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\section{Constant Relative Risk Aversion Utility}
\label{s:example}

Let $C_{t}$ denote the annual consumption endowment.  Let $P_{ct}$ denote the 
price of an asset that pays the consumption endowment.  Let 
\begin{equation}
\label{Rst}
R_{st}=(P_{st}+D_{st})/P_{s,t-1}
\end{equation}
denote the gross return on an asset $S$ that pays $D_{st}$ per period and has 
price $P_{st}$ at time $t$.  Prices and payoffs are real.  

The constant relative risk aversion utility function is
\begin{equation}
\label{crra_utility}
U = \sum^\infty_{t=0}\delta^t\left(\frac{c^\gamma_t-1}{\gamma}\right)
\end{equation}
where $\delta $ is the time preference parameter and
$\gamma $ is the coefficient of risk aversion (Lucus, 1978).
The agent's intertemporal marginal rate of substitution is
\begin{equation}
\label{crra_mrs}
\mbox{MRS}_{t+1} = 
\delta \left(\frac{C_{t+1}}{C_{t}}\right)^{\gamma -1}.
\end{equation}
The gross return on an asset $S$ that pays $D_{st}$ satisfies 
\begin{equation}
\label{euler_Md}
1 = {\cal E}_{t}\left(\mbox{MRS}_{t+1}R_{s,t+1}\right).
\end{equation}

The following variables were constructed for the 86 years 1930 to 2015 as
described in Subsection~\ref{ss:data} below.
\begin{itemize}
\item
$s_t$ = log real gross stock return (value weighted NYSE/AMEX/NASDAQ).
\item
$b_t$ = log real gross bond return (30 day T-bill return).
\item
$c_t$ = log real per capita consumption growth (nondurables and services).
\item
$\mbox{mrs}_t = \log(\mbox{MRS}_t) = \log(\delta) + (\gamma - 1)c_t$
\end{itemize}
Let $x$ denote an array of extent $n$ whose columns are
$x_t=(s_t,b_t,c_t)^\prime$.
The process 
$\left\{x_t\right\}^\infty_{t=-\infty}$ is assumed to be strictly
stationary; i.e., the distribution of $(x_{t+1},\ldots,x_{t+L})$ is the
same as the distribution of $(x_{1},\ldots,x_{L})$ for any $t$ and any $L$.

Given the parameters $\theta=(\gamma,\delta)$ and $x$, one
can compute the pricing errors
\begin{eqnarray*}
e_{1,t,t-1} &=& 1 - \exp(\mbox{mrs}_{t-1,t} + s_t) \\
e_{2,t,t-1} &=& 1 - \exp(\mbox{mrs}_{t-1,t} + b_t)
\end{eqnarray*}
and form the following moment equations for the estimation of
$\theta=(\gamma,\delta)$
\begin{eqnarray*}
m_{1}(x_t,x_{t-1},\theta) &=& e_{1,t,t-1} \label{e1} \\
m_{2}(x_t,x_{t-1},\theta) &=& e_{2,t,t-1} \label{e2} \\
m_{3}(x_t,x_{t-1},\theta) &=& e_{1,t,t-1} \times s_{t-1} \label{se1} \\
m_{4}(x_t,x_{t-1},\theta) &=& e_{1,t,t-1} \times b_{t-1} \label{be1} \\
m_{5}(x_t,x_{t-1},\theta) &=& e_{1,t,t-1} \times c_{t-1} \label{ce1} \\
m_{6}(x_t,x_{t-1},\theta) &=& e_{2,t,t-1} \times s_{t-1} \label{se2} \\
m_{7}(x_t,x_{t-1},\theta) &=& e_{2,t,t-1} \times b_{t-1} \label{be2} \\
m_{8}(x_t,x_{t-1},\theta) &=& e_{2,t,t-1} \times c_{t-1} \label{ce2} \\
\end{eqnarray*}
Abbreviating $m(x_t,x_{t-1},\theta)$ by $m_t$ and
$\bar{m}(x,\theta)$ by $\bar{m}$, define a heteroskedatic
autoregressive invariant (HAC) estimate of the variance of
$\bar{m}(x,\theta)$ by
\begin{equation}
\label{HACmatrix}
W\!\left(x,\theta\right)=\sum^{[n^{1/5}]}_{\tau=-[n^{1/5}]}
w\left(\frac{\tau}{[n^{1/5}]}\right)\bar{W}_{\tau }
\end{equation}
where \\
$$
w(u) = \left\{ \begin{array}{ll}
%
1-6|u|^{2}+6|u|^{3} & \mbox{ if } 0<u<\frac{1}{2} \\
%
2(1-|u|)^{3} & \mbox{ if } \frac{1}{2}\leq u<1 \\[1ex]
%
\end{array}
\right.
$$
\\[-1.5ex]
\begin{equation}
\label{CorrelationMatix}
\bar{W}_{\tau} = \left\{ \begin{array}{ll}
\frac{1}{n}\sum^{n}_{t=2+\tau }
\left(m_{t}-\bar{m}\right)\left(m_{t-\tau}-\bar{m}\right)^{\prime}
& \tau \geq 0 \\[1ex]
\tilde W^{\prime}_{n,-\tau} & \tau <0
\end{array}
\right.
\end{equation}
\\[-1.5ex]
See, e.g.,
Gallant (1987, p.\ 446, 533).

As per Chernozhukov and Hong (2003),
the Laplace criterion function for use with MCMC 
(Gamerman and Lopes, 2006) is
\begin{equation}
\label{laplace}
p(s,b,c\,|\,\theta)
=(2\pi )^{-\frac{M}{2}}\exp \left\{-\frac{n}{2}\bar m^{\prime}(x,\theta)
[W(x,\theta )]^{-1}\bar m(x,\theta )\right\},
\end{equation}
where $M=8$ in this case.

\subsection[data]{Data} 
\label{ss:data} 

\par
The raw data for stock returns are value weighted returns including
dividends for NYSE, AMEX, and NASDAQ from the Center for Research in 
Security Prices data at the Wharton Research Data Services web site 
(http://wrds.wharton.upenn.edu).

\par
The raw data for returns on U.S. Treasury 30 day debt are from 
the Center for Research in Security Prices data at the Wharton Research 
Data Services web site.

\par
The raw consumption data are personal consumption expenditures on 
nondurables and services obtained from Table 2.3.5 at the Bureau of 
Economic Analysis web site (http://www.bea.gov).  

\par
Raw data are converted from nominal to real using the annual 
consumer price index obtained from Table 2.3.4 at the Bureau 
of Economic Analysis web site.  Conversion of consumption to per 
capita is by means of the mid-year population data from
Table 7.1 at the Bureau of Economic Analysis web site. 

\begin{center}
{\bf \Large References}
\end{center}

\begin{description}

\item
Chernozhukov, Victor, and Han Hong (2003),
``An MCMC Approach to Classical Estimation,''
{\it Journal of Econometrics} 115, 293--346.

\item
BEA (2016),
Bureau of Economic Analysis,
``Table 7.1. Selected Per Capita Product and Income Series in Current
and Chained Dollars,'' www.bea.gov.

\item
CRSP (2016),
Center for Research in Security Prices,
Graduate School of Business, The University of Chicago,
Used with permission. All rights reserved.
www.crsp.uchicago.edu

\item
Gallant, A. R. (1987),
{\it Nonlinear Statistical Models,}
New York: Wiley.

\item
Gamerman, D., and H.\ F.\ Lopes (2006),
{\it Markov Chain Monte Carlo: Stochastic Simulation}
{\it for Bayesian Inference (2nd Edition),}
Chapman and Hall, Boca Raton, FL.

\item
Lucus, Robert E., Jr. (1978)
``Asset Prices in an Exchange Economy,''
{\it Econometrica} 46, 1429--1445.

\end{description}


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