Diego Perez, University of Manchester
Brian Kidd, Texas A&M
The following written contribution to the discussion has been accepted for publication by editor Michele Guindani:
Diego Perez (University of Manchester) and Tom Leonard (retired Wisconsin and Edinburgh)
Maybe the authors should refer to the Bayesian Econometric approach pioneered by Le Sage and Kelley (2002). Le Sage and Kelley proposed the matrix exponential spatial specification (MESS) as a way of simplification of the log-likelihood allowing a closed form solution to the problem of maximum likelihood estimation and simplification of the Bayesian estimation of the model. MESS can produce estimates and inferences similar to those from conventional spatial auto-regressive (AR) models, but has analytical, computational, and interpretive advantages.
Inference and estimation of traditional spatial autoregressive (SAR) models requires non-linear optimization for estimation and inference. The conventional spatial autoregressive approach introduces additional theoretical complexity relative to non-spatial autoregressive models and is difficult to implement in large samples.
MESS replaces the conventional geometric decay of influence over space with an exponential pattern of decay. It results in theoretical simplicity as well as improved numerical performance relative to the conventional spatial autoregression. MESS models the dependence of the covariances on explanatory variables by observing that for any real symmetric matrix A the matrix exponential transformation C is a positive definite matrix.
Le Sage and Kelley utilise an approach proposed by Chiu, Leonard, and Tsui (1996). Chiu et al develop a generalized linear model for covariance matrices together with a linear model for the means, using the matrix logarithmic transformation A=log C. This provides a very general paradigm for modeling a multitude of spatial processes, particularly when random effects are included with fixed effects, Why is another approach needed?
Perhaps the authors should also consider the large literature for Bayesian inference for a covariance matrix C that refers to the matrix transformation A =log C. Key papers include Leonard and Hsu (1996) and Hsu, Sinay, and Hsu (2012), who assume a matrix normal prior distribution for the upper triangular elements of A, In particular, Deng and Tsui (2012) address the estimation of large sparse covariance matrices. Most recently,Magnus, Pils and Sentana (2021) derive an explicit expression for the Jacobian of the matrix exponential transformation, with even further applications in Econometrics.
References included in : BACKGROUND MATERIAL