SDF_gmm {BayesianFactorZoo}

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 \times N, where t is the number of periods and N is the number of test assets;

f

A matrix of factors with dimension t \times k, where k is the number of factors and t 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 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.

Value

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.

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.