Estimation of spatial autoregressive models with measurement error for large data sets

Maximum likelihood (ML) estimation of spatial autoregressive models for large spatial data sets is well established by making use of the commonly sparse nature of the contiguity matrix on which spatial dependence is built. Adding a measurement error that naturally separates the spatial process from the measurement error process are not well established in the literature, however, and ML estimation of such models to large data sets is challenging. Recently a reduced rank approach was suggested which re-expresses and approximates such a model as a spatial random effects model (SRE) in order to achieve fast fitting of large data sets by fitting the corresponding SRE. In this paper we propose a fast and exact method to accomplish ML estimation and restricted ML estimation of complexity of $$O(n^{3/2})$$O(n3/2) operations when the contiguity matrix is based on a local neighbourhood. The methods are illustrated using the well known data set on house prices in Lucas County in Ohio.

[1]  Md Hamidul Huque,et al.  On the impact of covariate measurement error on spatial regression modelling , 2014, Environmetrics.

[2]  Noel Cressie,et al.  One-step estimation of spatial dependence parameters: Properties and extensions of the APLE statistic , 2012, J. Multivar. Anal..

[3]  Timothy A. Davis,et al.  Direct methods for sparse linear systems , 2006, Fundamentals of algorithms.

[4]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[5]  Klairung Samart,et al.  Linear regression with nested errors using probability-linked data , 2014 .

[6]  Guanpeng Dong,et al.  Multilevel Modelling with Spatial Interaction Effects with Application to an Emerging Land Market in Beijing, China , 2015, PloS one.

[7]  Mike Rees,et al.  5. Statistics for Spatial Data , 1993 .

[8]  Noel A Cressie,et al.  The SAR Model for Very Large Datasets: A Reduced Rank Approach , 2015 .

[9]  N. Cressie,et al.  Fixed rank kriging for very large spatial data sets , 2008 .

[10]  H. Rue,et al.  Spatial Data Analysis with R-INLA with Some Extensions , 2015 .

[11]  Roger Bivand,et al.  Comparing Estimation Methods for Spatial Econometrics Techniques Using R , 2010 .

[12]  J. Elhorst,et al.  The Slx Model , 2015 .

[13]  Christine Thomas-Agnan,et al.  About predictions in spatial autoregressive models: optimal and almost optimal strategies , 2017 .

[14]  Leonhard Held,et al.  Gaussian Markov Random Fields: Theory and Applications , 2005 .

[15]  B. Fingleton,et al.  Where is the Economics in Spatial Econometrics? , 2012 .

[16]  K. Ord.  Estimation Methods for Models of Spatial Interaction , 1975 .

[17]  J. Besag,et al.  Bayesian image restoration, with two applications in spatial statistics , 1991 .

[18]  Finn Lindgren,et al.  Bayesian Spatial Modelling with R-INLA , 2015 .

[19]  D. Bates,et al.  Newton-Raphson and EM Algorithms for Linear Mixed-Effects Models for Repeated-Measures Data , 1988 .

[20]  A Bayesian heterogeneous coefficients spatial autoregressive panel data model of retail fuel duopoly pricing , 2017 .

[21]  A. Verbyla,et al.  A CONDITIONAL DERIVATION OF RESIDUAL MAXIMUM LIKELIHOOD , 1990 .

[22]  N. Mitter,et al.  Silica Vesicle Nanovaccine Formulations Stimulate Long-Term Immune Responses to the Bovine Viral Diarrhoea Virus E2 Protein , 2015, PloS one.

[23]  H. Rue,et al.  Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations , 2009 .

[24]  Roger Bivand,et al.  Comparing Implementations of Estimation Methods for Spatial Econometrics , 2015 .

[25]  D. Rubinfeld,et al.  Hedonic housing prices and the demand for clean air , 1978 .

[26]  J. LeSage.  Introduction to spatial econometrics , 2009 .

[27]  Roger Bivand,et al.  Computing the Jacobian in Gaussian Spatial Autoregressive Models: An Illustrated Comparison of Available Methods , 2013 .

[28]  M. West,et al.  Spatially varying SAR models and Bayesian inference for high-resolution lattice data , 2014 .

[29]  Desheng Liu,et al.  Statistical analysis of small-area data based on independence, spatial, non-hierarchical, and hierarchical models , 2009, Comput. Stat. Data Anal..