# apca: Asymptotic PCA In FinTS: Companion to Tsay (2005) Analysis of Financial Time Series

## Description

Asymptotic Principal Components Analysis for a fixed number of factors

## Usage

 `1` ```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

Ruey Tsay

## References

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

`princomp`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29``` ```# 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) 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) # # (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). ```