Coefficient Decomposition of Spatial Regressive Models Based on Standardized Variables

TL;DR

This paper derives formulae linking spatial regression coefficients to statistical measures like Moran's index and cross-correlation, clarifying their deep structure and aiding spatial modeling.

Contribution

It introduces new formulae connecting spatial regression coefficients with spatial correlation measures, enhancing understanding of spatial autoregressive models.

Findings

Spatial autoregressive coefficient depends on Moran's index of independent variable.

Spatial lag-regressive coefficient depends on cross-correlation between variables.

Verified formulae with data from Beijing, Tianjin, and Hebei region.

Abstract

Spatial autocorrelation analysis is the basis for spatial autoregressive modeling. However, the relationships between spatial correlation coefficients and spatial regression models are not yet well clarified. The paper is devoted to explore the deep structure of spatial regression coefficients. By means of mathematical reasoning, a pair of formulae of canonical spatial regression coefficients are derived from a general spatial regression model based on standardized variables. The spatial auto- and lag-regression coefficients are reduced to a series of statistic parameters and measurements, including conventional regressive coefficient, Pearson correlation coefficient, Moran's indexes, spatial cross-correlation coefficients, and the variance of prediction residuals. The formulae show determinate inherent relationships between spatial correlation coefficients and spatial regression coefficients. New finding is as below: the spatial autoregressive coefficient mainly depends on the Moran's index of the independent variable, while the spatial lag-regressive coefficient chiefly depends on the cross-correlation coefficient of independent variable and dependent variable. The observational data of an urban system in Beijing, Tianjin, and Hebei region of China were employed to verify the newly derived formulae, and the results are satisfying. The new formulae and their variates are helpful for understand spatial regression models from the perspective of spatial correlation and can be used to assist spatial regression modeling.