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Covariance Matrix Estimation via Factor Models
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covFactorModel

Estimation of covariance matrix via factor models with application to financial data. Factor models decompose the asset returns into an exposure term to some factors and a residual idiosyncratic component. The resulting covariance matrix contains a low-rank term corresponding to the factors and another full-rank term corresponding to the residual component.

This package provides a function to separate the data into the factor component and residual component, as well as to estimate the corresponding covariance matrix. Different kind of factor models are considered, namely, macroeconomic factor models and statistical factor models. The estimation of the covariance matrix accepts different kinds of structure on the residual term: diagonal structure (implying that residual component is uncorrelated) and block diagonal structure (allowing correlation within sectors). The package includes a built-in database containing stock symbols and their sectors.

The package is based on the book: R. S. Tsay, Analysis of Financial Time Series. John Wiley & Sons, 2005.

Installation

# Installation from CRAN (not available yet)
#install.packages("covFactorModel")

# Installation from GitHub
# install.packages("devtools")
devtools::install_github("dppalomar/covFactorModel")

# Getting help
library(covFactorModel)
help(package = "covFactorModel")
package?covFactorModel
?factorModel
?covFactorModel
?getSectorInfo

Vignette

For more detailed information, please check the vignette: GitHub-html-vignette and GitHub-pdf-vignette.

Usage of factorModel()

The function factorModel() builds a factor model for the data, i.e., it decomposes the asset returns into a factor component and a residual component. The user can choose different types of factor models, namely, macroeconomic, BARRA, or statistical. We start by generating some synthetic data:

library(covFactorModel)
library(xts)
library(MASS)

# generate synthetic data
set.seed(234)
N <- 3  # number of stocks
T <- 5  # number of samples
mu <- rep(0, N)
Sigma <- diag(N)/1000

# generate asset returns TxN data matrix
X <- xts(mvrnorm(T, mu, Sigma), order.by = as.Date('2017-04-15') + 1:T) 
colnames(X) <- c("A", "B", "C")

# generate K=2 macroeconomic factors
econ_fact <- xts(mvrnorm(T, c(0, 0), diag(2)/1000), order.by = index(X))
colnames(econ_fact) <- c("factor1", "factor2")

We first build a macroeconomic factor model, which fits the data to the given macroeconomic factors:

macro_econ_model <- factorModel(X, type = "Macro", econ_fact = econ_fact)

# sanity check
X_ <- with(macro_econ_model, 
           matrix(alpha, T, N, byrow = TRUE) + factors %*% t(beta) + residual)
norm(X - X_, "F")
#> [1] 2.091133e-18

Next, we build a BARRA industry factor model (assuming assets A and C belong to sector 1 and asset B to sector 2):

stock_sector_info <- c(1, 2, 1)
barra_model <- factorModel(X, type = "Barra", stock_sector_info = stock_sector_info)

# sanity check
X_ <- with(barra_model, 
           matrix(alpha, T, N, byrow = TRUE) + factors %*% t(beta) + residual)
norm(X - X_, "F")
#> [1] 1.45461e-18

Finally, we build a statistical factor model, which is based on principal component analysis (PCA):

# set factor dimension as K=2
stat_model <- factorModel(X, K = 2)

# sanity check
X_ <- with(stat_model, 
           matrix(alpha, T, N, byrow = TRUE) + factors %*% t(beta) + residual)
norm(X - X_, "F")
#> [1] 1.414126e-17

Usage of covFactorModel()

The function covFactorModel() estimates the covariance matrix of the data based on factor models. The user can choose not only the type of factor model (i.e., macroeconomic, BARRA, or statistical) but also the structure of the residual covariance matrix (i.e., scaled identity, diagonal, block diagonal, and full). We start by preparing some synthetic data:

library(covFactorModel)
library(xts)
library(MASS)

# generate synthetic data
set.seed(234)
K <- 1   # number of factors
N <- 400  # number of stocks
mu <- rep(0, N)
beta <- mvrnorm(N, rep(1, K), diag(K)/10)
Sigma <- beta %*% t(beta) + diag(N)
print(eigen(Sigma)$values[1:10])
#>  [1] 438.757   1.000   1.000   1.000   1.000   1.000   1.000   1.000
#>  [9]   1.000   1.000

Then, we simply use function covFactorModel() (by default it uses a statistical factor model and a diagonal structure for the residual covariance matrix). We show the average error w.r.t number of observations:

# estimate error by loop
err_scm_vs_T <- err_statPCA_diag_vs_T <- c()
index_T <- N*seq(5)
for (T in index_T) {
  X <- xts(mvrnorm(T, mu, Sigma), order.by = as.Date('1995-03-15') + 1:T)
  # use statistical factor model
  cov_statPCA_diag <- covFactorModel(X, K = K, max_iter = 10)
  err_statPCA_diag_vs_T <- c(err_statPCA_diag_vs_T, norm(Sigma - cov_statPCA_diag, "F")^2)
  # use sample covariance matrix
  err_scm_vs_T <- c(err_scm_vs_T, norm(Sigma - cov(X), "F")^2)
}
res <- rbind(err_scm_vs_T, err_statPCA_diag_vs_T)
rownames(res) <- c("SCM", "stat + diag")
colnames(res) <- paste0("T/N=", index_T/N)
print(res)
#>                 T/N=1    T/N=2    T/N=3    T/N=4    T/N=5
#> SCM         1378.3156 689.3066 515.7518 322.9559 309.4131
#> stat + diag  967.7577 478.5742 368.6348 221.7183 215.2621

Usage of getSectorInfo()

The function getSectorInfo() provides sector information for a given set of stock symbols. The usage is rather simple:

library(covFactorModel)

mystocks <- c("AAPL",  "ABBV", "AET", "AMD", "APD", "AA","CF", "A", "ADI", "IBM")
getSectorInfo(mystocks)
#> $stock_sector_info
#> AAPL ABBV  AET  AMD  APD   AA   CF    A  ADI  IBM 
#>    1    2    2    1    3    3    3    2    1    1 
#> 
#> $sectors
#>                        1                        2                        3 
#> "Information Technology"            "Health Care"              "Materials"

The built-in sector database can be overidden by providing a stock-sector pairing:

my_stock_sector_database <- cbind(mystocks, c(rep("sector1", 3),
                                              rep("sector2", 4),
                                              rep("sector3", 3)))
getSectorInfo(mystocks, my_stock_sector_database)
#> $stock_sector_info
#> AAPL ABBV  AET  AMD  APD   AA   CF    A  ADI  IBM 
#>    1    1    1    2    2    2    2    3    3    3 
#> 
#> $sectors
#>         1         2         3 
#> "sector1" "sector2" "sector3"

Links

Package: GitHub.

README file: GitHub-readme.

Vignette: GitHub-html-vignette and GitHub-pdf-vignette.

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