matAR.RR.est {tensorTS}

Estimation for Reduced Rank MAR(1) Model

Description

Estimation of the reduced rank MAR(1) model, using least squares (RRLSE) or MLE (RRMLE), as determined by the value of method.

Usage

matAR.RR.est(xx, method, A1.init=NULL, A2.init=NULL,Sig1.init=NULL,Sig2.init=NULL,
k1=NULL, k2=NULL, niter=200,tol=1e-4)

Arguments

xx

T \times d_1 \times d_2 matrix-valued time series, T is the length of the series.

method

character string, specifying the method of the estimation to be used.

"RRLSE",

Least squares.

"RRMLE",

MLE under a separable cov(vec(E_t)).

A1.init

initial value of A_1. The default is the identity matrix.

A2.init

initial value of A_2. The default is the identity matrix.

Sig1.init

only if method=RRMLE, initial value of \Sigma_1. The default is the identity matrix.

Sig2.init

only if method=RRMLE, initial value of \Sigma_2. The default is the identity matrix.

k1

rank of A_1, a positive integer.

k2

rank of A_2, a positive integer.

niter

maximum number of iterations if error stays above tol.

tol

relative Frobenius norm error tolerance.

Details

The reduced rank MAR(1) model takes the form:

X_t = A_1 X_{t-1} A_2^{^\top} + E_t,

where A_i are d_i \times d_i coefficient matrices of ranks \mathrm{rank}(A_i) = k_i \le d_i, i=1,2. For the MLE method we also assume

\mathrm{Cov}(\mathrm{vec}(E_t))=\Sigma_2 \otimes \Sigma_1

Value

return a list containing the following:

A1

estimator of A_1, a d_1 by d_1 matrix.

A2

estimator of A_2, a d_2 by d_2 matrix.

loading

a list of estimated U_i, V_i, where we write A_i=U_iD_iV_i as the singular value decomposition (SVD) of A_i, i = 1,2.

Sig1

only if method=MLE, when \mathrm{Cov}(\mathrm{vec}(E_t))=\Sigma_2 \otimes \Sigma_1.

Sig2

only if method=MLE, when \mathrm{Cov}(\mathrm{vec}(E_t))=\Sigma_2 \otimes \Sigma_1.

res

residuals.

Sig

sample covariance matrix of the residuals vec(\hat E_t).

cov

a list containing

Sigma

asymptotic covariance matrix of (vec( \hat A_1),vec(\hat A_2^{\top})).

Theta1.u, Theta1.v

asymptotic covariance matrix of vec(\hat U_1), vec(\hat V_1).

Theta2.u, Theta2.v

asymptotic covariance matrix of vec(\hat U_2), vec(\hat V_2).

sd.A1

element-wise standard errors of \hat A_1, aligned with A1.

sd.A2

element-wise standard errors of \hat A_2, aligned with A2.

niter

number of iterations.

BIC

value of the extended Bayesian information criterion.

References

Reduced Rank Autoregressive Models for Matrix Time Series, by Han Xiao, Yuefeng Han, Rong Chen and Chengcheng Liu.

Examples

set.seed(333)
dim <- c(3,3)
xx <- tenAR.sim(t=500, dim, R=2, P=1, rho=0.5, cov='iid')
est <- matAR.RR.est(xx, method="RRLSE", k1=1, k2=1)