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Block-diagonal idiosyncratic covariance estimation in high-dimensional factor models for financial time series

Lucija Žignić, Stjepan Begušić, Zvonko Kostanjčar

Abstract

Estimation of high-dimensional covariance matrices in latent factor models is an important topic in many fields and especially in finance. Since the number of financial assets grows while the estimation window length remains of limited size, the often used sample estimator yields noisy estimates which are not even positive definite. Under the assumption of latent factor models, the covariance matrix is decomposed into a common low-rank component and a full-rank idiosyncratic component. In this paper we focus on the estimation of the idiosyncratic component, under the assumption of a grouped structure of the time series, which may arise due to specific factors such as industries, asset classes or countries. We propose a generalized methodology for estimation of the block-diagonal idiosyncratic component by clustering the residual series and applying shrinkage to the obtained blocks in order to ensure positive definiteness. We derive two different estimators based on different clustering methods and test their performance using simulation and historical data. The proposed methods are shown to provide reliable estimates and outperform other state-of-the-art estimators based on thresholding methods.

Block-diagonal idiosyncratic covariance estimation in high-dimensional factor models for financial time series

Abstract

Estimation of high-dimensional covariance matrices in latent factor models is an important topic in many fields and especially in finance. Since the number of financial assets grows while the estimation window length remains of limited size, the often used sample estimator yields noisy estimates which are not even positive definite. Under the assumption of latent factor models, the covariance matrix is decomposed into a common low-rank component and a full-rank idiosyncratic component. In this paper we focus on the estimation of the idiosyncratic component, under the assumption of a grouped structure of the time series, which may arise due to specific factors such as industries, asset classes or countries. We propose a generalized methodology for estimation of the block-diagonal idiosyncratic component by clustering the residual series and applying shrinkage to the obtained blocks in order to ensure positive definiteness. We derive two different estimators based on different clustering methods and test their performance using simulation and historical data. The proposed methods are shown to provide reliable estimates and outperform other state-of-the-art estimators based on thresholding methods.
Paper Structure (23 sections, 34 equations, 5 figures, 3 tables)

This paper contains 23 sections, 34 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Estimation
  4. Estimating the number of factors
  5. Estimating the idiosyncratic covariance via thresholding
  6. Block-diagonal idiosyncratic covariance estimation
  7. Estimating the blocks using predefined asset groups
  8. Clustering based estimation of the idiosyncratic covariance
  9. Estimating blocks using $k$-means clustering
  10. Estimating blocks using hierarchical clustering
  11. Iterative procedure for selecting the number of blocks
  12. Data and performance measures
  13. Simulation data
  14. Generating the common component
  15. Generating the idiosyncratic component
  16. ...and 8 more sections

Figures (5)

  • Figure 1: Validation errors through iterations over the hyperparameter space for both estimators performed on an example estimation window using historical market data. The red lines show the minimum value and the iteration it was reached in.
  • Figure 2: Out-of-sample portfolio risk $\sigma_p$ for different dimensionalities of the data. The sample window length is $T=250$ and the dimension varies from $p=250$ to $p=1000$ with a step size of $50$.
  • Figure 3: Plot of the simulated idiosyncratic covariance (ground truth) for a single simulation case, in comparison to the idiosyncratic covariance estimated by the CSH, CSK and the SCAD estimators. Top row shows the full block-diagonal case, and the bottom row shows the partial block-diagonal case. Blue areas on the matrices correspond to zero-valued entries.
  • Figure 4: Estimated number of factors throughout the historical time period, for $p=1000$ assets.
  • Figure 5: Block-diagonal idiosyncratic covariances obtained by using respective CSI, CSK and CSH estimators. The blue areas represent the zero-valued entries.