Asymptotic PCA

Asymptotic Principal Components Analysis for a fixed number of factors

Usage

apca(x,nf)

Arguments

x

a numeric matrix or other object for which 'as.matrix' will produce a numeric matrix.

nf

number of factors desired.

Details

NOTE: This is a preliminary version of this function, and it may be modified in the future.

Value

A list with four components:

eig

eigenvalues

factors

estimated factor scores

loadings

estimated factor loadings

rsq

R-squared from the regression of each variable on the factor space

Author

Ruey Tsay

References

Ruey Tsay (2005) Analysis of Financial Time Series, 2nd ed. (Wiley, sec. 9.6, pp. 436-440)

See also

princomp

Examples

# Consider the monthly simple returns of 40 stocks on NYSE and NASDAQ
# from 2001 to 2003 with 36 observations.
data(m.apca0103)
dim(m.apca0103)
#> [1] 1440    3
M.apca0103 <- with(m.apca0103,
    array(return, dim = c(36, 40),
          dimnames = list(as.character(date[1:36]),
                          paste("Co", CompanyID[seq(1, 1440, 36)], sep = ""))))

# The traditional PCA is not applicable to estimate the factor model
# because of the singularity of the covariance matrix.  The asymptotic
# PCA provides an approach to estimate factor model based on asymptotic
# properties. For the simple example considered, the sample size is
# $T$ = 36 and the dimension is $k$ = 40. If the number of factor is
# assumed to be 1, the APCA gives a summary of the factor loadings as
# below:
#
apca40 <- apca(M.apca0103, 1)
#> Asymptotic PCA:  Extracting 1 factors from 36 observations on 40 series
#> 
#> Factor Loading: Summary
#>     factor     min 1st Qu. median   mean    Q3   max
#> F.1      1 0.06947  0.4322 0.6286 0.6883 1.071 1.612
#> R-squared: Summary
#>     min 1st Qu.  median    mean      Q3     max 
#> 0.01733 0.21922 0.47494 0.43715 0.57393 0.82652 
#
# (min, 1st Quartile, median, mean, 3rd quartile, max) =
# (0.069, 0.432, 0.629, 0.688, 1.071, 1.612).
#
# Note that the sign of any loading vector is not uniquely determined
# in the same way as the sign of an eigenvector is not uniquely
# determined. The output also contains the summary statistics of the
# R-squares of individual returns, i.e. the R-squares measuring the
# total variation of individual return explained by the factors. For
# the simple case considered, the summary of R-squares is (min, 1st
# Quartile, median, mean, 3rd quartile, max) =
# (0.090,0.287,0.487,0.456,0.574,0.831).