\par
\begin{table}[t]
\begin{normalsize}
\begin{center}
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  \begin{center}
  {\textbf{\large Table~\ref{t:estimates}. 
  Unconstrained and Constrained Parameter Estimates}}
  \end{center}
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\vspace{2.5ex}
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\begin{tabular}{cccccccccccccccccccc}
\hline \hline \\[-2.5ex]
Parameter   & Unconstrained & Constrained \\
\hline   \\[-2.0ex]
$a_{01}$    &   0.13156 &   0.09396 \\
$a_{02}$    &  -0.00779 &  -0.01758 \\
$a_{03}$    &   0.01877 &   0.01659 \\
$a_{04}$    &   0.08176 &   0.07536 \\
$a_{05}$    &  -0.04785 &  -0.04523 \\
$a_{06}$    &  -0.02612 &  -0.03465 \\
$a_{07}$    &  -0.03515 &  -0.03961 \\
$a_{08}$    &   0.15387 &   0.15566 \\
$b_{0,1}$   &   0.12900 &   0.09102 \\
$b_{0,2}$   &  -0.25768 &  -0.20761 \\
$B_{1,1}$   &  -0.06766 &   0.03206 \\
$B_{2,1}$   &  -0.05357 &  -0.00676 \\
$B_{1,2}$   &  -0.01054 &  -0.00617 \\
$B_{2,2}$   &  -0.02601 &  -0.03625 \\
$R_{0,1,1}$ &   0.83192 &   0.84131 \\
$R_{0,1,2}$ &  -0.03808 &  -0.04482 \\
$R_{0,2,2}$ &   0.98477 &   0.98180 \\
$P_{1,1}$   &   0.49290 &   0.46244 \\
$P_{2,2}$   &  -0.12324 &  -0.05271 \\
$a_1$       &           &  -0.53822 \\
$a_2$       &           &   0.12894 \\
$a_3$       &           &   0.04643 \\
\hline \hline \\
\end{tabular}
\vspace{-1ex}
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The prior to determine the distribution of $Z$ (e.g.,
Figure~\ref{f:dcf_Z_0050_qq}), is over $\rho=(b_0,B,R,\beta,\gamma)$.  The
prior is multivariate normal 
with means and standard errors as shown here 
and correlations as shown in Table~\ref{t:dcf_heatmap}
subject to support conditions shown here.  The VAR
$y_t = b_0 + By_{t-1} + Ru$ with $u$ independent normal determines the
the distribution of $y$ = $\log$\,(marginal rate of substitution,
GDP/corporate profits, corporate profits).  For each $\tilde\rho$ drawn
from the prior, a realization $\{y_t\}^n_{t=1}$ of the VAR with initial
condition $y_0$ for 2015 is computed.  From each realization, the return
on corporate profits conditional on $y_t$ is computed analytically from 
the drawn $(\tilde b_0,\tilde B,\tilde R)$ and the
marginal rate of substitution is converted to consumption growth using
CRRA utility with drawn $(\tilde\beta,\tilde\gamma)$.  The
resulting bivariate geometric returns to corporate profits and log
consumption growth series is used to compute the $Z$ corresponding to
$\tilde\rho$. 
In a simulation of size 5,000 with $\rho$
set to the prior mean shown here, log consumption growth has mean 0.0020 
and standard deviation 0.1441, the geometric return on corporate profits 
has mean 0.06310 and standard deviation 0.07255.
To estimate $\theta=(\beta,\gamma)$ from bivariate returns and consumption 
growth data using the method described in 
Section~\ref{s:bayes} the prior is bivariate normal with means and 
standard deviations as shown here for $(\beta,\gamma)$, zero correlation, 
and support conditions $0.8 \leq \beta \leq 0.99$ and 
$0 \leq \gamma \leq 100$. 
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\end{center}
\end{normalsize}
\end{table}
