# Studies in covariance estimation and applications in finance

## Abstract/Contents

- Abstract
- This thesis examines estimation of covariance and correlation matrices. More specifically we will in the first part study dynamical properties of the top eigenvalue and eigenvector for sample estimators of covariance and correlation matrices. This is done under the assumption that the top eigenvalue is separated from the others, which is reasonable when the data comes from financial returns. We show exactly how these quantities behave when the true covariance or correlation is stationary and derive theoretical values of related quantities that can be useful when quantifying the amount of non-stationarity for real data. We also validate the results by using Monte-Carlo simulations. A major contribution from the analysis is that it shows how and under which regimes correlation matrices differ from covariance matrices from a dynamic viewpoint. This effect has been observed in financial data, but never explained. In the second part of the thesis we study modifications to covariance estimators that find the optimal estimator within a certain sub-class. This type of estimators is generally known as shrinkage estimators as they modify only eigenvalues of the original estimator. We will do this when the original estimator takes the form A1/2XBXT A1/2, where A and B are matrices and X is a matrix of i.i.d. variables. The analysis is done in the asymptotic limit where both the number of samples and variables approach infinity jointly so that random-matrix theory can be used. Our goal is to find the shrinkage estimator which minimizes expected value of the Frobenius norm between the estimator and the true covariance matrix. To do this we first derive a generalization to the Marchenko-Pastur equation for this class of estimators. This theorem allows us to calculate the asymptotic value of the projection of the sample eigenvectors onto the true covariance matrix. We then show how to use these to find the optimal covariance estimator. At last, we show with simulations that these estimators are close to the optimal bound when used on finite data sets.

## Description

Type of resource | text |
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Form | electronic; electronic resource; remote |

Extent | 1 online resource. |

Publication date | 2016 |

Issuance | monographic |

Language | English |

## Creators/Contributors

Associated with | Arndt, Carl-Fredrik | |
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Associated with | Stanford University, Institute for Computational and Mathematical Engineering. | |

Primary advisor | Papanicolaou, George | |

Thesis advisor | Papanicolaou, George | |

Thesis advisor | Johnstone, Iain | |

Thesis advisor | Ryzhik, Leonid | |

Advisor | Johnstone, Iain | |

Advisor | Ryzhik, Leonid |

## Subjects

Genre | Theses |
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## Bibliographic information

Statement of responsibility | Carl-Fredrik Arndt. |
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Note | Submitted to the Institute for Computational and Mathematical Engineering. |

Thesis | Thesis (Ph.D.)--Stanford University, 2016. |

Location | electronic resource |

## Access conditions

- Copyright
- © 2016 by Carl-Fredrik Arndt
- License
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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