Distinguishing substantively meaningful spillover effects from correlated residuals is of great importance in cross-sectional studies. Both forms of spatial dependence not only hold different implications for the choice of an unbiased estimator but also for the validity of inferences. Empirically, a prominent strategy is to estimate an unrestricted spatial Durbin model and use the Wald test to scrutinize the non-linear restriction of common factors implied by pure error dependence. However, the Wald test's sensitivity to algebraically equivalent formulations of the null hypothesis receives scant attention in the context of cross-sectional analyses. This article shows analytically that the Wald test's non-invariance to such reparameterizations stems from the application of a Taylor series expansion to approximate the restriction's sampling distribution. While asymptotic calculations demonstrate the validity of this approximation when the sample size approaches infinity, a series of Monte Carlo simulations find that alternative formulations of the common factor restriction frequently produce conflicting conclusions in finite samples. An empirical example further illustrates the consequences of this problem for applied research. By implication, researchers should avoid the Wald test and use alternative procedures that are invariant to such reparameterizations in order to adequately specify the underlying spatial process.