fitSfm: Fit a statistical factor model using principal component...
In sangeeuw/factorAnalytics_Avinash: Factor Analytics

Description Usage Arguments Details Value Author(s) References See Also Examples

Fits a statistical factor model using Principal Component Analysis (PCA) for one or more asset returns or excess returns. When the number of assets exceeds the number of time periods, Asymptotic Principal Component Analysis (APCA) is performed. An object of class "sfm" is returned. This function is based on the S+FinMetric function mfactor.

fitSfm(data, k = 1, max.k = NULL, refine = TRUE, sig = 0.05,
  check = FALSE, corr = FALSE, ...)

## S3 method for class 'sfm'
coef(object, ...)

## S3 method for class 'sfm'
fitted(object, ...)

## S3 method for class 'sfm'
residuals(object, ...)

`data`	vector, matrix, data.frame, xts, timeSeries or zoo object with asset returns. See details.
`k`	number of factors (or) a method for determining the optimal number of factors, one of "bn" or "ck". See details. Default is 1.
`max.k`	scalar; the maximum number of factors to be considered for methods "bn" or "ck". Default is `NULL`. See details.
`refine`	logical; whether to use the Connor-Korajczyk refinement for APCA. Default is `TRUE`.
`sig`	scalar; desired level of significance when "ck" method is specified. Default is 0.05.
`check`	logical; to check if any asset has identical observations. Default is `FALSE`.
`corr`	logical; whether to use the correlation instead of the covariance matrix when finding the principal components. Default is `FALSE`.
`...`	optional arguments passed to `lm`.
`object`	a fit object of class `sfm` which is returned by `fitSfm`

If data is not of class "xts", rownames must provide an "xts" compatible time index. Before model fitting, incomplete cases in data are removed using na.omit. Specifying check=TRUE, issues a warning if any asset is found to have identical observations.

Let N be the number of columns or assets and T be the number of rows or observations. When N < T, Principal Component Analysis (PCA) is performed. Any number of factors less than min(N,T) can be chosen via argument k. Default is 1. Refer to Zivot and Wang (2007) for more details and references.

When N >= T, Asymptotic Principal Component Analysis (APCA) is performed. The user can directly specify k similar to PCA above, or a method to automatically determine the number of factors can be specified: k="bn" corresponds to Bai and Ng (2002) and k="ck" corresponds to Connor and Korajczyk (1993). Users can choose the maximum number of factors, max.k, to consider with these methods. The default for max.k is set to be 10 or $T-1$, whichever is smaller.

refine specifies whether a refinement of the APCA procedure from Connor and Korajczyk (1988), that may improve efficiency, is to be used.

When corr=TRUE, the correlation matrix of returns are used for finding the principal components instead of the covariance matrix. This is typically decided by practioners on a case-by-case basis. The variable with the highest variance dominates the PCA when the covariance matrix is used. However, this may be justified if a volatile asset is more interesting for some reason and volatility information shouldn't be discarded. On the other hand, using the correlation matrix standardizes the variables and makes them comparable, avoiding penalizing variables with less dispersion.

Finally, if the median of the 1st principal component is negative, all it's factor realizations are automatically inverted to enable more meaningful interpretation.

fitTsfm returns an object of class "sfm" for which print, plot, predict and summary methods exist.

The generic accessor functions coef, fitted and residuals extract various useful features of the fit object. Additionally, fmCov computes the covariance matrix for asset returns based on the fitted factor model

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

`asset.fit`	fitted object of class `"mlm"` or `"lm"` from the time-series LS regression of asset returns on estimated factors.
`k`	number of factors; as input or determined by "ck" or "bn" methods.
`factors`	T x K xts object of estimated factor realizations.
`loadings`	N x K matrix of factor loadings estimated by regressing the asset returns on estimated factors.
`alpha`	length-N vector of estimated alphas.
`r2`	length-N vector of R-squared values.
`resid.sd`	length-N vector of residual standard deviations.
`residuals`	T x N xts object of residuals from the LS regression.
`Omega`	N x N return covariance matrix estimated by the factor model.
`eigen`	length-N (or length-T for APCA) vector of eigenvalues of the sample covariance matrix.
`mimic`	N x K matrix of factor mimicking portfolio weights.
`call`	the matched function call.
`data`	T x N xts data object containing the asset returns.
`asset.names`	length-N vector of column names from data.

Where N is the number of assets, K is the number of factors, and T is the number of observations.

Eric Zivot, Sangeetha Srinivasan and Yi-An Chen

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

Connor, G., & Korajczyk, R. A. (1988). Risk and return in an equilibrium APT: Application of a new test methodology. Journal of Financial Economics, 21(2), 255-289.

Connor, G., & Korajczyk, R. A. (1993). A test for the number of factors in an approximate factor model. The Journal of Finance, 48(4), 1263-1291.

Zivot, E., & Wang, J. (2007). Modeling Financial Time Series with S-PLUS (Vol. 191). Springer.

The sfm methods for generic functions: plot.sfm, predict.sfm, print.sfm and summary.sfm.

And, the following extractor functions: coef, fitted, residuals, fmCov, fmSdDecomp, fmVaRDecomp and fmEsDecomp.

paFm for Performance Attribution.

# load return data
data(StockReturns)

# PCA is performed on r.M and APCA on r.W
class(r.M)
dim(r.M)
range(rownames(r.M))
class(r.W)
dim(r.W)

# PCA
args(fitSfm)
fit.pca <- fitSfm(r.M, k=2)
class(fit.pca)
names(fit.pca)
head(fit.pca$factors)
head(fit.pca$loadings)
fit.pca$r2
fit.pca$resid.sd
fit.pca$mimic

# APCA with number of factors, k=15
fit.apca <- fitSfm(r.W, k=15, refine=TRUE)

# APCA with the Bai & Ng method
fit.apca.bn <- fitSfm(r.W, k="bn")

# APCA with the Connor-Korajczyk method
fit.apca.ck <- fitSfm(r.W, k="ck")