华裔计量经济学家 Changyou Sun 教授的 erer 工具包推荐

Here  we complicate the lean model of section \ref{sec:NeutStick} above, by
adding price stickiness and investment. There are adjustment costs for capital,
so that there is a variable price of investment goods.  In  a  model  where
transactions prices are not true shadow prices,  there is  some ambiguity as
to how to define the transactions costs that determine the demand for money. It
seems most natural to  use  a flow of actual contract payments to measure
transactions,  not  a notional value of production or purchases priced at
shadow  prices.   This  eliminates the neat conclusion that contract prices
have no effect on the real equilibrium.

Even without the complication of contract prices, a classical model with
transactions costs implies some real effects of monetary policy.  High
inflation and nominal interest rates reduces equilibrium real balances and
raises transactions costs.  When the model includes an investment
decision, it is possible in principle for investment decisions to be
noticeably affected by variations in monetary policy even though
transactions costs are not in equilibrium a large fraction of output
\cite{RebeloRbar}.

While it is therefore not a foregone conclusion that real effects of monetary policy
will be small, in the model presented here, with realistic parameter
values, the real effects of monetary policy do indeed emerge as small.

As with all the models we are considering, there is a representative
consumer and a representative firm.  Their objectives and constraints are
described below.

Consumer:

$$
\max_{C, L, V, M, B_C, Y_C, and Y_L} \int_0^\infty  {e^{-\beta t}{{C_t^{\mu _0}
      \left( {1-L} \right)^{\mu _1}} \over {\mu _0+\mu _1}}dt}
$$
subject to
$$
\begin{align}
  \lambda&\!: & Y_C+\dot B_C+\dot M+\tau &\le Y_L+\pi +rB_C \label{eq:CSClambda}\\
  \psi_V&\!: & V&\ge {{Y_C} \over M} \label{eq:CSCpsiV} \\
  \psi_C&\! & \Cstar &\ge C\cdot \left( {1+\gamma V} \right) \label{eq:CSCpsiC}\\
  \nu_{YC}&\!: & \dot{Y}_C-p\dot{\Cstar} & \geq -\eta_C\cdot(Y_C-p\Cstar)
  \label{eq:CSCnuYC}\\
  \nu_{YL}&\!: & \dot Y_L-w\dot L&\le -\eta _L\cdot \left( {Y_L-wL} \right)
  \label{eq:CSCnuYL}\\
   && B_C\ge0,&\: M\ge0 \:.
\end{align}
$$

$$
  \max_{C,L,V,M,B_C} \int\limits_0^\infty  {e^{-\beta t}{{C_t^{\mu _0}
\left( {1-L_t} \right)^{\mu _1}} \over {\mu _0+\mu _1}}dt}
$$

subject to

$$
\begin{align}
   \zeta&\text{:} &\pi &\le PC^*-WL+\dot B_F-rB_F \label{eq:CIFzeta}\\
   \omega&\text{:}& C^*+\left( {1+\xi {I \over K}} \right)I&\le AK^\alpha L^{1-\alpha
   }\label{eq:CIFomega}\\
   \sigma_K&\text{:}& \dot K&\le I-\delta K \:.\label{eq:CIFsigmaK}
\end{align}
$$

$$
begin{align}

   \frac{\dot{P}}{P}=-\eta_C \log\left(\frac{A\cdot(1-\alpha)\left({K}/{L}\right)^\alpha P }{W} \right)

$$

$$
PC^*+\dot B_C+\dot M+\tau +{1 \over 2}\chi \dot W^2\le WL+\pi +rB_C
$$

$$
\begin{align}
   \partial L &\!: & {{\mu _1U} \over {1-L}}&=W\lambda -\eta _W{{\psi _W} \over L}
   \label{eq:MCMCCdL}
\end{align}

$$

% We briefly discuss the two non-linearities here.

% # The Phillips curve has the following form:
    $$ \pi_t - \pi_{t-1} = \beta_t^\star \, \mathrm E_t \left[ \pi_{t+1} -
    \pi_t \right] + \gamma \tfrac{1}{\delta} \left( \exp\delta y_t - 1
    \right) + \epsilon_{\pi,t}, $$
% where the forward-lookingness of inflation, $\hat\beta$, is affected by
central-bank crediblity, $c_t \in [0,1]$:
$$ \beta^\star_t := c_t \cdot \beta. $$
and $\beta\in(0,1)$ is a fixed parameter (attained at full credibility).
% In other words, the lower the central bank credibility, the greater
% persistence in inflation, and vice versa.
%
% The function that describes the Phillips curve non-linearity, i.e.
$\left( \exp\delta y_t - 1 \right)/\delta$ is an m-file function saved
% under the name `nonlinpc.m`. It is adapted so as to be able to handle the
special case with $\delta=0$ (a linear Phillips curve).

# The law of motion for credibility is as follows:
    $$ c_t = \psi c_{t-1} + (1-\psi) s_t,
    $$
where $s_t$ is a ``signal'' by which the credibility is updated. The
signal is given by
     $$ s_t = \exp \left(\omega\hat\pi^4_{t-1}\right)^2
     $$
where $\hat\pi^4_t$ is the deviation of current four-quarter inflation,
$\pi^4_t$, from the target, $\tau_t$:
    $$ \hat\pi^4_t = \pi_t - \tau_t.
    $$
% In other words, credibility is damaged every time inflation deviates from
% the target (either way), and the extent of it increases with the square
% of the deviation.
%
% To be able to use an exact non-linear simulation mode in our later
% simulates and Kalman filter experiments, we need to earmark the equations

$$
\tau_k = \mu*t_k + \theta \sum_{t_i<t_k} \left( 1-e^{-\omega(t_k-t_i)} \right)
        = \mu*t_k + \theta*(k-1)-\theta * e^{-\omega*(t_{k}-t_{k-1})}\left(
          \sum_{t_{i}<t_{k-1}}e^{-\omega*(t_{k-1}-t_{i})}+1 \right)
$$

In general, we can rewrite the model equation Eq. (1) in the form of a multivariate regression as \[ \mathbf{y}_t = \mathbf{\Psi}^\top \mathbf{x}_t + \boldsymbol{\epsilon}_t, \qquad\qquad\qquad (5) \] where \(\mathbf{\Psi} = (\mathbf{A}_1, \mathbf{A}_2, \ldots, \mathbf{A}_p, \mathbf{C})^\top\) is a \((Kp + L) \times K\) matrix of coefficients, \(\mathbf{x}_t = (\mathbf{y}_{t-1}^\top, \ldots, \mathbf{y}_{t-p}^\top, \mathbf{d}_t^\top)^\top\) is a \((Kp + L)\times 1\) vector of regressors. For estimation of VAR parameters from the observed time series data \(\mathbf{y}_1, \mathbf{y}_2, \ldots, \mathbf{y}_T\), we define data matrices as \[\begin{equation} \begin{split} \mathbf{Y} &= ( \mathbf{y}_{p+1}, \mathbf{y}_{p+2}, \ldots, \mathbf{y}_{T} )^\top \in \mathbb{R}^{N \times K}, \\ \mathbf{X} &= ( \mathbf{x}_{p+1}, \mathbf{x}_{p+2}, \ldots, \mathbf{x}_{T} )^\top \in \mathbb{R}^{N \times (Kp + L)}, \end{split} \end{equation}\] with \(N = T-p\). Then, we can rewrite Eq. (5) in a matrix form as \[ \mathbf{Y} = \mathbf{X} \mathbf{\Psi} + \mathbf{E} \in \mathbb{R}^{N \times K}, \qquad\qquad\qquad (6) \] with \(\mathbf{E} = (\boldsymbol\epsilon_{p+1}, \ldots, \boldsymbol\epsilon_T)^\top\) and \(N = T-p\).

==============================================================================================================
                                                                                                            
                                                              Model: 1
                                                                                                            
                                                                                                            
                                       Prior Distribution                    Posterior Draws: 25500
                                                                                                            
==============================================================================================================
Variable Name                       5%         50%        95%              5%        50%       95%         
==============================================================================================================
$\mu$                               -0.0066     0.0015     0.0097          0.0012     0.0016     0.0020
$\rho$                              -0.9000     0.0000     0.9000          0.8634     0.9094     0.9499
$\varphi_x$                          0.0500     0.5000     0.9500          0.5999     0.8132     0.9738
$\sigma$                             0.0008     0.0019     0.0061          0.0013     0.0016     0.0019
$\sigma_{eps}$                       0.0008     0.0019     0.0061          0.0018     0.0019     0.0021
$\rho_{g}$                          -0.9000     0.0000     0.9000          0.9417     0.9616     0.9792
$\varphi_{g}$                        5.0000    50.0000    95.0000          8.4946    11.2300    15.5129
$\alpha$                           -90.0000     0.0000    90.0000         -1.9842    -1.7575    -1.5244
posterior                            NaN        NaN        NaN       3701.5356  3705.3229  3707.2187
mdd                                   NaN        NaN        NaN             NaN  3671.0438        NaN
==============================================================================================================
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