In some economic situations, observations are cross-sectionally correlated. One example of cross-sectional correlation is spatial correlation, which means the correlations come from the spatial closeness of different individuals. Spillover effect, externalities, network issues and so on are common causes for spatial correlation. For example, a supply shock in one region will result in production shocks in the regions nearby. This type of correlation reflects the correlations among individuals' unobservables.In Chapter 1, I study a linear regression model with a spatially correlated error term. Most current literature in econometrics assumes cluster sampling (independence between different clusters) in the population; however, this could be easily violated. I study the case in which spatial correlation exists between each pair of observations without assuming independent clusters. Generalized least squares (GLS) can be applied to the cross-sectional dimension but it is hard to account for all pairwise correlations for a large sample of spatial data. It is because the calculation of the huge error covariance matrix generally needs large computer memory. Instead I use a pseudo generalized least squares (PGLS) approach, which means it is a GLS procedure but uses a "tapered" error covariance matrix. Data could be divided into groups according to natural geographic areas, only correlations within groups are accounted for while ignoring the correlations between groups. Since correlations within groups account for most of the correlations among observations, the resulting PGLS estimator will not lose much efficiency compared to GLS. The PGLS estimator is consistent, asymptotically normal, and computationally easier than GLS. A spatial heteroskedasticity and autocorrelation consistent (HAC) covariance estimator for PGLS which is robust to both heteroskedasticity and spatial correlation is provided. Monte Carlo simulations show that PGLS becomes more efficient than ordinary least squares (OLS) as spatial correlation increases. Chapter 2 studies nonlinear estimation with spatial data.Generalized estimating equations (GEE) is applied to cross section data with spatial correlations in nonlinear models. I use a partial quasi-maximum likelihood estimator (PQMLE) in the first step and use GEE approach in the second step. Given some regularity conditions and assumptions, the asymptotic distribution of the two-step estimator is derived in the framework of M-estimation. I use a Probit model for binary data with a latent spatially error and a Poisson model for count data with a multiplicative spatial error to demonstrate the GEE procedures. As the spatial correlations in the underlying error term increase, those in the dependent variable also increase. Monte Carlo simulations show efficiency comparison of the PQMLE and GEE. The results show that correctly modeling the structure of the working correlation matrix is important in nonlinear models, which is quite different from the linear model. In addition, as spatial correlation increases, more efficiency estimation can be obtained by the GEE approach. Chapter 3 studies conditions for the Numerical Equality of the OLS, GLS and Amemiya-Cragg Estimators. Conditions under which the ordinary least squares (OLS) and generalized least squares (GLS) estimators are equal are well known. This chapter extends these results in two ways. First, it give conditions under which GLS based on one assumed error variance matrix equals GLS based on a different assumed variance matrix. Second, it give conditions under which GLS equals the GMM estimator of Amemiya (1983) and Cragg (1983).
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