SDF_gmm {BayesianFactorZoo} | R Documentation |
This function provides the GMM estimates of factors' risk prices under the linear SDF framework (including the common intercept).
SDF_gmm(R, f, W)
R |
A matrix of test assets with dimension |
f |
A matrix of factors with dimension |
W |
Weighting matrix in GMM estimation (see Details). |
We follow the notations in Section I of Bryzgalova et al. (2023).
Suppose that there are K
factors, f_t = (f_{1t},...,f_{Kt})^\top, t=1,...,T
.
The returns of N
test assets are denoted by R_t = (R_{1t},...,R_{Nt})^\top
.
Consider linear SDFs (M
), that is, models of the form M_t = 1- (f_t -E[f_t])^\top \lambda_f
.
The model is estimated via GMM with moment conditions
E[g_t (\lambda_c, \lambda_f, \mu_f)] =E\left(\begin{array}{c} R_t - \lambda_c 1_N - R_t (f_t - \mu_f)^\top \lambda_f \\ f_t - \mu_f \end{array} \right) =\left(\begin{array}{c} 0_N \\ 0_K \end{array} \right)
and the corresponding sample analog function g_T (\lambda_c, \lambda_f, \mu_f) = \frac{1}{T} \Sigma_{t=1}^T g_t (\lambda_c, \lambda_f, \mu_f)
. Different weighting matrices deliver different point estimates. Two popular choices are
W_{ols}=\left(\begin{array}{cc}I_N & 0_{N \times K} \\ 0_{K \times N} & \kappa I_K\end{array}\right), \ \ W_{gls}=\left(\begin{array}{cc} \Sigma_R^{-1} & 0_{N \times K} \\ 0_{K \times N} & \kappa I_K\end{array}\right),
where \Sigma_R
is the covariance matrix of returns and \kappa >0
is a large constant so that \hat{\mu}_f = \frac{1}{T} \Sigma_{t=1}^{T} f_t
.
The asymptotic covariance matrix of risk premia estimates, Avar_hat
, is based on the assumption that
g_t (\lambda_c, \lambda_f, \mu_f)
is independent over time.
The return of SDF_gmm
is a list of the following elements:
lambda_gmm
: Risk price estimates;
mu_f
: Sample means of factors;
Avar_hat
: Asymptotic covariance matrix of GMM estimates (see Details);
R2_adj
: Adjusted cross-sectional R^2
;
S_hat
: Spectral matrix.
Bryzgalova S, Huang J, Julliard C (2023). “Bayesian solutions for the factor zoo: We just ran two quadrillion models <https://doi.org/10.1111/jofi.13197>.” Journal of Finance, 78(1), 487–557.