A spatial interaction model with spatially structured origin and destination effects

Abstract

We introduce a Bayesian hierarchical regression model that extends the traditional least-squares regression model used to estimate gravity or spatial interaction relations involving origin-destination flows. Spatial interaction models attempt to explain variation in flows from n origin regions to n destination regions resulting in a sample of N = n ² observations that reflect an n by n flow matrix converted to a vector. Explanatory variables typically include origin and destination characteristics as well as distance between each region and all other regions. Our extension introduces latent spatial effects parameters structured to follow a spatial autoregressive process. Individual effects parameters are included in the model to reflect latent or unobservable influences at work that are unique to each region treated as an origin and destination. That is, we estimate 2n individual effects parameters using the sample of N = n ² observations. We illustrate the method using a sample of commodity flows between 18 Spanish regions during the 2002 period.

Appendix: Details regarding the MCMC sampler

First, we show that the conditional posterior for δ takes the multivariate form presented in the text.

$$ \begin{aligned} p(\delta | \theta, \phi, \rho_o, \rho_d, \sigma^2_o, \sigma^2_d, \sigma^2_{\varepsilon}) &\propto \pi(\delta) \cdot \hbox{exp} \left\{ - \frac{1}{2 \sigma^2{\varepsilon}} (y - Z \delta - V \theta - W \phi)^{\prime} (y - Z \delta - V \theta - W \phi ) \right\} \\ &\propto \hbox{exp} \left\{ - \frac{1} {2 \sigma^2{\varepsilon}} (y - Z \delta - V \theta - W \phi)^{\prime} (y - Z \delta - V \theta - W \phi ) \right\} \\ &\quad\cdot \hbox{exp} \left\{ - \frac{1}{2 \sigma^2{\varepsilon}} \delta^{\prime} T^{-1} \delta \right\} \\ &\propto \hbox{exp} \left( - \frac{1}{2 \sigma^2{\varepsilon}} [\delta' (Z^{\prime} Z + T^{-1}) \delta - 2 Z^{\prime} (y - V \theta - W \phi)' \delta] \right) \\ &\propto \hbox{exp} \left( - \frac{1}{2 \sigma^2{\varepsilon}} \left[\delta - \Upsigma_{\delta}^{-1} \mu_{\delta}\right]^{\prime}\Upsigma_{\delta} \left [\delta - \Upsigma_{\delta}^{-1} \mu_{\delta}\right] \right) \\ \end{aligned} $$

where as reported in the text:

$$ \begin{aligned} \mu_{\delta} &= \frac{1}{\sigma^2{\varepsilon}} Z^{\prime} (y - V \theta - W \phi) \\ \Upsigma_{\delta} &= \frac{1}{\sigma^2{\varepsilon}} (Z^{\prime} Z + T^{-1}) \\ \end{aligned} $$

In this appendix, we follow Smith and LeSage (2004) in deriving the conditional posterior for the spatial autoregressive effects parameters θ. They note that:

$$ \begin{aligned} p(\theta | \ldots ) &\propto \hbox{exp} \left( -\frac{1}{2 \sigma^2_{\varepsilon}} [ V \theta - (y - Z \delta - W \phi)]^{\prime} [V \theta - (y - Z \delta - W \phi)] \right) \\ &\quad\cdot \hbox{exp} \left( - \frac{1}{2 \sigma^2_o} \theta^{\prime} B_o^{\prime} B_o \theta \right) \\ &\propto \hbox{exp} \left( - \frac{1}{2 \sigma^2_{\varepsilon} }\left [ \theta^{\prime} V^{\prime} V \theta - 2 (y - Z \delta - W \phi)^{\prime} V \theta + \theta^{\prime} \left( \sigma^{-2}_o B_o^{\prime} B_o \right) \theta\right] \right) \\ &= \hbox{exp} \left( - \frac{1}{2 \sigma^2_{\varepsilon}} \left[ \theta^{\prime} \left(\sigma^{-2}_o B_o^{\prime} B_o + V^{\prime} V\right) \theta - 2 (y - Z \delta - W \phi)^{\prime} V \theta\right] \right) \\ \end{aligned} $$

from which it follows that:

$$ \begin{aligned} \theta | \delta, \phi, \rho_o, \rho_d, \sigma^2_o, \sigma^2_d, \sigma^2_{\varepsilon}, y,Z &\sim N_{n}[\Upsigma_{\theta}^{-1} \mu_{\theta},\Upsigma_{\theta}^{-1}] \\ \mu_{\theta} &= \sigma^{-2}_{\varepsilon} V^{\prime} (y - Z \delta - W \phi) \\ \Upsigma_{\theta} &= \left(\frac{1}{\sigma^2_o} B_o^{\prime} B_o + \frac{1}{\sigma^2_{\varepsilon}} V^{\prime} V\right) \\ \end{aligned} $$

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The conditional posteriors for σ ² _o , σ ² _d :

$$ \begin{aligned} p(\sigma^2_o | \ldots) &\propto \pi(\theta | \rho_o, \sigma^2_o) \pi(\sigma^2_o) \\ &\propto (\sigma^2_o)^{-n/2} \hbox{exp}\left( - \frac{1}{2 \sigma_o^2} \theta' B_o' B_o \theta \right) (\sigma^2_o)^{\nu_1 + 1} \hbox{exp} \left( -\frac{\nu_2}{\sigma_o^2} \right) \\ &\propto (\sigma^2_o)^{-(\frac{n}{2} + \nu_1 + 1)} \hbox{exp} \left[ - \theta' B_o' B_o \theta + \frac{2 \nu_2}{2 \sigma_o^2}\right ] \\ \end{aligned} $$

Which is proportional to the inverse gamma distribution reported in the text. A similar approach leads to $p(\sigma^2_d | \ldots),$ and the conditional posterior for $\sigma^2_{\varepsilon}:$

$$ \begin{aligned} p(\sigma^2_{\varepsilon} | \ldots) &\propto (\sigma^2_{\varepsilon})^{-n/2} \hbox{exp}\left( - \frac{1}{2 \sigma^2_{\varepsilon}} e' e \right) (\sigma^2_{\varepsilon})^{\nu_1 + 1} \hbox{exp} \left( - \frac{\nu_2}{\sigma^2_{\varepsilon}} \right) \\ &\propto (\sigma^2_{\varepsilon})^{-(\frac{n}{2} + \nu_1 + 1)} \hbox{exp} \left[ - e' e + \frac{2 \nu_2}{2 \sigma^2_{\varepsilon}} \right] \\ e &= y - Z \delta - V \theta - W \phi \\ \end{aligned} $$