17,330 research outputs found
Efficient Estimation of Approximate Factor Models via Regularized Maximum Likelihood
We study the estimation of a high dimensional approximate factor model in the
presence of both cross sectional dependence and heteroskedasticity. The
classical method of principal components analysis (PCA) does not efficiently
estimate the factor loadings or common factors because it essentially treats
the idiosyncratic error to be homoskedastic and cross sectionally uncorrelated.
For efficient estimation it is essential to estimate a large error covariance
matrix. We assume the model to be conditionally sparse, and propose two
approaches to estimating the common factors and factor loadings; both are based
on maximizing a Gaussian quasi-likelihood and involve regularizing a large
covariance sparse matrix. In the first approach the factor loadings and the
error covariance are estimated separately while in the second approach they are
estimated jointly. Extensive asymptotic analysis has been carried out. In
particular, we develop the inferential theory for the two-step estimation.
Because the proposed approaches take into account the large error covariance
matrix, they produce more efficient estimators than the classical PCA methods
or methods based on a strict factor model
Statistical Inferences Using Large Estimated Covariances for Panel Data and Factor Models
While most of the convergence results in the literature on high dimensional
covariance matrix are concerned about the accuracy of estimating the covariance
matrix (and precision matrix), relatively less is known about the effect of
estimating large covariances on statistical inferences. We study two important
models: factor analysis and panel data model with interactive effects, and
focus on the statistical inference and estimation efficiency of structural
parameters based on large covariance estimators. For efficient estimation, both
models call for a weighted principle components (WPC), which relies on a high
dimensional weight matrix. This paper derives an efficient and feasible WPC
using the covariance matrix estimator of Fan et al. (2013). However, we
demonstrate that existing results on large covariance estimation based on
absolute convergence are not suitable for statistical inferences of the
structural parameters. What is needed is some weighted consistency and the
associated rate of convergence, which are obtained in this paper. Finally, the
proposed method is applied to the US divorce rate data. We find that the
efficient WPC identifies the significant effects of divorce-law reforms on the
divorce rate, and it provides more accurate estimation and tighter confidence
intervals than existing methods
Large Covariance Estimation by Thresholding Principal Orthogonal Complements
This paper deals with the estimation of a high-dimensional covariance with a
conditional sparsity structure and fast-diverging eigenvalues. By assuming
sparse error covariance matrix in an approximate factor model, we allow for the
presence of some cross-sectional correlation even after taking out common but
unobservable factors. We introduce the Principal Orthogonal complEment
Thresholding (POET) method to explore such an approximate factor structure with
sparsity. The POET estimator includes the sample covariance matrix, the
factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding
estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator
(Cai and Liu, 2011) as specific examples. We provide mathematical insights when
the factor analysis is approximately the same as the principal component
analysis for high-dimensional data. The rates of convergence of the sparse
residual covariance matrix and the conditional sparse covariance matrix are
studied under various norms. It is shown that the impact of estimating the
unknown factors vanishes as the dimensionality increases. The uniform rates of
convergence for the unobserved factors and their factor loadings are derived.
The asymptotic results are also verified by extensive simulation studies.
Finally, a real data application on portfolio allocation is presented
Risks of Large Portfolios
Estimating and assessing the risk of a large portfolio is an important topic
in financial econometrics and risk management. The risk is often estimated by a
substitution of a good estimator of the volatility matrix. However, the
accuracy of such a risk estimator for large portfolios is largely unknown, and
a simple inequality in the previous literature gives an infeasible upper bound
for the estimation error. In addition, numerical studies illustrate that this
upper bound is very crude. In this paper, we propose factor-based risk
estimators under a large amount of assets, and introduce a high-confidence
level upper bound (H-CLUB) to assess the accuracy of the risk estimation. The
H-CLUB is constructed based on three different estimates of the volatility
matrix: sample covariance, approximate factor model with known factors, and
unknown factors (POET, Fan, Liao and Mincheva, 2013). For the first time in the
literature, we derive the limiting distribution of the estimated risks in high
dimensionality. Our numerical results demonstrate that the proposed upper
bounds significantly outperform the traditional crude bounds, and provide
insightful assessment of the estimation of the portfolio risks. In addition,
our simulated results quantify the relative error in the risk estimation, which
is usually negligible using 3-month daily data. Finally, the proposed methods
are applied to an empirical study
Projected principal component analysis in factor models
This paper introduces a Projected Principal Component Analysis
(Projected-PCA), which employs principal component analysis to the projected
(smoothed) data matrix onto a given linear space spanned by covariates. When it
applies to high-dimensional factor analysis, the projection removes noise
components. We show that the unobserved latent factors can be more accurately
estimated than the conventional PCA if the projection is genuine, or more
precisely, when the factor loading matrices are related to the projected linear
space. When the dimensionality is large, the factors can be estimated
accurately even when the sample size is finite. We propose a flexible
semiparametric factor model, which decomposes the factor loading matrix into
the component that can be explained by subject-specific covariates and the
orthogonal residual component. The covariates' effects on the factor loadings
are further modeled by the additive model via sieve approximations. By using
the newly proposed Projected-PCA, the rates of convergence of the smooth factor
loading matrices are obtained, which are much faster than those of the
conventional factor analysis. The convergence is achieved even when the sample
size is finite and is particularly appealing in the
high-dimension-low-sample-size situation. This leads us to developing
nonparametric tests on whether observed covariates have explaining powers on
the loadings and whether they fully explain the loadings. The proposed method
is illustrated by both simulated data and the returns of the components of the
S&P 500 index.Comment: Published at http://dx.doi.org/10.1214/15-AOS1364 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A lava attack on the recovery of sums of dense and sparse signals
Common high-dimensional methods for prediction rely on having either a sparse
signal model, a model in which most parameters are zero and there are a small
number of non-zero parameters that are large in magnitude, or a dense signal
model, a model with no large parameters and very many small non-zero
parameters. We consider a generalization of these two basic models, termed here
a "sparse+dense" model, in which the signal is given by the sum of a sparse
signal and a dense signal. Such a structure poses problems for traditional
sparse estimators, such as the lasso, and for traditional dense estimation
methods, such as ridge estimation. We propose a new penalization-based method,
called lava, which is computationally efficient. With suitable choices of
penalty parameters, the proposed method strictly dominates both lasso and
ridge. We derive analytic expressions for the finite-sample risk function of
the lava estimator in the Gaussian sequence model. We also provide an deviation
bound for the prediction risk in the Gaussian regression model with fixed
design. In both cases, we provide Stein's unbiased estimator for lava's
prediction risk. A simulation example compares the performance of lava to
lasso, ridge, and elastic net in a regression example using feasible,
data-dependent penalty parameters and illustrates lava's improved performance
relative to these benchmarks
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