# An Improved DCC Model Based on Large-Dimensional Covariance Matrices Estimation and Its Applications

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- We propose a new large covariance matrix estimator to realize the sparsity and positive-definiteness by constructing a nonconvex optimization model with smoothly clipped absolute deviation (SCAD) and hard threshold penalty functions based on the rotation-invariant estimator.
- To improve the performance of the DCC model, we use the new covariance matrix estimator to replace the unconditional covariance matrix in the DCC model.
- We show that the improved DCC model has a smaller loss and lower out-of-sample risk in portfolio optimization model.

## 2. Preliminary Work

#### 2.1. Covariance Matrix Estimation Based on Convex Combination

#### 2.2. Classical DCC Model

## 3. An Improved DCC Model Based on Nonconvex Combination

- Step 1: Solve the model (1) to obtain the covariance matrix estimation ${\Sigma}^{*}$.
- Step 2: Input ${\Sigma}^{*}$ into the nonconvex optimization model (7) and use ADM to solve it to obtain the covariance matrix estimation $\Theta $.
- Step 4: Use the covariance matrix to estimate ${\Sigma}^{*}$ to replace the unconditional covariance matrix $\overline{S}$ in Equation (6), and get the matrix ${Q}_{t}$ by maximizing the composite likelihood function.
- Step 5: Standardize ${Q}_{t}$ and calculate the dynamic conditional correlation matrix ${H}_{t}$ in Equation (2).

## 4. Numerical Experiments and Application

#### 4.1. The Simulation Datasets

#### 4.1.1. The Simulation of Typical Sparse Covariance Matrix

#### 4.1.2. The Monte Carlo Simulation

- DCC-S: The $\overline{S}$ in the DCC model is replaced by the sample covariance matrix ${\Sigma}_{SCM}$.
- DCC-L2: The $\overline{S}$ in the DCC model is replaced by the estimator obtained from the method of Engle and Ledoit [7].
- DCC-NL: The $\overline{S}$ in the DCC model is replaced by the estimator obtained from the method of [18].
- DCC-NCP1: The $\overline{S}$ in the DCC model is replaced by the estimator ${\Sigma}_{Hard}$ obtained from the model (7) based on the hard-threshold penalty function.
- DCC-NCP2: The $\overline{S}$ in the DCC model is replaced by the estimator ${\Sigma}_{SCAd}$ obtained from the model (7) based on the SCAD penalty function.

#### 4.2. The Application

#### 4.2.1. Global Minimum Variance Portfolio

#### 4.2.2. Analysis of Empirical Research

## 5. Discussion

## 6. Conlusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Engel, J.; Buydens, L.; Blanchet, L. An overview of large-dimensional covariance and precision matrix estimators with applications in chemometrics. J. Chemom.
**2017**, 31, e2880. [Google Scholar] [CrossRef] - Fan, J.; Liao, Y.; Liu, H. An overview of the estimation of large covariance and precision matrices. Econom. J.
**2016**, 19, C1–C32. [Google Scholar] [CrossRef] - Tong, T.; Wang, C.; Wang, Y. Estimation of variances and covariances for high-dimensional data: A selective review. Wiley Interdiscip. Rev. Comput. Stat.
**2014**, 6, 255–264. [Google Scholar] [CrossRef] - Stein, C. Lectures on the theory of estimation of many parameters. J. Sov. Math.
**1986**, 34, 1373–1403. [Google Scholar] [CrossRef] - Engle, R.F.; Ledoit, O.; Wolf, M. Large dynamic covariance matrices. J. Bus. Econom. Stat.
**2019**, 37, 363–375. [Google Scholar] [CrossRef] - Ledoit, O.; Wolf, M. Nonlinear shrinkage of the covariance matrix for portfolio selection: Markowitz meets goldilocks. Rev. Financ. Stud.
**2018**, 30, 4349–4388. [Google Scholar] [CrossRef] - Ledoit, O.; Wolf, M. Honey, I shrunk the sample covariance matrix. J. Portfolio. Mange.
**2004**, 30, 110–119. [Google Scholar] [CrossRef] - Bickel, P.J.; Elizaveta, L. Covariance regularization by thresholding. Ann. Stat.
**2008**, 36, 2577–2604. [Google Scholar] [CrossRef] - Rothman, A.J.; Bickel, P.J.; Levina, E.; Zhu, J. Sparse permutation invariant covariance estimation. Electron. J. Stat.
**2008**, 2, 494–515. [Google Scholar] - Balmand, S.; Dalalyan, A.S. On estimation of the diagonal elements of a sparse precision matrix. Electron. J. Stat.
**2016**, 10, 1551–1579. [Google Scholar] - Ravikumar, P.; Wainwright, M.J.; Raskutti, G.; Yu, B. High-dimensional covariance estimation by minimizing L
_{1}-penalized log-determinant divergence. Electron. J. Stat.**2011**, 5, 935–980. [Google Scholar] [CrossRef] - Zhou, S.; Xiu, N.H.; Luo, Z.; Kong, L.C. Sparse and low-rank covariance matrix estimation. J. Oper. Res. Soc. China
**2015**, 3, 231–250. [Google Scholar] [CrossRef] - Fan, J.; Li, R. Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc.
**2001**, 96, 1348–1360. [Google Scholar] [CrossRef] - Fan, J.; Peng, J. Nonconcave penalized likelihood with a diverging number of parameters. Ann. Stat.
**2004**, 32, 928–961. [Google Scholar] [CrossRef] - Engle, R. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. J. Bus. Econom. Stat.
**2002**, 20, 339–350. [Google Scholar] [CrossRef] - Bollerslev, T. Generalized autoregressive conditional heteroskedasticity. EERI Res. Paper.
**1986**, 31, 307–327. [Google Scholar] [CrossRef] - Bollerslev, T. Modeling the coherence in short-run nominal exchange rates: A multivariate generalized ARCH model. Rev. Econ. Stat.
**1990**, 72, 498–505. [Google Scholar] [CrossRef] - Ledoit, O.; Wolf, M. Nonlinear shrinkage estimation of large-dimensional covariance matrices. Ann. Stat.
**2012**, 40, 1024–1060. [Google Scholar] [CrossRef] - De Nard, G.; Engle, R.F.; Ledoit, O.; Wolf, M. Large dynamic covariance matrices: Enhancements based on intraday data. J. Bank. Financ.
**2022**, 138, 1–16. [Google Scholar] [CrossRef] - Jarjour, R.; Chan, K.S. Dynamic conditional angular correlation. J. Econom.
**2020**, 216, 137–150. [Google Scholar] [CrossRef] - Tse, Y.K.; Tsui, A.K.C. A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations. J. Bus. Econom. Stat.
**2002**, 20, 51–362. [Google Scholar] [CrossRef] - Yuan, X.; Yu, W.; Yin, Z.X.; Wang, G.Q. Improved large dynamic covariance matrix estimation with graphical lasso and its application in portfolio selection. IEEE Access
**2020**, 8, 189179–189188. [Google Scholar] [CrossRef] - Wen, F.; Yang, Y.; Liu, L.P.; Qiu, R.C. Positive definite estimation of large covariance matrix using generalized non-convex penalties. IEEE Access
**2016**, 4, 4168–4182. [Google Scholar] [CrossRef] - Zhang, Y.; Tao, J.Y.; Yin, Z.X.; Wang, G.Q. Improved large covariance matrix estimation based on efficient convex combination and its application in portfolio optimization. Mathematics
**2022**, 10, 4282. [Google Scholar] [CrossRef] - Ledoit, O.; Wolf, M. Quadratic shrinkage for large covariance matrices. Bernoulli
**2022**, 28, 1519–1547. [Google Scholar] [CrossRef] - Antoniadis, A. Wavelets in Statistics: A Review (with discussion). J. Ital. Stat. Soc.
**1997**, 6, 97–130. [Google Scholar] [CrossRef] - Pakel, C.; Shephard, N.; Sheppard, K.; Engle, R.F. Fitting vast dimensional time-Varying covariance models. J. Bus. Econ. Stat.
**2021**, 39, 652–668. [Google Scholar] [CrossRef] - Hafner, C.M.; Reznikova, O. On the estimation of dynamic conditional correlation models. Comput. Stat. Data Anal.
**2012**, 56, 3533–3545. [Google Scholar] [CrossRef]

**Figure 1.**The simulated matrices for $d=100$. (

**a**) Simulation matrix-Block matrix; (

**b**) Simulation matrix-Toeplitz matrix; (

**c**) Simulation matrix-Banded matrix.

**Figure 2.**The heat map of each estimation for $d=100$. (

**a**) Block matrix ($d=100$, $n=100$); (

**b**) Block matrix ($d=100$, $n=1000$); (

**c**) Block matrix ($d=100$, $n=100$); (

**d**) Block matrix ($d=100$, $n=1000$); (

**e**) Block matrix ($d=100$, $n=100$); (

**f**) Block matrix ($d=100$, $n=1000$).

**Figure 3.**The sum error of the estimator ${\Sigma}_{SCAD}$ in five folds cross-validation under the F-norm for $d=100$. (

**a**) Block matrix ($f=1$, $n=100$); (

**b**) Block matrix ($f=10$, $n=1000$); (

**c**) Block matrix ($f=1$, $n=100$); (

**d**) Block matrix ($f=1$, $n=100$); (

**e**) Block matrix ($f=10$, $n=1000$); (

**f**) Block matrix ($f=1$, $n=100$).

**Figure 4.**The heat map of each estimator for $d=400$. (

**a**) Block matrix ($d=400$, $n=100$); (

**b**) Block matrix ($d=400$, $n=1000$); (

**c**) Block matrix ($d=400$, $n=100$); (

**d**) Block matrix ($d=400$, $n=1000$); (

**e**) Block matrix ($d=400$, $n=100$); (

**f**) Block matrix ($d=400$, $n=1000$).

**Figure 5.**The sum error of the estimator ${\Sigma}_{SCAD}$ in five folds cross-validation under the F-norm for $d=400$. (

**a**) Block matrix ($f=1$, $n=100$); (

**b**) Block matrix ($f=10$, $n=1000$); (

**c**) Block matrix ($f=1$, $n=100$); (

**d**) Block matrix ($f=1$, $n=100$); (

**e**) Block matrix ($f=10$, $n=1000$); (

**f**) Block matrix ($f=1$, $n=100$).

**Figure 6.**The average relative error of each estimator for $d=100$. (

**a**) Simulation matrix-Block matrix; (

**b**) Simulation matrix-Toeplitz matrix; (

**c**) Simulation matrix-Banded matrix.

**Figure 7.**The average relative error of each estimator for $d=400$. (

**a**) Simulation matrix-Block matrix; (

**b**) Simulation matrix-Toeplitz matrix; (

**c**) Simulation matrix-Banded matrix.

**Table 1.**The loss of five estimators in different dimensions of the assets. (The unit is ${10}^{-6}$).

N | DCC-S | DCC-L2 | DCC-NL | DCC-NCP1 | DCC-NCP2 |
---|---|---|---|---|---|

100 | 4.6874 | 0.6223 | 0.4996 | 0.2279 | 2.4265 |

400 | 3.2222 | 2.4103 | 3.1089 | 2.6286 | 0.2583 |

800 | 5.6512 | 4.2802 | 4.9354 | 3.7852 | 1.4324 |

**Table 2.**The out-of-sample performance of five estimators of 41 assets based on SSE50 in the portfolio optimization model.

Model | MR * | SD * | SR | |||
---|---|---|---|---|---|---|

100 d | 200 d | 100 d | 200 d | 100 d | 200 d | |

DCC-S | 37.80 | 37.80 | 20.783 | 16.561 | 1.735 | 2.177 |

DCC-L2 | 37.80 | 37.80 | 20.700 | 16.417 | 1.742 | 2.196 |

DCC-NL | 37.80 | 37.80 | 20.701 | 16.470 | 1.741 | 2.189 |

DDC-NCP1 | 37.80 | 37.80 | 20.658 | 16.652 | 1.745 | 2.165 |

DCC-NCP2 | 37.80 | 37.80 | 20.488 | 16.543 | 1.760 | 2.179 |

**Table 3.**The out-of-sample performance of five estimators of 218 assets based on HS300 in the portfolio optimization model.

Model | MR * | SD * | SR | |||
---|---|---|---|---|---|---|

100 d | 200 d | 100 d | 200 d | 100 d | 200 d | |

DCC-S | 37.800 | 37.800 | 15.447 | 15.393 | 2.334 | 2.2.342 |

DCC-L2 | 37.800 | 37.800 | 15.305 | 15.174 | 2.355 | 2.376 |

DCC-NL | 37.800 | 37.800 | 15.319 | 15.377 | 2.353 | 2.346 |

DDC-NCP1 | 37.800 | 37.800 | 15.271 | 15.418 | 2.367 | 2.338 |

DCC-NCP2 | 37.800 | 37.800 | 15.559 | 15.079 | 2.317 | 2.391 |

**Table 4.**The out-of-sample performance of five estimators of 426 assets based on CSI500 in the portfolio optimization model.

Model | MR * | SD * | SR | |||
---|---|---|---|---|---|---|

100 d | 200 d | 100 d | 200 d | 100 d | 200 d | |

DCC-S | 37.800 | 37.800 | 13.132 | 13.452 | 2.745 | 2.680 |

DCC-L2 | 37.800 | 37.800 | 13.029 | 13.543 | 2.767 | 2.662 |

DCC-NL | 37.800 | 37.800 | 12.906 | 13.384 | 2.793 | 2.693 |

DDC-NCP1 | 37.800 | 37.800 | 12.883 | 13.219 | 2.798 | 2.727 |

DCC-NCP2 | 37.800 | 37.800 | 12.951 | 13.189 | 2.783 | 2.733 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhang, Y.; Tao, J.; Lv, Y.; Wang, G.
An Improved DCC Model Based on Large-Dimensional Covariance Matrices Estimation and Its Applications. *Symmetry* **2023**, *15*, 953.
https://doi.org/10.3390/sym15040953

**AMA Style**

Zhang Y, Tao J, Lv Y, Wang G.
An Improved DCC Model Based on Large-Dimensional Covariance Matrices Estimation and Its Applications. *Symmetry*. 2023; 15(4):953.
https://doi.org/10.3390/sym15040953

**Chicago/Turabian Style**

Zhang, Yan, Jiyuan Tao, Yongyao Lv, and Guoqiang Wang.
2023. "An Improved DCC Model Based on Large-Dimensional Covariance Matrices Estimation and Its Applications" *Symmetry* 15, no. 4: 953.
https://doi.org/10.3390/sym15040953