Incorporating spatial interactions in zero-inflated negative binomial models for freight trip generation

Abstract

This paper formulates a spatial autoregressive zero-inflated negative binomial model for freight trip productions and attractions. The model captures the following freight trip characteristics: count data type, positive trip rates, overdispersion, zero-inflation, and spatial autocorrelation. The spatial autoregressive structure is applied in the negative binomial part of the models to obtain unbiased estimates of the effects of different regressors. Further, we estimate parameters using the full information maximum likelihood estimator. We perform empirical analysis with an establishment based freight survey conducted in Chennai. Separate models are estimated for trips generated by motorised two-wheelers and three-wheelers, and pickups besides an aggregate model. Spatial variables such as road density and indicator of geolocation are insignificant in all the models. In contrast, the spatial autocorrelation is significant in all of the models except for the freight trips attracted and produced by pickups. From a policy standpoint, the elasticity results show the importance of considering spatial autocorrelation. We also highlight the bias due to aggregation of vehicle classes, based on the elasticities.

Appendix: Sample-wide average elasticities of SAR-ZINB model

Equation 11 gives the expected conditional mean of a random variable following a negative binomial distribution.

$$\begin{aligned} E(y|x) = exp({\bar{x}}'\hat{\beta }) \end{aligned}$$

(11)

The unconditional mean can be obtained as shown in Eq. 12.

$$\begin{aligned} E(y) = p(y_{i} >0) exp({\bar{x}}'\hat{\beta }) \end{aligned}$$

(12)

where,

$$\begin{aligned} p(y_{i} >0) = \frac{1}{1+exp({\bar{z}}'\hat{\gamma })} \end{aligned}$$

The elasticity of a variable is given by Eq. 13.

$$\begin{aligned} \varepsilon _{x_{m}}= & {} \frac{\partial }{\partial x_{m}}[E(y)] \times \frac{{\bar{x}}_{m}}{E(y)} \end{aligned}$$

(13)

$$\begin{aligned}= & {} \left( \frac{\partial }{\partial x_{m}}\left( \frac{1}{1+exp({\bar{z}}'\hat{\gamma })}\right) exp({\bar{x}}'\hat{\beta }) + \frac{\partial }{\partial x_{m}}\left( exp({\bar{x}}'\hat{\beta })\right) \frac{1}{1+exp({\bar{z}}'\hat{\gamma })} \right) \frac{{\bar{x}}_{m}}{E(y)} \end{aligned}$$

(14)

Solving the above equation and rearranging the terms gives the elasticities of independent variables of an non-spatial model (Eq. 9), similar to Isgin et al. (2008).

$$\begin{aligned} \varepsilon _{x_{m}} = \underbrace{{\bar{x}}_{m} \hat{\beta }_{m}}_\text {Count model}-\underbrace{\left[ \frac{exp({\bar{z}}\hat{\gamma })}{1+exp({\bar{z}}'\hat{\gamma })}\right] {\bar{x}}_{m} \hat{\gamma }_{m}}_\text {Inflation model} \end{aligned}$$

(15)

To account for the autoregressive nature of the model, the elasticities of count model are decomposed into direct and indirect effects (Eq. 10) for the SAR-ZINB model, following Lambert et al. (2010).

$$\begin{aligned} \varepsilon _{x_{m}}^{SAR} = \underbrace{\frac{\beta _{m}}{N} \sum _{j \ne i}^{N} a_{ii}^{-1}x_{mi}}_\text {Direct effects}+\underbrace{\frac{\beta _{m}}{N} \sum _{i=1}^{N} \sum _{j \ne i}^{N} a_{ij}^{-1}x_{mj}}_\text {Indirect effects}-\underbrace{\Bigg [\frac{exp({\bar{z}}'\hat{\gamma })}{1+exp({\bar{z}}'\hat{\gamma })}\Bigg ] {\bar{x}}_{m} \hat{\gamma }_{m}}_\text {Inflation model} \end{aligned}$$

(16)

where $a_{ii}^{-1}$ and $a_{ij}^{-1}$ refers to diagonal (off-diagonal) elements of the estimated spatial multiplier matrix $A^{-1}$.