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Author: Alberto Quaini
The intrinsicFRP
library implements functions
designed for comprehensive evaluation and testing of linear asset
pricing models, focusing on the estimation and assessment of
factor risk premia, factor SDF coefficients, selection of “useful” risk
factors (those displaying non-zero population correlation with test
asset returns), examination of model misspecification, validation of
model identification, and heteroskedasticity and autocorrelation robust
covariance matrix estimation.
Given \(T\) observations on \(N\) test asset excess returns collected in matrix \(R\in\mathbb R^{TxN}\), and \(T\) observations on \(K\) risk factors in \(F\in\mathbb R^{TxK}\), consider the linear asset pricing model: \[R - \iota_T * E[R_t]' = (F - \iota_T * E[F_t]')\beta + \varepsilon,\] where \(\iota_T\in\mathbb R^T\) is a unit vector, \(E[R_t]\in\mathbb R^N\) and \(E[F_t]\in\mathbb R^K\) are the vector of expected returns and factors, respectively, \(\beta\in\mathbb R^{NxK}\) is the matrix of regression coefficients, and \(\varepsilon\in\mathbb R^{TxN}\) is a matrix of residuals.
The vector of compensations for risk factor exposures \(\lambda\in\mathbb R^K\), i.e., factor risk premia, are determined by the second-stage regression of expected asset returns on regression coefficients: \[E[R_t] = \beta\lambda + \nu,\] where \(\nu\in\mathbb R^N\) is the vector of model’s pricing errors.
For estimating and testing factor risk premia, our toolkit incorporates the classic two-pass approach outlined by [@fama1973risk], the misspecification-robust methodology proposed by [@kan2013pricing], and the tradable and “Oracle” tradable approaches introduced by [@quaini2023tradable].
The classic two-pass factor risk premia are given by: \[\lambda=(\beta'\beta)^{-1}\beta'E[R_t].\] The misspecification-robust factor risk premia are given by: \[\lambda=(\beta'W\beta)^{-1}\beta'WE[R_t],\] where \(W\) is a symmetric and positive definite weighting matrix. The tradable factor risk premia are given by: \[\lambda=Cov[R_t,F_t]Var[R_t]^{-1}E[R_t].\] Details on the motivations and properties of these notions of factor risk premia are found in the aforementioned references.
If the law of one price holds, then there is a Stochastic Discount Factor (SDF) \(M\) pricing all excess returns, i.e., \(E[M_tR_t]=0\). A (linear) factor model for the SDF is of the form \(M_t=1-\gamma'(F_t-E[F_t])\), where the mean is normalized to one since we work with excess returns. This package implements the estimators in [@fama1973risk], which propose to find the candidate factor SDF such that \[\gamma=\arg\min_{g\in\mathbb R^K}E[R_tM_t]'E[R_tM_t],\] which is given by: \[\gamma=(Cov[R_t,F_t]'Cov[R_t,F_t])^{-1}Cov[R_t,F_t]'E[R_t].\] Furthermore, it implements the estimators in [@gospodinov2014misspecification], which propose to find the candidate factor SDF that minimizes the pricing errors, under a weighted \(L_2-\)distance: \[\gamma=\arg\min_{g\in\mathbb R^K}E[R_tM_t]'Var[R]^{-1}E[R_tM_t],\] which is given by: \[\gamma=(Cov[R_t,F_t]'Var[R_t]^{-1}Cov[R_t,F_t])^{-1}Cov[R_t,F_t]'Var[R_t]^{-1}E[R_t].\]
Many problems arise when some factors are useless or weak, that is, when their population correlations with test asset excess returns are zero or near-zero, or more generally when the matrix collecting such population correlations is reduced-rank. In that case, the matrix of regression coefficients \(\beta\) is also reduced rank, thereby the vector of factor risk premia \(\lambda\) is not identified. This lack of identification has serious repercussions in its estimation and inference. It is therefore important to filter out the problematic factors from the model.
For selecting the set of useful risk factors, our toolkit implements the iterative factor screening procedure of [@gospodinov2014misspecification], the one-step Oracle selection procedure of [@quaini2023tradable], and the three-step procedure of [@feng2020taming].
\[\gamma=\arg\min_{g\in\mathbb R^K}E[R_tM_t]'Var[R]^{-1}E[R_tM_t].\] The screening procedure of [@gospodinov2014misspecification] is based on the result that the asymptotic distribution of misspecification-robust t-statistics of the factor SDF coefficients are \(\chi^2(1)\), Chi-squared with one degree of freedom, for the coefficients of useless factors, and they are stochastically dominated by a \(\chi^2(1)\) for the useful factors. With this motivation, they suggest a sequential removal of factors associated with the smallest insignificant t-statistics of a nonzero misspecification-robust SDF coefficient. Note that this procedure gives rise to a conservative factor selection, it requires a correction for the multiple testing problem, and it is only robust to the presence of useless factors, and not to weak factors or more general linear dependence structures in the population correlation matrix between test asset returns and risk factors.
The screening methodology in [@quaini2023tradable] removes the factors associated to a zero Oracle tradable factor risk premia estimates, which arise from the one-step closed-form estimator: \[\check{\lambda}_k = sign(\hat\lambda_k)\max\{|\hat{\lambda}_k|-\tau/||\rho_k||_2^2,0\},\] where \(\hat{\lambda}=\widehat{Cov}[R_t,F_t]\widehat{Var}[R_t]^{-1}\widehat{E}[R_t]\) is the sample tradable factor risk premia estimator, which simply replaces population moments with empirical moments, \(\tau\) is a penalty parameter that can be tuned via, e.g., cross validation, and \(\rho_k=\widehat{Corr}[F_{tk}, R_t]\) is the estimated correlation between factor \(k\) and test asset excess returns. Borrowing the terminology of [@fan2001variable], this procedure achieves the so-called “Oracle” variable selection property, i.e., it consistently selects the useful factors. More precisely, the probability that the factors selected by the estimator are indeed useful factors tends to 1 as the sample size tends to infinity.
[@feng2020taming] propose a three-steps testing procedure that evaluates the contribution to cross-sectional pricing of any new factors on top of a set of control factors. The third step is a OLS regression of average returns on the covariances between asset returns and the new factors, as well as the control factors selected in either one of the first two steps. The first stwo steps consists in (i) a Lasso regression of average returns on the ovariances between asset returns and all control factors and (ii) a Lasso regression of the covariances between asset returns and the new factors on the ovariances between asset returns and all control factors. The second selection aims at correcting for potential omitted variables in the first selection.
For evaluating model misspecification, the toolkit implements point and confidence interval estimation for the HJ model misspecification distance formulated by [@kan2008specification], which is a modification of the prominent Hansen-Jagannathan misspecification distance of [@hansen1997assessing]. Such (squared) distance \(\delta_m^2\) is given by the minimum quadratic form in the model pricing errors, weighted by the inverse variance of test asset excess returns: \[\delta_m^2 = min_{\gamma\in\mathbb R^k} (E[R_t] - Cov[R_t,F_t] * \gamma)' * V[R_t]^{-1} * (E[R_t] - Cov[R_t,F_t] * \gamma).\] Clearly, computation of the confidence interval is obtained by means of an asymptotic analysis under potentially misspecified models, i.e., without assuming correct model specification.
Lastly, the functions for testing model identification are specialized versions of the rank tests proposed by [@kleibergen2006generalized] and [@chen2019improved]. These tests are specifically tailored to assess the regression coefficient matrix of test asset returns on risk factors.
Factor risk premia \(\lambda\), equivalently SDF coefficients \(\gamma\), are identified if the population correlation matrix between test asset excess returns and risk factors, or equivalently the regression coefficient matrix \(\beta\), is full column rank. The beta rank test of [@kleibergen2006generalized] is a procedure that tests the Null hypothesis \[H_0:rank(\beta)=q\] iteratively for \(q=0,\ldots,K-1\). It therefore requires a correction for multiple testing. Instead, the beta rank test of [@chen2019improved] is a bootstrap-based procedure designed to test directly the Null hypothesis \[H_0:rank(\beta)<= K-1.\]
For heteroskedasticity and autocorrelation robust covariance matrix estimation of a centred time series \(q_t\in\mathbb R^M\), with \(M\ge 1\), the package implements the [@newey1994automatic] estimator. This estimator aims at computing: \[\lim_{T\to\infty}\frac{1}{T}\sum_{t=1}^T\sum_{s=1}^TE[q_tq_s'],\] by estimating the cross-moments \(E[q_tq_s']\) up to an appropriate number of lags \(l=|t-s|\). This estimator is frequently used, e.g., for a robust estimation of the covariance matrix of the model’s pricing errors \(\varepsilon_t\).
To cite intrinsicFRP
in publications, please use:
Quaini, A. (2023).
intrinsicFRP
: An R Package for Factor Model Asset Pricing.R
package version 2.0.0. URL: https://CRAN.R-project.org/package=intrinsicFRP.
Package intrinsicFRP
is on CRAN (The Comprehensive R
Archive Network), hence the latest release can be easily installed from
the R
command line via
install.packages("intrinsicFRP")
To install the latest (possibly unstable) development version from
GitHub, you can pull this repository and install it from the
R
command line via
# if you already have package `devtools` installed, you can skip the next line
install.packages("devtools")
::install_github("a91quaini/intrinsicFRP") devtools
Package intrinsicFRP
contains C++
code that
needs to be compiled, so you may need to download and install the necessary tools for
MacOS or the necessary tools
for Windows.
R package intrinsicFRP
implements the following
functions:
FRP()
: Computes the [@fama1973risk] factor risk premia:
FMFRP = (beta' * beta)^{-1} * beta' * E[R]
where
beta = Cov[R, F] * V[F]^{-1}
or the misspecification-robust
factor risk premia of [@kan2013pricing]:
KRSFRP = (beta' * V[R]^{-1} * beta)^{-1} * beta' * V[R]^{-1} * E[R]
,
from data on factors F
and test asset excess returns
R
. These notions of factor risk premia are by construction
the negative covariance of factors F
with candidate SDF
M = 1 - d' * (F - E[F])
, where SDF coefficients
d
are obtained by minimizing pricing errors:
argmin_{d} (E[R] - Cov[R,F] * d)' * (E[R] - Cov[R,F] * d)
and
argmin_{d} (E[R] - Cov[R,F] * d)' * V[R]^{-1} * (E[R] - Cov[R,F] * d)
,
respectively. Optionally computes the corresponding heteroskedasticity
and autocorrelation robust standard errors (accounting for a potential
model misspecification) using the [@newey1994automatic] plug-in
procedure to select the number of relevant lags, i.e.,
n_lags = 4 * (n_observations/100)^(2/9)
. For the standard
error computations, the function allows to internally pre-whiten the
series by fitting a VAR(1), i.e., a vector autoregressive model of order
1. All the details can be found in [@kan2013pricing].TFRP()
: Computes tradable factor risk premia from data
on factors F
and test asset excess returns R
:
TFRP = Cov[F, R] * Var[R]^{-1} * E[R]
; which are by
construction the negative covariance of factors F
with the
SDF projection on asset returns, i.e., the minimum variance SDF.
Optionally computes the corresponding heteroskedasticity and
autocorrelation robust standard errors using the [@newey1994automatic] plug-in
procedure to select the number of relevant lags, i.e.,
n_lags = 4 * (n_observations/100)^(2/9)
. For the standard
error computations, the function allows to internally pre-whiten the
series by fitting a VAR(1), i.e., a vector autoregressive model of order
1. All details are found in [@quaini2023tradable].OracleTFRP()
: Computes Oracle tradable factor risk
premia of [@quaini2023tradable] from data
on K
factors F = [F_1,...,F_K]'
and test asset
excess returns R
:
OTFRP = argmin_x ||TFRP - x||_2^2 + tau * sum_{k=1}^K w_k * |x_k|
,
where TFRP
is the tradable factor risk premia estimator,
tau > 0
is a penalty parameter, and the Oracle weights
are given by w_k = 1 / ||corr[F_k, R]||_2^2
. This estimator
is called “Oracle” in the sense that the probability that the index set
of its nonzero estimated risk premia equals the index set of the true
strong factors tends to 1 (Oracle selection), and that on the strong
factors, the estimator reaches the optimal asymptotic Normal
distribution. Here, strong factors are those that have a nonzero
population marginal correlation with asset excess returns. Tuning of the
penalty parameter tau
is performed via Generalized Cross
Validation (GCV), Cross Validation (CV) or Rolling Validation (RV). GCV
tunes parameter tau
by minimizing the criterium:
||PE(tau)||_2^2 / (1-df(tau)/T)^2
where
PE(tau) = E[R] - beta_{S(tau)} * OTFRP(tau)
are the pricing
errors of the model for given tuning parameter tau
, with
S(tau)
being the index set of the nonzero Oracle TFRP
computed with tuning parameter tau
, and
beta_{S(tau)} = Cov[R, F_{S(tau)}] * (Cov[F_{S(tau)}, R] * V[R]^{-1} * Cov[R, F_{S(tau)}])^{-1}
the regression coefficients of the test assets excess returns on the
factor mimicking portfolios, and df(tau) = |S(tau)|
are the
degrees of freedom of the model, given by the number of nonzero Oracle
TFRP. CV and RV, instead, choose the value of tau
that
minimize the criterium: PE(tau)' * V[PE(tau)]^{-1} PE(tau)
where V[PE(tau)]
is the diagonal matrix collecting the
marginal variances of pricing errors PE(tau)
, and each of
these components are aggregated over k-fold cross-validated data or over
rolling windows of data, respectively. Oracle weights can be based on
the correlation between factors and returns (suggested approach), on the
regression coefficients of returns on factors or on the first-step
tradable risk premia estimator. Optionally computes the corresponding
heteroskedasticity and autocorrelation robust standard errors using the
[@newey1994automatic] plug-in
procedure to select the number of relevant lags, i.e.,
n_lags = 4 * (n_observations/100)^(2/9)
. For the standard
error computations, the function allows to internally pre-whiten the
series by fitting a VAR(1), i.e., a vector autoregressive model of order
1. All details are found in [@quaini2023tradable].SDFCoefficients
: Computes the SDF coefficients of [@fama1973risk]:
FMSDFcoefficients = (C' * C)^{-1} * C' * E[R]
, or the
misspecification-robust SDF coefficients of [@gospodinov2014misspecification]:
GKRSDFcoefficients = (C' * V[R]^{-1} * C)^{-1} * C' * V[R]^{-1} * E[R]
from data on factors F
and test asset excess returns
R
. These notions of SDF coefficients minimize pricing
errors:
argmin_{d} (E[R] - Cov[R,F] * d)' * W * (E[R] - Cov[R,F] * d)
,
with W=I
, i.e., the identity, and W=V[R]^{-1}
,
respectively. Optionally computes the corresponding heteroskedasticity
and autocorrelation robust standard errors (accounting for a potential
model misspecification) using the [@newey1994automatic] plug-in
procedure to select the number of relevant lags, i.e.,
n_lags = 4 * (n_observations/100)^(2/9)
.FGXFactorsTest()
: Computes the three-pass procedure of
[@feng2020taming], which evaluates
the contribution to cross-sectional pricing of any new factors on top of
a set of control factors. The third step is a OLS regression of average
returns on the covariances between asset returns and the new factors, as
well as the control factors selected in either one of the first two
steps. The first stwo steps consists in (i) a Lasso regression of
average returns on the ovariances between asset returns and all control
factors and (ii) a Lasso regression of the covariances between asset
returns and the new factors on the ovariances between asset returns and
all control factors. The second selection aims at correcting for
potential omitted variables in the first selection. Tuning of the
penalty parameters in the Lasso regressions is performed via Cross
Validation (CV). Standard errors are computed following Feng Giglio and
Xiu (2020) using the [@newey1994automatic] plug-in
procedure to select the number of relevant lags, i.e.,
n_lags = 4 * (n_observations/100)^(2/9)
. For the standard
error computations, the function allows to internally pre-whiten the
series by fitting a VAR(1), i.e., a vector autoregressive model of order
1.ChenFang2019BetaRankTest()
: Tests the null hypothesis
of reduced rank in the matrix of regression loadings for test asset
excess returns on risk factors using the [@chen2019improved] beta rank test.
The test applies the [@kleibergen2006generalized]
iterative rank test for initial rank estimation when
target_level_kp2006_rank_test > 0
, with an adjustment to
level = target_level_kp2006_rank_test / n_factors
. When
target_level_kp2006_rank_test <= 0
, the number of
singular values above n_observations^(-1/4)
is used
instead. It presumes that the number of factors is less than the number
of returns (n_factors < n_returns
). All the details can
be found in [@chen2019improved].IterativeKleibergenPaap2006BetaRankTest()
: Evaluates
the rank of regression loadings in an asset pricing model using the
iterative [@kleibergen2006generalized]
beta rank test. It systematically tests the null hypothesis for each
potential rank q = 0, ..., n_factors - 1
and estimates the
rank as the smallest q
that has a p-value below the
significance level, adjusted for the number of factors. The function
presupposes more returns than factors
(n_factors < n_returns
). All the details can be found in
[@kleibergen2006generalized].HACcovariance()
: estimates the long-run covariance
matrix of a multivariate centred time series accounting for
heteroskedasticity and autocorrelation using the [@newey1994automatic] estimator.
If the number of lags is not provided, they are selected using the
Newey-West plug-in procedure, where
n_lags = 4 * (n_observations/100)^(2/9)
. The function
allows to internally prewhiten the series by fitting a VAR(1). All the
details can be found in [@newey1994automatic].For usage details, type ?FunctionName
in the R console,
e.g.:
?TFRP
The intrinsicFRP
R package includes a dataset comprising
following test asset excess returns and risk factors frequently used in
the asset pricing literature:
returns
: Monthly observations from July 1963 to
February 2024, containing excess returns data for 25 Size/Book-to-Market
portfolios and 17 industry portfolios.factors
: Monthly observations from July 1963 to
February 2024, containing data for the Fama-French 5 factors and the
momentum factor.risk_free
: Monthly observations from July 1963 to
February 2024, containing returns data for the US T-bill, used as proxy
for the risk free asset.This dataset was sourced from the Kenneth French data library and processed so that the observations on the test assets and the factors are not expressed in percentage points, and that returns on the test assets are in excess of the risk-free rate.
Let us ompute various factor risk premia estimates and corresponding 95% confidence intervals for the Fama-French 6 factors and a (simulated) “useless” factor.
# if you already have package `stats` installed you can skip the next line
install.packages("stats")
# import package data on 6 risk factors and 42 test asset excess returns
# remove the first column containing the date
= intrinsicFRP::factors[,-1]
factors = intrinsicFRP::returns[,-1]
returns = intrinsicFRP::risk_free[,-1]
RF
# simulate a useless factor and add it to the matrix of factors
set.seed(23)
= cbind(
factors
factors,::rnorm(n = nrow(factors), sd = stats::sd(factors[,3]))
stats
)colnames(factors) = c(colnames(intrinsicFRP::factors[,2:7]), "Useless")
# index set of specific factor models
# Fama-French 3 factor model
= 1:3
ff3 # Fama-French 6 factor model
= 1:6 # "Mkt-RF" "SMB" "HML" "RMW" "CMA" "Mom"
ff6 # model comprising the Fama-French 6 factors and the simulated useless factor
= 1:7 # "Mkt-RF" "SMB" "HML" "RMW" "CMA" "Mom" "Useless"
ff6usl
# compute the factor SDF coefficients and their standard errors
# for the Fama-MacBeth two-pass procedure
= intrinsicFRP::SDFCoefficients(
fm_sdf
returns,
factors[,ff6usl],misspecification_robust = FALSE,
include_standard_errors = TRUE
)# and for the misspecification-robust procedure of Gospodinov Kan and Robottu
= intrinsicFRP::SDFCoefficients(
gkr_sdf
returns, factors[,ff6usl],include_standard_errors = TRUE
)
Visualization of the Fama MacBeth (FM) and the misspecification-robust (GKR) factor SDF coefficients estimators:
# compute tradable factor risk premia and their standard errors
= intrinsicFRP::TFRP(returns, factors[,ff6usl], include_standard_errors = TRUE)
tfrp
# compute the GLS factor risk premia of Kan Robotti and Shanken (2013) and their
# standard errors
= intrinsicFRP::FRP(returns, factors[,ff6usl], include_standard_errors = TRUE)
krs_frp
# set penalty parameters
= seq(1e-4, 4e-3, length.out = 1000)
penalty_parameters
# compute Oracle tradable factor risk premia and their standard errors
# for low factor models, no need for the "one standard deviation" tuning rule
= intrinsicFRP::OracleTFRP(
oracle_tfrp
returns,
factors[,ff6usl],
penalty_parameters,include_standard_errors = TRUE,
one_stddev_rule = FALSE
)
Tuning model score of the Oracle TFRP estimator:
Visualization of the Fama MacBeth (FM), misspecification-robust (KRS), tradable (TFRP) and Oracle TFRP (O-TFRP) factor risk premia estimates:
In this plot, we notice a number of features:
Let us now focus on the factor screening procedures under the Oracle TFRP estimator and the approach of [@gospodinov2014misspecification].
# recover the indices of the factors selected by the Oracle TFRP estimator
which(oracle_tfrp$risk_premia != 0)
# compute the GKR factor screening procedure
::GKRFactorScreening(returns, factors[,ff6]) intrinsicFRP
The results are:
# factor indices of the factors selected by the Oracle TFRP estimator
1] 1 2 3 4 5 6
[
# results of the GKR factor screening procedure
$sdf_coefficients
1]
[,1,] 6.401245
[2,] 8.597950
[3,] 12.483170
[
$standard_errors
1]
[,1,] 1.600189
[2,] 2.657617
[3,] 4.173861
[
$t_statistics
1]
[,1,] 4.000306
[2,] 3.235211
[3,] 2.990797
[
$selected_factor_indices
1]
[,1,] 1
[2,] 3
[3,] 6 [
While the Oracle TFRP only removes the simulated useless factor, the procedure by [@gospodinov2014misspecification] only retains the market factor.
Let us compute the HJ misspecification distance of the Fama-French 3 and 6 factor models.
# compute the HJ misspecification distance of the Fama-French 3 and 6 factor models
::HJMisspecificationDistance(returns, factors[,ff3])
intrinsicFRP::HJMisspecificationDistance(returns, factors[,ff6]) intrinsicFRP
The result of the HJ misspecification distance is:
# HJ misspecification test p-value for the Fama-French 3 factor model
$squared_distance
1] 0.2488529
[
$lower_bound
1] 0.1559376
[
$upper_bound
1] 0.3417683
[
# HJ misspecification test p-value for the Fama-French 6 factor model
$squared_distance
1] 0.1619658
[
$lower_bound
1] 0.08727944
[
$upper_bound
1] 0.2366521 [
Since the p-value of both HJ misspecification tests is below the standard thresholds of \(10\%\), \(5\%\) and \(1\%\), we reject the Null that the Fama-French 3 and 6 factor models are correctly specified.
Let us compute the iterative [@kleibergen2006generalized] and the [@chen2019improved] beta rank identification tests for the Fama-French 6 factor model.
# compute identification tests of the Fama-French 6 factor model
::IterativeKleibergenPaap2006BetaRankTest(returns, factors[,ff6])
intrinsicFRP::ChenFang2019BetaRankTest(returns, factors[,ff6]) intrinsicFRP
The results of the identification tests for the Fama-French 6 factor model are:
# output of the Iteraive Kleibergen Paap (2006) Beta Rank Test
$rank
1] 6
[
$q
1] [,2] [,3] [,4] [,5] [,6]
[,1,] 0 1 2 3 4 5
[
$statistics
1] 108162.2473 10291.4280 4710.5822 1015.9157 376.7892 112.8450
[
$pvalues
1] 0.000000e+00 0.000000e+00 0.000000e+00 3.510246e-143 2.062222e-41
[6] 1.340701e-09
[
# p-value of the Chen Fang (2019) Beta Rank Test
$statistic
1] 143.1936
[
$`p-value`
1] 0 [
Since the largest p-value of the Iteraive [@kleibergen2006generalized] Beta Rank Test and the p-value of the [@chen2019improved] Beta Rank Test are below the standard thresholds of \(10\%\), \(5\%\) and \(1\%\), we reject the Null that the Fama-French 6 factor model is not identified.
For sanity check, let us compute the same identification tests for the unidentified model comprising the Fama-French 6 factors and a (simulated) “useless” factor.
# compute identification tests of unidentified factor model comprising the
# Fama-French 6 factors and the simulated useless factor
::IterativeKleibergenPaap2006BetaRankTest(returns, factors[,ff6usl])
intrinsicFRP::ChenFang2019BetaRankTest(returns, factors[,ff6usl]) intrinsicFRP
The results are:
# output of the Iteraive Kleibergen Paap (2006) Beta Rank Test
$rank
1] 6
[
$q
1] [,2] [,3] [,4] [,5] [,6] [,7]
[,1,] 0 1 2 3 4 5 6
[
$statistics
1] 128024.51364 11595.52592 5480.53412 1127.71644 475.16366
[6] 190.55705 57.75449
[
$pvalues
1] 0.000000e+00 0.000000e+00 0.000000e+00 7.265767e-147 1.346504e-45
[6] 3.134998e-12 1.216120e-02
[
# p-value of the Chen Fang (2019) Beta Rank Test
$statistic
1] 41.87344
[
$`p-value`
1] 0.188 [
Since the largest p-value of the Iteraive [@kleibergen2006generalized] Beta Rank Test and the p-value of the [@chen2019improved] Beta Rank Test are above the standard thresholds of \(10\%\), \(5\%\) and \(1\%\), we do not reject the Null that the Fama-French 6 factor model augmented with a simulated useless factor is not identified.
To optimize computational performance, all methods implemented in
package intrinsicFRP
are written in C++ and make use of the
Armadillo [@sanderson2016armadillo]
library for efficient linear algebra calculations. However, for user
convenience, the interface of package intrinsicFRP
is
entirely implemented in R, with minimal dependencies, including:
Rcpp
[@eddelbuettel2018extending]
and RcppArmadillo
[@eddelbuettel2014rcpparmadillo]:
They facilitate seamless integration between R, C++, and the armadillo
C++ library.graphics
: It provides R functions for creating basic
graphics.You can raise issues, report bugs, seek for further help, or submit
your contribution to the R package intrinsicFRP
at the
github repository a91quaini/intrinsicFRP.
For bug reports, you are kindly asked to make a small and self-contained program which exposes the bug.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.