HDTSA (version 1.0.5)

HDSReg: Factor analysis with observed regressors for vector time series

Description

HDSReg() considers a multivariate time series model which represents a high-dimensional vector process as a sum of three terms: a linear regression of some observed regressors, a linear combination of some latent and serially correlated factors, and a vector white noise:$${\bf y}_t = {\bf Dz}_t + {\bf Ax}_t + {\boldsymbol {\epsilon}}_t,$$ where \({\bf y}_t\) and \({\bf z}_t\) are, respectively, observable \(p\times 1\) and \(m \times 1\) time series, \({\bf x}_t\) is an \(r \times 1\) latent factor process, \({\boldsymbol{\epsilon}}_t\) is a vector white noise process, \({\bf D}\) is an unknown regression coefficient matrix, and \({\bf A}\) is an unknown factor loading matrix. This procedure proposed in Chang, Guo and Yao (2015) aims to estimate the regression coefficient matrix \({\bf D}\), the number of factors \(r\) and the factor loading matrix \({\bf A}\).

Usage

HDSReg(
  Y,
  Z,
  D = NULL,
  lag.k = 5,
  thresh = FALSE,
  delta = 2 * sqrt(log(ncol(Y))/nrow(Y)),
  twostep = FALSE
)

Value

An object of class "factors", which contains the following components:

factor_num

The estimated number of factors \(\hat{r}\).

reg.coff.mat

The estimated \(p \times m\) regression coefficient matrix \(\tilde{\bf D}\).

loading.mat

The estimated \(p \times \hat{r}\) factor loading matrix \({\bf \hat{A}}\).

X

The \(n\times \hat{r}\) matrix \(\hat{\bf X}=(\hat{\bf x}_1,\dots,\hat{\bf x}_n)'\) with \(\hat{\mathbf{x}}_t=\hat{\mathbf{A}}'(\mathbf{y}_t-\tilde{\mathbf{D}} \mathbf{z}_t)\).

lag.k

The time lag used in function.

Arguments

Y

An \(n \times p\) data matrix \({\bf Y} = ({\bf y}_1, \dots , {\bf y}_n )'\), where \(n\) is the number of the observations of the \(p \times 1\) time series \(\{{\bf y}_t\}_{t=1}^n\).

Z

An \(n \times m\) data matrix \({\bf Z} = ({\bf z}_1, \dots , {\bf z}_n )'\) consisting of the observed regressors.

D

A \(p\times m\) regression coefficient matrix \(\tilde{\bf D}\). If D = NULL (the default), our procedure will estimate \({\bf D}\) first and let \(\tilde{\bf D}\) be the estimate of \({\bf D}\). If D is given by the users, then \(\tilde{\bf D}={\bf D}\).

lag.k

The time lag \(K\) used to calculate the nonnegative definte matrix \( \hat{\mathbf{M}}_{\eta}\): $$\hat{\mathbf{M}}_{\eta}\ =\ \sum_{k=1}^{K} T_\delta\{\hat{\mathbf{\Sigma}}_{\eta}(k)\} T_\delta\{\hat{\mathbf{\Sigma}}_{\eta}(k)\}', $$ where \(\hat{\bf \Sigma}_{\eta}(k)\) is the sample autocovariance of \( {\boldsymbol {\eta}}_t = {\bf y}_t - \tilde{\bf D}{\bf z}_t\) at lag \(k\) and \(T_\delta(\cdot)\) is a threshold operator with the threshold level \(\delta \geq 0\). See 'Details'. The default is 5.

thresh

Logical. If thresh = FALSE (the default), no thresholding will be applied to estimate \(\hat{\mathbf{M}}_{\eta}\). If thresh = TRUE, \(\delta\) will be set through delta. See 'Details'.

delta

The value of the threshold level \(\delta\). The default is \( \delta = 2 \sqrt{n^{-1}\log p}\).

twostep

Logical. The same as the argument twostep in Factors.

Details

The threshold operator \(T_\delta(\cdot)\) is defined as \(T_\delta({\bf W}) = \{w_{i,j}1(|w_{i,j}|\geq \delta)\}\) for any matrix \({\bf W}=(w_{i,j})\), with the threshold level \(\delta \geq 0\) and \(1(\cdot)\) representing the indicator function. We recommend to choose \(\delta=0\) when \(p\) is fixed and \(\delta>0\) when \(p \gg n\).

References

Chang, J., Guo, B., & Yao, Q. (2015). High dimensional stochastic regression with latent factors, endogeneity and nonlinearity. Journal of Econometrics, 189, 297--312. tools:::Rd_expr_doi("doi:10.1016/j.jeconom.2015.03.024").

See Also

Factors.

Examples

# Example 1 (Example 1 in Chang, Guo and Yao (2015)).
## Generate xt
n <- 400
p <- 200
m <- 2
r <- 3
X <- mat.or.vec(n,r)
x1 <- arima.sim(model = list(ar = c(0.6)), n = n)
x2 <- arima.sim(model = list(ar = c(-0.5)), n = n)
x3 <- arima.sim(model = list(ar = c(0.3)), n = n)
X <- cbind(x1, x2, x3)
X <- t(X)

## Generate yt
Z <- mat.or.vec(m,n)
S1 <- matrix(c(5/8, 1/8, 1/8, 5/8), 2, 2)
Z[,1] <- c(rnorm(m))
for(i in c(2:n)){
  Z[,i] <- S1%*%Z[, i-1] + c(rnorm(m))
}
D <- matrix(runif(p*m, -2, 2), ncol = m)
A <- matrix(runif(p*r, -2, 2), ncol = r)
eps <- mat.or.vec(n, p)
eps <- matrix(rnorm(n*p), p, n)
Y <- D %*% Z + A %*% X + eps
Y <- t(Y)
Z <- t(Z)

## D is known
res1 <- HDSReg(Y, Z, D, lag.k = 2)
## D is unknown
res2 <- HDSReg(Y, Z, lag.k = 2)