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  • Abhimanyu Gupta (a1)

We examine a higher-order spatial autoregressive model with stochastic, but exogenous, spatial weight matrices. Allowing a general spatial linear process form for the disturbances that permits many common types of error specifications as well as potential ‘long memory’, we provide sufficient conditions for consistency and asymptotic normality of instrumental variables, ordinary least squares, and pseudo maximum likelihood estimates. The implications of popular weight matrix normalizations and structures for our theoretical conditions are discussed. A set of Monte Carlo simulations examines the behaviour of the estimates in a variety of situations. Our results are especially pertinent in situations where spatial weights are functions of stochastic economic variables, and this type of setting is also studied in our simulations.

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*Address correspondence to Abhimanyu Gupta, Department of Economics, University of Essex; e-mail:
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I am grateful to co-editor Guido Kuersteiner and three anonymous referees for excellent comments that improved the article substantially. I thank Peter Robinson for several comments and am also grateful to Nicolas Debarsy, Javier Hidalgo, Ingmar Prucha, and Renata Rabovic for useful suggestions and discussions.

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Baltagi, B.H., Fingleton, B., & Pirotte, A. (2014) Spatial lag models with nested random effects: An instrumental variable procedure with an application to English house prices. Journal of Urban Economics 80, 7686.
Bell, K.P. & Bockstael, N.E. (2000) Applying the generalized-moments estimation approach to spatial problems involving micro-level data. Review of Economics and Statistics 82, 7282.
Boucher, V. & Fortin, B. (2016) Some challenges in the empirics of the effects of networks. In Bramoullé, Y., Galeotti, A., & Rogers, B. (eds.), The Oxford Handbook of the Economics of Networks, chapter 12. Oxford University Press.
Case, A.C. (1991) Spatial patterns in household demand. Econometrica 59, 953965.
Cliff, A.D. & Ord, J.K. (1973) Spatial Autocorrelation. Pion.
Conley, T.G. & Dupor, B. (2003) A spatial analysis of sectoral complementarity. Journal of Political Economy 111, 311352.
Conley, T.G. & Ligon, E. (2002) Economic distance and cross-country spillovers. Journal of Economic Growth 7, 157187.
Das, D., Kelejian, H.H., & Prucha, I.R. (2003) Finite sample properties of estimators of spatial autoregressive models with autoregressive disturbances. Papers in Regional Science 82, 126.
Davis, P.J. (1979) Circulant Matrices. Wiley Interscience.
Delgado, M. & Robinson, P.M. (2015) Non-nested testing of spatial correlation. Journal of Econometrics 187, 385401.
Gupta, A. & Robinson, P.M. (2015) Inference on higher-order spatial autoregressive models with increasingly many parameters. Journal of Econometrics 186, 1931.
Gupta, A. & Robinson, P.M. (2018) Pseudo maximum likelihood estimation of spatial autoregressive models with increasing dimension. Journal of Econometrics 202, 92107.
Hillier, G. & Martellosio, F. (2013) Properties of the Maximum Likelihood Estimator in Spatial Autoregressive Models. Mimeo.
Hillier, G. & Martellosio, F. (2018) Exact likelihood inference in group interaction network models. Econometric Theory 34, 383415.
Jenish, N. & Prucha, I.R. (2012) On spatial processes and asymptotic inference under near-epoch dependence. Journal of Econometrics 170, 178190.
Kelejian, H.H. & Piras, G. (2014) Estimation of spatial models with endogenous weighting matrices, and an application to a demand model for cigarettes. Regional Science and Urban Economics 46, 140149.
Kelejian, H.H. & Prucha, I.R. (1998). A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances. The Journal of Real Estate Finance and Economics 17, 99121.
Kelejian, H.H. & Prucha, I.R. (1999) A generalized moments estimator for the autoregressive parameter in a spatial model. International Economic Review 40, 509533.
Kelejian, H.H. & Prucha, I.R. (2001) On the asymptotic distribution of the Moran I test statistic with applications. Journal of Econometrics 104, 219257.
Kelejian, H.H. & Prucha, I.R. (2007) HAC estimation in a spatial framework. Journal of Econometrics 140, 131154.
Kelejian, H.H. & Prucha, I.R. (2010) Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances. Journal of Econometrics 157, 5367.
Kuersteiner, G.M. & Prucha, I.R. (2013) Limit theory for panel data models with cross sectional dependence and sequential exogeneity. Journal of Econometrics 174, 107126.
Kuersteiner, G.M. & Prucha, I.R. (2015) Dynamic Spatial Panel Models: Networks, Common Shocks, and Sequential Exogeneity. CESifo Working paper 5445.
Lee, L.F. (2002) Consistency and efficiency of least squares estimation for mixed regressive, spatial autoregressive models. Econometric Theory 18, 252277.
Lee, L.F. (2003) Best spatial two-stage least squares estimators for a spatial autoregressive model with autoregressive disturbances. Econometric Reviews 22, 307335.
Lee, L.F. (2004) Asymptotic distributions of quasi-maximum likelihood estimators for spatial autoregressive models. Econometrica 72, 18991925.
Lee, L.F. & Liu, X. (2010) Efficient GMM estimation of high order spatial autoregressive models with autoregressive disturbances. Econometric Theory 26, 187230.
Lee, L.F. & Yu, J. (2013) Near unit root in the spatial autoregressive model. Spatial Economic Analysis 8, 314351.
Lee, L.F. & Yu, J. (2014) Efficient GMM estimation of spatial dynamic panel data models with fixed effects. Journal of Econometrics 180, 174197.
Qu, X. & Lee, L.F. (2015) Estimating a spatial autoregressive model with an endogenous spatial weight matrix. Journal of Econometrics 184, 209232.
Robinson, P.M. (2008) Correlation testing in time series, spatial and cross-sectional data. Journal of Econometrics 147, 516.
Robinson, P.M. (2010) Efficient estimation of the semiparametric spatial autoregressive model. Journal of Econometrics 157, 617.
Robinson, P.M. & Hidalgo, F.J. (1997) Time series regression with long-range dependence. The Annals of Statistics 25, 77104.
Robinson, P.M. & Thawornkaiwong, S. (2012) Statistical inference on regression with spatial dependence. Journal of Econometrics 167, 521542.
Scott, D.J. (1973) Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach. Advances in Applied Probability 5, 119137.
Souza, P.C.L. (2015) Estimating Network Effects without Network Data. Mimeo.
Su, L. & Jin, S. (2010) Profile quasi-maximum likelihood estimation of partially linear spatial autoregressive models. Journal of Econometrics 157, 1833.
Xu, X. & Lee, L.F. (2015) A spatial autoregressive model with a nonlinear transformation of the dependent variable. Journal of Econometrics 186, 118.
Yuzefovich, Y.A. (2003) Two Essays on Spatial Econometrics. Ph.D. thesis, University of Maryland.
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Econometric Theory
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