Wald tests when restrictions are locally singular
研究了当约束由多项式函数给出且可能存在奇异性时，沃尔德型检验统计量的渐近零分布，发现其可能非枢轴化，导致检验过度拒绝或拒绝不足，并给出了单约束的极限分布和一般界限，以及多约束的极限分布存在条件。
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.