"A Multiple Testing Approach to the Regularisation of Large Sample Correlation Matrices", by Natalia Bailey, M. Hashem Pesaran and L. Vanessa Smith, CAFE Research Paper No. 14.05, forthcoming in Journal of Econometrics, October 2018

Abstract: This paper proposes a regularisation method for the estimation of large covariance matrices that uses insights from the multiple testing (MT) literature. The approach tests the statistical signi…cance of individual pair-wise correlations and sets to zero those elements that are not statistically signi…cant, taking account of the multiple testing nature of the problem. The effective p-values of the tests are set as a decreasing function of N (the cross section dimension), the rate of which is governed by the nature of dependence of the underlying observations, and the relative expansion rates of N and T (the time dimension). In this respect, the method speci…es the appropriate thresholding parameter to be used under Gaussian and non-Gaussian settings. The MT estimator of the sample correlation matrix is shown to be consistent in the spectral and Frobenius norms, and in terms of support recovery, so long as the true covariance matrix is sparse. The performance of the proposed MT estimator is compared to a number of other estimators in the literature using Monte Carlo experiments. It is shown that the MT estimator performs well and tends to outperform the other estimators, particularly when N is larger than T.
JEL Classifications: C13, C58.
Key Words: High-dimensional data, Multiple testing, Non-Gaussian observations, Sparsity, Thresholding, Shrinkage.
Full Text: http://www.econ.cam.ac.uk/people-files/emeritus/mhp1/fp18/BPS_October_2018_JoE.pdf
Supplementary Material: http://www.econ.cam.ac.uk/people-files/emeritus/mhp1/wp16/BPS_14_September_2016_Supplement.pdf

"A Bayesian Analysis of Linear Regression Models with Highly Collinear Regressors", by M. Hashem Pesaran and Ron P. Smith, forthcoming in Econometrics and Statistics, October 2018

Abstract: Exact collinearity between regressors makes their individual coefficients not identified. But, given an informative prior, their Bayesian posterior means are well defined. Just as exact collinearity causes non-identification of the parameters, high collinearity can be viewed as weak identification of the parameters, which is represented, in line with the weak instrument literature, by the correlation matrix being of full rank for a finite sample size T, but converging to a rank deficient matrix as T goes to infinity. The asymptotic behaviour of the posterior mean and precision of the parameters of a linear regression model are examined in the cases of exactly and highly collinear regressors. In both cases the posterior mean remains sensitive to the choice of prior means even if the sample size is sufficiently large, and that the precision rises at a slower rate than the sample size. In the highly collinear case, the posterior means converge to normally distributed random variables whose mean and variance depend on the prior means and prior precisions. The distribution degenerates to fixed points for either exact collinearity or strong identification. The analysis also suggests a diagnostic statistic for the highly collinear case. Monte Carlo simulations and an empirical example are used to illustrate the main …findings.
JEL Classifications: C11, C18
Key Words: Bayesian identi…cation, multicollinear regressions, weakly identi…ed regression coefficients, highly collinear regressors.
Full Text: http://www.econ.cam.ac.uk/emeritus/mhp1/fp18/PS_high_collinearity_6_October_2018.pdf

Note: A previous version of the paper was distributed as CESifo Working Paper 6785 under the title of "Posterior Means and Precisions of the Coefficients in Linear Models with Highly Collinear Regressors"