# Bayesian Fama-MacBeth This function provides the Bayesian Fama-MacBeth regression. ```r BayesianFM(f, R, sim_length) ``` ## 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 MCMCs; ## Returns The return of `BayesianFM` is a list of the following elements: * `lambda_ols_path`: A `sim_length`$\times (k+1)$ matrix of OLS risk premia estimates (Each row represents a draw. Note that the first column is $\lambda_c$ corresponding to the constant term. The next $k$ columns are the risk premia estimates of the $k$ factors); * `lambda_gls_path`: A `sim_length`$\times (k+1)$ matrix of the risk premia estimates $\lambda$ (GLS); * `R2_ols_path`: A `sim_length`$\times 1$ matrix of the $R^2_{OLS}$; * `R2_gls_path`: A `sim_length`$\times 1$ matrix of the $R^2_{GLS}$. ## Details `BayesianFM` is similar to another twin function in this package, `BayesianSDF`, except that we estimate factors' risk premia rather than risk prices in this function. Unlike `BayesianSDF`, we use factor loadings, $\beta_f$, instead of covariance exposures, $C_f$, in the Fama-MacBeth regression. In particular, after we obtain the posterior draws of $\mu_{Y}$ and $\Sigma_{Y}$ (details can be found in the section introducing `BayesianSDF` function), we calculate $\beta_f$ as follows: $\beta_f = C_f \Sigma_f^{-1}$, and $\beta = (1_N, \beta_f)$. Bayesian Fama-MacBeth (BFM) The posterior distribution of $\lambda$ conditional on $\mu_{Y}$, $\Sigma_{Y}$, and the data, is a Dirac distribution at $(\beta^\top \beta)^{-1} \beta^\top \mu_R$. Bayesian Fama-MacBeth GLS (BFM-GLS) The posterior distribution of $\lambda$ conditional on $\mu_{Y}$, $\Sigma_{Y}$, and the data, is a Dirac distribution at $ (\beta^\top \Sigma_R^{-1} \beta)^{-1} \beta^\top \Sigma_R^{-1} \mu_R $. ## Examples ```r ## <--------------------------------------------------------------------------------> ## Example: Bayesian Fama-MacBeth ## <--------------------------------------------------------------------------------> library(reshape2) library(ggplot2) # Load Data data("BFactor_zoo_example") HML <- BFactor_zoo_example$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$uf ## <-------------------Case 1: strong factor----------------------------------------> # the Frequentist Fama-MacBeth # sim_f: simulated factor, sim_R: simulated return # sim_f is the useful (i.e., strong) factor results.fm <- Two_Pass_Regression(sim_f, sim_R) # the Bayesian Fama-MacBeth with 10000 simulations results.bfm <- BayesianFM(sim_f, sim_R, 2000) # Note that the first element correspond to lambda of the constant term # So we choose k=2 to get lambda of the strong factor k <- 2 m1 <- results.fm$lambda[k] sd1 <- sqrt(results.fm$cov_lambda[k,k]) bfm<-results.bfm$lambda_ols_path[1001:2000,k] fm<-rnorm(20000,mean = m1, sd=sd1) data<-data.frame(cbind(fm, bfm)) colnames(data)<-c("Frequentist FM", "Bayesian FM") data.long<-melt(data) 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 premium ("~lambda[strong]~")")) + ylab("Density" ) ## <-------------------Case 2: useless factor---------------------------------------> # uf is the useless factor # the Frequentist Fama-MacBeth results.fm <- Two_Pass_Regression(uf, sim_R) # the Bayesian Fama-MacBeth with 10000 simulations results.bfm <- BayesianFM(uf, sim_R, 2000) # Note that the first element correspond to lambda of the constant term # So we choose k=2 to get lambda of the useless factor k <- 2 m1 <- results.fm$lambda[k] sd1 <- sqrt(results.fm$cov_lambda[k,k]) bfm<-results.bfm$lambda_ols_path[1001:2000,k] fm<-rnorm(20000,mean = m1, sd=sd1) data<-data.frame(cbind(fm, bfm)) colnames(data)<-c("Frequentist FM", "Bayesian FM") data.long<-melt(data) 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 premium ("~lambda[strong]~")")) + ylab("Density" ) ```

BayesianFM function