A cornerstone of the SUR model is the presence of contemporaneous correlation. When X is diagonal, joint estimation is not required, which simplifies computations. Shiba and Tsurumi (1988) provide a complete set of LM, W, and LR tests of the null hypothesis that X is block diagonal. When there are only two equations their LM test reduces to the popular test proposed by Breusch and Pagan (1980). They also derive a Bayesian test.
Classical tests such as the Breusch and Pagan (1980) LM test rely on large-T asymptotics. Frees (1995) recognizes that such tests may be inappropriate with other data configurations and explores alternative tests concentrating on issues that arise when N and possibly T are large.
Work in urban and regional economics is distinguished by consideration of spatial aspects. In SUR models where the observations within equations refer to regions, careful consideration needs to be given to the potential for spatial autocorrelation. Anselin (1990) stresses this point in the context of tests for regional heterogeneity. Neglecting the presence of spatial autocorrelation is shown to distort the size and power of conventional Chow-type tests of parameter constancy across equations. On the basis of his Monte Carlo simulations Anselin (1990) advocates a pre-test approach where the first step involves testing for spatial autocorrelation using the LM test proposed in Anselin (1988).