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<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">BMR: BVAR with Conditional Normal-inverse-Wishart 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 conditional Normal-inverse-Wishart 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(bvarcnw)
</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>alpha_hat</code> OLS estimate of $\alpha = \text{vec}(\beta)$.</li>
<li><code>Sigma_hat</code> OLS estimate of $\Sigma$.</li>
<br>
<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>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>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> two methods,</p>
<pre class="brush: R;">
bvar_obj$prior(coef_prior,HP_1,HP_3,gamma)
bvar_obj$prior(coef_prior,HP_1,HP_3,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>gamma</code> prior degrees of freedom for $\Sigma$.</li>
<li><code>HP_1,HP_3</code> hyperparameters; see below for more details.</li>
<li><code>full_cov_prior</code> a logical value (TRUE/FALSE) to switch between diagonal or full prior error covariance matrix construction.</li>
</ul>
<p><strong>Gibbs sampling:</strong></p>
<pre class="brush: R;">
bvar_obj$gibbs(n_draws)
</pre>
<ul>
<li><code>n_draws</code> number of posterior 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>
<p>The model:</p>
$$
y_t = \beta_0 + y_{t-1} \beta_1 + \cdots y_{t-p} \beta_p + e_t, \ \ e_t \stackrel{\text{iid}}{\sim} \mathcal{N} \left( \mathbf{0}, \Sigma \right)
$$
<p>where $y_t$ is a $1 \times M$ vector. We write this in stacked form as:</p>
$$
Y = X \beta + e
$$
<p>Priors take the following form:</p>
\begin{align}
p(\Sigma^{-1}) &= \mathcal{W} \left( \frac{1}{\gamma}\Xi_{\Sigma}^{-1} , \gamma \right) \\
p(\alpha | \Sigma) &= \mathcal{N} \left( \text{vec}(\bar{\beta}), \Sigma \otimes \Xi_{\beta} \right)
\end{align}
<p>Notes on priors and hyperparameters:</p>
<ul>
<li>$\gamma \geq M + 1$ is set by the <code>gamma</code> input in the prior function;</li>
<li>the prior mean of the coefficients, $\bar{\beta}$, is a $K \times M$ matrix of zeros, except for the first-order own-lag terms which are set by <code>coef_prior</code> the prior function.</li>
<li>The location matrix $\Xi_{\Sigma}$ is formed in an identical way to a Minnesota prior: as the residual variance from equation-by-equation estimation of AR($p$) models,</li>
$$
\Xi_{\Sigma} = \begin{bmatrix} \sigma_{1}^2 &0 & 0 & \cdots & 0 \\ 0& \sigma_2^2 & 0 & \cdots & 0 \\ 0& 0 & \sigma_3^2 & \ddots & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0& 0 & 0 & 0 & \sigma_M^2 \end{bmatrix}
$$
<li>$\Xi_\beta$ is a diagonal matrix, with elements given in the following order: the first element relates to a constant term (intercept); then $p$-number of $M \times M$ diagonal blocks, one for each lag; and finally a block for any additional exogenous variables (other than a constant term).
<br>The scaling of these variances is controlled by the hyperparameters <code>HP_1,HP_3</code>, with elements of each block given by:</li>
\begin{equation*}
\Xi_{\beta_{i,i}} (\ell) =
\begin{cases}
& H_1 / \left( \ell^2 \cdot \sigma_{i}^2 \right) \\
& H_1 \cdot H_3
\end{cases}
\end{equation*}
for lags and exogenous variables, respectively.</li>
</ul>
<p>The conditional posterior kernels are given by</p>
\begin{align}
p(\Sigma^{-1} | X,Y) &= \mathcal{W} \left( \gamma \Xi_{\Sigma} + \bar{\beta}^\top \Xi_{\beta}^{-1} \bar{\beta} + Y^\top Y - \widetilde{\beta}^\top \widetilde{\Sigma}_{\beta}^{-1} \widetilde{\beta}, n - p + \gamma \right) \\
p(\alpha | \Sigma,X,Y) &= \mathcal{N} \left( \text{vec}(\widetilde{\beta}), \Sigma \otimes \widetilde{\Sigma}_{\beta} \right)
\end{align}
<p>where</p>
\begin{align*}
\widetilde{\Sigma}_{\beta}^{-1} &= \Xi_{\beta}^{-1} + X^{\top} X \\
\widetilde{\beta} &= \widetilde{\Sigma}_{\beta} \left( \Xi_{\beta}^{-1} \bar{\beta} + X^\top Y \right)
\end{align*}
<p><strong>Computational notes:</strong> $\widetilde{\beta}$ and $\widetilde{\Sigma}_{\beta}$ are fixed over posterior draws, and the posterior distribution for $\Sigma$ does not depend on a $\beta$ draw. Thus the sampling steps can be performed in parallel:</p>
<ul>
<li>first, draw many values from the distribution $p(\Sigma^{-1} | X,Y)$; then</li>
<li>for each $\Sigma^{(i)}$ draw, sample from $p(\alpha | \Sigma^{(i)},X,Y)$.</li>
</ul>
<hr style="height:2px;border-width:0;background-color:black">
<h3 style="text-align: left;" id="examples"><strong style="font-size: 100%;">Examples</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)
HP_1 <- 1/2
HP_3 <- 1
gamma <- 4
bvar_obj <- new(bvarcnw)
#
bvar_obj$build(bvar_data,TRUE,4)
bvar_obj$prior(coef_prior,HP_1,HP_3,gamma)
bvar_obj$gibbs(10000)
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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