Forecasting High-Dimensional Covariance Matrices Using
High-Dimensional Principal Component Analysis
Hideto Shigemoto ∗, Takayuki Morimoto †November 17, 2023

Abstract
We modify the recently proposed forecasting model of high-dimensional covariancematrices (HDCM) of asset returns using high-dimensional principal component analysis (PCA). It is well-known that when the sample size is smaller than the dimension,
eigenvalues estimated by classical PCA have a bias. In particular, a very small number of
eigenvalues are extremely large and they are called spiked eigenvalues. High-dimensional
PCA gives eigenvalues which correct the biases of the spiked eigenvalues. This situation
also happens in the ﬁnancial ﬁeld, especially in situations where high-frequency and highdimensional data are handled. The research aims to estimate the HDCM of asset returns
using high-dimensional PCA for the realized covariance matrix using the Nikkei 225 data,
it estimates 5- and 10-minute intraday asset-returns intervals. We construct time-seriesmodels for eigenvalues which are estimated by each PCA, and forecast HDCM. Our simulation analysis shows that the high-dimensional PCA has better estimation performance
than classical PCA for the estimating integrated covariance matrix. In our empiricalanalysis, we show that we will be able to improve the forecasting performance using thehigh-dimensional PCA and make a portfolio with smaller variance.

Keywords: covariance forecasting; high-dimensional covariance; principal component analysis;
high-frequency data; time series

1 Introduction
Modeling and forecasting covariance matrices of asset returns have an essential role in portfolio
allocations and risk management. For estimating and forecasting covariance matrix, a lot of
papers are published on both low- and high-frequency data. Concerning the low-frequency data,
the multivariate GARCH models [1], for example, BEKK-GARCH [2] and DCC-GARCH [3, 4],
are usually used to estimate and forecast the covariance matrix as latent. On the other hand,the availability of high-frequency data recently enabled the direct estimation of the covariance∗ Group Risk Management Department, Nomura Holdings, Inc., 2-2-2 Otemachi, Chiyoda-ku, Tokyo 100-

8130, Japan.
† Corresponding author:

Department of Mathematical Sciences, Kwansei Gakuin University, 1, Gakuen
Uegahara, Sanda, Hyogo 669-1330, Japan.
morimot@kwansei.ac.jp

1matrix, for example, the realized covariance matrix estimator [5], and the multivariate realized
kernel estimator [6]. Additionally, some forecasting models such as the multivariate HAR [7],conditional autoregressive Wishart (CAW) [8], and realized DCC [9] models, use these covarianceestimators to forecast them. However, when the dimensions increase, these covariance estimatorsand forecasting models have less accurate performance and suﬀer from an increase in the numberof estimated parameters because of various reasons, such as the curse of dimensionality.
To solve these problems, the DCC-NL model which can overcome the curse of dimensionality using nonlinear shrinkage estimation is proposed [10]. To analyze the conditional highdimensional covariance matrix (HDCM), recent studies using some multivariate GARCH models
use the DCC-NL model instead of Tse and Tsui’s and Engle’s DCC-GARCH models [11, 12,
13, 14]. Then, to solve the curse of dimensionality, many studies assume that the covariancematrix process or the price process follows a factor structure. Wang and Zou [15] propose acovariance estimator assuming that the integrated covariance matrix is sparse. Considering asparse covariance matrix allows only important elements to remain and also reduces the numberof elements to be estimated. In addition, Tao et al. [16] introduce a covariance estimator whichuses the matrix factor structure for an HDCM. We can obtain not only a consistent estimatorof an HDCM but also a forecasted value using the vector autoregressive (VAR) model for a lowdimensional factor covariance matrix. Kim et al. [17] propose a threshold covariance estimator
to regularize some realized covariance measures under the same assumption as [15]. Shen etal. [18] apply the method proposed by [16] to a realized covariance matrix and consider the
CAW model instead of the VAR model for the factors. However, these studies assume sparsity
in the integrated covariance matrix itself, which represents the target to be estimated. If there
are some common factors across asset returns, the assumption that the integrated covariance issparse becomes unrealistic because there are correlations among all pairs of assets through thecommon factors [19, 20, 21, 22].
Fan et al. [19] propose the principal orthogonal complement thresholding (POET) method
which assumes sparsity, not for the covariance matrix itself, but for the covariance matrix ofthe residual process, and estimates the latent factor using principal component analysis (PCA)
to solve some problems. For high-frequency data, Fan et al. [20] assume the observable factor
structure inspired by [23], and propose the covariance estimator under the assumption that thecovariance matrix of the residual process is sparse. To estimate the latent factor structure,Aı̈t-Sahalia and Xiu [24] impose sparsity on the residual covariance matrix and apply POET tohigh-frequency data using PCA to estimate an HDCM. They show that even when the factor islatent, if the residual covariance matrix is suﬃciently sparse, the factor part can be estimated by
PCA on the consistent estimator of the integrated covariance matrix, like the realized covariancematrix. In addition, they show that their estimator is a consistent estimator even if the intervalof intraday return is ∆ → 0 and the dimension is d → ∞. In addition, Dai et al. [25] also
propose an estimation method of the sparse residual covariance matrix using thresholding, anda high-dimensional covariance estimator using the POET estimator. The diﬀerence between [24]and [25] is the sparse structure. While Aı̈t-Sahalia and Xiu [24] assume the block-diagonalize
2

structure instead of thresholding, Dai et al. [25] do not assume the block-diagonalize structure
but set a more general assumption, and use soft-, hard-, and adaptive-lasso (AL) [26], andsmoothly clipped absolute deviation (SCAD) [27] thresholding. For the sparse estimation of theresidual covariance matrix, Cai and Liu [28] propose the adaptive and hard thresholding method,
but this method cannot guarantee the positive deﬁniteness under the ﬁnite sample [29], and also
has less performance than [25]. Brownlees et al. [21] propose the realized network estimatorusing the graphical lasso to estimate the precision matrix. Jian et al. [29] build time-seriesmodels for estimated eigenvalues based on the estimator of [24], and forecast the HDCM. In
addition, they propose the regularized method to guarantee the positive deﬁniteness.
The classical PCA, which is used by these models, creates a bias under d > M ; d is thedimension of a covariance matrix and M is the sample size [15, 30, 24, 31]. Wang and Fan[31] characterize the asymptotic distribution of empirical eigenvalues under the i.i.d setting andd > M . They also propose the shrinkage POET (SPOET) method based on their asymptotic
distribution. The SPOET method corrects the biases of eigenvalues estimated by classical PCA.
In this paper, we estimate the HDCM under the factor structure for the high-frequency data,and create the forecasting models using its eigenvalues. It is well-known that the realized
covariance matrix is a consistent estimator of the integrated covariance matrix when the numberof intraday observations M goes to ∞. However, in the empirical situation, we consider the
microstructure noise, and often use the realized covariance matrix which is estimated using 5or 10-minute interval intraday returns. In this case, since the Japanese stock market opens from
9 a.m. to 3 p.m. with an hour break, the sample sizes are 60 and 30 per day. Under such asituation, although we want to consider a large portfolio including 100 or 200 stocks, the matrix
dimension is larger than the sample size, d > M . Therefore, we apply spoet corresponding
to d > M to the realized covariance matrix, rather than the POET using PCA as consideredin [24, 25]. Additionally, we construct the forecasting models similar to [29], by deriving the
eigenvalues of the realized covariance matrix estimated using SPOET.
There are two contributions to the literature. First, this paper shows through a simulation
study that SPOET considered in the i.i.d. setting has excellent performance for estimating the
integrated covariance matrix under the assumption of continuous Itô semi-martingale. Second,
our empirical analysis shows that the forecasting models using SPOET are more accurate covariance matrix than the models using the POET. Hence, using our proposed models gives us
a more accurate covariance estimator under the high-dimensional setting that results in bad
performance and unreliable results. This point is the largest diﬀerence between [29] and thispaper. Although Jian et al. [29] do not consider the relationship between the dimension of thecovariance matrix and the sample size of intraday, we focus on the relationship and make thesemodels forecast more accurately than their models.
The paper is organized as follows: Section 2 explains the factor model, the sparse estimations,and the principal component analysis to estimate the factor part. Section 3 introduces theforecasting model of estimated eigenvalues by PCA used in the empirical analysis.

3

2 Factor Model and PCA
2.1 Factor Structure
We assume that the log-price Y follows a continuous-time factor model,
Yt = βXt + Zt ,(1)

where Yt is a d-dimensional vector process, Xt is a r-dimensional latent common factor process,Zt is the d-dimensional idiosyncratic component, and β is a d × r constant-factor loading matrix.
In addition, Xt and Zt are independent. In this paper, the number of factors r is unknown.
Here, we assume that Xt and Zt are continuous Itô semi-martingale, as with [24, 25] as follows:
∫ t
Xt =

∫ t
hs ds +∫ t
ηs dWs , Zt =

0

0

∫ t
fs ds +

0γs dBs .
0Then, the integrated covariance matrices of Xt , Zt , and Yt are deﬁned under Assumptions 1,
2, and 3, and the sparsity assumption of [25] as follows:∫ t
ΣX t =∫ tηs ηs′ ds,

ΣZt =

0

γs γs′ ds,0ΣYt = βΣXt β ′ + ΣZt .

(2)

Although Jian et al. [29] consider the factor model following Assumption 1, 2, 3, 4, and 5 of[24], we assume more general sparsity of [25] and we do not assume that idiosyncratic componentis block diagonal.

2.2 Sparsity
To estimate an HDCM, a certain condition of sparsity is necessary for dimension reduction andfactor model. However, the sparsity assumption of the covariance matrix itself is inappropriate
from the viewpoint of the factor model. To solve this problem, we assume that the covariancematrix of the idiosyncratic component ΣZ is sparse, and then the form of Equation (2) becomesa low-rank plus sparse structure. A low-rank plus sparsity structure of the residual covariancematrix turns out to be a good match for asset high-frequency data [24] and guarantees a wellconditioned estimator as well as its precision matrix [25].
We use four types of thresholding functions, hard-, soft-, adaptive lasso (AL) and smoothly
clipped absolute deviation (SCAD) threshold, for ΣZ as following:
η+1
AL|z|−η )+ ,(z) = z1(|z| > λ), sSoft
sHard
λ (z) = sign(z)(|z| − λ)+ , sλ (z) = sign(z)(|z| − λλsign(z)(|z| − λ)+ , |z| ≤ 2λ;SCAD(z) = (a−1)z−sign(z)aλsλ,
2λ < |z| ≤ aλ;
a−2


z,
aλ < |z|.

4

where we set a = 3.7 and η = 1 same as [32]. We adopt these thresholding functions andestimate the residual covariance matrix as follows:{Σ̂Zt ,ij ,SΣ̃Zt ,ij =sλij (Σ̂Zt ,ij ),i = j;i ̸= j.

Dai et al. [25] denote that despite these estimations lead to the same convergence rate from theiranalysis, the results of ﬁnite sample performance of the covariance matrix in their simulation
study and empirical analysis are quite diﬀerent.
2.2.1

Thresholding Method

Following [25], the thresholding λij in sparse functions is estimated as follows:√
λij = τ Σ̂Zt ,ii Σ̂Zt ,jj ,where τ is a constant to be determined. Under the ﬁnite sample, we use a grid search to guarantee
positive semi-deﬁnite. We divide into K pieces in τ ∈ [0, 1] and gradually increase τ until theﬁnal high-dimensional covariance matrix becomes positive semi-deﬁnite. As τ becomes larger,
the degree of sparsity of the residual covariance increases, and, ﬁnally, the matrix becomes a
diagonal matrix [25]. Thus, an estimated HDCM always becomes positive semi-deﬁnite.
2.2.2

The Number of Factors

If the log-price is observed by latent common factors, we have to estimate the number offactors. The consistent estimator of the number of latent factors is proposed by [24] under
the continuous-time setting without random matrix theory. We adopt their estimator, which
minimizes the penalized function using an estimator of the integrated covariance matrix Σ̂t :
(
)
λ
(
Σ̂
)
j
Y
t
r̂t = arg min+ j × g(M, d) − 1,(3)
1≤j≤rmaxdwhere rmax is 20. In theory, the choice of rmax is not important. This is simply used to avoid
making economically meaningless choice of r in ﬁnite samples [24]. The function g(n, d) is
deﬁned as follows:(g(M, d) = 0.02 × λ̂tmin( d , M ) (Σ̂Yt )
2 2

log d
M

) 14
.

(4)

2.3 PCA for High-Frequency Data
To estimate an HDCM, we show the PCA for the realized covariance matrix estimated byhigh-frequency data following [29]. Here, yj,t is the j-th intraday log-return observed on day t.
The realized covariance matrix is deﬁned as follows:Σ̂Yt =

M∑′.
yj,t yj,t

j=1

We assume d > M ; thus, the realized covariance matrix is estimated under this assumption.
5

2.3.1

POET Method

The eigenvalues of the realized covariance matrix Σ̂Yt are λ̂t1 > λ̂t2 > · · · > λ̂td , and ξˆ1t , ξˆ2t , . . . , ξˆdt
denote the corresponding eigenvectors. If r̂ is the estimator of r, which is the number of factors,
Σ̂Yt has a spectral decomposition as follows:Σ̂Yt =r̂∑

λ̂tj ξˆjt ξˆjt′ + Σ̂Zt ,

(5)

j=1

where Σ̂Zt is the covariance matrix of the residual process, which is calculated by Σ̂Zt =∑dt ˆt ˆt′j=r̂+1 λ̂j ξj ξj . Here, even if the common factor Xt is an unobservable process, if ΣZ is suﬃciently sparse, βΣXt β ′ in Equation (2) can be estimated using the eigenvalues and eigenvectorsof Σ̂Yt [24]. Therefore, we estimate the sparse residual covariance matrix, and then estimate ahigh-dimensional covariance matrix Σ̂SYt as follows:Σ̂SYt =r̂∑

λ̂tj ξˆjt ξˆjt′ + Σ̂SZt ,

(6)

j=1

where Σ̂SZt is the estimated sparse residual covariance matrix. This high-dimensional covarianceestimator consists of the POET for low-frequency data of [19] and the PCA approach adopted
in [24, 25] for high-frequency data.
2.3.2

Shrinkage POET Method

The PCA which is used in Equation (5) is eﬀective, when dimension d is ﬁxed and the sample
size (the number of observations in a day) is suﬃciently large. However, it is well-known that insituations where d > M , the eigenvalues and eigenvectors of the realized covariance matrix are
not consistent estimators in the sense that they are quite far from the true values [16]. To deal
with this problem, we use shrinkage POET (SPOET), proposed by [31], which corrects biases
of empirical eigenvalues and estimates an HDCM as follows:Σ̃SYt =r∑′

λ̃tj ξˆjt ξˆjt + Σ̂SZt ,j=1

where λ̃tj = max{λ̂tj − cd/M, 0}. In addition, as c is unknown, we have to estimate it. In thispaper, we follow [31] to estimate as follows:ĉ = (tr(Σ̂tY ) −r∑λ̂j )/(d − r − dr/M ).

(7)

j=1

3 Forecasting Models
In this section, in order to forecast an HDCM, we introduce forecasting models based on the
PCA. We denote the eigenvalues as:σft = [λt1 , . . . , λtr ]′ .
6

Since these eigenvalues are the variances of factors, we can consider models similar to thetime-series model of the realized variance of asset returns [29]. To model the eigenvalues, we use
the exponentially weighted moving average (EWMA), (Vector) HAR, and (Vector) AR models,
the same as [29]. All models except the EWMA model can be easily estimated using OLS.

3.1 EWMA ModelIn this paper, we use the EWMA model developed by [33] as a benchmark model, as follows:
t+1|t

σf

t|t−1

= aσf

+ (1 − a)σ t ,where a is the decaying parameter that determines the weight of the observed value 1 period
before the forecast, and we set a = 0.94 following the framework of a RiskMetrics approach [33].
As this model is easy to implement to forecast volatility and covariance, a lot of studies use itin practice.

3.2 VAR Model
We introduce the AR(1) and VAR models based on high-frequency factor model as:
λti = a0,i + a1,i λt−1
+ εti , i = 1, . . . , r,i(8)σft = A0 + A1 σft−1 + εtf ,(9)where ak,i , Ak , k = 0, 1 are scalar parameters and parameter matrices, respectively. εti denotes
the innovation term.
Andersen et al. [34] pointed out that the logarithmic standard deviations are closer to a normal
distribution in general compared to the realized variance itself, and modeling and forecastinglog volatility guarantee that the ﬁtted and forecasted volatility are non-negative without any
constrains. Therefore, we also apply the logarithmic eigenvalues to these models.

3.3 V-HAR ModelIn this subsection, we introduce the HAR model and V-HAR model which are proposedby [7, 35], respectively. These models are usually applied to forecasting both univariate andmultivariate realized volatility. The HAR model is advantaged for approximating the longmemory properties using daily, weekly and monthly volatility. Also, given the multivariate
framework, the impact of the short- and long-term volatility of another asset can be included
in a forecast of the volatility of one asset. To use these models, we calculate the weekly andmonthly eigenvalues as follows:7

λti,W =1 ∑ t−jλ ,5 j=0 i(10)

λti,M =

1 ∑ t−jλ .
22 j=0 i

(11)

4

21

t
t
In addition, we deﬁne that σf.W
= [λt1,W , . . . , λtr,W ]′ and σf.M
= [λt1,M , . . . , λtr,M ]′ . We con-struct the HAR and V-HAR models using daily, weekly, and monthly eigenvalues as follows:t−1tλti = a0,i + a1,i λt−1
+ a2,i λi,W
+ a3,i λt−1ii,M + εi , i = 1, . . . , r,

(12)

t−1
t−1
σft = A0 + A1 σft−1 + A2 σf,W
+ A3 σf,M+ εtf .

(13)

where ak,i , Ak , k = 0, . . . , 3 are scalar parameters and parameter matrices, respectively. Similar
to AR and VAR models, these models are transformed into logarithmic models.
Using these models, we can obtain the forecasted HDCM, Ŝt+1 , as follows:Ŝt+1 =

r̂
∑

′

S
ˆt ˆt
λ̌t+1
j ξj ξj + Σ̂Zt ,

(14)

j=1

where λ̌t+1
denotes the forecasted j-th eigenvalues at t + 1, ξˆj is the eigenvectors correspondingjto the forecasted eigenvalues, and Σ̂SZt is the sparse residual covariance matrix at t. Hence, inthis model, we use the eigenvectors and sparse residual covariance matrix at t and the forecasted
eigenvalues at t + 1 to forecast an HDCM.

This study is partly supported by the Institute of Statistical Mathematics (ISM) cooperative
research program (2022-ISMCRP-2024), JSPS KAKENHI Grant Number 21K01433, and Grantin-Aid for JSPS Fellows Grant Number 22J10285. The views expressed in this study are those of
the authors and do not necessarily reﬂect the oﬃcial views of Nomura Holdings, Inc. or Kwansei
Gakuin University.

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