6,585 research outputs found
Improved estimation of the covariance matrix of stock returns with an application to portofolio selection
This paper proposes to estimate the covariance matrix of stock returns by an optimally weighted average of two existing estimators: the sample covariance matrix and single-index covariance matrix. This method is generally known as shrinkage, and it is standard in decision theory and in empirical Bayesian statistics. Our shrinkage estimator can be seen as a way to account for extra-market covariance without having to specify an arbitrary multi-factor structure. For NYSE and AMEX stock returns from 1972 to 1995, it can be used to select portfolios with significantly lower out-of-sample variance than a set of existing estimators, including multi-factor models.Covariance matrix estimation, factor models, portofolio selection, shrinkage
Optimal trading strategies - a time series approach
Motivated by recent advances in the spectral theory of auto-covariance
matrices, we are led to revisit a reformulation of Markowitz' mean-variance
portfolio optimization approach in the time domain. In its simplest incarnation
it applies to a single traded asset and allows to find an optimal trading
strategy which - for a given return - is minimally exposed to market price
fluctuations. The model is initially investigated for a range of synthetic
price processes, taken to be either second order stationary, or to exhibit
second order stationary increments. Attention is paid to consequences of
estimating auto-covariance matrices from small finite samples, and
auto-covariance matrix cleaning strategies to mitigate against these are
investigated. Finally we apply our framework to real world data
Covariance Estimation: The GLM and Regularization Perspectives
Finding an unconstrained and statistically interpretable reparameterization
of a covariance matrix is still an open problem in statistics. Its solution is
of central importance in covariance estimation, particularly in the recent
high-dimensional data environment where enforcing the positive-definiteness
constraint could be computationally expensive. We provide a survey of the
progress made in modeling covariance matrices from two relatively complementary
perspectives: (1) generalized linear models (GLM) or parsimony and use of
covariates in low dimensions, and (2) regularization or sparsity for
high-dimensional data. An emerging, unifying and powerful trend in both
perspectives is that of reducing a covariance estimation problem to that of
estimating a sequence of regression problems. We point out several instances of
the regression-based formulation. A notable case is in sparse estimation of a
precision matrix or a Gaussian graphical model leading to the fast graphical
LASSO algorithm. Some advantages and limitations of the regression-based
Cholesky decomposition relative to the classical spectral (eigenvalue) and
variance-correlation decompositions are highlighted. The former provides an
unconstrained and statistically interpretable reparameterization, and
guarantees the positive-definiteness of the estimated covariance matrix. It
reduces the unintuitive task of covariance estimation to that of modeling a
sequence of regressions at the cost of imposing an a priori order among the
variables. Elementwise regularization of the sample covariance matrix such as
banding, tapering and thresholding has desirable asymptotic properties and the
sparse estimated covariance matrix is positive definite with probability
tending to one for large samples and dimensions.Comment: Published in at http://dx.doi.org/10.1214/11-STS358 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Estimating High Dimensional Covariance Matrices and its Applications
Estimating covariance matrices is an important part of portfolio selection, risk management, and asset pricing. This paper reviews the recent development in estimating high dimensional covariance matrices, where the number of variables can be greater than the number of observations. The limitations of the sample covariance matrix are discussed. Several new approaches are presented, including the shrinkage method, the observable and latent factor method, the Bayesian approach, and the random matrix theory approach. For each method, the construction of covariance matrices is given. The relationships among these methods are discussed.Factor analysis, Principal components, Singular value decomposition, Random matrix theory, Empirical Bayes, Shrinkage method, Optimal portfolios, CAPM, APT, GMM
The generalized shrinkage estimator for the analysis of functional connectivity of brain signals
We develop a new statistical method for estimating functional connectivity
between neurophysiological signals represented by a multivariate time series.
We use partial coherence as the measure of functional connectivity. Partial
coherence identifies the frequency bands that drive the direct linear
association between any pair of channels. To estimate partial coherence, one
would first need an estimate of the spectral density matrix of the multivariate
time series. Parametric estimators of the spectral density matrix provide good
frequency resolution but could be sensitive when the parametric model is
misspecified. Smoothing-based nonparametric estimators are robust to model
misspecification and are consistent but may have poor frequency resolution. In
this work, we develop the generalized shrinkage estimator, which is a weighted
average of a parametric estimator and a nonparametric estimator. The optimal
weights are frequency-specific and derived under the quadratic risk criterion
so that the estimator, either the parametric estimator or the nonparametric
estimator, that performs better at a particular frequency receives heavier
weight. We validate the proposed estimator in a simulation study and apply it
on electroencephalogram recordings from a visual-motor experiment.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS396 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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