Elimination of circular routes in spatial dynamic regression equations

Spatial lag operators play an important role in such fields as biometrics, geography and spatial econometrics. Higher-order spatial interactions can be represented by powering spatial lag operators. The paper discusses the conditions under which higher-order spatial lag operators will contain circular routes. Next, it is demonstrated that valid causal inference with the help of spatial dynamic regression models, necessitates the elimination of circular routes. It is shown that the ML-search procedure proposed by [Ord, 1975] and generalized by [Blommestein, 1983], will generate spurious results in case of circular routes. A ML-procedure to obtain non-spurious estimated parameters in general dynamic spatial regression models is outlined. In addition some recent proposed approximate ML-search procedures [see [Blommestein, 1983]], based on the spectral decomposition of a matrix, are modified in a similar way. The last section demonstrates empirically the purport of the theoretical results derived in the paper.