It is well known that the traditional maximum likelihood estimation of GARCH model is severely biased in high dimensions. In this paper, the nonlinear shrinkage method proposed by Ledoit and Wolf is used to estimate DCC and BEKK models. In particular, the initial sample covariance estimator in maximum m-profile quasi-likelihood estimation (MMLE) proposed by Engle et al. is substituted by the nonlinear shrinkage estimator, which turns out to largely improve the estimation efficiency of high dimensional DCC and BEKK models, and for the first time, makes the valid estimation possible when the sample size is larger than the time series dimension. Based on the Percentage Relative Improvement in Average Loss (PRIAL), the Monte-Carlo simulations verify the obvious superiority of the nonlinear shrinkage substitution over the usual DCC and BEKK, which even strengthens as the ratio between sample size and time series dimension increases. Besides, for both DCC and BEKK, the performance of nonlinear shrinkage estimation is better than that of linear shrinkage, while linear shrinkage estimation is better than the usual estimation. Furthermore, the performance of DCC is better than BEKK, and the optimizing effect of nonlinear shrinkage on DCC is more significant than on BEKK. Finally, in the empirical part, using daily stock return data from the Center for Research in Security Prices (CRSP), the global minimum variance (GMV) portfolios of stocks traded in NYSE and NASDAQ are constructed based on various methods, and their real variances are compared. The empirical result supports the important role nonlinear shrinkage plays in promoting the estimation of high dimensional conditional covariance matrix, and thus in optimizing the portfolio selection.
ZHAO Zhao
. Nonlinear Shrinkage Estimation of High Dimensional Conditional Covariance Matrix and its Application in Portfolio Selection[J]. Chinese Journal of Management Science, 2017
, 25(8)
: 46
-57
.
DOI: 10.16381/j.cnki.issn1003-207x.2017.08.006
[1] Lam C, Fan Jianqing.Sparsistency and rates of convergence in large covariance matrix estimation[J]. Annals of Statistics, 2009,37(6B):4254-4278.
[2] Rigollet P, Tsybakov A B. Sparse estimation by exponential weighting[J]. Statistical Science, 2012,27(4):558-575.
[3] Fama E F, French K R. Common risk factors in the returns on stocks and bonds[J]. Journal of Financial Economics, 1993,33(1):3-56.
[4] Chen N F, Roll R, Ross S A.Economic forces and the stock market[J]. Journal of Business, 1986,59(3):383-403.
[5] Ross S A.The arbitrage theory of capital asset pricing[J]. Journal of Economic Theory, 1976,13(3):341-360.
[6] Ledoit O, Wolf M. A well-conditioned estimator for large-dimensional covariance matrices[J]. Journal of Multivariate Analysis, 2004,88(2):365-411.
[7] Ledoit O, Wolf M.Improved estimation of the covariance matrix of stock returns with an application to portfolio selection[J]. Journal of Empirical Finance, 2003,10(5):603-621.
[8] Ledoit O, Wolf M.Honey, I shrunk the sample covariance matrix[J]. Journal of Portfolio Management, 2004,30(4):110-119.
[9] Ledoit O, Wolf M.Nonlinear shrinkage estimation of large-dimensional covariance matrices[J]. The Annals of Statistics, 2012,40(2):1024-1060.
[10] Ledoit O, Wolf M.Optimal estimation of a large-dimensional covariance matrix under Stein's loss[J]. Working Paper, University of Zurich,2014.
[11] Ledoit O, Wolf M.Spectrum estimation:A unified framework for covariance matrix estimation and PCA in large dimensions[J]. Journal of Multivariate Analysis, 2015,139:360-384.
[12] Bollerslev T, Engle R F, Wooldridge J M.A capital asset pricing model with time-varying covariances[J]. The Journal of Political Economy, 1988,96(1):116-131.
[13] Engle R F, Kroner K F.Multivariate simultaneous generalized ARCH[J]. Econometric theory, 1995,11(1):122-150.
[14] Engle R.Dynamic conditional correlation:A simple class of multivariate generalized autoregressive conditional heteroskedasticity models[J]. Journal of Business & Economic Statistics, 2002,20(3):339-350.
[15] Ledoit O, Santa-Clara P, Wolf M.Flexible multivariate GARCH modeling with an application to international stock markets[J]. Review of Economics and Statistics,2003, 85(3):735-747.
[16] Cappiello L, Engle R F, Sheppard K.Asymmetric dynamics in the correlations of global equity and bond returns[J]. Journal of Financial Econometrics, 2006,4(4):537-572.
[17] Engle R, Kelly B.Dynamic equicorrelation[J]. Journal of Business & Economic Statistics, 2012,30(2):212-228.
[18] 刘志东,薛莉. 金融市场高维波动率的扩展广义正交GARCH模型与参数估计方法研究[J]. 中国管理科学, 2010, 18(6):33-41.
[19] 刘丽萍, 马丹, 白万平. 大维数据的动态条件协方差阵的估计及其应用[J]. 统计研究, 2015, 32(6):105-112.
[20] 张贵生, 张信东. 基于近邻互信息的SVM-GARCH股票价格预测模型研究[J]. 中国管理科学, 2016, 24(9):11-20.
[21] Hafner C M, Reznikova O.On the estimation of dynamic conditional correlation models[J]. Computational Statistics & Data Analysis, 2012,56(11):3533-3545.
[22] Engle R F, Shephard N, Sheppard K.Fitting and testing vast dimensional time-varying covariance models[R].Working Paper,New York University,2007.
[23] Engle R, Mezrich J. Garch for groups:A round-up of recent developments in Garch techniques for estimating correlation[J]. Risk:Managing Risk in the World's Financial Markets, 1996,9(8):36-40.
[24] Stein C.Estimation of a covariance matrix[R]. Conference Paper,Rietz Lecture,1975.
[25] Stein C.Lectures on the theory of estimation of many parameters[J]. Journal of Soviet Mathematics,1986,34(1):1373-1403.
[26] Silverstein J W, Choi S I.Analysis of the limiting spectral distribution of large dimensional random matrices[J]. Journal of Multivariate Analysis, 1995,54:295-309.
