# Bayesian estimation of Linear SDF (B-SDF) This function provides the Bayesian estimates of factors' risk prices. The estimates with the flat prior are given by Definitions 1 and 2 in \insertCite bryzgalova2023bayesian;textualBayesianFactorZoo. The estimates with the normal prior are used in Table I (see the footnote of Table I). ```r BayesianSDF( f, R, sim_length = 10000, intercept = TRUE, type = "OLS", prior = "Flat", psi0 = 5, d = 0.5 ) ``` ## Arguments - `f`: A $t \times k$ matrix of factors, where $k$ is the number of factors and $t$ is the number of periods - `R`: A $t \times N$ matrix of test assets, where $t$ is the number of periods and $N$ is the number of test assets - `sim_length`: The length of MCMCs - `intercept`: If `intercept = TRUE` (`intercept = FALSE`), the model includes (does not include) the intercept. The default is `intercept = TRUE` - `type`: If `type = 'OLS'` (`type = 'GLS'`), the function returns Bayesian OLS (GLS) estimates of risk prices $\lambda$. The default is 'OLS' - `prior`: If `type = 'Flat'` (`type = 'Normal'`), the function executes the Bayesian estimation with the flat prior (normal prior). The default is 'Flat' - `psi0`: The hyper-parameter of the prior distribution of risk prices $\lambda$ used in the normal prior (see Details ). This parameter is needed only when the user chooses the normal prior. The default value is 5 - `d`: The hyper-parameter of the prior distribution of risk prices $\lambda$ used in the normal prior (see Details ). The default value is 0.5 ## Returns The return of `BayesianSDF` is a list that contains the following elements: * `lambda_path`: A `sim_length`$\times (k+1)$ matrix if the intercept is included. NOTE: the first column $\lambda_c$ corresponds to the intercept. The next $k$ columns (i.e., the 2th -- $(k+1)$-th columns) are the risk prices of $k$ factors. If the intercept is excluded, the dimension of `lambda_path` is `sim_length`$\times k$. * `R2_path`: A `sim_length`$\times 1$ matrix, which contains the posterior draws of the OLS or GLS $R^2$. ## Details Intercept Consider the cross-sectional step. If one includes the intercept, the model is $$ \mu_R = \lambda_c 1_N + C_f \lambda_f = C \lambda, $$ where $C = (1_N, C_f)$ and $\lambda^\top = (\lambda_c^\top, \lambda_f^\top)^\top $. If one doesn't include the intercept, the model is $$ \mu_R = C_f \lambda_f = C \lambda, $$ where $C = C_f$ and $\lambda = \lambda_f$. Bayesian Estimation Let $Y_t = f_t \cup R_t$. Conditional on the data $Y = \{Y_t\}_{t=1}^T$, we can draw $\mu_{Y}$ and $\Sigma_{Y}$ from the Normal-inverse-Wishart system $$ \mu_Y | \Sigma_Y, Y \sim N (\hat{\mu}_Y , \Sigma_Y / T) , $$ $$ \Sigma_Y | Y \sim W^{-1} (T-1, \Sigma_{t=1}^{T} (Y_t - \hat{\mu}_Y ) ( Y_t - \hat{\mu}_Y )^\top ) , $$ where $W^{-1}$ is the inverse-Wishart distribution. We do not standardize $Y_t$ in the time-series regression. In the empirical implementation, after obtaining posterior draws for $\mu_{Y}$ and $\Sigma_{Y}$, we calculate $\mu_R$ and $C_f$ as the standardized expected returns of test assets and correlation between test assets and factors. It follows that $C$ is a matrix containing a vector of ones and $C_f$. The prior distribution of risk prices is either the flat prior or the normal prior. With `prior = 'Flat'` and `type = 'OLS'`, for each draw, the risk price estimate is $$ \hat{\lambda} = (C^{\top} C)^{-1}C^{T} \mu_{R} . $$ With `prior = 'Flat'` and `type = 'GLS'`, for each draw, the risk price estimate is $$ \hat{\lambda} = (C^{\top} \Sigma^{-1}_{R} C)^{-1} C^{\top} \Sigma^{-1}_{R} \mu_{R} $$ If one chooses `prior = 'Normal'`, the prior of factor $j$'s risk price is $$ \lambda_j | \sigma^2 \sim N(0, \sigma^2 \psi \tilde{\rho}_j^\top \tilde{\rho}_j T^d ) , $$ 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. Equivalently, $$ \lambda | \sigma^2 \sim N(0, \sigma^2 D^{-1}) , $$ $$ D = diag \{ (\psi \tilde{\rho}_1^\top \tilde{\rho}_1 T^d)^{-1}, ..., (\psi \tilde{\rho}_k^\top \tilde{\rho}_k T^d)^{-1} \} \ \ without \ intercept; $$ $$ D = diag \{ c, (\psi \tilde{\rho}_1^\top \tilde{\rho}_1 T^d)^{-1}, ..., (\psi \tilde{\rho}_k^\top \tilde{\rho}_k T^d)^{-1} \} \ \ with \ intercept; $$ where $c$ is a small positive number corresponding to the common cross-sectional intercept ($\lambda_c$). Default values for $\psi$ (`psi0`) and $d$ (`d`) are 5 and 0.5, respectively. With `prior = 'Normal'` and `type = 'OLS'`, for each draw, the risk price estimate is $$ \hat{\lambda} = ( C^{\top} C +D )^{-1} C^{\top} \mu_R . $$ With `prior = 'Normal'` and `type = 'GLS'`, for each draw, the risk price estimate is $$ \hat{\lambda} = ( C^{\top} \Sigma_R^{-1} C +D )^{-1} C^{\top} \Sigma_R^{-1} \mu_R . $$ ## Examples ```r ## <--------------------------------------------------------------------------------> ## Example: Bayesian estimates of risk prices and R2 ## This example is from the paper (see Section III. Simulation) ## <--------------------------------------------------------------------------------> library(reshape2) library(ggplot2) # 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 W_ols <- BFactor_zoo_example$W_ols cat("Load the simulated example \n") cat("Cross-section: Fama-French 25 size and value portfolios \n") cat("True pricing factor in simulations: HML \n") cat("Pseudo-true cross-sectional R-squared:", R2.ols.true, "\n") cat("Pseudo-true (monthly) risk price:", lambda_ols[2], "\n") cat("----------------------------- Bayesian SDF ----------------------------\n") cat("------------------------ See definitions 1 and 2 ----------------------\n") cat("--------------------- Bayesian SDF: Strong factor ---------------------\n") sim_result <- SDF_gmm(sim_R, sim_f, W_ols) # GMM estimation # sim_result$lambda_gmm # sqrt(sim_result$Avar_hat[2,2]) # sim_result$R2_adj ## Now estimate the model using Bayesian method two_step <- BayesianSDF(sim_f, sim_R, sim_length = 2000, psi0 = 5, d = 0.5) # apply(X = two_step$lambda_path, FUN = quantile, MARGIN = 2, probs = c(0.05, 0.95)) # quantile(two_step$R2_path, probs = c(0.05, 0.5, 0.95)) # Note that the first element correspond to lambda of the constant term # So we choose k=2 to get lambda of the strong factor k <- 2 m1 <- sim_result$lambda_gmm[k] sd1 <- sqrt(sim_result$Avar_hat[k,k]) bfm<-two_step$lambda_path[1001:2000, k] fm<-rnorm(5000,mean = m1, sd=sd1) data<-data.frame(cbind(fm, bfm)) colnames(data)<-c("GMM-OLS", "BSDF-OLS") data.long<-melt(data) # ### Figure 1(c) # p <- ggplot(aes(x=value, colour=variable, linetype=variable), data=data.long) p+ stat_density(aes(x=value, colour=variable), geom="line",position="identity", size = 2, adjust=1) + geom_vline(xintercept = lambda_ols[2], linetype="dotted", color = "#8c8c8c", size=1.5)+ guides(colour = guide_legend(override.aes=list(size=2), title.position = "top", title.hjust = 0.5, nrow=1,byrow=TRUE))+ theme_bw()+ labs(color=element_blank()) + labs(linetype=element_blank()) + theme(legend.key.width=unit(4,"line")) + theme(legend.position="bottom")+ theme(text = element_text(size = 26))+ xlab(bquote("Risk price ("~lambda[strong]~")")) + ylab("Density" ) cat("--------------------- Bayesian SDF: Useless factor --------------------\n") sim_result <- SDF_gmm(sim_R, uf, W_ols) # sim_result$lambda_gmm # sqrt(sim_result$Avar_hat[2,2]) # sim_result$R2_adj two_step <- BayesianSDF(uf, sim_R, sim_length = 2000, psi0 = 5, d = 0.5) #apply(X = two_step$lambda_path, FUN = quantile, MARGIN = 2, probs = c(0.05, 0.95)) ## Posterior (Asymptotic) Distribution of lambda k <- 2 m1 <- sim_result$lambda[k] sd1 <- sqrt(sim_result$Avar_hat[k,k]) bfm<-two_step$lambda_path[1001:2000, k] fm<-rnorm(5000,mean = m1, sd=sd1) data<-data.frame(cbind(fm, bfm)) colnames(data)<-c("GMM-OLS", "BSDF-OLS") data.long<-melt(data) # ### Figure 1(a) # p <- ggplot(aes(x=value, colour=variable, linetype=variable), data=data.long) p+ stat_density(aes(x=value, colour=variable), geom="line",position="identity", size = 2, adjust=2) + geom_vline(xintercept = 0, linetype="dotted", color = "#8c8c8c", size=1.5)+ guides(colour = guide_legend(override.aes=list(size=2), title.position = "top", title.hjust = 0.5, nrow=1,byrow=TRUE))+ theme_bw()+ labs(color=element_blank()) + labs(linetype=element_blank()) + theme(legend.key.width=unit(4,"line")) + theme(legend.position="bottom")+ theme(text = element_text(size = 26))+ xlab(bquote("Risk price ("~lambda[spurious]~")")) + ylab("Density" ) ``` ## References \insertRef bryzgalova2023bayesianBayesianFactorZoo

BayesianSDF function