When the Wald Test Misleads Spatial Model Choice

🔎 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.