Parameter-free conditions for equality of general ridge estimators under spatial error models

This paper investigates when two general ridge estimators coincide under spatial error models. First, in the general linear model, we derive a necessary and sufficient condition based on the commutativity of an extended dispersion matrix and an orthogonal projector, thereby extending the classical result of Zyskind. Next, we establish parameter-free conditions for first-order spatial autoregressive and spatial moving average processes. Also, we obtain a necessary and sufficient condition valid for all values of the spatial correlation coefficient in the specified parameter range and characterize all penalty matrices for which the two general ridge estimators coincide. A numerical experiment illustrates the theoretical results and shows that the proposed conditions can simplify the two-step estimation procedure by eliminating the need to estimate the spatial correlation coefficient.