and

$$ \hat{Z}_{t}=\xi_{Z}\hat{Z}_{t-1}+\epsilon_{Z,t}, $$

where $ \epsilon_{C,t}\sim N(0,\sigma_{C,t}^{2}) $ , $ \epsilon_{\pi,t}\sim N(0,\sigma_{\pi,t}^{2}) $ , $ \epsilon_{I,t}\sim N(0,\sigma_{I,t}^{2}) $ , $ \epsilon_{Q,t}\sim N(0,\sigma_{Q,t}^{2}) $ , $ \epsilon_{G,t}\sim N(0,\sigma_{G,t}^{2}) $ , and $ \epsilon_{Z,t}\sim N(0,\sigma_{Z,t}^{2}) $ . Note that $ \Psi_{C}=\bar{C}/\bar{Y} $ , $ \Psi_{I}=\bar{I}/\bar{Y} $ , and $ \Psi_{G}=\bar{G}/\bar{Y} $ . Thus, the linearized TR is given by

$$ \hat{i}_{t}=\rho_{i}\hat{i}_{t-1}+(1-\rho_{i})(\phi_{Y}\hat{Y}_{t}+\phi_{\pi}\hat{\pi}_{t})+\epsilon_{i,t}, $$

where $ r^{s} $ is an equilibrium real rate and $ \pi^{s} $ is the target rate of inflation.

### 2.8 State Space Model

Structural linear rational expectations models are given by

$$ \mathbf{\Gamma}_{0}\mathbf{\boldsymbol{x}}_{t}=\mathbf{\Gamma}_{1}\mathbf{\boldsymbol{x}}_{t-1}+\mathbf{\Psi}\mathbf{\boldsymbol{z}}_{t}+\mathbf{\Pi}\mathbf{\boldsymbol{\eta}}_{t}+\mathbf{\boldsymbol{C}}, $$

where $ \pmb{x}_{t}=[E_{t}\hat{C}_{t+1},E_{t}\hat{\pi}_{t+1},E_{t}\hat{r}_{t+1}^{K},E_{t}\hat{I}_{t+1},E_{t}\hat{Q}_{t+1},\hat{Y}_{t},\hat{C}_{t},\hat{i}_{t},\hat{\pi}_{t},\hat{W}_{t},\hat{r}_{t}^{K}.\hat{Z}_{t},\hat{L}_{t},\hat{K}_{t},\hat{I}_{t},\hat{Q}_{t},\hat{G}_{t}]^{t} $

$ \mathbf{z}_{t}=\left(\epsilon_{C,t},\epsilon_{\pi,t},\epsilon_{I,t},\epsilon_{Q,t},\epsilon_{i,t},\epsilon_{G,t},\epsilon_{z,t}\right)^{T}\sim N(\mathbf{0},\mathbf{\Sigma}_{t}) $

with $ \mathbf{\Sigma}_{t}=diag((\sigma_{C,t})^{2},(\sigma_{\pi,t})^{2},(\sigma_{I,t})^{2},(\sigma_{Q,t})^{2},(\sigma_{i,t})^{2},(\sigma_{G,t})^{2},(\sigma_{z,t})^{2}) $ , $ \mathbf{z}_{t}^{\prime}=\left(\epsilon_{C,t},\epsilon_{\pi,t},\epsilon_{I,t},\epsilon_{Q,t},\epsilon_{G,t},\epsilon_{z,t}\right)^{T} $

$ \sim N(\mathbf{0},\mathbf{\Sigma}_{t}) $ with $ \mathbf{\Sigma}_{t}=diag((\sigma_{C,t})^{2},(\sigma_{\pi,t})^{2},(\sigma_{I,t})^{2},(\sigma_{Q,t})^{2},(\sigma_{G,t})^{2},(\sigma_{z,t})^{2}) $ , $ \Pi=0 $ , and $ C=0^{9} $ . Sims (2002) proposes the solution of linear rational expectations models using QZ decomposition $ ^{10} $ . Following Sims (2002), reduced linear rational expectations models are obtained by

$$ \boldsymbol{x}_{t}=\boldsymbol{\Theta}_{1}\boldsymbol{x}_{t-1}+\boldsymbol{\epsilon}_{1,t}, $$

where $ \epsilon_{1,t}=\Theta_{0}z_{t} $ and $ \epsilon_{1,t}^{\prime}=\Theta_{0}^{\prime}z_{t}^{\prime} $ . The symbols, $ \Theta_{1} $ , $ \Theta_{0} $ , $ \Theta_{1}^{\prime} $ , and $ \Theta_{0}^{\prime} $ are described in Sims (2002).

The measurement equation of the model is

$$ \mathbf{Y}_{t}=\mathbf{Y}^{s}+\mathbf{H}\mathbf{x}+\mathbf{v}_{t}, $$

$$ \mathbf{Y}_{t}=\left[YGR_{t},CGR_{t},IGR_{t},WGR_{t},INFL_{t},LGR_{t},INT_{t}\right]^{t},\mathbf{Y}^{s}=\left[Y^{s},Y^{s},Y^{s}+I^{s},Y^{s},\pi^{s},L^{s},r^{s}+\pi^{s}\right]^{t}. $$

$$ \boldsymbol{v}_{t}=\left(\epsilon_{Y,t}^{v},\epsilon_{C,t}^{v},\epsilon_{\pi,t}^{v},\epsilon_{I,t}^{v}\epsilon_{W,t}^{v},\epsilon_{L,t}^{v},\epsilon_{i,t}^{v}\right)^{T}\sim N(\mathbf{0},\mathbf{\Sigma}_{v,t}) $$

with $ \mathbf{\Sigma}_{v,t}=diag((\sigma_{Y,t}^{v})^{2}(\sigma_{C,t}^{v})^{2},(\sigma_{I,t}^{v})^{2},(\sigma_{\pi,t}^{v})^{2},(\sigma_{W,t}^{v})^{2},(\sigma_{L,t}^{v})^{2},(\sigma_{i,t}^{v})^{2})^{11} $ . The growth rate of real variables, $ YGR_{t} $ , $ CGR_{t} $ , $ IGR_{t} $ , $ WGR_{t} $ , and $ LGR_{t} $ , are log differences of real GDP per capita, real consumption per capita, real investment per capita, real average wage, and average labor hours, respectively. The rate of inflation, $ INF_{t} $ , is the log difference of GDP deflator, and the nominal interest rate, $ INT_{t} $ , is the uncollateralized overnight call rate. Any observations are annualized. The symbols, $ Y^{s} $ , $ \pi^{s} $ , $ L^{s} $ and $ r^{s} $ are the growths of real output, the target rate of inflation, the trend of labor, and the trend of real interest rates, respectively. The term $ I^{s} $ denotes the investment-specific growth rate $ ^{12} $ .