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\title[ECB-BASE-MCE]{Model-consistent-expectations in the ECB-BASE: First results and Roadmap}
\author[Adjemian \and Bokan \and Darracq Pariès]{Stéphane Adjemian \and Nikola Bokan \and Matthieu Darracq Pariès}

\date{September, 2020}

\begin{document}
\maketitle

\section*{Context}

\begin{frame}
  \frametitle{Context}
  \framesubtitle{MCE activities within the ECB-MC project}

  \bigskip

  \begin{itemize}

  \item Starts from the ECB-BASE (backward) model and introduces further developments

    \medskip

    \begin{itemize}

    \item new "infrastructure": one-stop shop allowing

      \begin{itemize}
      \item pre-estimation data-generation
      \item estimation
      \item post-estimation data generation
      \item dynare .mod model production
      \item model simulations
      \end{itemize}

      \medskip

    \item re-estimation of the model on recent euro area data (ECB-BASE V2)\newline

    \item appropriate convergence towards the balance growth path\newline

    \item various other (bug) fixes

    \end{itemize}

  \end{itemize}

\end{frame}


\begin{frame}
  \frametitle{ECB-BASE V2 (beta)}
  \framesubtitle{Full re-estimation using the recent EA data}

  \begin{itemize}

  \item First draft of ECB-BASE V2 pointed out to potential issues in several blocks\newline

  \item Improved properties by fine-tuning towards ECB-BASE V1 along several dimensions
    \medskip
    \begin{itemize}

    \item Consumption block

      \begin{itemize}
      \item BOLS estimation of the target equation
      \item interest rate coeff in the PAC equation constrained as in ECB-BASE V1 (WP)
      \end{itemize}

    \item BOLS estimation of wage target equation

    \item BOLS estimation of trade equations

    \end{itemize}

    \medskip

  \item Some further work is still needed to fine tune the ECB-BASE V2 and make it operational in the (B)MPE process\newline

  \item MCE activities will proceed in parallel to this process

  \end{itemize}

\end{frame}


\section{Dealing with expectations}


\subsection{VAR expectations}


\begin{frame}
  \frametitle{Expectations in ECB BASE}
  \framesubtitle{VAR expectations}

  \begin{itemize}

  \item ECB BASE is estimated under the assumption that agents use a VAR model to form expectations.\newline

  \item Consumption depends on permanent income(s):
    \[
      y^P_t = \sum_{h=0}^{\infty}\beta^hy_{t+h}
    \]
    the flow of expected incomes can be computed from a VAR. Suppose $y$ is an element of $Y$ and that:
    \[
      Y_t = AY_{t-1} + \epsilon_t
    \]
    Then we have:
    \[
      y_{t+h} = ZA^hY_t \quad\text{ and }\quad y^P_t = Z \left( \sum_{h=0}^{\infty}\beta^hA^h\right)Y_{t} = Z\left(I-\beta A\right)^{-1}Y_t
    \]

  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Expectations in ECB BASE}
  \framesubtitle{VAR expectations (contd')}

  \begin{itemize}

  \item Implementation in Dynare is straightforward:
    \medskip
    \begin{itemize}

    \item[--] Declare a VAR model with the \verb+VAR_MODEL+ command
      (pointing to some equations in the \verb+model+ block),\newline

    \item[--] Alternatively, declare a trend component model with the
      \verb+TREND_COMPONENT_MODEL+ command,\newline

    \item[--] Use the \verb+VAR_EXPECTATION_MODEL+ command to specify
      the expected term to be used in the \verb+model+ block (which
      variable or linear combination of variables is to be expected,
      at which horizon, \ldots),\newline

    \item[--] Use the \verb+VAR_EXPECTATION+ command in the \verb+model+ block.

    \end{itemize}

  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Expectations in ECB BASE}
  \framesubtitle{VAR expectations in WAPRO}

  \begin{scaledverbatim}

    trend_component_model(model_name = base_var_model,
    eqtags = ['U2_WAPRO_A_YED', 'U2_WAPRO_G_YER', 'U2_WAPRO_ESTN',
              'U2_WAPRO_EHIC', 'U2_WAPRO_G_EYER', 'U2_WAPRO_STN'],
    targets=['U2_WAPRO_G_EYER', 'U2_WAPRO_EHIC', 'U2_WAPRO_ESTN']);

    var_expectation_model(model_name = U2_A_YED_OP_FCAST,
    expression = U2_WAPRO_A_YED,
    auxiliary_model_name = base_var_model,
    horizon = 1,
    discount = 1.0);

    model;
    ...
    U2_A_YED = (0.387271924091236*U2_A_YED(-1)
    + 0.629532709573858*var_expectation(U2_A_YED_OP_FCAST)
    + (1-0.629532709573858)*(1-0.387271924091236)*U2_EHIC
    + 0.421357609729608*(U2_G_C_CER + 0.3/(1-0.3)*U2_G_YER))
    / (1+0.387271924091236*0.629532709573858)+ res_U2_A_YED;

    end;

  \end{scaledverbatim}

\end{frame}


\subsection{PAC equation}


\begin{frame}
  \frametitle{PAC equation}
  \framesubtitle{}

  \begin{itemize}

  \item Penalize deviations of a variable from its target and the variations of the variable:
    \[
      C_t = \sum_{i=0}^{\infty}\beta^i\left[(y_{t+i}-y_{t+i}^{\star})^2 + \sum_{k=1}^m b_k\left(\Delta^ky_{t+i}\right)^2\right]
    \]

  \item Reduced form associated to the minimization of the cost $C_t$:
    \[
      \Delta y_t = a_0 (y_{t-1}^{\star}-y_{t-1}) + \sum_{k=1}^{m-1} a_k \Delta y_{t-k} + \underbrace{\sum_{j=0}^{\infty}d_j\Delta y_{t+j}^{\star}}_{\substack{\mathcal Z_t\\\text{Expected variations}\\\text{of the target}}}
    \]

  \item Where $\{a_i\}_{i=0}^{m-1}$ and $\{d_i\}_{i\in\mathbb N}$ are reduced form parameters depending on the parameters of the cost function.
  \end{itemize}

\end{frame}


\begin{frame}
  \frametitle{PAC equation}
  \framesubtitle{Estimation}

  \begin{itemize}

  \item The estimation of the parameters $\{a_i\}_{i=0}^{m-1}$ (not
    the deep parameters) requires the knowledge of $\mathcal Z_t$
    (expectations).\newline

  \item The estimation is done under the assumption that the
    expectations are VAR based.\newline

  \item Each $\Delta y_{t+i}$ (for $i=,\ldots,\infty$) can be
    expressed as a linear combination of the VAR variables, where the
    weights are a convolution of the parameters $\{a_i\}_{i=0}^{m-1}$
    and the VAR autorgressive matrices.\newline
    \[
      \Delta y_t = a_0 (y_{t-1}^{\star}-y_{t-1}) + \sum_{k=1}^{m-1} a_k \Delta y_{t-k} + h'Y_t + \epsilon_t
    \]

  \item Estimation is carried through NLS.

  \end{itemize}

\end{frame}


\begin{frame}
  \frametitle{PAC equation}
  \framesubtitle{MCE simulations}

  \begin{itemize}

  \item In the MCE version of ECB-BASE we cannot rely on a VAR model
    to determine the expected variations of the target,
    $\mathcal Z_t$.\newline

  \item Rewrite the expected infinite sum $\mathcal Z_t$ recursively:
    \[
      \begin{split}
        \mathcal Z_t = &-\sum_{i=1}^m\alpha_i\beta^i\mathcal Z_{t+i} + \left(1+\sum_{i=1}^m\alpha_i\right)\times\\
        &\left[\Delta y_t^{\star}+\sum_{k=1}^{m-1}\left(\sum_{j=k}^{m-1}\alpha_{j+1}\beta^{j+1}\Delta y_{t+k}^{\star}\right)\right]
      \end{split}
    \]
  \item  The parameters $\{\alpha_i\}_{i=1}^m$ depend on the previously estimated parameters $\{\alpha_i\}_{i=0}^{m-1}$.\newline

  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{PAC equation}
  \framesubtitle{PAC with Dynare}

  \begin{itemize}

  \item Dynare expands $\mathcal Z_t$ (depending on the context) for you and
    estimate the PAC equation.\newline

    \begin{scaledverbatim}
      var_model(model_name=toto, eqtags=['eq:x', 'eq:y']);
      pac_model(auxiliary_model_name=toto,
      discount=beta, model_name=pacman);
      model;
      [name='eq:y']
      y = a_y_1*y(-1) + a_y_2*diff(x(-1)) + b_y_1*y(-2)
      + b_y_2*diff(x(-2)) + ey ;
      [name='eq:x']
      diff(x) = b_x_1*y(-2) + b_x_2*diff(x(-1)) + ex ;

      [name='eq:pac']
      diff(z) = e_c_m*(x(-1)-z(-1)) + c_z_1*diff(z(-1))
      + c_z_2*diff(z(-2))
      + pac_expectation(pacman) + ez;
      end;
    \end{scaledverbatim}

  \item For the MCE just drop option \verb+auxiliary_model_name+

  \end{itemize}

\end{frame}


\section{MCE specification}


\subsection{Exchange rate determination}


\begin{frame}
  \frametitle{MCE specification: Exchange rate determination}
  \framesubtitle{MCE for the Uncovered Interest rate Parity equation (UIP)}

  \bigskip

  \begin{itemize}

  \item Replacing the "level" equation for the nominal exchange rate in the backward model by a UIP condition\newline

  \item The expected depreciation rate depends on the short-term interest rate differential...\newline

  \item ... augmented with a term premium effect to obtain a plausible "APP-like" transmission of term premium shocks\newline

  \item Given the consumption specification, adding a term on NFA in the UIP is not necessary to ensure stationary NFA dynamics\newline

  \end{itemize}

  \bigskip

  \begin{center}
    \textit{Empirical and theoretical challenges remain on the UIP specification}
  \end{center}

\end{frame}


\subsection{Financial block}


\begin{frame}
  \frametitle{MCE specification: financial block}
  \framesubtitle{MCE in the term structure of interest rate}

  \bigskip

  \begin{itemize}

  \item Long-term interest rates\newline

    \begin{itemize}

    \item Expectation theory for the risk-free 10-year OIS rate: we introduce a consol bond serving geometrically decaying coupons, discounted by the short-term policy rate and a duration corresponding the one of a 10-year zero coupon.\newline

    \item The 10-year OIS rate accounting for a term premium: we introduce a similar consol with a discounting of coupons augmented by a term premium.\newline

    \item Similar treatment for corporate bond\newline

    \end{itemize}

  \end{itemize}

\end{frame}


\begin{frame}
  \frametitle{MCE specification: financial block}
  \framesubtitle{MCE in financial spreads}

  \bigskip

  \begin{itemize}

  \item Replacing the VAR-based expectation by MCE expectation for 10-year average of expected output gap\newline

  \item this variable then loads into the financial spreads (term premium, corporate spread, lending rate spreads and cost-of-equity)\newline

  \item Consistent reformulation of the revaluation effects on households net financial wealth\newline

  \end{itemize}

  \bigskip

  \begin{center}
    \textit{Follow-up: stock prices may be formally specified}
  \end{center}

\end{frame}


\subsection{Price and wage setting blocks}


\begin{frame}
  \frametitle{MCE specification: wapro and wage blocks}
  \framesubtitle{MCE in the wage and price setting equations}

  \bigskip

  \begin{itemize}

  \item Replacing the VAR-based expectation by MCE expectation for the next period inflation term in the GDP-deflator equation\newline

  \item Replacing the VAR-based expectation by MCE expectation for the next period wage gap term in the wage gap equation\newline

  \item Long-term inflation expectations remain imperfectly anchored as in the backward model\newline

  \end{itemize}

\end{frame}


\section{Simulations}


\subsection{Benchmark backward model}


\begin{frame}[fragile]
  \frametitle{Infrastuctures: V1 vs V2}

  \begin{center}
    \includegraphics[page=2, scale=0.27]{plots/new_vs_wp.pdf}
  \end{center}
\end{frame}


\begin{frame}[fragile]
  \frametitle{Infrastuctures: V1 vs V2}

  \begin{center}
    \includegraphics[page=3, scale=0.27]{plots/new_vs_wp.pdf}
  \end{center}
\end{frame}

\subsection{Backward versus full MCE}

\begin{frame}[fragile]
  \frametitle{Responses to a one point shock on the nominal interest rate}

  \begin{center}
    \includegraphics[scale=1.0]{plots/backward-full-comparison-res_U2_STN.pdf}\\
    {\tiny{\color{black}---} Backward model, \tiny{\color{blue}---} Backward with UIP model, {\color{red}---} Full MCE model}
  \end{center}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Responses to a one point shock on the term premium}

  \begin{center}
    \includegraphics[scale=1.0]{plots/backward-full-comparison-res_U2_TP_10Y.pdf}\\
    {\tiny{\color{black}---} Backward model, \tiny{\color{blue}---} Backward with UIP model, {\color{red}---} Full MCE model}
  \end{center}

\end{frame}


\subsection{Standard shocks across MCE specifications}


\begin{frame}
  \frametitle{Simulations}

  \begin{itemize}

  \item[{\color{blue} ---}] Backward model with UIP,\newline

  \item[{\color{red} ---}] Full MCE,\newline

  \item[{\color{green} ---}] Hybrid mode (MCE in wage, wapro, financial and exchange rate blocks).\newline

  \end{itemize}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Responses to a one point shock on the nominal interest rate}

  \begin{center}
    \includegraphics[scale=1.0]{plots/irfs-res_U2_STN.pdf}\\

    \bigskip

    {\tiny {\color{blue}---} Backward with UIP model, {\color{red}---} Full MCE model, {\color{green}---} Hybrid model}

  \end{center}

\end{frame}


\begin{frame}[fragile]
  \frametitle{Responses to a one point shock on the term premium}

  \begin{center}
    \includegraphics[scale=1.0]{plots/irfs-res_U2_TP_10Y.pdf}\\

    \bigskip

    {\tiny {\color{blue}---} Backward with UIP model, {\color{red}---} Full MCE model, {\color{green}---} Hybrid model}

  \end{center}

\end{frame}

\subsection{Forward guidance shocks}


\begin{frame}
  \frametitle{Forward guidance (Backward with UIP model)}

  \begin{center}
    \includegraphics[scale=0.35]{plots/forward-guidance-backward-with-uip.pdf}\\
  \end{center}

\end{frame}


\begin{frame}
  \frametitle{Forward guidance (full MCE)}

  \begin{center}
    \includegraphics[scale=0.35]{plots/forward-guidance-full.pdf}\\
  \end{center}

\end{frame}


\begin{frame}
  \frametitle{Forward guidance (Hybrid model)}

  \begin{center}
    \includegraphics[scale=0.35]{plots/forward-guidance-hybrid.pdf}\\
  \end{center}

\end{frame}


\subsection{Anticipated productivity shocks}

\begin{frame}[fragile]
  \frametitle{Expected TFP growth shock in 10 years for one year (4$\times$1\%)}

  \begin{center}

    \includegraphics[scale=1.0]{plots/expected-tfp-growth-shock.pdf}\\

    \bigskip

    {\tiny {\color{blue}---} Backward with UIP model, {\color{red}---} Full MCE model, {\color{green}---} Hybrid model}

  \end{center}

\end{frame}


\section{Roadmap for MCE activities}


\begin{frame}
  \frametitle{Next steps}

  \textit{Main deliverables for the end-August milestone}

  \bigskip

  \begin{itemize}
  \item Freeze the benchmark backward model\newline

  \item Systematic comparison of shock transmission between the backward model, the hybrid expectations model and the full MCE model\newline

  \item Further examples of anticipated shocks\newline

  \item Policy application\newline

  \end{itemize}

\end{frame}


\begin{frame}
  \frametitle{Follow-up activities}

  \textit{Possible workstreams}

  \bigskip

  \begin{itemize}

  \item Stationary version of the model and linear approximation\newline

    \begin{itemize}
    \item Tractability of stochastic simulations

    \item Filtering\newline

    \end{itemize}

  \item Empirical validation of the MCE model\newline

    \begin{itemize}

    \item Indirect inference on the Wapro/wage blocks

    \item System-wide inference using the linear model\newline

    \end{itemize}

  \item Policy analysis with the MCE model\newline

    \begin{itemize}

    \item Nesting a term-structure model into the ECB-BASE-MCE

    \item Optimal policy projections

    \end{itemize}

  \end{itemize}

\end{frame}


\end{document}




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