HDSReg | R Documentation |
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}.
HDSReg(Y, Z, D = NULL, lag.k = 1, twostep = FALSE)
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 |
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 |
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 |
loading.mat |
The estimated p \times m factor loading matrix {\bf \widehat{A}}. |
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.
factors
.
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)
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