PC {GCCfactor}

Principal component (PC) estimation of the approximate factor model

Description

Perform PC estimation of the (2D) approximate factor model:

y_{it}=\boldsymbol{\lambda}_{i}^{\prime}\boldsymbol{F}_{t}+e_{it},

or in matrix notation:

\boldsymbol{Y}=\boldsymbol{F}\boldsymbol{\Lambda}^{\prime}+\boldsymbol{e}.

The factors \boldsymbol{F} is estimated as \sqrt{T} times the r eigenvectors of the matrix \boldsymbol{Y}\boldsymbol{Y}^{\prime} corresponding to the r largest eigenvalues in descending order, and the loading matrix is estimated by \boldsymbol{\Lambda}=T^{-1}\boldsymbol{Y}^{\prime}\boldsymbol{F}. See e.g. Bai and Ng (2002).

Usage

PC(Y, r)

Arguments

Y

A T \times N data matrix. T = number of time series observations, N = cross-sectional dimension.

r

= the number of factors.

Value

A list containing the factors and factor loadings:

• factor = a T \times r matrix of the estimated factors.

• loading = a N \times r matrix of the estimated factor loadings.

References

Bai, J. and Ng, S., 2002. Determining the number of factors in approximate factor models. Econometrica, 70(1), pp.191-221.

Examples

# simulate data

T <- 100
N <- 50
r <- 2
F <- matrix(stats::rnorm(T * r, 0, 1), nrow = T)
Lambda <- matrix(stats::rnorm(N * r, 0, 1), nrow = N)
err <- matrix(stats::rnorm(T * N, 0, 1), nrow = T)
Y <- F %*% t(Lambda) + err

# estimation

est_PC <- PC(Y, r)