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<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">BMR: BVAR with Steady-State Prior</strong></h4></a>
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<a href="#definition">Fields and Methods</a> <br>
<a href="#details">Details</a> <br>
<a href="#examples">Examples</a>
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<p>BVAR with steady-state prior.</p>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="definition"><strong style="font-size: 100%;">Fields and Methods</strong></h3>
<br>
<p><strong>Instantiation:</strong></p>
<pre class="brush: R;">
# create a new object
bvar_obj <- new(bvars)
</pre>
<p><strong>Fields:</strong></p>
<ul>
<li><code>cons_term</code> a logical value (TRUE/FALSE) indicating the presence of a constant term (intercept) in the model.</li>
<li><code>p</code> lag order (integer).</li>
<li><code>c_int</code> integer version of <code>cons_term</code>.</li>
<br>
<li><code>n</code> sample length.</li>
<li><code>M</code> number of variables.</li>
<li><code>n_ext_vars</code> number of exogenous variables.</li>
<li><code>K</code> number of right-hand-side terms in each equation.</li>
<br>
<li><code>Y</code> a $(n-p) \times M$ data matrix.</li>
<li><code>X</code> a $(n-p) \times K$ data matrix.</li>
<br>
<li><code>psi_hat</code> OLS estimate of $\psi = \text{vec}(\Psi)$.</li>
<li><code>alpha_hat</code> OLS estimate of $\alpha = \text{vec}(\beta)$.</li>
<li><code>Sigma_hat</code> OLS estimate of $\Sigma$.</li>
<br>
<li><code>psi_pr_mean</code> prior mean of $\psi = \text{vec}(\Psi)$.</li>
<li><code>psi_pr_var</code> prior covariance matrix of $\psi = \text{vec}(\Psi)$.</li>
<li><code>alpha_pr_mean</code> prior mean of $\alpha = \text{vec}(\beta)$.</li>
<li><code>alpha_pr_var</code> prior covariance matrix of $\alpha = \text{vec}(\beta)$.</li>
<li><code>Sigma_pr_scale</code> prior scale matrix of $\Sigma$.</li>
<li><code>Sigma_pr_dof</code> prior degrees of freedom of $\Sigma$.</li>
<br>
<li><code>psi_pt_mean</code> posterior mean of $\psi = \text{vec}(\Psi)$.</li>
<li><code>psi_pt_var</code> posterior covariance matrix of $\psi = \text{vec}(\Psi)$.</li>
<li><code>alpha_pt_mean</code> posterior mean of $\alpha = \text{vec}(\beta)$.</li>
<li><code>alpha_pt_var</code> posterior covariance matrix of $\alpha = \text{vec}(\beta)$.</li>
<li><code>Sigma_pt_mean</code> posterior mean of $\Sigma$.</li>
<li><code>Sigma_pt_dof</code> posterior degrees of freedom of $\Sigma$.</li>
<br>
<li><code>Psi_draws</code> an array of posterior draws for $\Psi$.</li>
<li><code>beta_draws</code> an array of posterior draws for $\beta$.</li>
<li><code>Sigma_draws</code> an array of posterior draws for $\Sigma$.</li>
</ul>
<hr>
<p><strong>Build:</strong> two methods,</p>
<pre class="brush: R;">
bvar_obj$build(data_endog,cons_term,p)
bvar_obj$build(data_endog,data_exog,cons_term,p)
</pre>
<ul>
<li><code>data_endog</code> a $n \times m$ matrix of endogenous varibles.</li>
<li><code>data_exog</code> a $n \times q$ matrix of exogenous varibles.</li>
<li><code>cons_term</code> a logical value (TRUE/FALSE) indicating the presence of a constant term (intercept) in the model.</li>
<li><code>p</code> lag order (integer).</li>
</ul>
<p><strong>Prior:</strong></p>
<pre class="brush: R;">
bvar_obj$prior(coef_prior,HP_1,HP_2,Psi_prior,Xi_psi,gamma,full_cov_prior)
</pre>
<ul>
<li><code>coef_prior</code> a $m \times 1$ vector containing the prior mean for each first own-lag coefficient.</li>
<li><code>HP_1,HP_2</code> hyperparameters.</li>
<li><code>Psi_prior</code> prior mean of $\Psi$.</li>
<li><code>Xi_psi</code> scaling value for the prior covariance matrix of $\psi = \text{vec}(\Psi)$.</li>
<li><code>gamma</code> prior degrees of freedom for $\Sigma$.</li>
<li><code>full_cov_prior</code> a logical value (TRUE/FALSE) to switch between using the full MLE covariance matrix as the diffuse prior for Sigma (TRUE) or the classical Minnesota prior (FALSE).</li>
</ul>
<p><strong>Gibbs sampling:</strong></p>
<pre class="brush: R;">
bvar_obj$gibbs(n_draws,n_burnin)
</pre>
<ul>
<li><code>n_draws</code> number of posterior draws.</li>
<li><code>n_burnin</code> number of burnin draws.</li>
</ul>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="details"><strong style="font-size: 100%;">Details</strong></h3>
<br>
The model:
$$
( Y_t - d_t \Psi^\top) \beta(L) = \varepsilon_t,
$$
<p>The prior on $\beta$ is constructed as
\begin{equation*}
\Xi_{\beta_{i,j}} (\ell) =
\begin{cases}
&H_1 / \ell^2 \\
&H_2 \cdot \sigma_{i}^2 / \left( \ell^2 \cdot \sigma_{j}^2 \right)
\end{cases}
\end{equation*}
for own lags and cross-variable lags, respectively.</p>
<p>Let
\begin{align*}
\xi &= Y - X \beta \\
Y_d &= Y - d \Psi^\top \\
X_d &= X - d_X ( \mathbb{I}_p \otimes \Psi^\top ) \\
\varepsilon &= Y_d - X_d \beta
\end{align*}
The conditional posterior kernels are given by:
\begin{align}
p(\psi | \beta,\Sigma,X,Y) &= \mathcal{N} \left( \widetilde{\psi}, \widetilde{\Sigma}_{\psi} \right) \\
p(\alpha | \Psi,\Sigma,X,Y) &= \mathcal{N} \left( \widetilde{\alpha}, \widetilde{\Sigma}_{\alpha} \right) \\
p(\Sigma | \Psi,\beta,X,Y) &= \mathcal{IW} \left( \Xi_{\Sigma} + \varepsilon^{\top} \varepsilon, n - p + \gamma \right)
\end{align}
where
\begin{align*}
\widetilde{\Sigma}_{\psi}^{-1} &= \Xi_{\psi}^{-1} + U^\top(D^{\top} D \otimes \Sigma^{-1}) U \\
\widetilde{\psi} &= \widetilde{\Sigma}_{\psi} \left( \Xi_{\alpha}^{-1} \bar{\alpha} + \text{vec}(\Sigma^{-1} \xi^{\top} D) \right) \\
\widetilde{\Sigma}_{\alpha}^{-1} &= \Xi_{\alpha}^{-1} + (\Sigma^{-1} \otimes X_d^{\top} X_d) \\
\widetilde{\alpha} &= \widetilde{\Sigma}_{\alpha} \left( \Xi_{\alpha}^{-1} \bar{\alpha} + \text{vec}(X_d^{\top} Y_d \Sigma^{-1}) \right)
\end{align*}
and $U^\top = (\mathbb{I}_{Mp}, \mathbb{I}_{q} \otimes \beta_1^\top, \ldots, \mathbb{I}_{q} \otimes \beta_p^\top)$.</p>
</p>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="examples"><strong style="font-size: 100%;">Example</strong></h3>
<br>
<pre class="brush: ruby;">
rm(list=ls())
library(BMR)
#
data(BMRVARData)
bvar_data <- data.matrix(USMacroData[,2:4])
#
coef_prior <- c(0.9,0.9,0.9)
psi_prior <- matrix(c(3,6,5),nrow=1)
HP1 <- 0.5
HP2 <- 0.5
XiPsi <- 1
gamma <- 4
bvar_obj <- new(bvars)
#
bvar_obj$build(bvar_data,TRUE,1)
bvar_obj$prior(coef_prior,HP1,HP2,psi_prior,XiPsi,gamma,FALSE)
bvar_obj$gibbs(10000,5000)
IRF(bvar_obj,20,var_names=colnames(bvar_data),save=FALSE)
plot(bvar_obj,var_names=colnames(bvar_data),save=FALSE)
forecast(bvar_obj,shocks=TRUE,var_names=colnames(bvar_data),back_data=10,save=FALSE)
</pre>
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