Size distortion in the analysis of volatility and covolatility effects

Let us assume that is a consistent, asymptotically normal estimator of a matrix A (where T is the sample size), this paper shows that test statistics used in empirical work to test 1) the noninvertibility of A, i.e. det A = 0, 2) the positivite semi-definiteness A > > 0, have a different asymptotic distribution in the case where A = 0 than in the case where A ≠ 0. Moreover, the paper shows that an estimator of A constrained by symmetry or reduced rank has a different asymptotic distribution when A = 0 than when A ≠ 0. The implication is that inference procedures that use critical values equal to appropriate quantiles from the distribution when A ≠ 0 may be size distorted. The paper points out how the above statistical problems arise in standard models in Finance in the analysis of risk effects.A Monte Carlo study explores how the asymptotic results are reflected in finite sample.