The following papers comprise a four-part series that discusses introductory concepts of spatial econometrics. The texts were written in Portuguese and intend to present this field of study to students at upper undergraduate to graduate levels in Economy and in Regional Sciences.

How to Estimate Spatial Models with the use of the Likelihood Function

Abstract: This main objective of the first paper in this series is designed to present some of the very basic topics of spatial econometrics, such as, spatial dependency and spatial heterogeneity, introduction to spatial models, and weights matrix. I present different types of spatial models, including some of the motivations to use them. In order to provide a broader perspective of the applicability of the models, I include a brief discussion and empirical applications for each spatial model, which were applied in different fields of interests. Moreover, the present discussion also serves as a basis for the subsequent discussion presented in the following papers in this series.

How to Interpret the Coefficients of Spatial Models

Abstract: This second paper addresses the question of how to interpret the coefficients of spatial models, such as, the spatial error model, the spatial lag model, the Kelejia-Prucha model, the spatial Durbin error model and the spatial Durbin model. In many empirical studies, the spatial models’ coefficients were interpreted as partial derivatives, what might be correct in some, but not in all cases. In order to address these differences, I discuss each model separately, when I present topics such as partial derivatives, approximations, direct, indirect and total effects, and different orders of interactions. So as to give a broader perspective of the applicability of these concepts, I include a brief discussion of empirical applications that address these topics. Finally, I show some simulations using Matlab when I compare the different types of spatial models regarding direct, indirect and total effects. Lastly, I present approximations concerning spillovers among regions.

How to Estimate Spatial Models with the use of the Likelihood Function

Abstract: The main objective of this third text is to discuss how to apply the likelihood function to estimate different spatial models. First, I present this function in a general perspective and obtain the maximum likelihood estimator for the normal distribution. Then, I show some of the proprieties of this estimator that makes it very popular as an estimation method, in particular for spatial models. Afterwards, I develop the mathematical expressions of the maximum likelihood estimators for the OLS, spatial error and spatial lag models. Next, I present some illustrative simulations using Matlab that address the concepts discussed in the text. Finally, I obtain the covariance matrix for the spatial lag model as an example for the other spatial models.

How to Choose Among the Many Spatial Models

Abstract: This fourth paper discusses how to choose among the many possibilities of spatial models in a particular setting. The focus of the presentation is on models applied to cross-section data and estimated with the use of the likelihood function. First, I present some theoretical issues that might help to guide the selection of a specific model. Then, the data generating process of different models are compared, indicating when there will be bias and/or efficiency lost in the estimates. Afterwards, I discuss the specific to general and general to specific strategies of model selection, suggesting that the first one is superior to the second in some aspects. Finally, I address these theoretical issues in two groups of Monte Carlo simulations: the first compares the data generating process of different models; the second evaluates different empirical strategies of model choice.