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

Mynbaev, Kairat and Ullah, Aman (2006): A Remark on the Asymptotic Distribution of the OLS Estimator for a Purely Autoregressive Spatial Model.

Abstract

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.

• Item Type: MPRA Paper
• Institution: Economics Department, University of California at Riverside
• Language: English
• Keywords: spatial model; OLS estimator; asymptotic distribution; maximum likelihood; method of moments
• Subjects: C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C21 - Cross-Sectional Models ; Spatial Models ; Treatment Effect Models ; Quantile Regressions; C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General

References

Anderson, T.W. and D. A. Darling (1952) Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes. Annals of Math. Statist. 23, 193-212.

Anselin, L. (1988) Spatial Econometrics: Methods and Models. Boston, Kluwer Academic Publishers.

Anselin, L. and A.K. Bera (1998) Spatial Dependence in Linear Regression Models with an Introduction to Spatial Econometrics, in: Handbook of Applied Economics Statistics ed. by A. Ullah and D.E.A. Giles. New York, Marcel Dekker.

Billingsley, P. (1968) Convergence of Probability Measures. John Wiley & Sons, New York.

Cliff, A., Ord, J. (1981) Spatial Processes, Models and Applications. London, Pion.

Cressie, N.A.C. (1993) Statistics of Spatial Data. New York, Wiley.

Gohberg, I.C. and M.G. Krein (1969) Introduction to the Theory of Linear Nonselfadjoint Operators. American Mathematical Society, Providence, Rhode Island.

Hardy, G.H. (1956) Divergent Series. Oxford, Clarendon Press.

Hoque, A. The Exact Moments of Forecast Error in the General Dynamic Model. Sankhya: The Indian Journal of Statistics, 47, Series B, Pt. 1, 128-143.

Horn, R., and C. Johnson (1985) Matrix Analysis. New York, Cambridge University Press.

Jones, M.C. (1986) Expressions for Inverse Moments of Positive Quadratic Forms in Normal Variables, Australian Journal of Statistics, 28(2), 242-250.

Kelejian, H.H. and I.R. Prucha (1999) A Generalized Moments Estimator for the Autoregressive Parameter in a Spatial Model. International Economic Review 40, 509-533.

Kelejian, H.H. and I.R. Prucha (2002) 2SLS and OLS in a Spatial Autoregressive Model with Equal Spatial Weights, Regional Science and Urban Economics, 32, 691-707.

Kelejian, H.H., I.R. Prucha and Y. Yuzefovich (2004) Instrumental Variable Estimation of a Spatial Autoregressive Model with Autoregressive Disturbances: Large and Small Sample Results. Forthcoming in J. LeSage and R.K. Pace, Advances in Econometrics, New York: Elsevier.

Lee, L.F. (2001) Asymptotic Distributions of Quasi-Maximum Likelihood Estimators for Spatial Autoregressive Models I: Spatial Autoregressive Processes. Manuscript, Department of Economics, The Ohio State University, Columbus, Ohio.

Lee, L.F. (2002) Consistency and Efficiency of Least Squares Estimation for Mixed Regressive, Spatial Autoregressive Models. Econometric Theory 18, 252-277.

Lee, L.F. (2003) Best Spatial Two-Stage Least Squares Estimators for a Spatial Autoregressive Model with Autoregressive Disturbances. Econometric Reviews 22, No. 4, 307-335.

Lee, L.F. (2004) Asymptotic Distributions of Quasi-Maximum Likelihood Estimators for Spatial Autoregressive Models. Econometrica 72, No. 6, 1899-1925.

Mynbaev, K.T. (2001) Lp-approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables. Advances in Applied Mathematics 26, 302-329.

Ord, J.K. (1975) Estimation Methods for Models of Spatial Interaction. Journal of American Statistical Association, 70, 120-126.

Varberg, D.E. (1966) Convergence of Quadratic Forms in Independent Random Variables. Ann. Math. Stat. 37, 567-576.

White, H. (1994) Estimation, Inference and Specification Analysis. New York, Cambridge University Press.