How to call the estimation component separately in dynare?

Estimation ML/Bayesian estimation
1

Dear all,

I want to estimate a modified model declared by the ‘model’ module. If the linear rational expectation model imply by the ‘model’ module is

\hat{y}_t^w = T \hat{y}_{t+1}^w + R \omega_t

then the model that i need to estimate is

\hat{\mathbf{y}}_t^w = \mathbf{T}\hat{\mathbf{y}}_{t-1}^w + \mathbf{R}\omega_t + \mathbf{R}[ \mathbf{0}_{n\times1} \ a_{t-1} \ 0 ]^{\prime}

This idea comes from the paper “Ambiguous Business Cycles” of Ilut and Schneider (AER, 2014).

Can this be achieved by relying on dynare’s built-in functions?

2

So you just want to run the Kalman filter and an MCMC, providing the solution matrices yourself without solving the model in Dynare?

3

Dear professor jpfeifer

Thank you for your reply.

If possible, I would like to solve the model using dynare and manually modify the dynare’s solution matrixes and then run the Kalman filter and an MCMC.

4

Can you describe the exact problem in more detail. How do you want to modify the solution matrices?

5

Dear professor jpfeifer

Thank you for taking precious time to help me.

My aim is to characterize agents’ cautious expectations of future TFP, i.e., the “worst case belief”. Under this expectation, total factor productivity Z obeys,

\log Z_t = \rho_z \log Z_{t-1} -a_{t-1} + \varepsilon_{z,t}\\ a_t -\bar{a} = \rho_a(a_{t-1}-\bar{a}) +\varepsilon_{a,t}

But the true DGP of TFP in the eyes of econometricians is,

\log Z_t = \rho_z \log Z_{t-1} + \varepsilon_{z,t}\\

So the relationship between expectation of future TFP and the true is:

z_t = \mathbb{E}^*_{t-1}z_{t} + a_{t-1} +\varepsilon_{z,t}\qquad (a1)\\ z_t = \log Z_t

As illustrated by Ilut and Schneider (AER, 2014), the solution can be:

  • step1: Find the deterministic “worst case steady state” .
\bar{s}_t^0 := (\bar{s}_{n \times 1}, \bar{z}, \bar{a})\\ \bar{z} = \frac{-\bar{a}}{1-\rho_z }

\bar{s}_t^0 denotes steady state values of exogenous variables. And there are other endogenous variables.

  • step2: Linearize the model around the “worst case steady state.”（first order）

Based on Sims (2002), the canonical form is,

\Gamma_0 \hat{\mathbf{y}}_t^0 = \Gamma_1 \hat{\mathbf{y}}_{t-1}^0 + \Psi \omega_t + \Upsilon\eta_t,

then the solution is,

\hat{\mathbf{y}}_{t}^{0} = \mathbf{T}\hat{\mathbf{y}}_{t-1}^{0} + \mathbf{R}\omega_{t},
  • step3: Consider now the dynamics of the model from the perspective of the
    econometrician.

Based on equation (a1) ,the equilibrium law of motion under the econometrician’s data
generating process (DGP) is given by

\hat{\mathbf{y}}_t^0 = \mathbf{T}\hat{\mathbf{y}}_{t-1}^0 + \mathbf{R}\omega_t + \mathbf{R}[\mathbf{0}_{n\times1} \ a_{t-1} \ 0]^{\prime}.
  • step4: Find the Dynamics around the so called “zero risk steady state.”

Solve the following equation to obtain \overline{\mathbf{y}},

\mathbf{\overline{y}}-\mathbf{\overline{y}}^0 = \mathbf{T}(\mathbf{\overline{y}}-\mathbf{\overline{y}}^0) + \mathbf{R}[\mathbf{0}_{n\times1} \ \bar{a}\quad0]^{\prime}.

Finally, the following equation is obtained.

\hat{\mathbf{y}}_t = \mathbf{T}\hat{\mathbf{y}}_{t-1} + \mathbf{R}\omega_t + \mathbf{R} [ \mathbf{0}_{n\times1}\ \hat{a}_{t-1}\ 0 ]^{\prime}.\\ \hat{\mathbf{y}}_t = \mathbf{y}_t - \mathbf{\overline{y}}

I’m not sure I’ve made the problem and the goal of the matrix treatment clear. Please let me know if there are any issues that I have failed to clarify.