Computing the Jacobian in Spatial Models: An Applied Survey

Abstract

Despite attempts to get around the Jacobian in fitting spatial econometric models by using GMM and other approximations, it remains a central problem for maximum likelihood estimation. In principle, and for smaller data sets, the use of the eigenvalues of the spatial weights matrix provides a very rapid and satisfactory resolution. For somewhat larger problems, including those induced in spatial panel and dyadic (network) problems, solving the eigenproblem is not as attractive, and a number of alternatives have been proposed. This paper will survey chosen alternatives, and comment on their relative usefulness.

Comparing Estimation Methods for Spatial Econometrics Techniques Using R

“…In all cases, a simple line search may be used to find ρ or λ, and other coefficients may be calculated using an ancilliary regression once this has been done. Detailed reviews of methods for computing the Jacobian may be found in LeSage and Pace (2009); Smirnov and Anselin (2009);Bivand (2010), and interested readers are refered to these. The comparisons within spdep made here use methods for computing the Jacobian presented in full in Bivand (2010), and include the dense matrix eigenvalue method eigen (Ord, 1975, p. 121), the updating Cholesky decomposition method Matrix using functions in the R Matrix package for sparse matrix operations, the Monte Carlo method MC using the R Matrix package introduced by Barry and Pace (1999), and the Chebyshev method again using the R Matrix package (Pace and LeSage, 2004).…”

“…8 Collaborative development using platforms of this kind is very beneficial, for a description see Theussl et al (2010). Within spdep itself, provision is being made through modularization to permit users to choose between different ways of calculating the Jacobian (Bivand, 2010). It is also intended to provide a function to fit a general spatial regression model using different fitting techniques, which is needed to contrast with possibly more appropriate modelling strategies, such as the spatial Durbin model.…”

splm: Spatial Panel Data Models inR

“…The default method to compute the Jacobian is based on the eigenvalues decomposition using the functions eigenw. In line with the changes and improvements recently made in spdep (Bivand 2010), other methods are available, including the use of sparse matrices, and the Chebyshev and Monte Carlo approximations (LeSage and Pace 2009).…”

After 'Raising the Bar': Applied Maximum Likelihood Estimation of Families of Models in Spatial Econometrics

“…Detailed reviews of methods for computing the Jacobian may be found in LeSage and Pace (2009); Smirnov and Anselin (2009);Bivand (2010b), and interested readers are refered to these. The methods used for computing the Jacobian in spdep are presented in full in Bivand (2010b); here we use the dense matrix eigenvalue method eigen (Ord, 1975, p. 121) for the English garbage data set, and the updating Cholesky decomposition method Matrix, using sparse matrix functions in the R Matrix package (Bates and Maechler, 2011), and based on Pace and Barry (1997), for the two larger data sets.…”