# Hypothesis testing for risk prices (Bayesian p-values) with Dirac spike-and-slab prior This function tests the null hypothesis, $H_0: \lambda = \lambda_0$, when $\gamma=0$. When $\lambda_0 = 0$, we compare factor models using the algorithm in Proposition 1 of \insertCite bryzgalova2023bayesian;textualBayesianFactorZoo. When $\lambda_0 \neq 0$, this function corresponds to Corollary 2 in Section II.A.2 of \insertCite bryzgalova2023bayesian;textualBayesianFactorZoo. The function can also be used to compute the posterior probabilities of all possible models with up to a given maximum number of factors (see examples). ```r dirac_ss_sdf_pvalue(f, R, sim_length, lambda0, psi0 = 1, max_k = NULL) ``` ## 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 Monte-Carlo simulations; - `lambda0`: A $k \times 1$ vector of risk prices under the null hypothesis ($\gamma=0$); - `psi0`: The hyper-parameter in the prior distribution of risk price $\lambda$ (see Details ); - `max_k`: The maximal number of factors in models (`max_k` is a positive integer or `NULL` if the user does not impose any restriction on the model dimension). ## Returns The return of `dirac_ss_sdf_pvalue` 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; * `model_probs`: A $2^k \times (k+1)$ matrix of posterior model probabilities, where the first k columns are the model indices and the final column is a vector of model probabilities. ## Details Let $D$ denote a diagonal matrix with elements $c, \psi_1^{-1},..., \psi_K^{-1}$, and $D_\gamma$ the submatrix of $D$ corresponding to model $\gamma$, where $c$ is a small positive number corresponding to the common cross-sectional intercept ($\lambda_c$). The prior for the prices of risk ($\lambda_\gamma$) of model $\gamma$ is then $$ \lambda_\gamma | \sigma^2, \gamma \sim N (0, \sigma^2, D_{\gamma}^{-1}). $$ 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`. ## Examples ```r ## <--------------------------------------------------------------------------------> ## Example: Bayesian p-value (with the dirac spike-and-slab prior) ## <--------------------------------------------------------------------------------> # 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 ### Now we estimate the Bayesian p-values defined in Corollary 2. # ### Prior Sharpe ratio of factor model for different values of psi: see equation (27): # cat("--------------- Choose psi based on prior Sharpe ratio ----------------\n") cat("if psi = 1, prior Sharpe ratio is", psi_to_priorSR(sim_R, sim_f, psi0=1), "\n") cat("if psi = 2, prior Sharpe ratio is", psi_to_priorSR(sim_R, sim_f, psi0=2), "\n") cat("if psi = 5, prior Sharpe ratio is", psi_to_priorSR(sim_R, sim_f, psi0=5), "\n") ## Test whether factors' risk prices equal 'matrix(lambda_ols[2]*sd(HML),ncol=1)' ## Bayesian p-value is given by mean(shrinkage$gamma_path) shrinkage <- dirac_ss_sdf_pvalue(sim_f, sim_R, 1000, matrix(lambda_ols[2]*sd(HML),ncol=1)) cat("Null hypothesis: lambda =", matrix(lambda_ols[2]*sd(HML)), "\n") cat("Posterior probability of rejecting the above null hypothesis is:", mean(shrinkage$gamma_path), "\n") ## Test whether the risk price of factor 'sim_f' is equal to 0 shrinkage <- dirac_ss_sdf_pvalue(sim_f, sim_R, 1000, 0, psi0=1) cat("Null hypothesis: lambda =", 0, "\n") cat("Posterior probability of rejecting the above null hypothesis is:", mean(shrinkage$gamma_path), "\n") ## One can also put more than one factor into the test two_f = cbind(sim_f,uf) # sim_f is the strong factor while uf is the useless factor # Test1: lambda of sim_f = 0, Test2: lambda of uf = 0 lambda0_null_vec = t(cbind(0,0)) # 2x1 vector shrinkage <- dirac_ss_sdf_pvalue(two_f, sim_R, 1000, lambda0_null_vec, psi0=1) cat("Null hypothesis: lambda =", 0, "for each factor", "\n") cat("Posterior probabilities of rejecting the above null hypothesis are:", colMeans(shrinkage$gamma_path), "\n") ## We can also print the posterior model probabilities: cat('Posterior model probabilities are:\n') print(shrinkage$model_probs) ## One can compute the posterior probabilities of all possible models with up to ## a given maximum number of factors. For example, we consider two factors, but ## the number of factors is restricted to be less than two. lambda0_null_vec = t(cbind(0,0)) # 2x1 vector shrinkage <- dirac_ss_sdf_pvalue(two_f, sim_R, 1000, lambda0_null_vec, psi0=1, max_k=1) cat('Posterior model probabilities are:\n') print(shrinkage$model_probs) ## Comment: You may notice that the model with index (1, 1) has a posterior probability ## of exactly zero since the maximal number of factors is one. ``` ## References \insertRef bryzgalova2023bayesianBayesianFactorZoo

dirac_ss_sdf_pvalue function