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<meta name="author" content="Keith O'Hara">
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<title>BMR: cvar</title>
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<a data-toggle="collapse" href="#collapse1"><h4><strong style="font-size: 120%;">BMR: Classical VAR</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>Classical (frequentist) VAR model.</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
var_obj <- new(cvar)
</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>beta_hat</code> OLS estimate of $\beta$.</li>
<li><code>Sigma_hat</code> OLS estimate of $\Sigma$.</li>
<br>
<li><code>beta_draws</code> an array of bootstrap draws for $\beta$.</li>
<li><code>Sigma_draws</code> an array of bootstrap draws for $\Sigma$.</li>
</ul>
<hr>
<p><strong>Build:</strong> two methods,</p>
<pre class="brush: R;">
var_obj$build(data_endog,cons_term,p)
var_obj$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>OLS estimation:</strong></p>
<pre class="brush: R;">
var_obj$estim()
</pre>
<p><strong>Bootstrap:</strong></p>
<pre class="brush: R;">
var_obj$boot(n_draws)
</pre>
<ul>
<li><code>n_draws</code> number of bootstrap 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>
For the VAR model,
\begin{equation*}
Y = X \beta + \varepsilon
\end{equation*}
the bootstrapping algorithm is as follows. First, obtain the OLS estimates of the coefficients, $\widehat{\beta}$, and the estimated disturbance terms $\widehat{\varepsilon} = y - X\widehat{\beta}$.
Then, sample with replacement from $\widehat{\varepsilon}$, <code>n_draws</code> number of times, and build a new series of $Y$ by the following method.
<br> <br>
<ol>
<li> Let $h \in \{1, \ldots, $ <code>n_draws</code>$\}$, $t \in \{ 1, \ldots, T \}$, $X = [\iota_T \ \ Y_{t-1} \ \ Y_{t-2} \ \ \cdots \ \ Y_{t-p}]$, and fix $Y_0, Y_{-1}, \ldots, Y_{-p+1}$. </li>
<li> Using $\widehat{\beta}$, calculate
\begin{equation*}
Y_1^{(h)} = \widehat{\Phi} + Y_0 \widehat{\beta}_1 + Y_{-1} \widehat{\beta}_2 + \cdots + Y_{-p+1} \widehat{\beta}_p + \widehat{\varepsilon}_1^{(h)}
\end{equation*}</li>
<li> Using $Y_1^{(h)}$,
\begin{equation*}
Y_2^{(h)} = \widehat{\Phi} + Y_1^{(h)} \widehat{\beta}_1 + Y_{0} \widehat{\beta}_2 + \cdots + Y_{-p+2} \widehat{\beta}_p + \widehat{\varepsilon}_2^{(h)}
\end{equation*}</li>
<li> Continuing in this fashion:
\begin{align*}
Y_3^{(h)} &= \widehat{\Phi} + Y_2^{(h)} \widehat{\beta}_1 + Y_1^{(h)} \widehat{\beta}_2 + \cdots + Y_{-p+3} \widehat{\beta}_p + \widehat{\varepsilon}_3^{(h)} \\
\vdots \ \ \ \ &= \ \vdots \ \ \ \ \ \ \ \ \ \vdots \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots \ \ \ \ \ \ \ \ \ \ \ \vdots \\
Y_T^{(h)} &= \widehat{\Phi} + Y_{T-1}^{(h)} \widehat{\beta}_1 + Y_{T-2}^{(h)} \widehat{\beta}_2 + \cdots + Y_{T-p} \widehat{\beta}_p + \widehat{\varepsilon}_T^{(h)}
\end{align*}</li>
<li> Then, with the $Y^{(h)}$ series, estimate $\widehat{\beta}^{(h)}$ and $\widehat{\Sigma}^{(h)}$ and store them.</li>
<li> Go back to part 1 and do it all again for a new $h = h+1$.</li>
<li> After repeating the process above <code>n_draws</code> times, estimate the IRFs.</li>
</ol>
</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)
var_data <- data.matrix(USMacroData[,2:4])
var_obj <- new(cvar)
#
var_obj$build(var_data,TRUE,1)
var_obj$estim()
var_obj$boot(10000)
IRF(var_obj,20,var_names=colnames(USMacroData),save=FALSE)
plot(var_obj,var_names=colnames(USMacroData),save=FALSE)
forecast(var_obj,shocks=TRUE,var_names=colnames(USMacroData),back_data=10,save=FALSE)
</pre>
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