HDTSA (version 1.0.4)

Factors: Factor modeling: Inference for the number of factors

Description

Factors() deals with factor modeling for high-dimensional time series proposed in Lam and Yao (2012):$${\bf y}_t = {\bf Ax}_t + {\boldsymbol{\epsilon}}_t, $$ where \({\bf x}_t\) is an \(r \times 1\) latent process with (unknown) \(r \leq p\), \({\bf A}\) is a \(p \times r\) unknown constant matrix, and \( {\boldsymbol{\epsilon}}_t \sim \mathrm{WN}({\boldsymbol{\mu}}_{\epsilon}, {\bf \Sigma}_{\epsilon})\) is a vector white noise process. The number of factors \(r\) and the factor loadings \({\bf A}\) can be estimated in terms of an eigenanalysis for a nonnegative definite matrix, and is therefore applicable when the dimension of \({\bf y}_t\) is on the order of a few thousands. This function aims to estimate the number of factors \(r\) and the factor loading matrix \({\bf A}\).

Usage

Factors(Y, lag.k = 5, twostep = FALSE)

Value

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

factor_num

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

loading.mat

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

lag.k

the time lag used in function.

method

a character string indicating what method was performed.

Arguments

Y

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

lag.k

Time lag \(k_0\) used to calculate the nonnegative definte matrix \( \widehat{\mathbf{M}}\): $$\widehat{\mathbf{M}}\ =\ \sum_{k=1}^{k_0}\widehat{\mathbf{\Sigma}}_y(k)\widehat{\mathbf{\Sigma}}_y(k)', $$ where \(\widehat{\bf \Sigma}_y(k)\) is the sample autocovariance of \( {\bf y}_t\) at lag \(k\).

twostep

Logical. If FALSE (the default), then standard procedures [See Section 2.2 in Lam and Yao (2012)] for estimating \(r\) and \({\bf A}\) will be implemented. If TRUE, then a two step estimation procedure [See Section 4 in Lam and Yao (2012)] will be implemented for estimating \(r\) and \({\bf A}\).

References

Lam, C. & Yao, Q. (2012). Factor modelling for high-dimensional time series: Inference for the number of factors, The Annals of Statistics, Vol. 40, pp. 694--726.

Examples

## Generate x_t
p <- 400
n <- 400
r <- 3
X <- mat.or.vec(n, r)
A <- matrix(runif(p*r, -1, 1), ncol=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)
eps <- matrix(rnorm(n*p), p, n)
X <- t(cbind(x1, x2, x3))
Y <- A %*% X + eps
Y <- t(Y)
fac <- Factors(Y,lag.k=2)
r_hat <- fac$factor_num
loading_Mat <- fac$loading.mat