A Remark on the Asymptotic Distribution of the OLS Estimator for a Purely Autoregressive Spatial Model

We derive the asymptotics of the OLS estimator for a purely autoregressive spatial model. Only low-level conditions are used. As the sample size increases, the spatial matrix is assumed to approach a square-integrable function on the square $(0,1)^2$. The asymptotic distribution is a ratio of two infinite linear combinations of $\chi$-square variables. The formula involves eigenvalues of an integral operator associated with the function approached by the spatial matrices. Under the conditions imposed identification conditions for the maximum likelihood method and methods of moments fail. A remedial iterative procedure using the OLS estimator is proposed. Additional Information: This paper has been delivered at North American Summer Meeting of the Econometric Society in June 2006, see http://gemini.econ.umd.edu/conference/NASM2006/program/NASM2006.html. A revised and extended version (with computer simulations) has been accepted for publication as Mynbaev, K.T. and A. Ullah (2007) Asymptotic distribution of the OLS estimator for a purely autoregressive spatial model. Journal of Multivariate Analysis, doi:10.1016/j.jmva.2007.04.002.