### 2.4 Transformation to an estimated state space model

It is difficult directly to estimate state space representation (2.7), (2.8) and (2.9) as shown above applied to a large panel data set, as the size of the matrix in transition equations (2.8) and (2.9) is equal to the total number of state variables $ \bar{S}_{t} $ and measurement errors $ e_{t} $ . This framework induces a dramatic increase in the matrix as the number of data $ X_{t} $ increases. To avoid this situation, we transform to a small size for the transition equations. That is, we eliminate the AR process of measurement errors of (2.9) and express only $ \nu_{t} $ using the independent and identically distributed (i.i.d.) random variable process for measurement errors. Inserting (2.10) into (2.7), the measurement equation is transformed as follows:

$$ \begin{array}{r}{(\mathbf{I}-\mathbf{\Psi}L)\mathbf{X}_{t}=(\mathbf{I}-\mathbf{\Psi}L)\mathbf{\Lambda}\bar{S}_{t}+\mathbf{\nu}_{t},\qquad\mathbf{\nu}_{t}\sim\mathbf{i.i.d.}N(\mathbf{0},\mathbf{R}).}\end{array} $$

where L is the lag operator. By using the notations $ \tilde{X}_{t} = X_{t} - \Psi X_{t-1} $ , and $ \tilde{S}_{t} = [\bar{S}_{t}^{\prime} \bar{S}_{t-1}^{\prime}]^{\prime} $ , this equation can be rewritten as

$$ \tilde{\boldsymbol{X}}_{t}=\underbrace{[\boldsymbol{\Lambda}\quad-\boldsymbol{\Psi}\boldsymbol{\Lambda}]}_{\tilde{\boldsymbol{\Lambda}}}\underbrace{\left[\begin{array}{c}\bar{\boldsymbol{S}}_{t}\\ \bar{\boldsymbol{S}}_{t-1}\end{array}\right]}_{\tilde{\boldsymbol{S}}_{t}}+\boldsymbol{\nu}_{t},\qquad\boldsymbol{\nu}_{t}\sim\mathrm{i.i.d.}N(\mathbf{0},\mathbf{R}). $$

In a similar way, transition equation (2.2) is also rewritten as:

$$ \underbrace{\left[\begin{array}{c}\bar{\boldsymbol{S}}_{t}\\ \bar{\boldsymbol{S}}_{t-1}\end{array}\right]}_{\hat{\boldsymbol{S}}_{t}}=\underbrace{\left[\begin{array}{c c}\mathbf{G}(\boldsymbol{\theta})&\mathbf{O}\\ \mathbf{I}&\mathbf{O}\end{array}\right]}_{\hat{\mathbf{G}}}\underbrace{\left[\begin{array}{c}\bar{\boldsymbol{S}}_{t-1}\\ \bar{\boldsymbol{S}}_{t-2}\end{array}\right]}_{\hat{\boldsymbol{S}}_{t-1}}+\underbrace{\left[\begin{array}{c}\mathbf{H}(\boldsymbol{\theta})\\ \mathbf{O}\end{array}\right]}_{\hat{\mathbf{H}}}\boldsymbol{\varepsilon}_{t},\qquad\boldsymbol{\varepsilon}_{t}\sim\mathrm{i.i.d.}N(\mathbf{0},\mathbf{Q}(\boldsymbol{\theta})). $$

where I is a $ N \times N $ identity matrix. The estimation method of the data-rich DSGE model is explained using a state space model (2.12) and (2.13), and we estimate this model using the Bayesian method via the MCMC as described in Section 4. For convenience, we set the parameters of the measurement equation (2.12) as $ \Gamma = \{\Lambda, \Psi, R\} $ . The Bayesian estimation of parameters $ \Gamma $ follows Chib and Greenberg (1994), which is described in Section 4 and Appendix 2.

## 3 The DSGE Model

We use a medium-scale new Keynesian model with price and wage rigidities, capital accumulation, investment adjustment cost, and habit formation. The model is based on the work of Smets and Wouters (2003), Christiano et al. (2005) and Iiboshi et al. (2006). We present only the loglinearized equilibrium conditions, and refer the reader to the above-referenced papers for the derivation of these conditions.

For the sake of the Bayesian estimation, which requires the model to be in the linear state-space form, we log-linearize the above model around the steady state. Without laboring the derivation of the steady states and the log-linearization, we simply state the results of Smets and Wouters (2003) and Onatski and Williams (2010) here. The hat above a variable denotes log derivation from the steady state: i.e. $ \hat{x} = \ln x - \ln x^{ss} $ where $ x^{ss} $ is the steady state.