The Econometric Analysis of Non-Stationary Spatial Panel Data by Michael Beenstock & Daniel Felsenstein

Author:Michael Beenstock & Daniel Felsenstein
Language: eng
Format: epub
ISBN: 9783030036140
Publisher: Springer International Publishing

6.2 SpVAR Theory

Terminology

In what follows spatial units are labeled by i = 1,2,…,N, time periods are labeled by t = 1,2,…,T, state variables are labeled by ym where m = 1,2,…,M, exogenous variables are labeled by xk where k = 1,2,…,K, and innovations are denoted by εmit, assumed to be iid and uncorrelated unless otherwise stated. Temporal lag orders are labeled by p = 1,2,…,P, and spatial lags are labelled by tilde; e.g. is the first-order spatial lag of y and denotes the second-order spatial lag. The vectors ymt and xkt stack the observations on ymit and xkit by spatial unit and are therefore vectors of length N. The contemporaneous vector of innovations is denoted by εmt. Note that first-order spatial lags are defined as and second-order spatial lags are defined as , where W denotes the N × N connectivity matrix. Yt is an MN vector of all the current observations of the state variables stacked by m, i.e. the first N elements ordered by i refer to m = 1 and the last N elements refer to m = M. The spatial lag of Yt is . Finally, L denotes a temporal lag operator e.g. yt − j = Ljyt.

Contemporaneous SAR coefficients are denoted by λ. Lagged SAR coefficients (temporally-lagged spatial lags) are denoted by ϕ. Temporal AR coefficients are denoted by π. Simultaneous structural parameters between endogenous (state) variables are denoted by γ. Coefficients of the exogenous variables are denoted by β. Λ, Π, Θ, and Γ are N × N diagonal matrices with elements λi, πi, ϕi and γi on the leading diagonal.

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