Type: Package
Title: High-Dimensional Robust Factor Analysis
Version: 0.1.5
Author: Yong He [aut], Lingxiao Li [aut], Dong Liu [aut, cre], Wenxin Zhou [aut]
Maintainer: Dong Liu <liudong_stat@163.com>
Description: Factor models have been widely applied in areas such as economics and finance, and the well-known heavy-tailedness of macroeconomic/financial data should be taken into account when conducting factor analysis. We propose two algorithms to do robust factor analysis by considering the Huber loss. One is based on minimizing the Huber loss of the idiosyncratic error's L2 norm, which turns out to do Principal Component Analysis (PCA) on the weighted sample covariance matrix and thereby named as Huber PCA. The other one is based on minimizing the element-wise Huber loss, which can be solved by an iterative Huber regression algorithm. In this package we also provide the code for traditional PCA, the Robust Two Step (RTS) method by He et al. (2022) and the Quantile Factor Analysis (QFA) method by Chen et al. (2021) and He et al. (2023).
License: GPL-2 | GPL-3
Encoding: UTF-8
Imports: quantreg, pracma
Depends: R (≥ 3.5.0)
NeedsCompilation: no
Packaged: 2024-07-22 04:20:32 UTC; hey12
Repository: CRAN
Date/Publication: 2024-07-22 04:50:02 UTC

Huber Principal Component Analysis for Large-Dimensional Factor Models

Description

This function is to fit the factor models via the Huber Principal Component Analysis (HPCA) method. One is based on minimizing the Huber loss of the idiosyncratic error's \ell_2 norm, which turns out to do Principal Component Analysis (PCA) on the weighted sample covariance matrix and thereby named as Huber PCA. The other one is based on minimizing the elementwise Huber loss, which can be solved by an iterative Huber regression algorithm.

Usage

HPCA(X, r, Method = "E", tau = NULL, scale_est="MAD", L_init = NULL, 
     F_init = NULL, maxiter_HPCA = 100, maxiter_HLM = 100, eps = 0.001)

Arguments

X

Input matrix, of dimension T\times N. Each row is an observation with N features at time point t.

r

A positive integer indicating the factor numbers.

Method

Method="P" indicates minimizing the Huber loss of the idiosyncratic error's \ell_2 norm while Method="E" indicates minimizing the elementwise Huber loss. The default is the elementwise Huber loss.

tau

Optional user-supplied parameter for Huber loss; default is NULL, and \tau is provided by default.

scale_est

A parameter for the elementwise Huber loss. scale_est="MAD" indicates robust variance estimation in each iteration, while scale_est="const" indicates fixing user-supplied \tau. The default is scale_est="MAD".

L_init

User-supplied inital value of loadings; default is the PCA estimator.

F_init

User-supplied inital value of factors; default is the PCA estimator.

maxiter_HPCA

The maximum number of iterations in the HPCA. The default is 100.

maxiter_HLM

The maximum number of iterations in the iterative Huber regression algorithm. The default is 100.

eps

The stopping critetion parameter in the HPCA. The default is 1e-3.

Details

See He et al. (2023) for details.

Value

The return value is a list. In this list, it contains the following:

Fhat

The estimated factor matrix of dimension T\times r.

Lhat

The estimated loading matrix of dimension N\times r.

m

The number of iterations.

Author(s)

Yong He, Lingxiao Li, Dong Liu, Wenxin Zhou.

References

He Y, Li L, Liu D, Zhou W., 2023 Huber Principal Component Analysis for Large-dimensional Factor Models.

Examples

set.seed(1)
T=50;N=50;r=3
L=matrix(rnorm(N*r,0,1),N,r);F=matrix(rnorm(T*r,0,1),T,r)
E=matrix(rnorm(T*N,0,1),T,N)
X=F%*%t(L)+E

fit=HPCA(X,r,Method = "E")
fit$Fhat;fit$Lhat

fit=HPCA(X,r,Method = "P")
fit$Fhat;fit$Lhat

Estimating Factor Numbers via Rank Minimization Corresponding to Huber PCA

Description

This function is to estimate factor numbers via rank minimization corresponding to Huber Principal Component Analysis (HPCA).

Usage

HPCA_FN(X, rmax, Method = "E", tau = NULL, scale_est="MAD", threshold = NULL,
        L_init = NULL, F_init = NULL, maxiter_HPCA = 100, maxiter_HLM = 100, 
        eps = 0.001)

Arguments

X

Input matrix, of dimension T\times N. Each row is an observation with N features at time point t.

rmax

The user-supplied maximum factor numbers.

Method

Method="P" indicates minimizing the Huber loss of the idiosyncratic error's \ell_2 norm while Method="E" indicates minimizing the elementwise Huber loss. The default is the elementwise Huber loss.

tau

Optional user-supplied parameter for Huber loss; default is NULL, and \tau is provided by default.

scale_est

A parameter for the elementwise Huber loss. scale_est="MAD" indicates robust variance estimation in each iteration, while scale_est="const" indicates fixing user-supplied \tau. The default is scale_est="MAD".

threshold

The threshold of rank minimization; default is NULL.

L_init

User-supplied inital value of loadings in the HPCA; default is the PCA estimator.

F_init

User-supplied inital value of factors in the HPCA; default is the PCA estimator.

maxiter_HPCA

The maximum number of iterations in the HPCA. The default is 100.

maxiter_HLM

The maximum number of iterations in the iterative Huber regression algorithm. The default is 100.

eps

The stopping critetion parameter in the HPCA. The default is 1e-3.

Details

See He et al. (2023) for details.

Value

rhat

The estimated factor number.

Author(s)

Yong He, Lingxiao Li, Dong Liu, Wenxin Zhou.

References

He Y, Li L, Liu D, Zhou W., 2023 Huber Principal Component Analysis for Large-dimensional Factor Models.

Examples

set.seed(1)
T=50;N=50;r=3
L=matrix(rnorm(N*r,0,1),N,r);F=matrix(rnorm(T*r,0,1),T,r)
E=matrix(rnorm(T*N,0,1),T,N)
X=F%*%t(L)+E


HPCA_FN(X,8,Method="E")

HPCA_FN(X,8,Method="P")

Iterative Quantile Regression Methods for Quantile Factor Models

Description

This function is to fit the quantile factor model via the Iterative Quantile Regression (IQR) algorithm.

Usage

IQR(X, r, tau, L_init = NULL, F_init = NULL, max_iter = 100, eps = 0.001)

Arguments

X

Input matrix, of dimension T\times N. Each row is an observation with N features at time point t.

r

A positive integer indicating the factor numbers.

tau

The user-supplied quantile level.

L_init

User-supplied inital value of loadings; default is the PCA estimator.

F_init

User-supplied inital value of factors; default is the PCA estimator.

max_iter

The maximum number of iterations. The default is 100.

eps

The stopping critetion parameter. The default is 1e-06.

Details

See Chen et al. (2021) and He et al. (2023) for details.

Value

The return value is a list. In this list, it contains the following:

Fhat

The estimated factor matrix of dimension T\times r.

Lhat

The estimated loading matrix of dimension N\times r.

t

The number of iterations.

Author(s)

Yong He, Lingxiao Li, Dong Liu, Wenxin Zhou.

References

Chen, L., Dolado, J.J., Gonzalo, J., 2021. Quantile factor models. Econometrica 89, 875–910.

He Y, Kong X, Yu L, Zhao P., 2023 Quantile factor analysis for large-dimensional time series with statistical guarantee <arXiv:2006.08214>.

Examples

set.seed(1)
T=50;N=50;r=3
L=matrix(rnorm(N*r,0,1),N,r);F=matrix(rnorm(T*r,0,1),T,r)
E=matrix(rnorm(T*N,0,1),T,N)
X=F%*%t(L)+E

tau=0.5
fit=IQR(X,r,tau)
fit$Fhat;fit$Lhat

Estimating Factor Numbers via Rank Minimization Corresponding to IQR

Description

This function is to estimate factor numbers via rank minimization corresponding to Iterative Quantile Regression (IQR).

Usage

IQR_FN(X, rmax, tau, threshold = NULL, L_init = NULL, F_init = NULL, 
       max_iter = 100, eps = 10^(-6))

Arguments

X

Input matrix, of dimension T\times N. Each row is an observation with N features at time point t.

rmax

The user-supplied maximum factor numbers.

tau

The user-supplied quantile level.

threshold

The threshold of rank minimization; default is NULL.

L_init

User-supplied inital value of loadings in the IQR; default is the PCA estimator.

F_init

User-supplied inital value of factors in the IQR; default is the PCA estimator.

max_iter

The maximum number of iterations. The default is 100.

eps

The stopping critetion parameter of the IQR method. The default is 1e-06.

Details

See Chen et al. (2021) for more details.

Value

rhat

The estimated factor number.

Author(s)

Yong He, Lingxiao Li, Dong Liu, Wenxin Zhou.

References

Chen, L., Dolado, J.J., Gonzalo, J., 2021. Quantile factor models. Econometrica 89, 875–910.

Examples

set.seed(1)
T=50;N=50;r=3
L=matrix(rnorm(N*r,0,1),N,r);F=matrix(rnorm(T*r,0,1),T,r)
E=matrix(rnorm(T*N,0,1),T,N)
X=F%*%t(L)+E

tau=0.5
IQR_FN(X,8,tau)

Principal Component Analysis for Large-Dimensional Factor Models

Description

This function is to fit the factor models via Principal Component Analysis (PCA) methods.

Usage

PCA(X, r, constraint = "L")

Arguments

X

Input matrix, of dimension T\times N. Each row is an observation with N features at time point t.

r

A positive integer indicating the factor numbers.

constraint

The type of identification condition. If constraint="L", the columns of the estimated loading matrix are orthogonal and constraint="F" indicates the columns of the estimated factor matrix are orthogonal.

Details

See Bai (2003) for details.

Value

The return value is a list. In this list, it contains the following:

Fhat

The estimated factor matrix of dimension T\times r.

Lhat

The estimated loading matrix of dimension N\times r.

Author(s)

Yong He, Lingxiao Li, Dong Liu, Wenxin Zhou.

References

Bai, J., 2003. Inferential theory for factor models of large dimensions. Econometrica 71, 135–171.

Examples

set.seed(1)
T=50;N=50;r=3
L=matrix(rnorm(N*r,0,1),N,r);F=matrix(rnorm(T*r,0,1),T,r)
E=matrix(rnorm(T*N,0,1),T,N)
X=F%*%t(L)+E

fit=PCA(X,3,"L")
t(fit$Lhat)%*%fit$Lhat/N

fit=PCA(X,3,"F")
t(fit$Fhat)%*%fit$Fhat/T

Estimating Factor Numbers via Eigenvalue Ratios Corresponding to PCA

Description

This function is to estimate factor numbers via eigenvalue ratios corresponding to Principal Component Analysis (PCA).

Usage

PCA_FN(X, rmax)

Arguments

X

Input matrix, of dimension T\times N. Each row is an observation with N features at time point t.

rmax

The user-supplied maximum factor numbers.

Details

See Ahn and Horenstein (2013) for details.

Value

rhat

The estimated factor numbers.

Author(s)

Yong He, Lingxiao Li, Dong Liu, Wenxin Zhou.

References

Ahn, S.C., Horenstein, A.R., 2013. Eigenvalue ratio test for the number of factors. Econometrica 81, 1203–1227.

Examples

set.seed(1)
T=50;N=50;r=3
L=matrix(rnorm(N*r,0,1),N,r);F=matrix(rnorm(T*r,0,1),T,r)
E=matrix(rnorm(T*N,0,1),T,N)
X=F%*%t(L)+E

PCA_FN(X,8)

Robust Two Step Algorithm for Large-Dimensional Elliptical Factor Models

Description

This function is to fit the large-dimensional elliptical factor models via the Robust Two Step (RTS) algorithm.

Usage

RTS(X, r)

Arguments

X

Input matrix, of dimension T\times N. Each row is an observation with N features at time point t.

r

A positive integer indicating the factor numbers.

Details

See He et al. (2022) for details.

Value

The return value is a list. In this list, it contains the following:

Fhat

The estimated factor matrix of dimension T\times r.

Lhat

The estimated loading matrix of dimension N\times r.

Author(s)

Yong He, Lingxiao Li, Dong Liu, Wenxin Zhou.

References

He, Y., Kong, X., Yu, L., Zhang, X., 2022. Large-dimensional factor analysis without moment constraints. Journal of Business & Economic Statistics 40, 302–312.

Examples

set.seed(1)
T=50;N=50;r=3
L=matrix(rnorm(N*r,0,1),N,r);F=matrix(rnorm(T*r,0,1),T,r)
E=matrix(rnorm(T*N,0,1),T,N)
X=F%*%t(L)+E

fit=RTS(X,3)
fit$Fhat;fit$Lhat

Estimating Factor Numbers Robustly via Multivariate Kendall’s Tau Eigenvalue Ratios

Description

This function is to estimate factor numbers robustly via multivariate Kendall’s tau eigenvalue ratios.

Usage

RTS_FN(X, rmax)

Arguments

X

Input matrix, of dimension T\times N. Each row is an observation with N features at time point t.

rmax

The user-supplied maximum factor numbers.

Details

See Yu et al. (2019) for details.

Value

rhat

The estimated factor number.

Author(s)

Yong He, Lingxiao Li, Dong Liu, Wenxin Zhou.

References

Yu, L., He, Y., Zhang, X., 2019. Robust factor number specification for large-dimensional elliptical factor model. Journal of Multivariate analysis 174, 104543.

Examples

set.seed(1)
T=50;N=50;r=3
L=matrix(rnorm(N*r,0,1),N,r);F=matrix(rnorm(T*r,0,1),T,r)
E=matrix(rnorm(T*N,0,1),T,N)
X=F%*%t(L)+E

RTS_FN(X,8)