We introduce the matrix exponential as a way of modelling spatially dependent data. The matrix exponential spatial specification simplifies the loglikelihood allowing a closed form solution to the problem of maximum likelihood estimation, and greatly simplifies Bayesian estimation of the model. The matrix exponential spatial specification can produce estimates and inferences similar to those from conventional spatial autoregressive models, but has analytical, computational, and interpretive advantages. We present maximum likelihood and Bayesian approaches to estimation for this spatial model specification along with model diagnostic and comparison methods

MODELING AN EXPONENTIAL PATTERN OF DECAY

Data collected from geographic regions such as countries, states, and counties or individual points in space such as houses often exhibit spatial dependence. 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. We advocate use of a matrix exponential spatial specification (MESS) of dependence that replaces the conventional geometric decay of influence over space with an exponential pattern of decay. We show that this results in theoretical simplicity as well as improved numerical performance relative to the conventional spatial autoregression.

Chiu, Leonard, and Tsui (1996) proposed the use of the matrix exponential for covariance matrix modelling and discussed several of its advantages. One advantage is that the matrix exponential always leads to positive definite covariance matrices, eliminating the need to restrict the parameter space or test for positive definiteness during optimization. A second advantage is that inversion of the matrix exponential takes a simple mathematical form that is easy to implement in applied practice. Finally, use of the matrix exponential spatial specification leads to a log-likelihood where a troublesome term involving the log-determinant of an nxn covariance matrix vanishes. Collectively, these aspects of the matrix exponential spatial specification greatly simplify maximum likelihood as well as Bayesian estimation and inference. Specifically, we are able to provide a closed-form solution for maximum likelihood estimates, and produce Bayesian estimates using univariate integration over a scalar polynomial expression. In addition, we show how MESS can be used for model diagnostics and comparison of models based on different spatial weight structures or sets of explanatory variables. We demonstrate these procedures using a number of data sets that vary in size and area of application.

KIDD AND KATZFUSS OF TEXAS A&M SEEM TO BE UNAWARE OF THIS APPROACH

---SEE THEIR FORTHCOMING DISCUSSION PAPER TO ISBA WHICH IS SCHEDULED TO APPEAR IN "BAYESIAN ANALYSIS".

Please click on https://bit.ly/3ykbLr2.

** The paper by Chiu et al was published in JASA and proposes a generalised linear model for covariance matrices, together with a linear model for the means. It provides a very general paradigm, in its own right, for modelling a multitude of spatial processes, particularly when random effects are incorporated with fixed effects. Why do we need anything else? SEMANTIC SCHOLAR**

.

MATTHIAS KATZFUSS'S PUBLICATIONS

*Brian Kidd, Graduate Student, Texas A&M*

****************

Tom,

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There has been a bit of work on the matrix exponential in econometrics/spatial econometrics following our initial work.

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LeSage, James P. and R. Kelley Pace,

A matrix exponential spatial specification,

Journal of Econometrics, September, 2007, Volume 140, Issue 1, pp. 190-214.

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Debarsy, Nicolas, Fei Jin, and Lung-Fei Lee

Large sample properties of the matrix exponential spatial specification with an application to FDI.

Journal of Econometrics 2015, Volume 188 Issue 1, pp. 1–21.

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The second article discusses non-stationarity issues specifically and produces some results showing that estimates are robust to non-constant variance in the disturbance structure.

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By the way, I saw you present some work on the matrix exponential at one of the Zellner Seminars on the Interface between Bayesian econometrics and statistics, (or whatever the title of the NSF funded conferences that Arnold ran for quite a few years).

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Kelley and I also devote a chapter of our book on spatial econometrics to the matrix exponential,

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LeSage, James P. and R. Kelley Pace, *Introduction to Spatial Econometrics, *CRC Press, Taylor & Francis Group: Boca Raton, FL, January 2009.

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James P. LeSage, Professor Emeritus

University of Toledo

Department of Economics

***************

SEE ALSO

COVARIANCE MATRIX ESTIMATION AND ITS APPLICATIONS

By Xinwei Deng and co-authors, including Kam Wah Tsui

IN PARTICULAR

* Xinwei Deng, Virginia Tech*

Refers to Leonard and Hsu (1992) *Annals of Statistics *P.D.F

(Bayesian Inference for a Covariance Matrix)

which was extended by Hsu, Sinay, and Hsu (2012) PDF, using more easily computable conditional Laplacian approximations to various marginal posterior distributions. Appeared in *Annals of Math Stat*

See also Yang and Berger (1994) *Annals of Statistics* (who use matrix logarithmic transformation when creating reference prior for a covariance matrix). Good risk properties.

Sinay, Hsu, and Hsu (2013)

Sinay and Hsu (2014)

*John and Serene Hsu*

*Marick Sinay, Finance One Inc, L.A,*

* *

Jim Berger, Duke

Jim Berger, Duke

*I (Tom) first learnt about the matrix exponential (ME) transformation from Dennis Cox at UW Madison during the early 1980s, at which time I was advised about Richard Bellman's mathematics at the UW Army Math Research Center, by a scholar whose name I don't remember.*

** It was not until 1990 that John Hsu and I finished deriving our matrix normal approximation to the likelihood of the log of the covariance matrix. I never realised what a large literature ME would generate, much of it subsequent to my early retirement in 2001**,

**for example some of the multivariate stochastic volatility models developed by Manabu Asai and others, in particular the highly effective MEGARCH model proposed by the eminent co-authors*** who regard Chiu et al (1996) as seminal, and*

*Michael McAleer (1951-2021)*

**. My Ph.D. supervisor Dennis Lindley advised me in 1973 that the problem with smoothing a covariance matrix would be keeping it positive definite, and the matrix exponential transformation well handles this.**

.

Kam Wah Tsui

The following recent result was derived by Jan Magnus et al, It will be

very useful in various Bayesian and Econometric applications, as the authors fully demonstrate:

THE JACOBIAN OF THE MATRIX EXPONENTIAL FUNCTION

*Jan Magnus, Vrige University*

I believe that it's possible to derive a simpler expression for the Jacobian, as a function of the eigenvalues of the covariance matrix, but I was never quite sure that my algebra was correct, and I didn't preserve my scribblings.

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