Who's afraid of reduced-rank parameterizations of multivariate models? Theory and example

Abstract

Reduced-rank restrictions can add useful parsimony to coefficient matrices of multivariate models, but their use is limited by the daunting complexity of the methods and their theory. The present work takes the easy road, focusing on unifying themes and simplified methods. For Gaussian and non-Gaussian (GLM, GAM, mixed normal, etc.) multivariate models, the present work gives a unified, explicit theory for the general asymptotic (normal) distribution of maximum likelihood estimators (MLE). MLE can be complex and computationally hard, but we show a strong asymptotic equivalence between MLE and a relatively simple minimum (Mahalanobis) distance estimator. The latter method yields particularly simple tests of rank, and we describe its asymptotic behavior in detail. We also examine the method's performance in simulation and via analytical and empirical examples.

Testing for Latent Factors in Models with Autocorrelation and Heteroskedasticity of Unknows Form

“…In small samples, asymptotic significance levels may be poor approximations, and in that case, bootstrap/simulation methods may be a useful substitute. Gilbert and Zemčík (2004) report some such simulations and, while we have not encountered serious test distortions in simulations fit to the sample sizes and…”

“…The RAD test is a minimum-distance test, in that it is based on the minimum squared distance betweenb and the reduced-rank approximations tob. This minimum-squared distance can also be interpreted as (proportional to the log of) the ratio of constrained and unconstrained approximate (normal) densities for the parameter estimatorb, with constraint given by the reduced-rank restriction (see Gilbert and Zemčík 2004). Hence, the label RAD (ratio of asymptotic densities) is fitting and also intuitive in its similarity to the likelihood LR test statistic.…”

“…For the RAD test, Szroeter (1983) provides some general (but quite abstract) theory, and Gouriéroux and Monfort (1989) give a somewhat more streamlined and intuitive version of this broad theory. In specific application to reduced-rank linear models, Gilbert and Zemčík (2004) give an extensive theoretical description of the RAD test.…”

“…We can then view the constrained estimatorb of b as a function ofb and vecðbÞ , and applying the Delta (asymptotic expansion) method (see, for example, van der Vaart [1998]) tob, we can readily obtain a first-order normal approximation to the differencẽ b Àb, and from this conclude that W is asymptotically chi square. For other, more exhaustive proofs of the asymptotic normality of the minimized (square) distance objective function, see Dahm and Fuller (1986), Cragg and Donald (1995), and Gilbert and Zemčík (2004).…”

A fully Bayesian approach to sparse reduced-rank multivariate regression

“…Then, we have C=ABT=A*B*T while A*≠A and B*≠B. To achieve a unique decomposition, following (Geweke, 1996) and (Gilbert and Zemcik, 2006), we assume …”