臺灣博碩士論文加值系統

近幾年來透過因子分析(factor analysis)來估計高維度下的共變異矩陣(high dimensional covariance matrix)是越來越受歡迎，但要應用因子分析來估計在高維度下的共變異矩陣的反矩陣(high dimensional precision matrix)是非常的困難，因為在估計高維度誤差的共變異矩陣的反矩陣(high dimensional error precision matrix)當中，通常都含有稀疏(sparse)的限制。這篇論文結合了 modified Cholesky decomposition 以及 orthogonal greedy algorithm (OGA)的方法來估計在稀疏限制下高維度誤差的共變異矩陣的反矩陣，並應用在財務上 mean-variance portfolio optimization 的問題。在模擬的結果中，我們所提的方法比傳統的 threshold 來的更好。

Recently, it has drawn attention on estimation of high-dimensional covariance matrices by using factor analysis. However, it is very difficult to apply factor analysis estimation of high-dimensional precision matrices. Because one of the commonly used conditions for estimating high-dimensional error precision matrix is to assume the covariance matrix to be sparse. This study combine modified Cholesky decomposition and orthogonal greedy algorithm (OGA) approaches to estimate the high-dimensional precision matrix under the constraint that the covariance matrix is sparse. The result can be used to deal with the mean-variance portfolio optimization problem. According to the simulation results, the proposed approach outperforms the adaptive thresholding method.

Contents
1 Introduction 3
2 Estimating large precision matrix 4
2.1 Adaptive thresholding estimation . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Main estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Modied Cholesky decomposition . . . . . . . . . . . . . . . . . . . 5
2.2.2 OGA method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 MV optimization and Factor model 10
3.1 MV portfolio optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Factor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 The factor analysis-based estimator . . . . . . . . . . . . . . . . . . . . . . 11
4 Simulation study 13
4.1 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Conclusion 16
6 References 16

Bouchaud, J.P., Cizeau, P., Laloux, L. and Potters, M. (1999). Noise dressing of nancial
correlation matrices. Physical Review Letters 83, 1467-1470.
Bai, Z.D., Liu, H. andWong, W.K. (2009). Enhancement of the applicability of Markowitz0s
portfolio optimization by utilizing random matrix theory. Mathematical Finance 19, 639-
667.
Chiou, H.T., Guo, M. and Ing, C.K. (2014). Estimation of inverse autocovariance matrices
for long memory processes. Bernoulli, to appear.
Chen, B.B., Huang, S.F. and Pan, G. (2015). High dimensional meanvariance optimization through factor analysis. Journal of Multivariate Analysis 133, 140-159.
Cai, T. and Liu, W. (2011). Adaptive thresholding for sparse covariance matrix estimation.
Journal of the American Statistical Association 106:494, 672-684.
Fan, F., Fan, Y. and Lv, J. (2008). High dimensional covariance matrix estimation using
a factor model. Journal of Econometrics 147, 186-197.
Fan, J., Han, F. and Liu, H. (2014). Challenges of big data analysis. National sci-
ence review 1, 293-314.
Fan, F., Liu, H. and Liao Y. (2015). An overview on the estimation of large covariance
and precision matrices. Working paper.
Fan, F., Liao, Y. and Mincheva, M. (2011). Large covariance estimation by thresholding
principal orthogonal complements. Journal of the Royal Statistical Society: Series
B (Statistical Methodology) 75, 603-680.
Kroll, Y., Levy, H. and Markowitz, H.M. (1984). Mean-variance versus direct utility
maximization. Journal of Finance 39, 47-61.
Lai, T.L. and Ing, C.K. (2011). A stepwise regression method and consistent model
selection for high-dimensional sparse linear models. Statistica Sinica 21, 1473-1513.
Levina, E., Rothman, A. and Zhu, J. (2008). Sparse estimation of large covariance matrices
via a nested lasso penalty. The Annals of Applied Statistics 2, 245-263.
Ledoit, O. and Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance
matrices. Journal of Multivariate Analysis 88, 365-411.
Markowitz, H.M. (1952). Portfolio selection. Journal of Finance 7, 77-91.
McNamara, J.R. (1998). Portfolio selection using stochastic dominance criteria. Deci-
sion Sciences 29, 785-801.
Merton, R.C. (1972). An analytic derivation of the ecient portfolio frontier. The Jour-
nal of Financial and Quantitative Analysis 7, 1851-1872.
Onatski, A. (2010). Determining the number of factors from empirical distribution of
eigenvalues. The Review of Economics and Statistics 92 1004-1016.
Ross, S.A. (1976). The arbitrage theory of capital asset pricing. Journal of Economic
Theory 12, 341-360.