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The asymptotic spectrum of the EWMA covariance estimator
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2007 (English)In: Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, E-ISSN 1873-2119, no 385, 621-630 p.Article in journal (Refereed) Published
Abstract [en]

The exponentially weighted moving average (EWMA) covariance estimator is a standard estimator for financial time series, and its spectrum can be used for so-called random matrix filtering. Random matrix filtering using the spectrum of the sample covariance matrix is an established tool in finance and signal detection and the EWMA spectrum can be used analogously. In this paper, the asymptotic spectrum of the EWMA covariance estimator is calculated using the Mar enko-Pastur theorem. Equations for the spectrum and the boundaries of the support of the spectrum are obtained and solved numerically. The spectrum is compared with covariance estimates using simulated i.i.d. data and log-returns from a subset of stocks from the S&P 500. The behaviour of the EWMA estimator in this limited empirical study is similar to the results in previous studies of sample covariance matrices. Correlations in the data are found to only affect a small part of the EWMA spectrum, suggesting that a large part may be filtered out

Place, publisher, year, edition, pages
2007. no 385, 621-630 p.
Keyword [en]
EWMA; random matrix theory; covariance estimation; noise
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-7578DOI: 10.1016/j.physa.2007.07.030ISI: 000250491900021Scopus ID: 2-s2.0-34548759639OAI: oai:DiVA.org:kth-7578DiVA: diva2:12648
Note
QC 20100811Available from: 2007-11-07 Created: 2007-11-07 Last updated: 2010-11-19Bibliographically approved
In thesis
1. Some asymptotic results in dependence modelling
Open this publication in new window or tab >>Some asymptotic results in dependence modelling
2007 (English)Licentiate thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis consists of two papers, both devoted to the study of asymptotics in dependence modelling.

The first paper studies large deviation probabilities for a sum of dependent random variables, where the dependence stems from a few underlying random variables, so-called factors. Each summand is composed of two parts: an idiosyncratic part and a part given by the factors. Conditions under which both factors and idiosyncratic components contribute to the large deviation behaviour are found and the resulting approximation is evaluated in a simple special case. The results are then applied to stochastic processes with the same structure. Based on the results of the first part of the paper, it is concluded that large deviations on a finite time interval are due to one large jump that can come from either the factor or the idiosyncratic part of the process.

The second paper studies the asymptotic eigenvalue distribution of the exponentially weighted moving average (EWMA) covariance estimator. Equations for the limiting eigenvalue density and the boundaries of its support are found using the Marchenko-Pastur theorem.

Place, publisher, year, edition, pages
Stockholm: KTH, 2007. iii p.
Series
Trita-MAT, ISSN 1401-2286 ; 2007:01
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-4519 (URN)978-91-7178-792-7 (ISBN)
Presentation
2007-11-05, 3733, Matematiska Institutionen, Lindstedtsvägen 25, Stockholm, 15:30
Opponent
Supervisors
Note
QC 20101119Available from: 2007-11-07 Created: 2007-11-07 Last updated: 2010-11-19Bibliographically approved
2. On Importance Sampling and Dependence Modeling
Open this publication in new window or tab >>On Importance Sampling and Dependence Modeling
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers.

In the first paper, Monte Carlo simulation for tail probabilities of heavy-tailed random walks is considered. Importance sampling algorithms are constructed by using mixtures of the original distribution with some other state-dependent distributions. Sufficient conditions under which the relative error of such algorithms is bounded are found, and the bound is calculated. A new mixture algorithm based on scaling of the original distribution is presented and compared to existing algorithms.

In the second paper, Monte Carlo simulation of quantiles is treated. It is shown that by using importance sampling algorithms developed for tail probability estimation, efficient quantile estimators can be obtained. A functional limit of the quantile process under the importance sampling measure is found, and the variance of the limit process is calculated for regularly varying distributions. The procedure is also applied to the calculation of expected shortfall. The algorithms are illustrated numerically for a heavy-tailed random walk.

In the third paper, large deviation probabilities for a sum of dependent random variables are derived. The dependence stems from a few underlying random variables, so-called factors. Each summand is composed of two parts: an idiosyncratic part and a part given by the factors. Conditions under which both factors and idiosyncratic components contribute to the large deviation behavior are found, and the resulting approximation is evaluated in a simple example.

In the fourth paper, the asymptotic eigenvalue distribution of the exponentially weighted moving average covariance estimator is studied. Equations for the asymptotic spectral density and the boundaries of its support are found using the Marchenko-Pastur theorem.

Place, publisher, year, edition, pages
Stockholm: KTH, 2009. vi, 13 p.
Series
Trita-MAT. MS, 09:11
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-11272 (URN)978-91-7415-433-7 (ISBN)
Public defence
2009-10-23, D2, Lindstedtsvägen 5, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note
QC 20100811Available from: 2009-10-14 Created: 2009-10-13 Last updated: 2010-08-11Bibliographically approved

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