Spatio-Temporal Instrumental Variables Regression with Missing Data: A Bayesian Approach

Abstract

This paper proposes an extension of the Bayesian instrumental variables regression which allows spatial and temporal correlation among observations. For that, we introduce a double separable covariance matrix, adopting a Conditional Autoregressive structure for the spatial component, and a first-order autoregressive process for the temporal component. We also introduce a Bayesian multiple imputation to handle missing data considering uncertainty. The inference procedure is described joint with a step by step Monte Carlo Markov Chain algorithm for parameters estimation. We illustrate our methodology through a simulation study and a real application that investigates how broadband affects the Gross Domestic Product of municipalities in the state of Mato Grosso do Sul from 2010 to 2017.

Appendices

Appendix A: Summary Statistics

See Table 4

Table 4 Mean, standard deviation and quartiles of dependent, exogenous and endogenous variables in the logarithm scale for all years under analysis

Appendix B: Assessment of MCMC Convergence

See Figs. 4 and 5.

Fig. 4

Trace plots of MCMC draws for parameters $\beta$, $\beta ^{*}_{0}$, $\beta ^{*}_{1}$, $\beta ^{*}_{2}$ and $\beta ^{*}_{3}$. The solid lines represent the true values and the dashed lines represent the 95% HPD intervals. The left panels display how quickly the parameters converge to the true values, and the right panels display the chain after considering the burn-in period and the thin

Fig. 5

Trace plots of MCMC draws for parameters $\delta _{0}$, $\delta _{1}$, $\phi$ and $\rho$. The solid lines represent the true values and the dashed lines represent the 95% HPD intervals. The left panels display how quickly the parameters converge to the true values, and the right panels display the chain after considering the burn-in period and the thin