Distinguishing substantively meaningful spillover effects from correlated residuals is of great importance in cross-sectional studies as both forms of spatial dependence not only hold different implications for the choice of an estimator but also for the validity of inferences. Empirically, a prominent strategy is to estimate a spatial Durbin model and use either of the asymptotically equivalent likelihood ratio, Lagrange multiplier, or Wald test to scrutinize the non-linear restriction of common factors implied by pure error dependence. While previous research reveals important disparities in the test statistics' finite sample behaviors, the Wald test's sensitivity to algebraically equivalent alternative formulations of the null hypothesis receives scant attention in the context of cross-sectional analyses. In a series of Monte Carlo simulations, I investigate the performance of the Wald test under different expressions of the null hypothesis. As the results illustrate, the alternative formulations produce different test statistics and, at times, conflicting conclusions in small to medium sized samples. By implication, researchers need to exercise caution when using the Wald test to discriminate between substantive and residual dependence and should habitually perform robustness tests.