HDSRegR Documentation

High dimensional stochastic regression with latent factors

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 {ε}}_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{ε}}_t \sim \mathrm{WN}({\boldsymbol{0}},{\bf Σ}_{ε}) is a white noise with zero mean and covariance matrix {\bf Σ}_{ε} and {\boldsymbol{ε}}_t is uncorrelated with ({\bf z}_t, {\bf x}_t), {\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 unknown 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 = 1, twostep = FALSE)

Arguments

Y

{\bf Y} = \{{\bf y}_1, … , {\bf y}_n \}', a data matrix with n rows and p columns, where n is the sample size and p is the dimension of {\bf y}_t.

Z

{\bf Z} = \{{\bf z}_1, … , {\bf z}_n \}', a data matrix representing some observed regressors with n rows and m columns, where n is the sample size and m is the dimension of {\bf z}_t.

D

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

lag.k

Time lag k_0 used to calculate the nonnegative definte matrix \widehat{\mathbf{M}}:

\widehat{\mathbf{M}}\ =\ ∑_{k=1}^{k_0}\widehat{\mathbf{Σ}}_{η}(k)\widehat{\mathbf{Σ}}_{η}(k)',

where \widehat{\bf Σ}_{η}(k) is the sample autocovariance of {\boldsymbol {η}}_t = {\bf y}_t - \widetilde{\bf D}{\bf z}_t at lag k.

twostep

Logical. If FALSE (the default), then standard procedures (see factors) will be implemented to estimate r and {\bf A}. If TRUE, then a two step estimation procedure (see factors) will be implemented to estimate r and {\bf A}.

Value

An object of class "HDSReg" is a list containing the following components:

factor_num

The estimated number of factors \hat{r}.

reg.coff.mat

The estimated p \times m regression coefficient matrix \widetilde{\bf D} if D is not given.

loading.mat

The estimated p \times m factor loading matrix {\bf \widehat{A}}.

References

Chang, J., Guo, B. & Yao, Q. (2015). High dimensional stochastic regression with latent factors, endogeneity and nonlinearity, Journal of Econometrics, Vol. 189, pp. 297–312.

See Also

factors.

Examples

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)

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