continuous_ss_sdf: SDF model selection with continuous spike-and-slab prior

In BayesianFactorZoo: Bayesian Solutions for the Factor Zoo: We Just Ran Two Quadrillion Models

SDF model selection with continuous spike-and-slab prior

Description

This function provides the SDF model selection procedure using the continuous spike-and-slab prior. See Propositions 3 and 4 in \insertCitebryzgalova2023bayesian;textualBayesianFactorZoo.

Usage

continuous_ss_sdf(
  f,
  R,
  sim_length,
  psi0 = 1,
  r = 0.001,
  aw = 1,
  bw = 1,
  type = "OLS",
  intercept = TRUE
)

Arguments

f	A matrix of factors with dimension t \times k, where k is the number of factors and t is the number of periods;
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;
sim_length	The length of monte-carlo simulations;
psi0	The hyper-parameter in the prior distribution of risk prices (see Details);
r	The hyper-parameter related to the prior of risk prices (see Details);
aw	The hyper-parameter related to the prior of \gamma (see Details);
bw	The hyper-parameter related to the prior of \gamma (see Details);
type	If type = 'OLS' (type = 'GLS'), the function returns Bayesian OLS (GLS) estimates of risk prices. The default is 'OLS'.
intercept	If intercept = TRUE (intercept = FALSE), we include (exclude) the common intercept in the cross-sectional regression. The default is intercept = TRUE.

Details

To model the variable selection procedure, we introduce a vector of binary latent variables \gamma^\top = (\gamma_0,\gamma_1,...,\gamma_K), where \gamma_j \in \{0,1\} . When \gamma_j = 1, factor j (with associated loadings C_j) should be included in the model and vice verse.

The continuous spike-and-slab prior of risk prices \lambda is

\lambda_j | \gamma_j, \sigma^2 \sim N (0, r(\gamma_j) \psi_j \sigma^2 ) .

When the factor j is included, we have r(\gamma_j = 1)=1 . When the factor is excluded from the model, r(\gamma_j = 0) =r \ll 1 . Hence, the Dirac "spike" is replaced by a Gaussian spike, which is extremely concentrated at zero (the default value for r is 0.001). If intercept = TRUE, we choose \psi_j = \psi \tilde{\rho}_j^\top \tilde{\rho}_j , where \tilde{\rho}_j = \rho_j - (\frac{1}{N} \Sigma_{i=1}^{N} \rho_{j,i} ) \times 1_N is the cross-sectionally demeaned vector of factor j's correlations with asset returns. Instead, if intercept = FALSE, we choose \psi_j = \psi \rho_j^\top \rho_j . In the codes, \psi is equal to the value of psi0.

The prior \pi (\omega) encoded the belief about the sparsity of the true model using the prior distribution \pi (\gamma_j = 1 | \omega_j) = \omega_j . Following the literature on the variable selection, we set

\pi (\gamma_j = 1 | \omega_j) = \omega_j, \ \ \omega_j \sim Beta(a_\omega, b_\omega) .

Different hyperparameters a_\omega and b_\omega determine whether one a priori favors more parsimonious models or not. We choose a_\omega = 1 (aw) and b_\omega=1 (bw) as the default values.

For each posterior draw of factors' risk prices \lambda^{(j)}_f, we can define the SDF as m^{(j)}_t = 1 - (f_t - \mu_f)^\top \lambda^{(j)}_f.The Bayesian model averaging of the SDF (BMA-SDF) over J draws is

m^{bma}_t = \frac{1}{J} \sum^J_{j=1} m^{(j)}_t.

Value

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

• gamma_path: A sim_length\times k matrix of the posterior draws of \gamma. Each row represents a draw. If \gamma_j = 1 in one draw, factor j is included in the model in this draw and vice verse.

• lambda_path: A sim_length\times (k+1) matrix of the risk prices \lambda if intercept = TRUE. Each row represents a draw. Note that the first column is \lambda_c corresponding to the constant term. The next k columns (i.e., the 2-th – (k+1)-th columns) are the risk prices of the k factors. If intercept = FALSE, lambda_path is a sim_length\times k matrix of the risk prices, without the estimates of \lambda_c.

• sdf_path: A sim_length\times t matrix of posterior draws of SDFs. Each row represents a draw.

• bma_sdf: BMA-SDF.

References

\insertRef

bryzgalova2023bayesianBayesianFactorZoo

Examples

## Load the example data
data("BFactor_zoo_example")
HML <- BFactor_zoo_example$HML
lambda_ols <- BFactor_zoo_example$lambda_ols
R2.ols.true <- BFactor_zoo_example$R2.ols.true
sim_f <- BFactor_zoo_example$sim_f
sim_R <- BFactor_zoo_example$sim_R
uf <- BFactor_zoo_example$uf

## sim_f: simulated strong factor
## uf: simulated useless factor

psi_hat <- psi_to_priorSR(sim_R, cbind(sim_f,uf), priorSR=0.1)
shrinkage <- continuous_ss_sdf(cbind(sim_f,uf), sim_R, 5000, psi0=psi_hat, r=0.001, aw=1, bw=1)
cat("Null hypothesis: lambda =", 0, "for each factor", "\n")
cat("Posterior probabilities of rejecting the above null hypotheses are:",
    colMeans(shrinkage$gamma_path), "\n")

## We also have the posterior draws of SDF: m(t) = 1 - lambda_g %*% (f(t) - mu_f)
sdf_path <- shrinkage$sdf_path

## We also provide the Bayesian model averaging of the SDF (BMA-SDF)
bma_sdf <- shrinkage$bma_sdf

BayesianFactorZoo documentation built on Oct. 4, 2024, 5:11 p.m.