How household transportation expenditures have evolved in Canada: a long term perspective

Abstract

In recent years, increasing recognition of the challenges associated with global climate change and inequity in developed countries have revived researcher’s interest towards analyzing transportation related expenditure of households. The current research contributes to travel behaviour literature by developing an econometric model of household budgetary allocations with a particular focus on transportation expenditure. Towards this end, we employ the public-use micro-data extracted from the Survey of Household Spending (SHS) for the years 1997–2009. The proposed econometric modeling approach is built on the multiple discrete continuous extreme value model (MDCEV) framework. Specifically, in our analysis, the scaled version of the MDCEV model outperformed its other counterparts. Broadly, the model results indicated that a host of household socio-economic and demographic attributes along with the residential location characteristics affect the apportioning of income to various expenditure categories and savings. We also observed a relatively stable transportation spending behaviour over time. Additionally, a policy analysis exercise is conducted where we observed that with increase in health expenses and reduction in savings results in adjustments in all expenditure categories.

Appendices

Appendix 1: Mathematical formulation of MDCEV models

We provide a brief formulation of the econometric structure of the traditional MDCEV model and then extend the discussion to the formulation for MMDCEV and SMDCEV.

Basic structure and traditional MDCEV model

It is reasonable to expect that the money allocated to different expenditure categories depends on the marginal utility that households derive from spending in those categories. Let us consider that there are K different expenditure categories that a household can potentially allocate its money to. If x _k represents the allotted non-negative amount of the total budget to each expenditure category k (including savings), the total utility derived from such allocation can be expressed in the following additive non-linear functional form (Bhat 2008):

$$U\left( \varvec{x} \right) = \mathop \sum \limits_{k = 1}^{K} \frac{{\gamma_{k} }}{{\alpha_{k} }}\psi_{k} \left\{ {\left( {\frac{{x_{k} }}{{\gamma_{k} }} + 1} \right)^{{\alpha_{k} }} - 1} \right\}; \psi_{k} > 0, \alpha_{k} \le 1, \gamma_{k} > 0$$

(1)

where $U\left( \varvec{x} \right)$ is a quasi-concave, increasing, and continuously differentiable function with respect to the expenditure quantity (K × 1)—vector x (x _k  ≥ 0 for all k alternatives), γ _k and α _k are parameters associated with alternative k. ψ _k represents the baseline marginal utility. Through this term, the effect of observed and unobserved alternative attributes, decision-maker attributes, and the choice environment attributes may be introduced as ψ _k  = exp(β ^′ z _k  + ɛ _k ), where z _k represents the vector of exogenous variables and ɛ _k captures the idiosyncratic characteristics that affect the baseline utility. γ _k enables corner solutions while simultaneously influencing satiation and α _k influences satiation only. Note that the above utility function is formulated considering absence of outside goods (goods that is always consumed).

If, however, an outside goods is present, the utility function can be modified as follows using the same notational preliminaries:

$$U\left( \varvec{x} \right) = \frac{1}{{\alpha_{1} }}\exp \left( {\varepsilon_{1} } \right)\left\{ {\left( {x_{1} + \gamma_{1} } \right)^{{\alpha_{1} }} } \right\} + \mathop \sum \limits_{k = 2}^{K} \frac{{\gamma_{k} }}{{\alpha_{k} }}exp\left( {\beta^{\prime}z_{k} + \varepsilon_{k} } \right)\left\{ {\left( {\frac{{x_{k} }}{{\gamma_{k} }} + 1} \right)^{{\alpha_{k} }} - 1} \right\}$$

(2)

In the above formula, we need γ ₁ ≤ 0, while γ _k  > 0 for k > 1. Also, we need (x ₁ + γ ₁) > 0. The magnitude of γ ₁ may be interpreted as the required lower bound (or a “subsistence value”) for consumption of the outside goods. In the above baseline parameter expression, the term ɛ ₁ is an idiosyncratic term assumed to be identically and independently standard type I extreme-value distributed across households, as well as independent of the terms in the baseline parameter expression for other alternatives (inside goods).

It is very challenging to identify γ _k and α _k simultaneously in empirical applications for the outside and inside goods (see, Bhat 2008 and Bhat and Eluru 2010 for an elaborate discussion on the issue). Usually, the analyst can choose to estimate satiation using either γ _k or α _k , since these two parameters have similar role in terms of allowing for satiation. Depending on the chosen parameter structure for estimation, different utility structures can be estimated and the selection of the most appropriate form is based on statistical considerations.

If only γ _k parameters are estimated the utility simplifies to γ-profile

$$U\left( \varvec{x} \right) = \exp \left( {\varepsilon_{1} } \right)\ln \left\{ {\left( {x_{1} + \gamma_{1} } \right)} \right\} + \mathop \sum \limits_{k = 2}^{K} \gamma_{k} exp\left( {\beta^{\prime}z_{k} + \varepsilon_{k} } \right)\left\{ {ln\left( {\frac{{x_{k} }}{{\gamma_{k} }} + 1} \right)} \right\}$$

(3)

Similarly, if only α _k parameter are estimated, the corresponding utility expression collapses to α-profile

$$U\left( \varvec{x} \right) = \frac{1}{{\alpha_{1} }}\exp \left( {\varepsilon_{1} } \right)\left\{ {x_{1}^{{\alpha_{1} }} } \right\} + \mathop \sum \limits_{k = 2}^{K} \frac{1}{{\alpha_{k} }}exp\left( {\beta^{\prime}z_{k} + \varepsilon_{k} } \right)\left\{ {\left( {x_{k} + 1} \right)^{{\alpha_{k} }} - 1} \right\}$$

(4)

Let V _k be the alternative utility. The expressions for V _k for γ-profile and α-profile utility forms are as below:

$$V_{k} = \beta^{\prime}z_{k} - ln\left( {\frac{{x_{k}^{*} }}{{\gamma_{k} }} + 1} \right) - ln p_{k} \left( {k \ge 2} \right); V_{1} = - ln\left( {x_{1} + \gamma_{1} } \right)$$

(5)

$$V_{k} = \beta^{\prime}z_{k} + \left( {\alpha_{k} - 1} \right)\ln \left( {x_{k}^{*} + 1} \right) - ln p_{k} \left( {k \ge 2} \right); V_{1} = \left( {\alpha_{1} - 1} \right)ln\left( {x_{1}^{*} } \right)$$

(6)

We would assume (following Bhat 2005, 2008) that ɛ _k ’s are independently and identically distributed across alternatives with a scale parameter of σ. Given the values of the alternative utilities for the two profiles, the probability expression for the expenditure allocation to the first M of the K goods (M ≥ 1) is:

$$P\left( {e_{1}^{*} , e_{2}^{*} , e_{3}^{*} , \ldots ,e_{M}^{*} ,0,0,0, \ldots ,0} \right) = \frac{1}{{\sigma^{M - 1} }}\left[ {\mathop \prod \limits_{i = 1}^{M} C_{i} } \right]\left[ {\mathop \sum \limits_{i = 1}^{M} \frac{1}{{C_{i} }}} \right]\left[ {\frac{{\mathop \prod \nolimits_{i = 1}^{M} \exp^{{\frac{{V_{i} }}{\sigma }}} }}{{\left( {\mathop \sum \nolimits_{k = 1}^{K} \exp^{{\frac{{V_{k} }}{\sigma }}} } \right)^{M} }}} \right]\left( {M - 1} \right)!$$

(7)

where $C_{i} = \frac{{1 - \alpha_{i} }}{{e_{i}^{*} + \gamma_{i} p_{i} }}$. In the traditional MDCEV model, the scale parameter $\sigma$ is set to 1 for normalization.

Scaled MDCEV model

In our context, due to the inherent differences across the expenditure databases across years and different economic conditions, we can estimate the scale parameter provided we normalize σ for 1 year. The σ is parameterized as ${ \exp }\left( {\delta y} \right)$ where y is the vector of time elapsed variable as well as the annual economic indicators and δ is the corresponding coefficient vector to be estimated. The δ parameters are significant when they are different from 0 as that would imply that the scale parameter will be different from 1. The same expression in Eq. 7 is adopted with the appropriate σ for probability and likelihood computations.

Mixed MDCEV model

The mixed MDCEV model accommodates unobserved heterogeneity in the effect of exogenous variables (random coefficients structure) and correlations across alternatives (error correlations structure). The baseline parameter expression for the inside alternatives in Eq. 2 can be expressed as follows:

$$\psi_{k} = exp\left\{ {(\beta_{k}^{'} + \alpha_{k}^{'} ) z_{k} + \eta^{\prime}w_{k} + \xi_{k} } \right\}$$

(8)

In the above equation, β ^′ and α ^′ are column vector of parameters, where β ^′ represents the mean effect and α ^′ represents household level disturbance of the coefficient. The term η ^′ w _k constitutes the mechanism to generate household level correlation across unobserved utility components of the alternatives. In this component, w _k is specified to be a column vector of dimension H with each row representing a group h (h = 1, 2, …, H) of alternatives sharing common household-specific unobserved components and the vector η ^′ may be specified as a H-dimensional realization from a multivariate normally distributed random vector η, $\eta \,\sim\,N\left( {0,\varOmega } \right).$ As before, the component, ξ _k is assumed to be independently and identically Gumbel distributed across households. For complete formulation of likelihood for the MMDCEV model see, Bhat and Eluru (2010).

The parameters of MMDCEV model are estimated using maximum simulated likelihood procedure. Specifically, scrambled Halton sequence is used to draw realizations from the population normal distribution. In this research, the stability of the parameter estimates was tested using varying number of Halton draws per observation for the specifications considered, and the results were found to be stable with 100 draws.

Appendix 2: Dependent variable definition

See Table 3.

Table 3 Dependent variables