continuous_ss_sdf {BayesianFactorZoo} | R Documentation |
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).
continuous_ss_sdf(
f,
R,
sim_length,
psi0 = 1,
r = 0.001,
aw = 1,
bw = 1,
type = "OLS"
)
f |
A matrix of factors with dimension |
R |
A matrix of test assets with dimension |
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 |
bw |
The hyper-parameter related to the prior of |
type |
If |
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).
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.
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.
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
. 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.
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.
## 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