# SDF model selection with continuous spike-and-slab prior (tradable factors are treated as test assets) This function provides the SDF model selection procedure using the continuous spike-and-slab prior. See Propositions 3 and 4 in \insertCite bryzgalova2023bayesian;textualBayesianFactorZoo. Unlike `continuous_ss_sdf`, tradable factors are treated as test assets in this function. ```r continuous_ss_sdf_v2( f1, f2, R, sim_length, psi0 = 1, r = 0.001, aw = 1, bw = 1, type = "OLS", intercept = TRUE ) ``` ## 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'. - `intercept`: If `intercept = TRUE` (`intercept = FALSE`), we include (exclude) the common intercept in the cross-sectional regression. The default is `intercept = TRUE`. ## Returns 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 $\lambda$ 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. ## Details See the description in the twin function `continuous_ss_sdf`. ## Examples ```r 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") ``` ## References \insertRef bryzgalova2023bayesianBayesianFactorZoo

continuous_ss_sdf_v2 function