A Bayesian spatial autoregressive logit model with an empirical application to European regional FDI flows

Abstract

In this paper, we propose a Bayesian estimation approach for a spatial autoregressive logit specification. Our approach relies on recent advances in Bayesian computing, making use of Pólya–Gamma sampling for Bayesian Markov-chain Monte Carlo algorithms. The proposed specification assumes that the involved log-odds of the model follow a spatial autoregressive process. Pólya–Gamma sampling involves a computationally efficient treatment of the spatial autoregressive logit model, allowing for extensions to the existing baseline specification in an elegant and straightforward way. In a Monte Carlo study we demonstrate that our proposed approach markedly outperforms alternative specifications in terms of parameter precision. The paper moreover illustrates the performance of the proposed spatial autoregressive logit specification using pan-European regional data on foreign direct investments. Our empirical results highlight the importance of accounting for spatial dependence when modelling European regional FDI flows.

Appendix

See Tables 4 and 5.

Table 4 Classification of fDi Markets business functions

Table 5 List of regions in the study

1.1 Marginal effects

In our spatial Durbin logit model, the interpretation of marginal effects of the k-th explanatory variable (with $k = 1,\ldots ,K$) differs from those in linear models. This is due to the fact that (i) the logit model is nonlinear in nature and marginal effects differ by the level of the k-th variable, and (ii) the presence of spatial autocorrelation gives rise to an $N \times N$ matrix of partial derivatives, which makes interpretation of marginal effects richer, but also more complicated (see also LeSage and Pace 2009).

The first issue, where the marginal effect of the probability of $p(y_i = 1)$ varies with the level of the explanatory variable $z_{ik}$, is usually addressed in the logit literature by providing marginal effects in reference to the mean value of the k-th explanatory variable, which is denoted as $\overline{z_k} = \sum _{i=1}^N z_{ik} /N$. The marginal effects can thus be interpreted as the change in probability of observing $y=1$ associated with a change in the average sample observation of the k-th explanatory variable. Note that this also implies that marginal effects depend on the distribution of the explanatory variable itself.

Partial derivatives of the model in Eq. (2.1), with respect to the k-th coefficient can be written as:

$$\begin{aligned} \varvec{\mu }_k&= \varvec{A}^{-1} \varvec{I}_N \overline{z_k} \gamma _k + \varvec{A}^{-1} \varvec{W} \overline{z_{Wk}} \theta _k, \nonumber \\ \frac{\partial p({y}=1|\overline{z}_k )}{\partial \overline{z}_k'}&= \frac{\exp \varvec{\mu }_k}{1 + \exp \left( \varvec{\mu }_k \right) } \odot \left( \varvec{A}^{-1} \varvec{I}_N \beta _k + \varvec{A}^{-1} \varvec{W} \theta _k \right) \nonumber \\&= \varvec{\Lambda }_k, \end{aligned}$$

(A.1)

where $\beta _k$ and $\theta _k$ denote the k-th element of $\varvec{\beta }$ and $\varvec{\theta }$, respectively. $\overline{z_{Wk}}$ denotes the average value of the k-th spatially lagged explanatory variable, and $\odot$ is the Hadamard product. Note that marginal effects of the k-th coefficient, denoted as $\varvec{\Lambda }_k$, are an $N\times N$ matrix due to the presence of the $N \times N$ spatial multiplier $\varvec{A}^{-1}$.

Since interpreting $N \times N$ marginal effects proves cumbersome, we define in accordance with LeSage and Pace (2009) summary impact effects. These can be readily calculated from $\varvec{\Lambda }_k$:

$$\begin{aligned} \hbox {direct}_k&= \frac{1}{N}\varvec{\iota }_N' \hbox {diag}(\varvec{\Lambda }_k) \end{aligned}$$

(A.2)

$$\begin{aligned} \hbox {total}_k&= \frac{1}{N}\varvec{\iota }_N' \varvec{\Lambda }_k \varvec{\iota }_N \end{aligned}$$

(A.3)

$$\begin{aligned} \hbox {indirect}_k&= \hbox {total}_k - \hbox {direct}_k, \end{aligned}$$

(A.4)

where $\varvec{\iota }_N$ denotes an $N \times 1$ vector of ones. The average direct effects summarize the average effect of a marginal change in the k-th explanatory variable on the log-odds in the own region. Average indirect effects, on the other hand, summarize the average impact due to a marginal change in all other regions. A third measure is given by the average total effects, which summarizes the own regional change in log-odds due to marginal change of the k-th variable in all regions.