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).

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);