KPE {HDMFA}

Estimating the Pair of Factor Numbers via Eigenvalue Ratios Corresponding to PE

Description

The function is to estimate the pair of factor numbers via eigenvalue ratios corresponding to PE method.

Usage

KPE(X, kmax, c = 0)

Arguments

X

Input an array with T \times p_1 \times p_2, where T is the sample size, p_1 is the the row dimension of each matrix observation and p_2 is the the column dimension of each matrix observation.

kmax

The user-supplied maximum factor numbers. Here it means the upper bound of the number of row factors and column factors.

c

A constant to avoid vanishing denominators. The default is 0.

Details

The function KPE uses the eigenvalue-ratio idea to estimate the number of factors. First, obtain the initial estimators \hat{\bold{R}} and \hat{\bold{C}}. Second, define

\hat{\bold{Y}}_t=\frac{1}{p_2}\bold{X}_t\hat{\bold{C}}, \hat{\bold{Z}}_t=\frac{1}{p_1}\bold{X}_t^\top\hat{\bold{R}},

and

\tilde{\bold{M}}_1=\frac{1}{Tp_1}\hat{\bold{Y}}_t\hat{\bold{Y}}_t^\top, \tilde{\bold{M}}_2=\frac{1}{Tp_2}\sum_{t=1}^T\hat{\bold{Z}}_t\hat{\bold{Z}}_t^\top,

the number of factors k_1 is estimated by

\hat{k}_1 = \arg \max_{j \leq k_{max}} \frac{\lambda_j (\tilde{\bold{M}}_1)}{\lambda _{j+1} (\tilde{\bold{M}}_1)},

where k_{max} is a predetermined upper bound for k_1. The estimation of k_2 is defined similarly with respect to \tilde{\bold{M}}_2. For details, see Yu et al. (2022).

Value

\eqn{k_1}

The estimated row factor number.

\eqn{k_2}

The estimated column factor number.

Author(s)

Yong He, Changwei Zhao, Ran Zhao.

References

Yu, L., He, Y., Kong, X., & Zhang, X. (2022). Projected estimation for large-dimensional matrix factor models. Journal of Econometrics, 229(1), 201-217.

Examples

   set.seed(11111)
   T=20;p1=20;p2=20;k1=3;k2=3
   R=matrix(runif(p1*k1,min=-1,max=1),p1,k1)
   C=matrix(runif(p2*k2,min=-1,max=1),p2,k2)
   X=array(0,c(T,p1,p2))
   Y=X;E=Y
   F=array(0,c(T,k1,k2))
   for(t in 1:T){
     F[t,,]=matrix(rnorm(k1*k2),k1,k2)
     E[t,,]=matrix(rnorm(p1*p2),p1,p2)
     Y[t,,]=R%*%F[t,,]%*%t(C)
   }
   X=Y+E
   
   KPE(X, 8, c = 0)