Wald tests when restrictions are locally singular
Jean-Marie Dufour, Eric Renault, Victoria Zinde-Walsh
The Annals of Statistics · 2025 · Volume 53 · Issue 2
Abstract
This paper provides an exhaustive characterization of the asymptotic null distribution of Wald-type statistics for testing restrictions given by polynomial functions—which may involve singularities—when the limiting distribution of the parameter estimator is absolutely continuous (e.g., Gaussian). In addition to the well-known finite-sample noninvariance, there is also an asymptotic noninvariance (nonpivotality): with standard critical values, the test may either under-reject or over-reject, and even diverge under the null hypothesis. The asymptotic distribution of the test statistic can vary under the null hypothesis and depends on the true unknown parameter value. All these situations are possible in testing restrictions which arise in the statistical and econometric literatures, for example, for examining the specification of ARMA models, causality at different horizons, indirect effects, zero determinant hypotheses on matrices of coefficients, and other situations where singularities cannot be excluded. We provide the limit distribution and general bounds for the single restriction case. For multiple restrictions, we give a necessary and sufficient condition for the existence of a limit distribution, and its form if it exists.
Keywords
62D20, 62E20, 62F03, 62F05, bound, deficient rank, indirect effect, mediation, Nonlinear restriction, nonstandard asymptotic theory, singular covariance matrix, Wald test
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