\input{./econtexRoot.tex}\input{\econtexRoot/econtexPaths.tex} \documentclass{\econtex}\usepackage{\econtexSetup}\usepackage{\econtexShortcuts}\input{\econtexRoot/Switches.tex}\provideboolean{StandAlone}\setboolean{StandAlone}{true}\setboolean{BigAndWide}{true}   \begin{document}\bibliographystyle{\econtexBibStyle} \setboolean{DocVersion}{false}  \newcommand{\texname}{Muth}\hfill{\tiny \today} \large
  \section{Formulae Derived from \cite{muthOptimal}}\label{appendix:Muth}

  \begin{verbatimwrite}{\econtexRoot/LaTeX/appendices/MuthPre-muthOptimal.texinput}
  \cite{muthOptimal}, pp.\ 303--304,  shows that the signal-extracted estimate of permanent income is
  \begin{eqnarray}
  \perc{\PLev}_{t} & = & v_{1}Y_{t}+v_{2}Y_{t-1}+v_{3}Y_{t-2}+...
  \end{eqnarray}
  for a sequence of $v$'s given by
  \begin{eqnarray}
    v_{k} & = & (1-\lambda_{1})\lambda_{1}^{k-1}
  \end{eqnarray}
  for $k = 1,2,3,...$.  So:
  \begin{eqnarray}
    \perc{\PLev}_{t} & = & (1-\lambda_{1})(\phantom{\YLev_{t+1}+\lambda_{1}^{0}}\YLev_{t}+\lambda_{1} \YLev_{t-1} + \lambda_{1}^{2} \YLev_{t-2} ...)
\\    \perc{\PLev}_{t+1} & = & (1-\lambda_{1})(\YLev_{t+1}+\lambda_{1} \YLev_{t} + \lambda_{1}^{2} \YLev_{t-1}+ \lambda_{1}^{3} \YLev_{t-2} ...)
\\     & = & (1-\lambda_{1})\phantom{(}\YLev_{t+1}+\lambda_{1} \underbrace{(1-\lambda_{1})(\YLev_{t} + \lambda_{1}^{2} \YLev_{t-1}+ \lambda_{1}^{3} \YLev_{t-2} ...)}_{\perc{\PLev}_{t}}
\\    & = & (1-\lambda_{1})\YLev_{t+1}+\lambda_{1} \perc{\PLev}_{t}
  \end{eqnarray}

This compares with \eqref{eq:PischkeP} in the main text
  \begin{eqnarray}
    \perc{\PLev}_{t+1} & = & \Pi \YLev_{t+1} + (1-\Pi) \perc{\PLev}_{t}
  \end{eqnarray}
so the relationship between our $\Pi$ and Muth's $\lambda_{1}$ is:
  \begin{eqnarray}
    \lambda_{1} & = & 1-\Pi
  \end{eqnarray}

  Defining the signal-to-noise ratio $\varphi = \sigma_{\pmb{\psi}}/\sigma_{\pmb{\theta}}$, starting with equation (3.10) in \cite{muthOptimal} we have
  \begin{eqnarray}
\notag    \lambda_{1} & = & 1 + (1/2) \varphi^{2} - \varphi \sqrt{1+\varphi^{2}/4}
\notag \\ (1-\Pi) & = & 1 + (1/2) \varphi^{2} - \varphi \sqrt{1+\varphi^{2}/4}
                        \notag\\ -\Pi & = & (1/2) \varphi^{2} - \varphi \sqrt{1+\varphi^{2}/4}
  \end{eqnarray}
  yielding equation \eqref{eq:muthOptimal} in the main text.
\end{verbatimwrite}
\input{\econtexRoot/LaTeX/appendices/MuthPre-muthOptimal.texinput}
  \begin{verbatimwrite}{\eq/muthOptimal.tex}
  \begin{eqnarray}
\Pi & = & \varphi \sqrt{1+\varphi^{2}/4} - (1/2) \varphi^{2}
  \end{eqnarray}
\end{verbatimwrite}
\input{\eq/muthOptimal.tex}

\bibliography{economics,\texname,\texname-Add}

\end{document}
