High-dimensional Factor Models Research Articles

On the statistical analysis of high-dimensional factor models

On the statistical analysis of high-dimensional factor models

Block-diagonal idiosyncratic covariance estimation in high-dimensional factor models for financial time series

Estimation of high-dimensional covariance matrices in latent factor models is an important topic in many fields and especially in finance. Since the number of financial assets grows while the estimation window length remains of limited size, the often used sample estimator yields noisy estimates which are not even positive definite. Under the assumption of latent factor models, the covariance matrix is decomposed into a common low-rank component and a full-rank idiosyncratic component. In this paper we focus on the estimation of the idiosyncratic component, under the assumption of a grouped structure of the time series, which may arise due to specific factors such as industries, asset classes or countries. We propose a generalized methodology for estimation of the block-diagonal idiosyncratic component by clustering the residual series and applying shrinkage to the obtained blocks in order to ensure positive definiteness. We derive two different estimators based on different clustering methods and test their performance using simulation and historical data. The proposed methods are shown to provide reliable estimates and outperform other state-of-the-art estimators based on thresholding methods.

Testing the number of common factors by bootstrapped sample covariance matrix in high-dimensional factor models

This paper studies the impact of bootstrap procedure on the eigenvalue distributions of the sample covariance matrix under a high-dimensional factor structure. We provide asymptotic distributions for the top eigenvalues of bootstrapped sample covariance matrix under mild conditions. After bootstrap, the spiked eigenvalues which are driven by common factors will converge weakly to Gaussian limits after proper scaling and centralization. However, the largest non-spiked eigenvalue is mainly determined by the order statistics of the bootstrap resampling weights, and follows extreme value distribution. Based on the disparate behavior of the spiked and non-spiked eigenvalues, we propose innovative methods to test the number of common factors. Indicated by extensive numerical and empirical studies, the proposed methods perform reliably and convincingly under the existence of both weak factors and cross-sectionally correlated errors. Our technical details contribute to random matrix theory on spiked covariance model with convexly decaying density and unbounded support, or with general elliptical distributions.

Estimation and inference for high dimensional factor model with regime switching

This paper proposes maximum (quasi)likelihood estimation for high dimensional factor models with regime switching in the loadings. The model parameters are estimated jointly by the EM (expectation maximization) algorithm, which in the current context only requires iteratively calculating regime probabilities and principal components of the weighted sample covariance matrix. When regime dynamics are taken into account, smoothed regime probabilities are calculated using a recursive algorithm. Consistency, convergence rates and limit distributions of the estimated loadings and the estimated factors are established under weak cross-sectional and temporal dependence as well as heteroscedasticity. It is worth noting that due to high dimension, regime switching can be identified consistently after the switching point with only one observation period. Simulation results show good performance of the proposed method. An application to the FRED-MD dataset illustrates the potential of the proposed method for detection of business cycle turning points.

Factor correlation and the cross section of asset returns: A correlation-robust machine learning approach

This paper investigates high-dimensional factor models for cross-sectional asset returns, with a specific focus on robust estimation in the presence of (highly) correlated factors. Factor correlations can significantly compromise the robustness and credibility of commonly employed analytical methods. To address this, we utilize the stochastic discount factor (SDF) and integrate it with a recently developed Machine Learning methodology (Figueiredo and Nowak, 2016). This novel approach allows us to select factors while accounting for factor correlations and to disentangle correlated factors without imposing rigid assumptions. Our empirical findings consistently highlight the paramount role of the ‘market’ factor in driving cross-sectional asset returns. In contrast, other benchmarks, including the LASSO, the Elastic-Net, and the Fama–MacBeth regression, are adversely impacted by factor correlations, rendering the ‘market’ factor redundant. Additionally, our findings underscore the importance of ‘profitability’, ‘momentum’, and ‘liquidity’-related factors in driving cross-sectional asset returns.

Regularized covariance matrix estimation in high dimensional approximate factor models

We propose a novel factor-based regularized covariance matrix estimator when the number of factors is large compared to the sample size and derive the convergence rates of our estimator. Empirical results demonstrate our proposed estimator performs well in finite samples.

Adaptive Tests for Bandedness of High-dimensional Covariance Matrices

Estimation of the high-dimensional banded covariance matrix is widely used in multivariate statistical analysis. To ensure the validity of estimation, we aim to test the hypothesis that the covariance matrix is banded with a certain bandwidth under the high-dimensional framework. Though several testing methods have been proposed in the literature, the existing tests are only powerful for some alternatives with certain sparsity levels, whereas they may not be powerful for alternatives with other sparsity structures. The goal of this paper is to propose a new test for the bandedness of high-dimensional covariance matrix, which is powerful for alternatives with various sparsity levels. The proposed new test also be used for testing the banded structure of covariance matrices of error vectors in high-dimensional factor models. Based on these statistics, a consistent bandwidth estimator is also introduced for a banded high dimensional covariance matrix. Extensive simulation studies and an application to a prostate cancer dataset from protein mass spectroscopy are conducted for evaluating the effectiveness of the proposed adaptive tests blue and bandwidth estimator for the banded covariance matrix.

Determination of the Number of Breaks in High-Dimensional Factor Models via Cross-Validation

Abstract This paper proposes a cross-validation method to estimate the number of breaks in high-dimensional factor models. To preserve the original change structure, the parity-splitting strategy is adopted when employing the cross-validation method. The consistency of the estimator is established under some mild conditions. Simulation results show desired finite sample performance of the proposed method, especially in comparison with methods that need to predetermine the tuning parameters.

Determining the number of change-points in high-dimensional factor models by cross-validation with matrix completion

This paper focuses on the determination of the number of change-points in high-dimensional factor models via cross-validation with matrix completion. An imputed method is proposed to predict the validation data set which is seen as the “missing” data of the training set. The number of change-points can be determined by minimizing the prediction error on the validation set. The consistency of the estimator is established under some mild conditions. Monte Carlo simulation results show desired performance of the proposed method compared to the existing competitors.

Tests for group-specific heterogeneity in high-dimensional factor models

Standard high-dimensional factor models assume that the comovements in a large set of variables could be modeled using a small number of latent factors that affect all variables. In many relevant applications in economics and finance, heterogeneous comovements specific to some known groups of variables naturally arise, and reflect distinct cyclical movements within those groups. This paper develops two new statistical tests that can be used to investigate whether there is evidence supporting group-specific heterogeneity in the data. The paper also proposes and proves the validity of a permutation approach for approximating the asymptotic distributions of the two test statistics.

On singular values of large dimensional lag-[formula omitted] sample auto-correlation matrices

We study the limiting behavior of singular values of a lag-τ sample auto-correlation matrix Rτϵ of large dimensional vector white noise process, the error term ϵ in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) that characterizes the global spectrum of Rτϵ, and derive the limit of its largest singular value. All the asymptotic results are derived under the high-dimensional asymptotic regime where the data dimension and sample size go to infinity proportionally. Under mild assumptions, we show that the LSD of Rτϵ is the same as that of the lag-τ sample auto-covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of Rτϵ converges almost surely to the right end point of the support of its LSD. Based on these results, we further propose two estimators of total number of factors with lag-τ sample auto-correlation matrices in a factor model. Our theoretical results are fully supported by numerical experiments as well.

High-Dimensional Factor Models and the Factor Zoo

High-Dimensional Factor Models and the Factor Zoo

High-Dimensional Factor Models and the Factor Zoo

High-Dimensional Factor Models and the Factor Zoo

Estimation and Inference for High Dimensional Factor Model with Regime Switching

Estimation and Inference for High Dimensional Factor Model with Regime Switching

SpaceX: gene co-expression network estimation for spatial transcriptomics.

The analysis of spatially resolved transcriptome enables the understanding of the spatial interactions between the cellular environment and transcriptional regulation. In particular, the characterization of the gene-gene co-expression at distinct spatial locations or cell types in the tissue enables delineation of spatial co-regulatory patterns as opposed to standard differential single gene analyses. To enhance the ability and potential of spatial transcriptomics technologies to drive biological discovery, we develop a statistical framework to detect gene co-expression patterns in a spatially structured tissue consisting of different clusters in the form of cell classes or tissue domains. We develop SpaceX (spatially dependent gene co-expression network), a Bayesian methodology to identify both shared and cluster-specific co-expression network across genes. SpaceX uses an over-dispersed spatial Poisson model coupled with a high-dimensional factor model which is based on a dimension reduction technique for computational efficiency. We show via simulations, accuracy gains in co-expression network estimation and structure by accounting for (increasing) spatial correlation and appropriate noise distributions. In-depth analysis of two spatial transcriptomics datasets in mouse hypothalamus and human breast cancer using SpaceX, detected multiple hub genes which are related to cognitive abilities for the hypothalamus data and multiple cancer genes (e.g. collagen family) from the tumor region for the breast cancer data. The SpaceX R-package is available at github.com/bayesrx/SpaceX. Supplementary data are available at Bioinformatics online.

Testing for time-varying factor loadings in high-dimensional factor models

This paper proposes a test for structural changes in factor loadings in high-dimensional factor models under weak serial and cross-sectional dependence. The test is an aggregate statistic in the form of the maximum of the variable-specific statistics whose asymptotic null distribution and local power property are studied. Two approaches including extreme value theory and Bonferroni correction are adopted to compute the critical values of the aggregate test statistic. Monte Carlo simulations reveal the non-trivial power of the proposed test against various types of structural changes, including abrupt changes, nonrandom smooth changes, random-walk variations and stationary variations. Additionally, our test can be more powerful than some alternative tests in the considered scenarios. The usefulness of the test is illustrated by an empirical application to Stock and Watson’s U.S. data set.

Quasi-maximum likelihood estimation of break point in high-dimensional factor models

This paper proposes a quasi-maximum likelihood (QML) estimator of the break point for large-dimensional factor models with a single structural break in the factor loading matrix. We show that the QML estimator is consistent for the true break point when the covariance matrix of the pre- or post-break factor loading (or both) is singular. Consistency here means that the deviation of the estimated break date from the true break date k0 converges to zero as the sample size grows. This is a much stronger result than the break fraction kˆ/T being T-consistent (super-consistent) for k0/T. Also, singularity occurs for most types of structural changes, except for a rotational change. Even for a rotational change, the QML estimator is still T-consistent in terms of the break fraction. Simulation results confirm the theoretical properties of our estimator, and it significantly outperforms existing estimators for change points in factor models.

High Dimensional Factor Models with an Application to Mutual Fund Characteristics

High Dimensional Factor Models with an Application to Mutual Fund Characteristics

High-Dimensional Factor Models with an Application to Mutual Fund Characteristics

High-Dimensional Factor Models with an Application to Mutual Fund Characteristics

Estimation and Inference for High Dimensional Factor Model with Regime Switching

This paper proposes maximum (quasi)likelihood estimation for high dimensional factor models with regime switching in the loadings. The model parameters are estimated jointly by EM algorithm, which in the current context only requires iteratively calculating regime probabilities and principal components of the weighted sample covariance matrix. When regime dynamics are taken into account, smoothed regime probabilities are calculated using a recursive algorithm. Consistency, convergence rates and limit distributions of the estimated loadings and the estimated factors are established under weak cross-sectional and temporal dependence as well as heteroscedasticity. It is worth noting that due to high dimension, regime switching can be identified consistently right after the switching point with only one observation. Simulation results show good performance of the proposed method. An application to the FRED-MD dataset demonstrates the potential of the proposed method for quick detection of business cycle turning points.