🔎 The Problem
Distinguishing substantively meaningful spillover effects from correlated residuals is crucial in cross-sectional studies. The two forms of spatial dependence imply different unbiased estimators and different validity conditions for inference, so choosing the correct model specification matters for substantive conclusions.
📊 How This Was Investigated
Common empirical practice fits an unrestricted spatial Durbin model and applies the Wald test to assess the nonlinear restriction that pure error dependence implies a common-factor structure. Attention to the Wald test's sensitivity to algebraically equivalent reparameterizations of that null hypothesis is limited in cross-sectional work. The investigation combines:
- An analytical derivation tracing the Wald test's noninvariance to the Taylor series approximation used to derive the restriction's sampling distribution.
- Monte Carlo simulations assessing finite-sample behavior across alternative, algebraically equivalent formulations of the common-factor restriction.
- An empirical example that demonstrates the substantive consequences of conflicting test conclusions.
📈 Key Findings
- The Wald test is asymptotically valid but not invariant to algebraically equivalent reparameterizations of the null hypothesis in finite samples.
- The source of this noninvariance is the Taylor expansion approximation applied to the restriction's sampling distribution.
- Monte Carlo evidence shows that different—but algebraically equivalent—formulations of the common-factor restriction often lead to conflicting conclusions about whether residual dependence or substantive spillovers dominate.
- The empirical illustration highlights how these conflicting test outcomes can change substantive model choice and interpretation.
⚖️ What This Means for Practice
Because reparameterization-sensitive Wald tests can produce misleading finite-sample decisions, researchers should either:
- Base inference on bootstrap critical values for the Wald statistic, or
- Use the likelihood ratio test, which is invariant to such reparameterizations,
when deciding which spatial specification best reflects the data-generating spatial process.
