continuous_ss_sdf_v2 {BayesianFactorZoo}

SDF model selection with continuous spike-and-slab prior (tradable factors are treated as test assets)

Description

This function provides the SDF model selection procedure using the continuous spike-and-slab prior. See Propositions 3 and 4 in Bryzgalova et al. (2023). Unlike continuous_ss_sdf, tradable factors are treated as test assets in this function.

Usage

continuous_ss_sdf_v2(
  f1,
  f2,
  R,
  sim_length,
  psi0 = 1,
  r = 0.001,
  aw = 1,
  bw = 1,
  type = "OLS"
)

Arguments

f1

A matrix of nontradable factors with dimension t \times k_1, where k_1 is the number of nontradable factors and t is the number of periods.

f2

A matrix of tradable factors with dimension t \times k_2, where k_2 is the number of tradable 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 (R should NOT contain tradable factors f2);

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'.

Details

See the description in the twin function continuous_ss_sdf.

Value

The return of continuous_ss_sdf_v2 is a list of the following elements:

• gamma_path: A sim_length\times k matrix of the posterior draws of \gamma (k = k_1 + k_2). 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. 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.

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

• bma_sdf: BMA-SDF.

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.

Examples

library(timeSeries)

## 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,sim_R[,1]), priorSR=0.1)

## We include the first test asset, sim_R[,1], into factors, so f2 = sim_R[,1,drop=FALSE].
## Also remember excluding sim_R[,1,drop=FALSE] from test assets, so R = sim_R[,-1].
shrinkage <- continuous_ss_sdf_v2(cbind(sim_f,uf), sim_R[,1,drop=FALSE], sim_R[,-1], 1000,
                                  psi0=psi_hat, r=0.001, aw=1, bw=1)
cat("Null hypothesis: lambda =", 0, "for each of these three factors", "\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

## We can further estimate the posterior distributions of model-implied Sharpe ratios:
cat("The 5th, 50th, and 95th quantiles of model-implied Sharpe ratios:",
    quantile(colSds(t(sdf_path)), probs=c(0.05, 0.5, 0.95)), "\n")

## Finally, we can estimate the posterior distribution of model dimensions:
cat("The posterior distribution of model dimensions (= 0, 1, 2, 3):",
    prop.table(table(rowSums(shrinkage$gamma_path))), "\n")

## We now use the 17th test asset, sim_R[,17,drop=FALSE], as the tradable factor,
## so f2 = sim_R[,17,drop=FALSE].
## Also remember excluding sim_R[,17,drop=FALSE] from test assets, so R = sim_R[,-17].
psi_hat <- psi_to_priorSR(sim_R, cbind(sim_f,uf,sim_R[,17]), priorSR=0.1)
shrinkage <- continuous_ss_sdf_v2(cbind(sim_f,uf), sim_R[,17,drop=FALSE], sim_R[,-17],
                                  1000, psi0=psi_hat, r=0.001, aw=1, bw=1)
cat("Null hypothesis: lambda =", 0, "for each of these three factors", "\n")
cat("Posterior probabilities of rejecting the above null hypotheses are:",
    colMeans(shrinkage$gamma_path), "\n")