Robust discrete choice models with t-distributed kernel errors

Abstract

Outliers in discrete choice response data may result from misclassification and misreporting of the response variable and from choice behaviour that is inconsistent with modelling assumptions (e.g. random utility maximisation). In the presence of outliers, standard discrete choice models produce biased estimates and suffer from compromised predictive accuracy. Robust statistical models are less sensitive to outliers than standard non-robust models. This paper analyses two robust alternatives to the multinomial probit (MNP) model. The two models are robit models whose kernel error distributions are heavy-tailed t-distributions to moderate the influence of outliers. The first model is the multinomial robit (MNR) model, in which a generic degrees of freedom parameter controls the heavy-tailedness of the kernel error distribution. The second model, the generalised multinomial robit (Gen-MNR) model, is more flexible than MNR, as it allows for distinct heavy-tailedness in each dimension of the kernel error distribution. For both models, we derive Gibbs samplers for posterior inference. In a simulation study, we illustrate the finite sample properties of the proposed Bayes estimators and show that MNR and Gen-MNR produce more accurate estimates if the choice data contain outliers through the lens of the non-robust MNP model. In a case study on transport mode choice behaviour, MNR and Gen-MNR outperform MNP by substantial margins in terms of in-sample fit and out-of-sample predictive accuracy. The case study also highlights differences in elasticity estimates across models.

Appendices

Appendix A Gibbs sampling details

1.1 A.1 Sampling $\varvec{w}$

To update $\varvec{w}$, we iteratively sample from univariate truncated normal distributions. We have

$$\begin{aligned} w_{ij} \sim TN(\mu _{ij}, \tau _{ij}^{2}),\text {for } i = 1, \ldots , N, j = 1, \ldots , J-1. \nonumber \\ \end{aligned}$$

(10)

For MNP, $\mu _{ij} = \varvec{X}_{ij}^{\top } \varvec{\beta } + \varvec{\Sigma }_{j,-j} \varvec{\Sigma }_{-j,-j}^{-1} (w_{i,-j} - \varvec{X}_{i,-j} \varvec{\beta })$ and $\tau _{ij}^{2} = \varvec{\Sigma }_{jj} - \varvec{\Sigma }_{j,-j} \varvec{\Sigma }_{-j,-j}^{-1} \varvec{\Sigma }_{-j,j}$. For MNR, $\mu _{ij} = \varvec{X}_{ij}^{\top } \varvec{\beta } + \varvec{\Sigma }_{j,-j} \varvec{\Sigma }_{-j,-j}^{-1} (w_{i,-j} - \varvec{X}_{i,-j} \varvec{\beta })$ and $\tau _{ij}^{2} = (\varvec{\Sigma }_{jj} - \varvec{\Sigma }_{j,-j} \varvec{\Sigma }_{-j,-j}^{-1} \varvec{\Sigma }_{-j,j}) / q_{i}$. For Gen-MNR, $\mu _{ij} = \varvec{X}_{ij}^{\top } \varvec{\beta } + \varvec{Q}_{ijj}^{-1/2} \varvec{\Sigma }_{j,-j} \varvec{\Sigma }_{-j,-j}^{-1} \varvec{Q}_{i,-j,-j}^{1/2} (w_{i,-j} - \varvec{X}_{i,-j} \varvec{\beta })$ and $\tau _{ij}^{2} = (\varvec{\Sigma }_{jj} - \varvec{\Sigma }_{j,-j} \varvec{\Sigma }_{-j,-j}^{-1} \varvec{\Sigma }_{-j,j}) / q_{ij}$. Here, the index $-l$ denotes the vector without the lth element. For all models, the constraint on $w_{ij}$ is $w_{ij} \ge \max \{ 0, w_{i,-j} \}$, if $y_{ij} = j$; $w_{ij} < 0$, if $y_{ij} = J$; $w_{ij} \le \max \{ 0, w_{ij'} \}$, if $y_{ij} = j' \ne j$.

1.2 A.2 Sampling $\nu$

The full conditional distribution of $\nu$ is nonstandard. Ding (2014) shows that

$$\begin{aligned}{} & {} p(\nu \vert \cdot ) \propto \exp \nonumber \\{} & {} \quad \left\{ \frac{N \nu }{2} \log \left( \frac{\nu }{2} \right) - N \log \Gamma \left( \frac{\nu }{2} \right) + (\alpha _{0} - 1) \log \nu - \xi \nu \right\} ,\nonumber \\ \end{aligned}$$

(11)

where $\xi = \beta _{0} + \frac{1}{2} \sum _{i = 1}^{N} q_{i} - \frac{1}{2} \sum _{i = 1}^{N} \log q_{i}$. $\Gamma (x)$ denotes the Gamma function. Ding (2014) proposes to sample from (11) using a Metropolised Independence sampler (Liu 2008) with an approximate Gamma proposal. The shape parameter $\alpha ^{*}$ and the rate parameter $\beta ^{*}$ of the proposal density are obtained as follows. The log conditional density of $\nu$ up to an additive constant is

$$\begin{aligned} l(\nu ) \!=\! \frac{N \nu }{2} \log \left( \frac{\nu }{2} \right) \!-\! N \log \Gamma \left( \frac{\nu }{2} \right) \!+\! (\alpha _{0} \!-\! 1) \log \nu - \xi \nu .\nonumber \\ \end{aligned}$$

(12)

The log density of the Gamma proposal is

$$\begin{aligned} h(\nu ) = (\alpha ^{*} - 1) \log \nu - \beta ^{*} \nu . \end{aligned}$$

(13)

The first and second derivates of $l(\nu )$ and $h(\nu )$ are

$$\begin{aligned} l'(\nu )= & {} \frac{N}{2} \left[ \log \left( \frac{\nu }{2} \right) + 1 - \psi \left( \frac{\nu }{2} \right) \right] + \frac{\alpha _{0} - 1}{\nu } - \xi , \nonumber \\ h'(\nu )= & {} \frac{\alpha ^{*} - 1}{\nu } - \beta ^{*}, \end{aligned}$$

(14)

$$\begin{aligned} l''(\nu )= & {} \frac{N}{2} \left[ \frac{1}{\nu } - \frac{1}{2} \psi ' \left( \frac{\nu }{2} \right) \right] + \frac{\alpha _{0} - 1}{\nu ^{2}}, \nonumber \\ h''(\nu )= & {} - \frac{\alpha ^{*} - 1}{\nu ^{2}}, \end{aligned}$$

(15)

where $\psi (x)$ and $\psi '(x)$ are the di- and trigamma functions, respectively. The mode of $h(\nu )$ is $\frac{\alpha ^{*} - 1}{\beta ^{*}}$ and the corresponding curvature is $\frac{(\beta ^{*})^{2}}{\alpha ^{*} - 1}$. We numerically find the mode $\nu ^{*}$ of $l(\nu )$ and its corresponding curvature $l^{*} = l''(\nu ^{*})$. Ultimately, we match the modes and the corresponding curvatures of $l(\nu )$ and $h(\nu )$ to obtain

$$\begin{aligned} \alpha ^{*} = 1 - (\nu ^{*})^{2} l^{*}, \quad \beta ^{*} = - \nu ^{*} l^{*}. \end{aligned}$$

(16)

Fig. 6

Estimated posterior distribution and true values of the taste parameters $\{ \beta _{4}, \beta _{5} \}$ for MNR in simulation example I

1.3 A.3 Sampling $q_{ij}$

The full conditional distribution of $q_{ij}$ is nonstandard. Jiang and Ding (2016) show that

$$\begin{aligned} p(q_{ij} \vert \cdot ) \propto \exp \left\{ - \frac{q_{ij} u_{ij}}{2} - \sqrt{q_{ij}} c_{ij} + \frac{\nu _{j} - 1}{2} \log q_{ij} \right\} ,\nonumber \\ \end{aligned}$$

(17)

where $u_{ij} = \nu _{j} + ( \varvec{\Sigma }^{-1} )_{jj} (w_{ij} - \varvec{X}_{ij}^{\top } \varvec{\beta })^{2}$ and $c_{ij} = (w_{ij} - \varvec{X}_{ij}^{\top } \varvec{\beta }) \sum _{j' \ne j} \left( \sqrt{ q_{ij'} ( \varvec{\Sigma }^{-1} )_{jj'}} (w_{ij} - \varvec{X}_{ij}^{\top } \varvec{\beta }) \right)$. Jiang and Ding (2016) propose to sample from (17) using a Metropolised Independence sampler (Liu 2008) with an approximate Gamma proposal. The shape parameter $\alpha ^{*}$ and the rate parameter $\beta ^{*}$ of the proposal density are obtained as follows. For $\nu _{j} \le 1$, we set $\alpha ^{*} = 1$ and $\beta ^{*} = \frac{u_{ij}}{2}$. For $\nu _{j} > 1$, $\alpha ^{*}$ and $\beta ^{*}$ are obtained through matching the modes and the corresponding curvatures of the target and the proposal densities. The log conditional density of $q_{ij}$ up to an additive constant is

$$\begin{aligned} f(q_{ij}) = - \frac{q_{ij} u_{ij}}{2} - \sqrt{q_{ij}} c_{ij} + \frac{\nu _{j} - 1}{2} \log q_{ij}. \end{aligned}$$

(18)

The log density of the Gamma proposal is

$$\begin{aligned} g(q_{ij} ) = (\alpha ^{*} - 1) \log q_{ij} - \beta ^{*} q_{ij}. \end{aligned}$$

(19)

The mode of (19) and its corresponding curvature are $\frac{\alpha ^{*} - 1}{\beta ^{*}} = m_{ij}^{*}$ and $\frac{(\beta ^{*})^{2}}{\alpha ^{*} - 1} = l_{ij}^{*}$, respectively. The first and second derivatives of (18) are

$$\begin{aligned} f'(q_{ij})= & {} - \frac{u_{ij}}{2} - \frac{c_{ij}}{2 \sqrt{q_{ij}}} + \frac{\nu _{j} - 1}{2 q_{ij}}, \nonumber \\ f''(q_{ij})= & {} \frac{c_{ij}}{4 \sqrt{q_{ij}^{3}}} - \frac{\nu _{j} - 1}{2 q_{ij}^{2}}. \end{aligned}$$

(20)

The mode of (18) is $m_{ij}^{*} = \left( \frac{ \frac{c_{ij}}{2} + \sqrt{ \left( \frac{c_{ij}}{2} \right) ^{2} + u_{ij} (\nu _{j} - 1)}}{\nu _{j} - 1} \right) ^{-2}$, and the corresponding curvature is $l_{ij}^{*} = f''(m_{ij}^{*})$. After matching the modes and corresponding curvatures of the log target and the log proposal densities, we obtain

$$\begin{aligned} \alpha ^{*} = 1 - (m_{ij}^{*})^{2} l_{ij}^{*}, \quad \beta ^{*} = - m_{ij}^{*} l_{ij}^{*}. \end{aligned}$$

(21)

B Additional results for the simulation study

1.1 B.1 Example I

Fig. 7

Estimated posterior distribution and true values of the unique elements of the covariance matrix $\varvec{\Sigma }$ for MNR in simulation example I

Fig. 8

Estimated posterior distribution and true values of the taste parameters $\{ \beta _{4}, \beta _{5} \}$ for the Gen-MNR model in simulation example II

Fig. 9

Estimated posterior distribution and true values of the unique elements of the covariance matrix $\varvec{\Sigma }$ for the Gen-MNR model in simulation example II

1.2 B.2 Example II