Essays On Robust Estimators For Non-Identically Distributed Observations In Spatial Econometric And Time Series Models

This thesis proposal consists of three essays on the estimation methods and applications of spatial econometric models and one essay on the generalized autoregressive conditionally heteroskedastic (GARCH)-type models in financial time series. The first essay discusses the heteroskedasticity robust generalized method of moments estimator (RGMME) for the spatial models that allow for spatial dependence in both the dependent variable and the disturbance term (SARAR(1,1)). First, we show that the maximum likelihood estimator (MLE) is generally inconsistent in the presence of unknown heteroskedasticity. Then, we extend robust GMM approach in Lin and Lee (2010) to SARAR(1,1). The large sample properties are rigorously studied and presented for the RGMME. Through a comprehensive Monte Carlo study, we compare the finite sample properties of the RGMME with some other estimators proposed in the literature. The second essay focuses on the GMM estimation of the spatial autoregressive models which impose a moving average process for the disturbance term (SARMA). We extend the best GMM estimator (BGMME) of Liu et al. (2010) to the SARMA models and provide the best set of instruments for the SARMA(1,1) and the SARMA(0,1) specifications. The large sample properties are rigorously studied and presented for the BGMME. The finite sample properties are investigated through an extensive Monte Carlo study. To confirm our results from the Monte Carlo study, we replicate the results for the SARMA(1,1) specification in Behrens et al. (2012) in an empirical illustration. The third essay investigates the effect of foreign direct investment (FDI) on economic growth through a spatially augmented Solow growth model. The current literature on the relationship between FDI and economic growth uses canonical cross-country growth regression specifications that are derived from the textbook Solow growth model for closed economies. We claim that these specifications cannot reflect the relationship between economic growth and FDI, because they model each country as an isolated island that does not interact with the rest of the world. On the other hand, a spatially augmented Solow growth model allows for technological interdependence among countries through spatial externalities. The modified growth model yields regression specifications that properly account for spatial autocorrelations. We construct a panel of 85 countries for the period 1980-2010 and estimate the modified specifications with the tools from spatial econometrics. Our findings indicate that FDI inflows have a significant positive effect on the growth rate of host countries. The final essay proposes a flexible distribution for the maximum likelihood estimation of the GARCH-type time series models. The new distribution can better account for the potential skewness and leptokurticity in the driving noise sequence. We study the large sample properties of the new estimator following the methodology presented in Francq and Zakoïan (2004). To investigate the finite sample properties of the new estimator, we first conduct a Monte Carlo study. Furthermore, to test the relative out-of-sample predictive power of the new estimator, we test for its prediction power on two data sets using the methods described in White (2000) and Hansen et al. (2003)

Teaching Size and Power Properties of Hypothesis Tests Through Simulations

In this study, we review the graphical methods suggested in Davidson and MacKinnon (Davidson, Russell, and James G. MacKinnon. 1998. “Graphical Methods for Investigating the Size and Power of Hypothesis Tests.” The Manchester School 66 (1): 1–26.) that can be used to investigate size and power properties of hypothesis tests for undergraduate and graduate econometrics courses. These methods can be used to assess finite sample properties of various hypothesis tests through simulation studies. In addition, these methods can be effectively used in classrooms to reinforce students’ understanding of basic hypothesis testing concepts such as Type I error, Type II error, size, power, p-values and under-or-over-sized tests. We illustrate the procedural aspects of these graphical methods through Monte Carlo experiments, and provide the implementation codes written in Matlab and R for the classroom applications

Bayesian Inference in Spatial Sample Selection Models

In this study, we consider Bayesian methods for the estimation of a sample selection model with spatially correlated disturbance terms. We design a set of Markov chain Monte Carlo (MCMC) algorithms based on the method of data augmentation. The natural parameterization for the covariance structure of our model involves an unidentified parameter that complicates posterior analysis. The unidentified parameter -- the variance of the disturbance term in the selection equation -- is handled in different ways in these algorithms to achieve identification for other parameters. The Bayesian estimator based on these algorithms can account for the selection bias and the full covariance structure implied by the spatial correlation. We illustrate the implementation of these algorithms through a simulation study

Bayesian Inference in Spatial Sample Selection Models

In this study, we consider Bayesian methods for the estimation of a sample selection model with spatially correlated disturbance terms. We design a set of Markov chain Monte Carlo (MCMC) algorithms based on the method of data augmentation. The natural parameterization for the covariance structure of our model involves an unidentified parameter that complicates posterior analysis. The unidentified parameter -- the variance of the disturbance term in the selection equation -- is handled in different ways in these algorithms to achieve identification for other parameters. The Bayesian estimator based on these algorithms can account for the selection bias and the full covariance structure implied by the spatial correlation. We illustrate the implementation of these algorithms through a simulation study

A Review of Cross-Sectional Matrix Exponential Spatial Models

The matrix exponential spatial models exhibit similarities to the conventional spatial autoregressive model in spatial econometrics but offer analytical, computational, and interpretive advantages. This paper provides a comprehensive review of the literature on the estimation, inference, and model selection approaches for the cross-sectional matrix exponential spatial models. We discuss summary measures for the marginal effects of regressors and detail the matrix-vector product method for efficient estimation. Our aim is not only to summarize the main findings from the spatial econometric literature but also to make them more accessible to applied researchers. Additionally, we contribute to the literature by introducing some new results. We propose an M-estimation approach for models with heteroskedastic error terms and demonstrate that the resulting M-estimator is consistent and has an asymptotic normal distribution. We also consider some new results for model selection exercises. In a Monte Carlo study, we examine the finite sample properties of various estimators from the literature alongside the M-estimator.Comment: 60 pages, 4 table

GMM Gradient Tests for Spatial Dynamic Panel Data Models

In this study, we formulate the adjusted gradient tests when the alternative model used to construct tests deviates from the true data generating process for a spatial dynamic panel data model (SDPD). Following Bera et. al. (2010), we introduce these adjusted gradient tests along with the standard ones within a GMM framework. These tests can be used to detect the presence of (i) the contemporaneous spatial lag terms, (ii) the time lag term, and (iii) the spatial time lag terms in an higher order SDPD model. These adjusted tests have two advantages: (i) their null asymptotic distribution is a central chi-squared distribution irrespective of the mis-specified alternative model, and (ii) their test statistics are computationally simple and require only the ordinary least-squares (OLS) estimates from a non-spatial two-way panel data model. We investigate the finite sample size and power properties of these tests through Monte Carlo studies. Our results indicates that the adjusted gradient tests have good finite sample properties

Simple Tests for Social Interaction Models with Network Structures

We consider an extended spatial autoregressive model that can incorporate possible endogenous interactions, exogenous interactions, unobserved group fixed effects and correlation of unobservables. In the generalized method of moments (GMM) and the maximum likelihood (ML) frameworks, we introduce simple gradient based tests that can be used to test the presence of endogenous effects, the correlation of unobservables and the contextual effects. We show the asymptotic distributions of tests, and formulate robust tests that have central chi-square distributions under both the null and local misspecification. The proposed tests are easy to compute and only require the estimates from a transformed linear regression model. We carry out an extensive Monte Carlo study to investigate the size and power properties of the proposed tests. Our results show that the proposed tests have good finite sample properties and are useful for testing the presence of endogenous effects, correlation of unobservables and contextual effects in a social interaction model

GMM Gradient Tests for Spatial Dynamic Panel Data Models

In this study, we formulate the adjusted gradient tests when the alternative model used to construct tests deviates from the true data generating process for a spatial dynamic panel data model (SDPD). Following Bera et. al. (2010), we introduce these adjusted gradient tests along with the standard ones within a GMM framework. These tests can be used to detect the presence of (i) the contemporaneous spatial lag terms, (ii) the time lag term, and (iii) the spatial time lag terms in an higher order SDPD model. These adjusted tests have two advantages: (i) their null asymptotic distribution is a central chi-squared distribution irrespective of the misspecified alternative model, and (ii) their test statistics are computationally simple and require only the ordinary least-squares (OLS) estimates from a non-spatial two-way panel data model. We investigate the finite sample size and power properties of these tests through Monte Carlo studies. Our results indicate that the adjusted gradient tests have good finite sample properties

GMM Gradient Tests for Spatial Dynamic Panel Data Models

In this study, we formulate the adjusted gradient tests when the alternative model used to construct tests deviates from the true data generating process for a spatial dynamic panel data model (SDPD). Following Bera et. al. (2010), we introduce these adjusted gradient tests along with the standard ones within a GMM framework. These tests can be used to detect the presence of (i) the contemporaneous spatial lag terms, (ii) the time lag term, and (iii) the spatial time lag terms in an higher order SDPD model. These adjusted tests have two advantages: (i) their null asymptotic distribution is a central chi-squared distribution irrespective of the misspecified alternative model, and (ii) their test statistics are computationally simple and require only the ordinary least-squares (OLS) estimates from a non-spatial two-way panel data model. We investigate the finite sample size and power properties of these tests through Monte Carlo studies. Our results indicate that the adjusted gradient tests have good finite sample properties