Testing for spatial correlation under a complete bipartite network

This note shows that for a spatial regression with a weight matrix depicting a complete bipartite network, the Moran I test for zero spatial correlation is never rejected when the alternative is positive spatial correlation no matter how large the true value of the spatial correlation coefficient. In contrast, the null hypothesis of zero spatial correlation is always rejected (with probability one asymptotically) when the alternative is negative spatial correlation and the true value of the spatial correlation coefficient is near −1.