Silviase/jgov_v1.4_full_ann huggingface dataset:
Solving for an equilibrium thus requires a search for a fixed point of the three policy functions, $ y(g(i), u(j)) $ , $ \pi(g(i), u(j)) $ and $ i(g(i), u(j)) $ at each node $ (i, j) $ of states $ u(i) $ and $ g(j) $ , using the global numerical procedure proposed by Billi (2011).

### 3.3 Application of an SREE to an MS-DSGE Model

## (1) Expectations Function of an MS-DSGE Model

The expectations function, $ E_{t}x_{t+1} $ , shown in the previous subsection, is derived from state vector $ z_{t} $ characterized by a continuous-valued random variable. Since we extend a standard DSGE model fixed under the unique regime into an MS-DSGE model explained in Section 2.1, one more discrete-valued state such as $ S_{t} $ , denoting the monetary policy regime in period t, has to be introduced into the domains of the expectations as well as the policy function. Accordingly, the expectations function in an MS-DSGE model can be rewritten as

$$ \begin{aligned}E_{t}x_{t+1}(z_{t+1},S_{t}=s)=&\sum_{r=1}^{L}p_{sr}\int_{\varepsilon_{t+1}}x(z_{t+1},S_{t+1}=r)f(d\varepsilon_{t+1})\\&\text{for}s,r=1,\cdots,L\end{aligned} $$

where $ x(z_{t}, S_{t}) $ is a policy function including the current regime state $ S_{t} $ , and $ p_{sr} $ denotes a transition probability such as $ p_{sr} = \operatorname*{Pr}(S_{t+1} = r \mid S_{t} = s) $ . L is the number of regimes. In the case of my model, the value of L is set to 2. In a similar way, the stochastic rational expectations of inflation and output shown in (7) and (8) are also rewritten as

$$ E_{t}\pi_{t+1}\big(u_{t},g_{t},S_{t}=s\big)=\sum_{r=1}^{L}p_{sr}\int_{\varepsilon_{gt+1}}\int_{\varepsilon_{ut+1}}\pi\big(u_{t+1},g_{t+1},S_{t+1}=r\big)f(d\varepsilon_{u,t+1})f(d\varepsilon_{g,t+1}), $$

$$ E_{t}y_{t+1}(u_{t},g_{t},S_{t}=s)=\sum_{r=1}^{L}p_{sr}\int_{\varepsilon_{gt+1}}\int_{\varepsilon_{ut+1}}y(u_{t+1},g_{t+1},S_{t+1}=r)f(d\varepsilon_{u,t+1})f(d\varepsilon_{g,t+1}). $$

$$ for s,r=1,\cdots,L $$

where exogenous shocks $ \varepsilon_{u,t} $ and $ \varepsilon_{g,t} $ are assumed to be independent of each other in order to make the model simple, so that the density functions are multiplied such as $ f(d\varepsilon_{u,t+1})f(d\varepsilon_{g,t+1}) $ on the right-hand side.

Meanwhile, perfect foresight expressions of expectations in an MS-DSGE model are obtained by

$$ E_{t}\pi_{t+1}(u_{t},g_{t},S_{t}=s)=\sum_{r=1}^{L}p_{s r}\pi(u_{t+1},g_{t+1},S_{t+1}=r), $$
$$ \begin{aligned}E_{t}y_{t+1}(u_{t},g_{t},S_{t}=s)&=\sum_{r=1}^{L}p_{sr}y\left(u_{t+1},g_{t+1},S_{t+1}=r\right),\\ for s,&r=1,\cdots,L\end{aligned} $$

where $ u_{t+1} = \rho_{u} u_{t} $ , $ g_{t+1} = \rho_{g} g_{t} $ , and the realized values of future shocks are set as $ \varepsilon_{t+1} = 0 $ . Davig and Leeper (2007) and Farmer et al (2009, 2011) solve an MS-DSGE model using the concept of perfect foresight. As shown in Section 4, there is no discrepancy in the calibration results between stochastic expectations and perfect foresight in the case using log-linearized models. However, if we add the model with the ZLB constraint, we can observe a non-negligible gap between them.

## (2) Discretization of the Expectations Function

Following the previous subsection, we transform the above expectations functions into a discrete version of those functions. Using the notation of policy functions with pointwise elements $ x(i,j) $ and letting $ x^{e} $ , denote one period ahead of expectations $ E_{t}x_{t+1} $ , the stochastic expectations conditional on the current regime S = s can be written as

$$ y^{e}(\mathbf{\alpha}(i),u(j)\mathbf{\alpha}S=s)=\sum_{r=1}^{L}p_{sr}\sum_{k=1}^{M}\sum_{l=1}^{M}\omega(k)\omega(l)y(\mathbf{\alpha}u^{e}(k,i),g^{e}(l,j),S^{e}=r) $$

$$ for s,r=1,\cdots,L and k,l=1,2,\cdots,M. $$

where $ g^{e} $ , $ u^{e} $ , and $ S^{e} $ are one period ahead of the expectations of the state variables. $ u^{e}(k,i)=\rho_{u}u(i)+\varepsilon_{u}(k) $ , and $ g^{e}(k,i)=\rho_{g}g(i)+\varepsilon_{g}(k) $ are calculated by the Gauss-Hermite quadrature scheme. Perfect foresight under the current regime S=s is similarly given by

$$ \begin{aligned}x^{e}(g(i),u(j),S=s)&=\sum_{r=1}^{L}p_{sr}x(u^{e}(i),g^{e}(j),S^{e}=r)\\&for s,r=1,\cdots,L\end{aligned} $$

where $ u^{e}(i) = \rho_{u}u(i) $ , and $ g^{e}(i) = \rho_{g}g(i) $ . The left-hand side in (22) is one period ahead of the expectations of the control variables, and the right-hand side is the policy function including one period ahead of the expectations of the exogenous shocks and the regime state as the domain.

## (3) Static One-Period Problem of an MS-DSGE Model Subject to the ZLB

Using discrete expectation functions (21) or (22) based on stochastic rational expectations or perfect foresight, we numerically determine a fixed point in the discrete space, $ N \otimes N $ , of policy functions. Using the notation of policy functions with pointwise elements and expectations functions, the new Keynesian
DSGE model with a two-state regime switching Taylor monetary policy rule constrained by the ZLB, i.e., equations (1) through (4) and (6) conditional on the current regime S = s, is transformed into a simple static one-period problem in which the first-order conditions are represented as

$$ y(g(i),u(j),S=s)=y^{e}(g(i),u(j),S=s)-\sigma\bigg(i(g(i),u(j),S=s)-\pi^{e}(g(i),u(j),S=s)\bigg)+g(i),\\ (23) $$

$$ \pi(g(i),u(j),S=s)=\beta\pi^{e}(g(i),u(j),S=s)+\kappa y(g(i),u(j),S=s)+u(j), $$

$$ i(g(i),u(j),S=s)=\max\Big\{-r^{*},\phi_{\pi}\pi(g(i),u(j),S=s)+\phi_{y}y(g(i),u(j),S=s)\Big\}, $$

$$ for s=1,\cdots,L. $$

Solving for an equilibrium thus requires a search for fixed point of the three policy functions, $ y(g(i), u(j), S = s) $ , $ \pi(g(i), u(j), S = s) $ and $ i(g(i), u(j), S = s) $ at each node $ (i, j, s) $ of states $ u(i) $ $ g(j) $ and S = s, using the global numerical procedure proposed by Billi (2011).

## 4 Calibration

### 4.1 Method

Based on the global numerical procedure $ ^{8} $ as described in Appendix I of Billi (2011), I calibrate the model explained in Section 2. However, I modify his algorithm to a more efficient method, which consists of the following three steps, to produce the convergence of policy functions much faster and more accurately.

- Step 1. Solve an MS-DSGE model without the ZLB and with perfect foresight.

- Step 2. Solve an MS-DSGE model with the ZLB and perfect foresight using the results of Step 1 as the initial values.
- Step 3. Solve an MS-DSGE model with the ZLB and stochastic rational expectations using the results of Step 2 as the initial values.

When calculating the policy functions, I fix the values of the policy functions for all three control variables at no shock as steady states of these variables, i.e., $ x(u(i)=0, g(j)=0) $ . Furthermore, I set the number of grids of the nodes in the state variables, such as $ g_{t} $ and $ u_{t} $ , as N=21, and the number of grids of the nodes in the Gauss-Hermite quadrature scheme as M=3. The convergence tolerance of the prediction error of the three controlled variables is set to $ 2.5\times10^{-3}\% $ at each grid in their policy functions. $ ^{9} $

### 4.2 Calibration Parameters

I calibrate the model to the US economy employing the main structural parameters $ (\beta, \sigma, \kappa) $ and shock parameters $ (\rho_{u}, \rho_{g}, \sigma_{u}) $ of Adam and Billi (2006, 2007) except for the standard deviation of the aggregate demand shock (or real rate shock), $ \sigma_{g} $ , which is twice their values so as to hit the ZLB frequently. $ ^{10} $ The steady states of these three variables are fixed at the same values as zero for all the regimes and models. The parameter values are summarized in Table 1 and serve as the baseline calibration of the model. The implied steady-state real interest rate for parameterization is 3.5 % annually.

Throughout this study, all the variables are expressed in terms of percentage point deviations from deterministic steady-state values. The interest rates and inflation rates are expressed in annualized percentage deviations, while the aggregate demand shock (or real rate shock) and the aggregate supply shock (or markup shock) are stated as quarterly percentages.

The time-varying coefficients $ (\phi_{\pi0}, \phi_{y0}, \phi_{\pi1}, \phi_{y1}) $ of the Taylor rule changed by regime shifts, except for transition probabilities $ (p_{11}, p_{22}) $ , are selected from the empirical results by Lubik and Schorfheide (2004). These values are used for calibrations by Davig and Leeper (2007) and Farmer et al. (2009, 2011). Under the fixed regime used for comparisons as shown in the next section, the policy parameters of the non-switching Taylor rule are adopted as the same values as those of the active regime of the MS-Taylor rule. The transition probabilities,

$$ e r r=\gamma^{k+1}-\gamma^{k} $$

$$ \gamma^{k+1}=\left(1-\lambda\right)\gamma^{k}+\lambda\Gamma(\gamma^{k}) $$
Table 1: Parameter Values

parameter	Economic interpretation	assigned value
$ \beta $	quarterly discount factor	0.9913 = $ (1 + \frac{3.5\%}{4})^{-1} $
$ \sigma $	real rate elasticity of output	6.25
$ \kappa $	slope of the Phillips curve	0.024
$ \phi_{\pi0} $	reaction coefficient of inflation under active regime	2.2
$ \phi_{y0} $	reaction coefficient of output under active regime	0.5
$ \phi_{\pi1} $	reaction coefficient of inflation under passive regime	0.8
$ \phi_{y1} $	reaction coefficient of output under passive regime	0.15
$ \rho_{u} $	AR-coefficient aggregate supply shocks	0.0
$ \rho_{g} $	AR-coefficient aggregate demand shocks	0.8
$ \sigma_{u} $	S.d. aggregate supply shock innovations (quarterly %)	0.154
$ \sigma_{g} $	S.d. aggregate demand shock innovations (quarterly %)	3.048
$ p_{11} $	transition probability from active regime to active regime	0.75
$ p_{22} $	transition probability from passive regime to passive regime	0.75

Notes: S.d. denotes standard deviation. The main structural parameters $ (\beta,\sigma,\kappa) $ and shock parameters $ (\rho_{u},\rho_{g},\sigma_{u}) $ are set based on Adam and Billi (2006, and 2007) except the standard deviation of aggregate demand shock (or real rate shock), $ \sigma_{g} $ , which is twice their values so as to hit the ZLB frequently. The time-varying coefficients $ (\phi_{\pi0},\phi_{y0},\phi_{\pi1},\phi_{y1}) $ of the Taylor rule changed by regime shifts are selected from empirical result by Lubik and Schorfheide (2004). Transition probabilities, $ p_{11} $ , $ p_{22} $ are set to 0.75 so as to provoke frequent regime changes in which average shifts are one out of four periods.
$ p_{11}, p_{22} $ , are set to 0.75 so as to provoke frequent regime changes in which average shifts are one out of four periods.

## 5 Policy Evaluation

### 5.1 Analyses of Policy Functions

Figures 3 (a) and (b) show the policy functions of three control variables, output, inflation and nominal interest rate, reacting to the aggregate demand shock differentiated by the absence or the presence of the ZLB. Panel (a) of this figure focuses on the active regime based on stochastic rational expectations, and panel (b) focuses on the passive regime under the same expectations. The solid line represents the policy functions under the ZLB, while the dashed line represents those not subject to the ZLB. As can be seen from Figures 3 (a) and (b), all the policy functions of both regimes under the non-ZLB dashed line are expressed as linear functions. In contrast, when the nominal interest rate hits the ZLB, i.e., $ i = -r^{*} $ , the reactions of output and inflation to the demand shock are kinked and induce a deep decline in both regimes; this is far from the case of linear RE models, as the solid lines show in both the figures.

Figures 4 (a) and (b) show how the gap in the policy functions reacted to the aggregate demand shock between stochastic rational expectations and perfect foresight under each regime. The solid line represents the policy functions based on stochastic expectations, while the dashed line embodies those based on perfect foresight. As seen in panel (a), the gaps in the output and inflation between them increase when the interest rate reaches the ZLB under the active regime. On the other hand, these big gaps are revealed around the lower nominal interest rate, even though it reaches the bound under the passive regime as shown in panel (b).

Figure 5 shows a comparison of the functions against the aggregate demand shock between the active policy regime (solid line) and the passive policy regime (dashed line). Panel (a) deals with the case of stochastic expectations. Panel (b) deals with that of perfect foresight. Despite the forms of expectations, the fluctuations of the two variables are much larger for both positive and negative shocks under the passive policy regime than under the active policy regime. However, a decrease in the two variables (the first and second graphs in both panels) under the active policy regime is likely to present the same degree as under the passive regime, when the aggregate shock falls below the -8% level at which the nominal interest rate reaches the ZLB, as shown in the third graph of both panels.

### 5.2 Comparison of the Impulse Response Functions

Using the policy functions represented in Figures 3 through 5, the impulse responses in the model are calculated. Figure 6 shows a comparison of three kinds of impulse response functions of the three variables to an aggregate demand shock conditional on the specified regimes. The solid line indicates the response
### Figure 3: Policy functions; non-ZLB vs. ZLB

(a) Under Active Regime

(b) Under Passive Regime

Notes: Solid and dashed lines denote reactions of the three endogenous variables derived from policy functions to aggregate demand shock conditional on that aggregate supply shock is set to zero, with the ZLB and without the ZLB, respectively. Horizontal axis shows the size of aggregate demand shock stated as quarterly percentages. Vertical axis shows the size of reaction of three endogenous variables expressed in terms of percentage point deviations from deterministic steady state values. Interest rates and inflation rates are expressed in annualized percentage deviations.
Figure 4: Policy functions; Stochastic Expectations vs. Perfect Foresight (a) Under Active regime

(b) Under Passive Regime

Notes: Solid and dashed lines denote reactions derived from policy functions of the three endogenous variables to aggregate demand shock conditional on that aggregate supply shock is set to zero, under stochastic rational expectations and under perfect foresight, respectively. Horizontal axis shows the size of aggregate demand shock stated as quarterly percentages. Vertical axis shows the size of reaction of three endogenous variables expressed in terms of percentage point deviations from the deterministic steady state values. Interest rates and inflation rates are expressed in annualized percentage deviations.
Figure 5: Policy functions; Active Policy Regime vs. Passive Policy Regime

(a) Stochastic Rational Expectation

(b) Perfect Foresight

Notes: Solid and dashed lines denote reactions derived from policy functions of the three endogenous variables to aggregate demand shock conditional on that aggregate supply shock is set to zero, under the active regime and under the passive regime, respectively. Horizontal axis shows the size of aggregate demand shock stated as quarterly percentages. Vertical axis shows the size of reaction of three endogenous variables expressed in terms of percentage point deviations from the deterministic steady state values. Interest rates and inflation rates are expressed in annualized percentage deviations.
conditional on the active regime, the dashed line the passive regime and the dotted line the fixed regime, in which the coefficients of the Taylor rule are fixed at the same values without regime switching. The responses to a positive demand shock are drawn in panel (a), in which the reactions of output and inflation are the largest under the passive regime. It is shown that the reactions under the active regime are rather larger than those under the fixed policy regimes. Although the descending order of the three regimes for these reactions to a positive shock is the same as in the case without the ZLB, which was pointed out by Davig and Leeper (2007), it indicates that a regime-switching monetary policy makes the economy fluctuate more than a fixed monetary policy. Meanwhile, in the case of the negative demand shock hitting the ZLB, the responses of those variables would drop by almost the same amount under all three regimes, as shown in panel (b).

Figure 7 shows a comparison of the impulse response functions of the three variables to the aggregate demand shock conditional on active regimes differentiated between stochastic rational expectations and perfect foresight. In panel (a), there is no gap in the reactions of all three variables to a positive shock between stochastic expectations and perfect foresight. This is because the model can be expressed as a linear one under this condition without the ZLB constraint. In contrast, we can observe that the reaction of these functions to the negative aggregate demand shock, which makes the interest rate affect the ZLB, obviously shows a deeper decline in the case of stochastic expectations than in the case of perfect foresight, as can be seen in panel (b).

### 5.3 Monte Carlo Study of the Model

### Active Regime vs. Passive Regime

Table 2 summarizes the results of 100,000 replications derived from policy functions under three regimes. Panels (a) and (b) of this table represent the results under the non-ZLB and the ZLB constraints, respectively. The third column shows the means and standard deviations of the three variables of Monte Carlo simulation under the active regime, while the fourth column shows the passive regime case. The fifth column shows the difference between the two regimes. The second row shows those with stochastic rational expectations and the third row those with perfect foresight. In panel (a), we can see that the means of all these variables are close to zero and that the standard deviations of output and inflation under the passive regime are much larger than those under the active regime. This indicates that their means tend to be consistent with their steady states, which are set as zero without the ZLB constraint. In addition, there are no differences in the means and the standard deviations between stochastic rational expectations and perfect foresight.

In Panel (b) of Table 2, the simulation results with the ZLB constraint are presented. The standard deviations of output and inflation are almost double the values appearing in panel (a). In addition, the means of output and inflation tend to be below those of their steady state. In particular, the means under the
Figure 6: Impulse response of Aggregate Demand Shock conditional on each Regime

(a) Positive Shock under Stochastic Rational Expectations

(b) Negative Shock under Stochastic Rational Expectations

Notes: Horizontal axis shows horizon of periods expressed in terms of quarterly in which the unity size of aggregate demand shock arises at the period 1. Vertical axis shows the size of reaction of three endogenous variables expressed in terms of percentage point deviations from deterministic steady state values. Interest rates and inflation rates are expressed in annualized percentage deviations.
Figure 7: Impulse response of Aggregate Demand Shock under each expectations (a) Positive Shock conditional on Active Policy Regime

(b) Negative Shock conditional on Active Policy Regime

Notes: Horizontal axis shows horizon of periods expressed in terms of quarterly in which the unity size of aggregate demand shock arises at the period 1. Vertical axis shows the size of reaction of three endogenous variables expressed in terms of percentage point deviations from deterministic steady state values. Interest rates and inflation rates are expressed in annualized percentage deviations.
Table 2: Monte Carlo Simulations conditional on Specified Regimes (a) the Case in absence of the ZLB constraint

Expectations	variables	Active	Passive	Differences
mean	Std Dev	mean	Std Dev	mean	Std Dev
Stoc. Expect.	output	0.00	1.27	0.00	2.87	0.00	-1.60
inflation	0.00	0.13	0.00	0.16	0.00	-0.03
interest rate	0.00	0.87	0.00	0.62	0.00	+0.25
Perfect Fore.	output	0.00	1.29	0.01	2.92	-0.01	-1.63
inflation	0.00	0.13	0.00	0.16	0.00	-0.03
interest rate	0.00	0.89	0.00	0.64	0.00	+0.25

(b) the Case in presence of the ZLB constraint

Expectations	variables	Active	Passive	Differences
mean	Std Dev	mean	Std Dev	mean	Std Dev
Stoc. Expect.	output	-0.93	3.01	-0.54	4.00	-0.39	-0.99
inflation	-0.02	0.20	-0.01	0.22	-0.01	-0.02
interest rate	0.13	0.80	0.07	0.63	+0.06	+0.17
Perfect Fore.	output	-0.57	2.85	-0.12	3.90	-0.45	-1.05
inflation	-0.01	0.19	0.00	0.21	-0.01	-0.02
interest rate	0.12	0.78	0.06	0.62	+0.06	+0.16

Notes: Means and standard deviations are calculated from 100,000 replications derived from policy functions of the three variables under each regime and under each expectations form. Stoc. Expect. and Perfect Fore. denote stochastic rational expectations and perfect foresight, respectively. Differences means subtraction of the value under the active regime from under the passive regime in each expectations form.
Figure 8: Intuitive Interpretation of Calibration Results; Active Policy vs. Passive Policy

(a) Case in Absence of the ZLB constraint

Notes: The above illustrations are summarized from the results of Table 2. The stripe shaded area and the dotted shaded area denote the distributions of endogenous variables under active regime and passive regime, respectively. The solid line and the dotted line denote the means of endogenous variables under active regime and passive regime, respectively.
Figure 9: Calibrations; Active Policy vs. Passive Policy (a) Case in Absence of the ZLB constraint

(b) Case in Presence of the ZLB constraint

Notes: Artificial data conditional to the specified policy regimes generated from the 100 iterations of Monte Carlo simulations are depicted by distinguishing them according to the presence or the absence of the ZLB, respectively. The same random seed is adopted in panel (a) and (b) for comparing the effect of the ZLB constraint, so that the same structural shocks would determine the endogenous variables in both graphs. Solid line and dashed line represent the active policy regime and the passive policy regime, respectively.
Table 3: Monte Carlo Simulations under Regime Switching and Fixed Policies (a) the Case in absence of the ZLB constraint

Expectations	variables	R.S. Policy	Fixed Policy	Differences
mean	Std Dev	mean	Std Dev	mean	Std Dev
Stoc. Expect.	output	-0.01	2.22	0.00	1.21	-0.01	+1.01
inflation	0.00	0.15	0.00	0.12	0.00	+0.03
interest rate	0.00	0.76	0.00	0.79	0.00	-0.03
Perfect Fore.	output	-0.01	2.21	0.00	1.21	-0.01	+1.01
inflation	0.00	0.14	0.00	0.12	0.00	+0.02
interest rate	0.00	0.75	0.00	0.79	0.00	-0.04

(b) the Case in presence of the ZLB constraint

Policy	variables	R.S. Policy	Fixed Policy	Differences
mean	Std Dev	mean	Std Dev	mean	Std Dev
Stoc. Expect.	output	-0.71	3.60	-0.91	2.91	+0.20	+0.69
inflation	-0.01	0.21	-0.02	0.18	+0.01	+0.03
interest rate	0.12	0.73	0.12	0.74	0.00	-0.01
Perfect Fore.	output	-0.38	3.40	-0.51	2.64	+0.13	+0.76
inflation	-0.01	0.20	-0.01	0.17	0.00	+0.03
interest rate	0.08	0.70	0.09	0.69	-0.01	+0.01

Notes: Means and standard deviations are calculated from 100,000 replications derived from policy functions of the three variables under each expectation form. R.S. Policy is a regime switching policy in which the Taylor rule switches between the active and the passive regime based on setting parameters. Fixed Policy is a policy in which coefficients of the Taylor rule are fixed at the same values as active regime without regime switching. Stoc. Expect. and Perfect Fore. denote stochastic rational expectations and perfect foresight, respectively. Differences means subtraction of the value under the active regime from under the passive regime in each expectation form.