Abstract: This paper studies Wald-type tests in the presence of possibly
rank-deficient covariance matrices, allowing for singular covariance matrices, either in finite samples or asymptotically. Such difficulties occur in many statistical and econometric problems, such as causality and cointegration analysis in time series, (locally) redundant restrictions, (locally) redundant moment equations in GMM, tests on the determinant of a coefficient matrix (reduced rank hypotheses), tests of linear restrictions on Average Treatment Effects in regression discontinuity designs, etc. Two different types of singularity are considered. First, the estimated
covariance matrix has full rank but converges to a singular covariance matrix, so the Wald statistic can be computed as usual, but regularity conditions for the standard asymptotic chi-square distribution do not hold. Second, the estimated covariance matrix does not have full
rank but converges to a population matrix whose rank may differ from the
finite-sample rank. The proposed procedure works in all cases regardless of the finite-sample and asymptotic ranks. To address such difficulties, we introduce a novel mathematical object: the regularized inverse which
is related to generalized inverses, although different. The
regularized inverse exploits the spectral decomposition of the covariance matrix; its unique representation follows from the Spectral Theorem, see Theorem 1.2a, p.53, Eaton(2007). Results on total eigenprojections (that is the sum of eigenprojections over a subset of the spectral set) are combined with a variance regularizing function; the latter modifies small eigenvalues (using a threshold). The continuity property of the total eigenprojection technique ensures a valid asymptotic theory for the regularized inverse; it always exists and is unique. The proposed class of regularized inverse matrices includes both continuous and discontinuous regularized inverses; the Tikhonov-type inverse is continuous, the spectral cut-off regularized inverse as proposed by Lütkepohl and Burda
(1997) is discontinuous, and the full-rank regularized inverse we propose is continuous. Under general regularity conditions, we show that sample regularized inverse matrices converge to their regularized asymptotic counterparts. Regularized Wald statistics are then obtained through
replacement of the usual inverse of the estimated covariance matrix (or the
generalized inverse) by a regularized inverse. Both Gaussian and non-Gaussian distributions are allowed for the parameter estimates. Two classes of regularized Wald statistics are studied in relative detail. The
first one admits a nonstandard asymptotic distribution, which corresponds to
a linear combination of chi-square variables when the estimator used is asymptotically Gaussian. In this case, we show that the asymptotic distribution is bounded by the usual (full-rank) chi-square distribution, so standard critical values yield valid tests. In more general cases, we show that the asymptotic distribution can be simulated or bounded by simulation. The second class allows the threshold to vary with the sample size, but additional information is needed. This class of test statistics includes the spectral cut-off statistic proposed by Lütkepohl and Burda (1997, J. Econometrics) as a special case. The regularized statistics are consistent against global alternatives, with a loss of power (in certain directions) for the spectral cut-off Wald statistic. An application to U.S. data illustrates how the procedure works when testing for noncausality between saving, investment, growth and foreign direct investment.