Estimation of high-dimensional dynamic conditional precision matrices with an application to forecast combination

Abstract

The estimation of a large covariance matrix is challenging when the dimension p is large relative to the sample size n. Common approaches to deal with the challenge have been based on thresholding or shrinkage methods in estimating covariance matrices. However, in many applications (e.g., regression, forecast combination, portfolio selection), what we need is not the covariance matrix but its inverse (the precision matrix). In this paper we introduce a method of estimating the high-dimensional “dynamic conditional precision” (DCP) matrices. The proposed DCP algorithm is based on the estimator of a large unconditional precision matrix to deal with the high-dimension and the dynamic conditional correlation (DCC) model to embed a dynamic structure to the conditional precision matrix. The simulation results show that the DCP method performs substantially better than the methods of estimating covariance matrices based on thresholding or shrinkage methods. Finally, we examine the “forecast combination puzzle” using the DCP, thresholding, and shrinkage methods.

Keywords:

• DCP
• forecast combination puzzle
• high-dimensional conditional precision matrix; ISEE

JEL CLASSIFICATIONS:

• C3
• C4
• C5

Acknowledgment

The authors are thankful to the editor, associate editor, two anonymous referees, Ekaterina Seregina, and seminar participants at the conference in honor of Professor Cheng Hsiao in Beihang University in Beijing, the Joint Statistical Meetings (JSM2019) in Denver, and the seminar at Amazon Inc in Seattle, for many valuable comments.

Notes

1 There are other papers that use the random matrix theory (RMT) to estimate the covariance matrix, such as Karoui (2008) who develops an estimator of eigenvectors and eigenvalues of covariance matrices by discretizing and inverting the Stieltjes transform of a limiting sample spectral distribution. However, as Ledoit and Wolf (LW, 2015) point out, this method does not exploit the natural discreteness of the population spectral distribution for finite number of variables (p). Ledoit and Wolf (2012) use the same discretization strategy as in Karoui (2008), but they match population eigenvalues to sample eigenvalues on the real line. The drawback of this approach is that they only consider the case when p < n. LW (2015) extend Ledoit and Wolf (2012) and develop an estimator of the population eigenvalues that works also when $p>n.$ They use a different discretization strategy and show that their estimator works better than Ledoit and Wolf (2012) even for p < n. Another related paper is by Mestre (2008) who proposes an estimator of eigenvalues and eigenvectors of covariance matrices using contour integration of analytic functions in the complex plane. The review paper by Bun, Bouchaud, and Potter (2017) provides a comprehensive overview of the modern techniques in RMT and their usefulness for estimating large correlation matrices. Recently, Engle et al. (2019) use non-linear shrinkage method which is based on RMT by Ledoit and Wolf (2012, 2015) to develop an improved estimation of large dynamic covariance matrices (to be more precise, RMT is used for estimating the unconditional correlation matrix and then use it for correlation targeting).

2 COV$({\mathbf{e}}_A,{\mathbf{x}}_{A^c})=$ COV$({\mathbf{x}}_A+{\Omega }_{A,A}^{-1}{\Omega }_{A,A^c}{\mathbf{x}}_{A^c},{\mathbf{x}}_{A^c})={\Sigma }_{A,A^c}+{\Omega }_{A,A}^{-1}{\Omega }_{A,A^c}{\Sigma }_{A^c,A^c}.$ Based on the property of the inverse of a partitioned matrix, ${\Sigma }_{A,A^c}=-{\Omega }_{A,A}^{-1}{\Omega }_{A,A^c}{\Sigma }_{A^c,A^c}.$ Thus, COV$({\mathbf{e}}_A,{\mathbf{x}}_{A^c})=0.$ Since ${\mathbf{e}}_A$ and ${\mathbf{x}}_{A^c}$ have joint normality, they are independent.

3 The DCP algorithm is component-wise for the estimation of the conditional precision matrix Ω_t, i.e., element by element for each pair $B=\left\{j,k\right\}\subset \left\{1,\dots ,p\right\}.$ We recently found that a similar method was used in Pakel et al. (2020) who did the component-wise estimation of the conditional covariance matrix ${\Sigma }_t.$

4 We conjecture that the consistency of the DCP estimator for Ω_t may be established under some assumptions that the maximum number of nonzeros in a row in Ω_t (the degree of non-sparsity) and the dimensionality p grow at certain rates slow enough relative to the sample size n. We leave this for future research.