Silviase/jgov_v1.4_full_ann huggingface dataset:
DSGE model extended from DFM and its application to measuring the effects of monetary policy.

#### 2.1.1 State Space Representation of Dynamic Factor Model (DFM)

DFMs, which are a statistical model estimating common factors of business cycles, are proposed by Sargent and Sims (1977) and empirically applied by Stock and Watson (1989), who extract one unobserved common factor of business fluctuation from many macroeconomic time series using a Kalman filter. $ ^{3} $

The DFMs are represented by state space models composed from the following three linear equations. Let $ F_{t} $ denote the $ N \times 1 $ vector of unobserved common factor, and $ X_{t} $ denote the $ J \times 1 $ vector of massive panel of macroeconomic and financial data. Notice that the number of data is much more than those of factors; i.e., $ J \gg N $ .

$$ \underbrace{\boldsymbol{X}_{t}}_{J\times1}=\underbrace{\boldsymbol{\Lambda}}_{J\times N}\underbrace{\boldsymbol{F}_{t}}_{N\times1}+\underbrace{\boldsymbol{e}_{t}}_{J\times1}, $$

$$ \underbrace{\boldsymbol{F}_{t}}_{N\times1}=\underbrace{\boldsymbol{\mathbf{G}}}_{N\times N}\underbrace{\boldsymbol{F}_{t-1}}_{N\times1}+\underbrace{\boldsymbol{\varepsilon}_{t}}_{N\times1}\qquad\boldsymbol{\varepsilon}_{t}\sim\mathbf{i}.\mathbf{i}.\mathbf{d}.N(\mathbf{0},\mathbf{Q}), $$

$$ \underbrace{e_{t}}_{J\times1}=\underbrace{\Psi}_{J\times J}\underbrace{e_{t-1}}_{J\times1}+\underbrace{\nu_{t}}_{J\times1}\qquad\nu_{t}\sim\mathbf{i.i.d.}N(\mathbf{0},\mathbf{R}), $$

where $ \Lambda $ is $ J \times N $ matrix of factor loadings, $ e_{t} $ is the idiosyncratic errors which is allowed to be serially correlated as equation (2.3). G is $ N \times N $ matrix, and common factor $ F_{t} $ is following AR process (2.3). Matrices, $ \Psi $ , Q and R are assumed to be diagonal in an exact DFM as Stock and Watson (2005). Equation (2.1) is a measurement equation, and equations (2.2) and (2.3) are transition equations. A state space model is composed from the measurement and transition equations (2.1), (2.2) and (2.3).

The advantage of the model is to extract common components, $ \LambdaF_{t} $ , and idiosyncratic component $ e_{t} $ from the massive panel of macroeconomic and financial data $ X_{t} $ . Meanwhile, it is difficult to make an interpretation of factor $ F_{t} $ economically, since the above equations are statistically estimated like reduced models and the parameters are not derived from economic foundation like structural models are. A data-rich type DSGE-DFM, however, holds the
advantage of DFM and overcomes the drawback of DFM, as explained below.

#### 2.1.2 FAVAR Approach for Measuring the Effects of Monetary Policy

To defeat the drawbacks of VAR analysis for measuring the effect of monetary policy as mentioned in Section 1, Bernanke, Boivin and Eliasz (2005) proposed a Factor Augmented VAR (FAVAR) approach for measuring this by modifying from a DFM (2.1), (2.2), (2.3), to a FAVAR model (2.1') and (2.2') in which unobservable common factors $ F_{t} $ are put in a VAR model. And the effect of monetary policy on affluent kinds of macroeconomic series $ X_{t} $ are measured from innovation analysis using tools such as impulse response functions and variance decomposition of a FAVAR model.

$$ \underbrace{\mathbf{X}_{t}}_{J\times1}=\underbrace{\boldsymbol{\Lambda}^{\mathbf{f}}}_{J\times N}\underbrace{\mathbf{F}_{\mathbf{t}}}_{N\times1}+\boldsymbol{\Lambda}^{y}\mathbf{Y}_{\mathbf{t}}+\mathbf{e}_{t}, $$

$$ \left[\begin{array}{l}F_{t}\\ Y_{t}\end{array}\right]=\beta_{0}+\sum_{i=1}^{k}\beta_{i}\mathbf{F}_{\mathbf{t}-\mathbf{i}}+\sum_{j=1}^{p}\gamma_{j}\mathbf{Y}_{\mathbf{t}-\mathbf{j}}+\boldsymbol{\varepsilon}_{t},\boldsymbol{\varepsilon}_{t}\sim\mathrm{i.i.d.}N(0,\mathbf{Q}), $$

where $ Y_{t} $ is observable macroeconomic variables such as output gap, money stock, interest rate, inflation rate and unemployment rate, excluding macroeconomic indicator $ X_{t} $ . What variables to select as $ Y_{t} $ depends on what macroeconomic model is adopted as a underlying structural VAR model. To measure the effect of monetary policy on macroeconomic series $ X_{t} $ , they estimated a reduced VAR (2.2') and transformed a structural VAR based on macroeconomic model. Bernanke, Boivin and Eliasz (2005) employed a Cholesky decomposition for conversion to a structural VAR following Christiano, Eichenbaum and Evans (1999). From the structural VAR, they calculated impulse response function of monetary shock $ \varepsilon_{t}^{R} $ on observable variables $ Y_{t} $ and common factors $ F_{t} $ . And also, impulse response functions of $ \varepsilon_{t}^{R} $ on macroeconomic indicators $ X_{t} $ are derived through influence of $ Y_{t} $ and $ F_{t} $ on $ X_{t} $ using measurement equation (2.1). According to Bernanke, Boivin and Eliasz (2005), price puzzle, which means that impulse response of upward interest rate shock $ \varepsilon_{t}^{R} $ shows an increase of price level in contradiction to macroeconomic theory, cannot be found in FAVAR, whereas it is generally found in a VAR. Accordingly, they concluded that common factors $ F_{t} $ include useful information that observable variables $ Y_{t} $ used for a macroeconomic model do not include. Shibamoto (2007) applied their methods to the Japanese economy.

The FAVAR approach, however, has two shortcomings by caused from re-
remaining largely non-structural. First, it does not provide the appropriate strategy for identifying policy shocks in terms of consistent economic views. It implies that wrong identification could lead to wrong impulse response function of policy shocks if FAVAR fails to identify the appropriate policy shocks. Second, the FAVAR, like the standard VAR, addresses only the effects of unanticipated changes in monetary policy, not the arguably more important effects of the systematic portion of monetary policy or the choice of monetary policy rule. However, these two drawbacks are able to be resolved if a VAR model is replaced by a DSGE model constructed from structural way in a data rich environment.

#### 2.1.3 State Space Representation of DSGE-DFM

A DSGE model is also known to be a state space model and estimates using a Kalman filter as well as the DFM. So the framework of the DFM can be applied to estimating a DSGE model. But the discrepancy between a DFM and a DSGE model is the interpretation of their parameters and state variables (or common factors) $ F_{t} $ . The parameters of the DSGE model have economic meanings based on forward-looking macroeconomic model in which economic agents such as consumers and firms solve their intertemporal optimization problems with their rational expectations. In particular, its core parameters are referred to as deep parameters ( $ \theta $ ) which govern the rational behaviors of economic agents. In a DSGE model, transition equation (2.5) shows the law of equilibrium motion around steady state solved from log-linear approximation. So state variables $ S_{t} $ used in the DSGE model denote “model concepts” abstracted from macroeconomic aggregates variables such as output gap, inflation, consumption, etc. In contrast, non-deep parameters are based solely on a statistical backward-looking time-series model, and state variables $ F_{t} $ is represented as just common factors extracted statistically from panel data $ X_{t} $ in a DFM.

The statistical framework of a data-rich DSGE is shown as (2.4), (2.5) and (2.6), similar to the state space form of a DFM (2.1), (2.2) and (2.3), but including deep parameters $ \theta $ in (2.5).
variables $ S_{t} $ and measurement errors $ e_{t} $ . This framework induces a dramatically increase of the matrix as the number of data $ X_{t} $ is increasing. To avoid this situation, we try to transform to small size for the transition equations as following. That is, we eliminate AR process of measurement errors of (2.6) and express from only $ \nu_{t} $ with iid process for measurement errors. Inserting (2.6) into (2.4), measurement equation is transformed as

$$ \left(\mathbf{I}-\mathbf{\Psi}L\right)X_{t}=\left(\mathbf{I}-\mathbf{\Psi}L\right)\mathbf{\Lambda}\mathbf{S}_{t}+\nu_{t},\qquad\nu_{t}\sim\mathrm{i.i.d.}N(0,R). $$

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

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

In the similar way, transition equation $ (2.5) $ is also rewritten as

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

where I is a $ J \times J $ identity matrix. Estimation method of data rich DSGE model is explained using state space model (2.7) and (2.8) in Section 2.3. For convenience, we set parameters of measurement equation (2.7) as $ \Gamma = \{\Lambda, \Psi, R\} $ . And Bayesian estimation of parameters $ \Gamma $ are following Chib and Greenberg (1994) which is described in Section 2.3 and the appendix A2.

#### 2.1.4 Two Types of Data Indicator $ X_{t} $

For the data rich framework, it is important that how to match data indicator to common factors $ F_{t} $ (or model concepts $ S_{t} $ ) in measurement equation (2.1) and (2.4). In (2.1) of the DFM, state variables $ F_{t} $ is regarded as common factor from the point of views of statistics, so that it is not needed to specify coefficient matrix $ \Lambda $ which connects between $ X_{t} $ and $ F_{t} $ to identify the magnitude and sign of common factors $ F_{t} $ . In contrast, we need to specify a part of coefficient matrix $ \Lambda $ to identify economic concepts $ S_{t} $ in (2.4) of the data-rich DSGE model. To do so, data indicator $ X_{t} $ is divided into two types in terms of specification of $ \Lambda $ . The first type of data indicator $ X_{t} $ is referred to as sensor series
which is likely to be embodied for specific model concept and link to just one variable of the concepts $ \bar{S}_{t} $ . For example, for model concepts of output and inflation, real GDP and GDP deflator are selected. The other type is referred to as information series which do not need to specify the magnitude and sign of $ \Lambda $ from economical view and are not directly connected with specific one of model concepts $ \bar{S}_{t} $ but hold useful information on a number of model concepts $ S_{t} $ in the DSGE model. In contrast to common factors $ F_{t} $ in DFM, there are two types in model concepts $ S_{t} $ ; one is specific concept directly linking to data indicators, and the other does not link to data indicator. In order to represent characteristics of the types of data indicator and model concepts, we rewrite(2.4) as (2.9).

$$ \underbrace{\left[\begin{array}{c}{\mathbf{X}_{t}^{s e n s o r}}\\ {\quad\ldots\quad\ldots\quad\ldots\quad\ldots}\\ {\mathbf{X}_{t}^{i n f o}}\end{array}\right]}_{J\times1}=\underbrace{\left[\begin{array}{c c c c}{\mathbf{\Lambda}^{s e n s o r}}&{|}&{\mathbf{0}}\\ {\quad\ldots\quad\ldots\quad\ldots}&{\quad\ldots}&{\quad\ldots\quad\ldots}\\ {\mathbf{\Lambda}^{i n f o}}&{|}&{\mathbf{0}}\end{array}\right]}_{J\times N}\underbrace{\left[\begin{array}{c}{\mathbf{S}_{t}^{l i n k}}\\ {\mathbf{S}_{t}^{n o n}}\end{array}\right]}_{N\times1}+\underbrace{\mathbf{e}_{t}}_{J\times1}, $$

where $ X_{t}^{sensor} $ and $ X_{t}^{info} $ denote sensor series and information series, respectively. Matrices $ \Lambda^{sensor} $ and $ \Lambda^{info} $ are part of coefficient matrix $ \Lambda $ which connects state variables $ S_{t} $ with sensor series and information series, respectively. Let $ S_{t}^{link} $ and $ S_{t}^{non} $ denote $ N^{link} \times 1 $ and $ N^{non} \times 1 $ vectors of model concepts depending on whether they link with data or not, respectively.

Here focus on identification of model concepts (or state variables) $ S_{t} $ using sensor series. Since sensor series $ X_{t}^{sensor} $ correspond with model concepts $ S_{t}^{link} $ through one-to-one relation, we set a part of coefficient matrix $ \Lambda^{sensor} $ to unity in order to normalize the size of i-th model concept $ S_{t}^{link} $ as

$$ X_{it}^{senseor}=S_{it}^{link}+e_{it},\quad i=1,2,\cdots,N^{link}. $$

And for sensor series $ X_{j}^{sensor} $ , $ j > N^{link} $ , we specify i-th linkage model concept $ S_{it}^{link} $ for j-th data indicator as

$$ X_{j t}^{s e n s o r}=\Lambda_{j i}^{s e n s o r}S_{i t}^{l i n k}+e_{j t},\quad j>N^{l i n k}. $$

Accordingly, for a vector of sensor series $ X_{t}^{sensor} = [X_{1,t}^{sensor}, X_{2,t}^{sensor}, \cdots, X_{p,t}^{sensor}]' $ , we can represent the matrix of coefficients $ \Lambda^{sensor} $ as
$$ \mathbf{\Lambda}^{sensor}=\underbrace{\left[\begin{array}{c}\mathbf{I}\\ \mathbf{\Lambda}_{21}\\ \vdots\\ \mathbf{\Lambda}_{P1}\end{array}\right]}_{L\times N^{link}}, $$

where I is $ N^{link} \times N^{link} $ identity matrix, $ \Lambda_{P1} $ is $ N^{link} \times N^{link} $ diagonal matrix, P is the number of set matching model concepts to sensor series and L denotes the number of sensor series $ X_{t}^{sensor} $ so that $ L = P \times N^{link} $ . On the other hand, the relationship with data indicator of information series $ X_{t}^{info} $ and state variables $ S_{t} $ are similar to a DFM so that matrix $ \Lambda^{info} $ has full elements without zeros estimated from principle component analysis of statistical theory and no restriction of the sign and size of $ \Lambda^{info} $ from economic theory.

Alternatively, representing matrix-base equation (2.9) as element-base equation (2.10), the framework of a data-rich DSGE model might be more understandable. As can be seen from the second row of matrix $\Lambda(\theta): [\lambda_{y2} \ 0 \ \cdots \ 0]$, sensor series of data is directly related with only one linkage model concept $S_{t}^{link}$. And in order to identify the magnitude of each linkage model concept $S_{t}^{link}$, the value of $\lambda$ of just one variable of sensor series $X_{t}^{sensor}$ is unity as the first $N^{link}$ rows of matrix $\Lambda: N^{link} \times N^{link}$ identity matrix. Meanwhile, the parameters of information series can be represented as full elements in matrix $\Lambda$ such as $\left[\lambda_{11} \lambda_{12} \cdots \lambda_{1n}\right]$ which connects all of data indicators $X_{t}$ through many-to-many relation with all model variable $S_{t}$.
particularly important to determine more accurately the state of the economy, and helps in forecasting.

Third, a DSGE-DFM model can be interpreted as a DFM in which we impose the full structure of the DSGE model on the transition equation of the latent common factors. This has the added benefit of allowing us to provide a very clear economic interpretation of all estimated latent common factors, i.e., these factors correspond to model variables of the DSGE model. In contrast, in standard DFMs, the latent factors do not have a clear interpretation, since they are identified only “up to a rotation”.

#### 2.1.5 DSGE-DFM Approach for Measuring the Effects of Monetary Policy

Instead of FAVAR (2.1') and (2.2'), using deep and non-deep parameters, endogenous variables and exogenous shocks estimated from a DSGE-DFM (2.1), (2.2) and (2.3), I propose to measure the effects of monetary policy as explained in Section 2.4. DSGE-DFM can resolve two problems of the FAVAR approach pointed out by Bernanke, Boivin and Eliasz (2005), as mentioned above, since DSGE-DFM estimates and identifies deep parameters and structural shocks in terms of theoretical coherence. This point is the distinction of the DSGE-DFM approach against the FAVAR approach. That is, since DSGE-DFM identifies not only monetary policy shocks but also all structural shocks in terms of macroeconomic model, this model can illustrate historical decomposition and variance decomposition as well as impulse responses of all structural shocks. This outstanding feature makes it possible to analyze business cycles in terms of economic theoretical views, and I show this example in Section 3.2. As a benchmark model I adopt a Smets and Wouters type DSGE model, which is described in Section 2.2. The estimation method is dealt with in Section 2.3. In Section 2.4, innovations analysis such as impulse response functions and variance decomposition, and historical decomposition derived from exogenous shocks are explained as policy simulation for measuring the effects.

### 2.2 The DSGE model

We use a medium-scale new Keynesian DSGE model, which is derived from a microfoundation that combines forward-looking rational behavior of households and firms with backward-looking institutional frictions. My DSGE model is based on the works of Smets and Wouters (2003, 2007), Christiano et al.
(2005) and Iiboshi, Nishiyama and Watanabe (2006), in which there is assumed to be the presences of (1) price and wage rigidities of Calvo style, (2) capital accumulation, and adjustment cost of investment (an increment of capital), and (3) habit formation of consumption, as backward-looking factors. 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 Bayesian estimation of structural parameters of our model, we log-linearize the above model around the steady state. Without explaining the derivation of the steady states and the log-linearization, we simply state the results from Smets and Wouters (2003, 2007) and Onatski and Williams (2010) as the following Sections 2.2.1 through 2.2.4. The hat above a variable denotes log derivation from steady state: i.e. $ \hat{x} = \ln x - \ln x^{ss} $ where $ x^{ss} $ is steady state. And using Sims' (2002) solving method to rational expectation equilibrium as described in Section 2.2.5, the law of motions converged to equilibrium is derived from simultaneous equations of first-order approximation around the steady state. These reduced forms are transition equations (2.5) in Section 2.1.

#### 2.2.1 Equilibrium Conditions from Housing/Investor Sector

(1) Consumption Euler equation:

$$ \hat{c}_{t}=\frac{h}{1+h}\hat{c}_{t-1}+\frac{1}{1+h}E_{t}\hat{c}_{t+1}-\frac{1-h}{(1+h)\sigma_{c}}(\hat{R}_{t}-E_{t}\hat{\Pi}_{t+1})+\frac{1-h}{(1+h)\sigma_{c}}(u_{t}^{c}-E_{t}u_{t+1}^{c}), $$

where an additional assumption that the external habit stock $ H_{t} $ is proportional to aggregate past consumption: $ H_{t} = h c_{t-1} $ , is introduced where h denotes the habit persistence parameter. When h = 0, equation (2.11) reduces to the traditional forward-looking consumption equation. With external habit formation, consumption depends on a weighted average of past and expected future consumption. Note that in this case the interest elasticity of consumption depends not only on the inter-temporal elasticity of substitution ( $ \sigma_{c} $ ), but also on habit persistence parameter. A high degree of habit persistence will tend to reduce the impact of the real rate on consumption for a given elasticity of substitution. A persistent shock $ u_{t}^{c} $ is AR(1) process with coefficient $ \rho^{c} $ , and, therefore, the expected value $ E_{t}u_{t+1}^{c} $ can be rewritten as $ \rho^{c}u_{t}^{c} $ .
(2) Investment Euler equation

$$ \widehat{i n v}_{t}=\frac{1}{1+\beta}\widehat{i n v}_{t-1}+\frac{\beta}{1+\beta}E_{t}\widehat{i n v}_{t+1}+\frac{\varphi}{1+\beta}\hat{q}_{t}+\frac{\beta}{1+\beta}(E_{t}u_{t+1}^{i n v}-u_{t}^{i n v}) $$

where we set $ \varphi = 1/\bar{S}'' $ and the inverse, $ 1/\varphi $ , implies the elasticity of investment on the price of capital. Modeling the capital adjustment cost as a function of the change in investment rather than its level introduces additional dynamics into the investment equation, which is useful in capturing the hump-shaped response of investment to various shocks including monetary policy shocks. A positive shock to the adjustment cost function, $ u_{t}^{inv} $ , temporarily reduces investment. The expected value $ E_{t}u_{t+1}^{inv} $ is set as $ \rho^{inv}u_{t}^{inv} $ using AR(1) coefficient $ \rho^{inv} $ .

(3) Asset Pricing Euler equation:

$$ \hat{q}_{t}=-(\hat{R}_{t}-E_{t}\hat{\Pi}_{t+1})+\frac{1-\tau}{1-\tau+\overline{r}^{k}}E_{t}\hat{q}_{t+1}+\frac{\overline{r}^{k}}{1-\tau+\overline{r}^{k}}E_{t}\hat{r}_{t+1}^{k}+\varepsilon_{t}^{q} $$

where $ \beta = 1/(1 - \tau - \bar{r}_{k}) $ , $ \tau $ is the depreciation rate, and $ \bar{r}_{k} $ is steady-state rental rate. The current value of the capital stock, Tobin's q, depends negatively on the ex ante real interest rate, and positively on its expected future value and the expected rental rate. The introduction of a white noise shock to the required rate of return on equity investment, $ \varepsilon_{t}^{q} $ , is meant as a shortcut to capture changes in the cost of capital that may be due to stochastic variations in the external finance premium.

(4) Real wage law of motion:

$$ \begin{aligned}\hat{w}_{t}&=\frac{\beta}{1+\beta}E_{t}\hat{w}_{t+1}+\frac{1}{1+\beta}\hat{w}_{t-1}+\frac{\beta}{1+\beta}E_{t}\hat{\Pi}_{t+1}-\frac{1+\beta\gamma_{w}}{1+\beta}\hat{\Pi}_{t}+\frac{\gamma_{w}}{1+\beta}\hat{\Pi}_{t-1}\\&-\frac{\lambda_{w}(1-\beta\xi_{w})(1-\xi_{w})}{(1+\beta)(\lambda_{w}+(1+\lambda_{w})\sigma_{L})\xi_{w}}\left[\hat{w}_{t}-\sigma_{L}\hat{l}_{t}-\frac{\sigma_{c}}{1-h}(\hat{c}_{t}-h\hat{c}_{t-1})-u_{t}^{L}-\varepsilon_{t}^{w}\right]\end{aligned} $$

The real wage, $ \hat{w}_{t} $ , is a function of expected and past real wages and the expected, current, and past inflation rate, where the relative weight depends on the degree of indexation of the non-optimized wages, $ \gamma_{w} $ . When $ \gamma_{w} = 0 $ , real wages do not depend on the lagged inflation rate. The last term implies a negative effect of the deviation of the actual real wage from the wage that would prevail in a flexible labor market. The size of this effect will be greater, the smaller the degree of wage rigidity, the lower the demand elasticity for