Numerical implementation of the QuEST function

Abstract

Certain estimation problems involving the covariance matrix in large dimensions are considered. Due to the breakdown of finite-dimensional asymptotic theory when the dimension is not negligible with respect to the sample size, it is necessary to resort to an alternative framework known as large-dimensional asymptotics. Recently, an estimator of the eigenvalues of the population covariance matrix has been proposed that is consistent according to a mean-squared criterion under large-dimensional asymptotics. It requires numerical inversion of a multivariate nonrandom function called the QuEST function. The numerical implementation of this QuEST function in practice is explained through a series of six successive steps. An algorithm is provided in order to compute the Jacobian of the QuEST function analytically, which is necessary for numerical inversion via a nonlinear optimizer. Monte Carlo simulations document the effectiveness of the code.

Risk Budgeting Portfolios from Simulations

“…, r d,t ) are the vectors of prices and returns of all d assets at time t, respectively, and r i,t = p i,t −p i,t−1 p i,t−1 . Due to the small number of assets under consideration, the non-linear shrinkage estimator for the covariance matrix (see, e.g., Ledoit and Wolf (2017) and Ramprasad (2016)) did not provide any significantly different allocations, and thus are not reported.…”

Random matrix improved covariance estimation for a large class of metrics*

“…However, QuESTx has a larger variance, which is in part due to the occurrence of outlying unstable solutions (which our approach does not seem to suffer from). This unstable behavior can be traced back to QuEST being founded on a computationally intense and involved numerical procedure composed of six intricate steps of finely-tuned optimization schemes (see the implementation details in Ledoit and Wolf (2017)) aiming at stabilizing El Karoui's initial approach (El Karoui et al 2008), where our approach is a much more stable gradient descent method.…”

An overview of large‐dimensional covariance and precision matrix estimators with applications in chemometrics

“…We will refer to this approach as Ledoit–Wolf nonlinear shrinker (LW‐NONLIN). The QuEST solver requires nonconvex optimization, which makes the LW‐NONLIN method computationally quite intensive . Recently, Lam proposed the nonparametric eigenvalue‐regularized covariance matrix estimator (NERCOME) and showed that its eigenvalues asymptotically approach those found by LW‐NONLIN (it is questionable how useful this asymptotical property is in practice given the low sample sizes encountered in chemometrics) .…”