SDF_gmm {BayesianFactorZoo}R Documentation

GMM Estimates of Factors' Risk Prices under the Linear SDF Framework

Description

This function provides the GMM estimates of factors' risk prices under the linear SDF framework (including the common intercept).

Usage

SDF_gmm(R, f, W)

Arguments

R

A matrix of test assets with dimension t×Nt \times N, where tt is the number of periods and NN is the number of test assets;

f

A matrix of factors with dimension t×kt \times k, where kk is the number of factors and tt is the number of periods;

W

Weighting matrix in GMM estimation (see Details).

Details

We follow the notations in Section I of Bryzgalova et al. (2023). Suppose that there are KK factors, ft=(f1t,...,fKt),t=1,...,Tf_t = (f_{1t},...,f_{Kt})^\top, t=1,...,T. The returns of NN test assets are denoted by Rt=(R1t,...,RNt)R_t = (R_{1t},...,R_{Nt})^\top.

Consider linear SDFs (MM), that is, models of the form Mt=1(ftE[ft])λfM_t = 1- (f_t -E[f_t])^\top \lambda_f.

The model is estimated via GMM with moment conditions

E[gt(λc,λf,μf)]=E(Rtλc1NRt(ftμf)λfftμf)=(0N0K)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 gT(λc,λf,μf)=1TΣt=1Tgt(λc,λf,μf) 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

Wols=(IN0N×K0K×NκIK),  Wgls=(ΣR10N×K0K×NκIK), 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 ΣR\Sigma_R is the covariance matrix of returns and κ>0\kappa >0 is a large constant so that μ^f=1TΣt=1Tft\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 gt(λc,λf,μf)g_t (\lambda_c, \lambda_f, \mu_f) is independent over time.

Value

The return of SDF_gmm is a list of the following elements:

References

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.


[Package BayesianFactorZoo version 0.0.0.2 Index]