Is regularization necessary? A Wald-type test under non-regular conditions

We study hypotheses testing in the presence of a possibly singular covariance matrix. We propose an alternative way to handle possible non-regularity in a covariance matrix of a Wald test, using the identity matrix as the weighting matrix when calculating the quadratic form. The resulting test statistic is not pivotal, but its asymptotic distribution can be approximated using bootstrap methods. In order to prove the validity of the approximations, we show that the square root of a positive semi-definite matrix is a continuously differentiable transformation with respect to the elements of the matrix. This result is important for the continuous mapping theorem to be applicable. We use two types of approximations. The first uses the parametric bootstrap and draws from the asymptotic distribution of the restriction with an estimated covariance matrix. The second applies the residual bootstrap to obtain the distribution of the test and delivers critical values, which control size and show good empirical power even in small samples. In contrast to regularization approaches, the test statistic considered in this paper does not involve arbitrary truncation parameters for which no practical guidelines are available and does not modify the information in the data.