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  • C. Alan Bester (a1), Timothy G. Conley (a1), Christian B. Hansen (a2) and Timothy J. Vogelsang (a3)

This paper develops a method for performing inference using spatially dependent data. We consider test statistics formed using nonparametric covariance matrix estimators that account for heteroskedasticity and spatial correlation (spatial HAC). We provide distributions of commonly used test statistics under “fixed-b” asymptotics, in which HAC smoothing parameters are proportional to the sample size. Under this sequence, spatial HAC estimators are not consistent but converge to nondegenerate limiting random variables that depend on the HAC smoothing parameters, the HAC kernel, and the shape of the spatial region in which the data are located. We illustrate the performance of the “fixed-b” approximation in the spatial context through a simulation example.

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*Address correspondence to Christian Hansen, University of Chicago Booth School of Business, 5807 S Woodlawn Ave, Chicago, IL 60637, USA. e-mail:
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Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59(3), 817858.
Arellano, M. (1987) Computing robust standard errors for within-groups estimators. Oxford Bulletin of Economics and Statistics 49(4), 431434.
Basu, A.K. & Dorea, C.C.Y. (1979) On functional central limit theorem for stationary martingale random fields. Acta Mathematica Academia Scientiarum Hungaricae 33, 307316.
Bertrand, M., Duflo, E., & Mullainathan, S. (2004) How much should we trust differences-in-differences estimates? Quarterly Journal of Economics 119, 249275.
Bester, C.A., Conley, T.G., & Hansen, C.B. (2010) Inference for dependent data using cluster covariance estimators. Available at SSRN:
Conley, T.G. (1996) Econometric modelling of cross-sectional dependence. Ph.D. dissertation, University of Chicago.
Conley, T.G. (1999) GMM estimation with cross sectional dependence. Journal of Econometrics 92, 145.
Dedecker, J. (2001) Exponential inequalities and functional central limit theorems for random fields. ESAIM: Probability and Statistics 5, 77104.
Deo, C. (1975) A functional central limit theorem for stationary random fields. Annals of Probability 3(4), 708715.
Dudley, R.M. (1973) Sample functions of the Gaussian process. Annals of Probability 1, 66103.
Goldie, C.M. & Greenwood, P.E. (1986) Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes. Annals of Probability 14(3), 817839.
Gonçalves, S. & Vogelsang, T.J. (2011) Block bootstrap HAC robust tests: The sophistication of the naive bootstrap. Econometric Theory 27(4), 745791.
Hansen, C.B. (2007) Asymptotic properties of a robust variance matrix estimator for panel data when T is large. Journal of Econometrics 141, 597620.
Ibragimov, R. & Müller, U.K. (2010) t-Statistic based correlation and heterogeneity robust inference. Journal of Business and Economic Statistics 28, 453468.
Jansson, M. (2004) The error in rejection probability of simple autocorrelation robust tests. Econometrica 72(3), 937946.
Kelegian, H. & Prucha, I. (2007) HAC estimation in a spatial framework. Journal of Econometrics 140, 131154.
Kelejian, H.H. & Prucha, I. (1999) A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, 509533.
Kelejian, H.H. & Prucha, I. (2001) On the asymptotic distribution of the Moran I test statistic with applications. Journal of Econometrics 104, 219257.
Kiefer, N.M. & Vogelsang, T.J. (2002) Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory 18, 13501366.
Kiefer, N.M. & Vogelsang, T.J. (2005) A new asymptotic theory for heteroskedasticity-autocorrelation robust tests. Econometric Theory 21, 11301164.
Kim, M.S. & Sun, Y. (2011) Spatial heteroskedasticity and autocorrelation consistent estimation of covariance matrix. Journal of Econometrics 160, 349371.
Lee, L.-F. (2004) Asymptotic distributions of quasi-maximum likelihood estimators for spatial econometric models. Econometrica 72, 18991926.
Lee, L.-F. (2007a) GMM and 2SLS estimation of mixed regressive, spatial autoregressive models. Journal of Econometrics 137, 489514.
Lee, L.-F. (2007b) Identification and estimation of econometric models with group interactions, contextual factors and fixed effects. Journal of Econometrics 140, 333374.
Liang, K.-Y. & Zeger, S. (1986) Longitudinal data analysis using generalized linear models. Biometrika 73(1), 1322.
Newey, W.K. & West, K.D. (1987) A simple, positive semi-definite heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3), 703708.
Sun, Y., Phillips, P.C.B., & Jin, S. (2008) Optimal bandwidth selection in heteroskedasticity-autocorrelation robust testing. Econometrica 76(1), 175194.
Wooldridge, J.M. (2003) Cluster-sample methods in applied econometrics. American Economic Review 93(2), 133188.
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Econometric Theory
  • ISSN: 0266-4666
  • EISSN: 1469-4360
  • URL: /core/journals/econometric-theory
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