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.