.26 -0.41 -0.09 -0.02 0.16

Age (years/100) -0.07 -2.98 0.40 0.98

Disability status

Yes [base: no] -0.22 -1.06 0.21 -0.31


Yes [base: no] -0.02 0.20 0.19 0.17 -0.01


Yes [base: no] 0.20

Immigration status

US-born citizen [base: immigrant] -0.04



Vehicle sufficiency

Less cars than drivers [base: equal] -0.07

Vehicle sufficiency

More cars than drivers [base: equal] 0.29



Works full time [base: does not] 0.50 -0.06 -0.02


Commute distance (mi/100) 0.19 -0.79 Job category

Sales and services [base: other jobs] 0.16


Scale parameter ln(𝜆𝑘) -0.12 -1.25 -1.42 -2.58 -2.06 -3.01 -2.27

Log likelihood -65061.4



The MDCEC under heteroscedastic conditions performs the best of all models, more

importantly, the MDCEC outperforms the MDCEV when compared in equal conditions, providing

additional empirical proof of its more adequate specifications for multiple discrete-continuous

31

choice modeling, beyond the theoretical properties and better behavioral modeling from the

MDCEC.

Chapter 6

Concluding remarks

The standard model for multiple discrete-continuous choices, MDCEV, is based on a

Gumbel distribution that is positively skewed and does not describe an adequate choice behavior

from consumers with good information about the available alternatives, which is the most common

case in choice modeling. Multiple discrete-continuous choices models can be improved by

including a more realistic stochastic distribution that reflects the expected behavior of individuals

regarding their willingness to pay (or perceived value of alternatives) when they have access to

more and better information about a different set of alternatives. This study used a negative

exponential distribution for the error components that leads to an exponomial choice (EC) and

attains these better behavioral properties.

The proposed model was verified to successfully retrieve the true value of the parameters

being estimated under different conditions, including and ignoring outside goods and under

homoscedastic and heteroscedastic stochasticity across alternatives.

The MDCEV and MDCEC homoscedastic models were compared in an empirical case of

time allocation for daily activity demand in households with high time poverty, where both spouses

work and at least one of them has the option to telecommute. The MDCEC significantly

outperformed the standard MDCEV and the best overall results were obtained with an MDCEC

considering heteroscedastic scale parameters for the stochastic component, which highlights the

importance of having heteroscedastic MDC models that are easy to implement and estimate.

The proposed model is kept simple and does not account for many other conditions that

may be needed for advanced analyses with the intention to present the basic properties and base

case, however, it has an elegant and convenient closed form that unlike the MDCEV, allows for

33

easy modifications to include to more sophisticated conditions (like heteroscedasticity of the scale

parameters), but can be expanded to include other relevant cases. Future research should focus on

extending the power of the MDCEC models to more advanced conditions, e.g., random (as opposed

to fixed) parameters, panel data, and nested MDCEC. Additionally, the elasticities for MDC models

are not as straightforward as for single choice models, but because of their importance, they also

need to be estimated for the proposed MDCEC.

It is necessary to apply this methodology to other empirical cases, including, but not limited

to, time allocation in daily activities to confirm the empirical evidence of better performance of the

MDCEC compared to MDCEV.

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