Abstract
In this paper we consider a portfolio selection problem under the global minimum variance model where the optimal portfolio weights only depend on the covariance matrix of asset returns. First, to reflect the rapid changes of financial markets, we incorporate a time-varying factor in the covariance matrix. Second, to improve the estimation of the covariance matrix we use the shrinkage method. Based on these two key aspects, we propose a framework for shrinking the time-varying inverse conditional covariance matrix in order to enhance the performance of the portfolio selection. Furthermore, given the shortcoming that the inverse covariance matrix is inaccurate in a number of cases, we develop a new method that transforms the inverse of the covariance matrix into a product to improve the performance of the inverse covariance matrix, and prove its theoretical availability. The proposed portfolio selection strategy is applied to analyze real-world data and the numerical studies show it performs well.
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Acknowledgements
We would like to thank the Editor and Referees very much for their constructive comments, which significantly helped us to improve the manuscript. The first two authors’ research was supported by the Fundamental Research Funds for Central Universities, China (Nos. JBK1607121, JBK120509, JBK140507, JBK141111). This study was also supported by the National Natural Science Foundation of China (Nos. 11471264, 11401148, 51437003).
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Appendix
Appendix
1.1 A1. Complete expression of (8)
Assume that G, M and \(G+M\) are invertible matrices, then we have
Using the above formula, we have
where
1.2 A2. Proof of Proposition 1
Taking the derivative of \({\widehat{\beta }}_{i}\) with respect to \(\beta _{1}\), we obtain
Similarly, we have
We can reformulate the above formulas in terms of D and b, \(DK=b\), where \(K=(\beta _{1},\beta _{2},\beta _{3})^\top \). So, \(\varXi =D^{-1}b\). The proof is complete. \(\square \)
1.3 A3. Proof of Proposition 2
Similar to the proof of Proposition 1, we can get \(DK=b\), where \(K=(\beta _{1},\beta _{2},\beta _{3},\beta _{4})^\top \). Then, we have \(\varXi =D^{-1}b\). The proof is complete. \(\square \)
1.4 A4. Proof of Proposition 3
Let \(\varPhi =\varOmega -I\), then \(\varPhi \) and \(I-\varPhi +\varPhi ^{2}\) are symmetric.
So, \(I-\varPhi +\varPhi ^{2}\) is positive definite. The proof is complete. \(\square \)
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Sun, R., Ma, T. & Liu, S. Portfolio selection: shrinking the time-varying inverse conditional covariance matrix. Stat Papers 61, 2583–2604 (2020). https://doi.org/10.1007/s00362-018-1059-0
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DOI: https://doi.org/10.1007/s00362-018-1059-0