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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, I}

  \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}

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

  \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\\ dashed lines are simulated from the same models without TP in exchange rate equation}
  \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{Follow-up activities}
  \framesubtitle{Possible workstreams}

  \bigskip

  \begin{itemize}
  \item Codes and Documentation of the "as-is" MCE version of ECB-BASE\newline

  \item Stationary version of the model and linear approximation

    \medskip

    \begin{itemize}

    \item Tractability of stochastic simulations

    \item Stabilisation properties of alternative monetary policy conduct\newline

    \end{itemize}

  \item Other policy analysis

    \medskip

    \begin{itemize}

    \item Fiscal-Monetary interactions\newline

    \end{itemize}

  \item Empirical validation of the MCE model

    \medskip

    \begin{itemize}

    \item Filtering

    \item Indirect inference on the Wapro/wage blocks

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

    \end{itemize}

  \end{itemize}

\end{frame}

\end{document}




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