Silviase/jgov_v1.4_full_ann huggingface dataset:
### Appendix B. Monthly Survey of Japanese Economic Forecasts (ESP Forecast, or ESPF)

The ESPF, the first regular publication to cover economic forecasts produced by business and academic economists in Japan, was launched in May 2004 after a trial survey in April. $ ^{A2-1} $ The Economic Planning Association, a public-service corporation established with the authorization of the Japanese government in 1965, distributes questionnaires to participants around the $ 25^{th} $ of each month and publishes the result around the $ 10^{th} $ of the following month. Participants are requested to provide their annual forecasts of 16 variables for the current and next fiscal year (from April to March) and their quarterly forecasts of three macro variables during the coming two fiscal years. In addition, the survey contains a number of questions asking for respondents' judgment on certain topical issues (See Table A.2. for details on the questions). The number of participants was 38 at the start and as of early 2009 had remained more or less unchanged.

The design of the ESPF was modeled on the Blue Chip Economic Indicators in the United States. This is reflected in the frequency of publication, the choice of forecasted variables (especially in the annual forecast), and the forecast period of two years. A difference is that the number of variables forecasted quarterly is much smaller in the ESPF than in the Blue Chip Economic Indicators. This is to lighten the burden on forecasters participating in the survey. Another difference is that, in contrast with the Blue Chip survey, the ESPF maintains respondents' anonymity, based on the reasoning that anonymity may make it more likely that participants reveal their true forecasts.

We converted the quarterly forecasts into our monthly forecasts in this paper as follows: first, we assume a quarterly forecast to be a monthly one for the second month of the quarter; second, we calculate monthly figures for the other months in the quarter by taking weighted averages of two consecutive quarterly forecasts. To be more specific, suppose t is February 2008. Then $ \pi_{t,t+12} $ is set to equal $ \pi_{08Q1,09Q1} $ available in the ESPF. As for the forecast for January 2008, $ \pi_{t,t+12} $ is calculated as $ (1/3)\times\pi_{07Q4,08Q4}+(2/3)\times\pi_{08Q1,09Q1} $ .
### Appendix C. Timing of MCCS and ESPF Publication

Please refer to Table A.3. for the timing of the publication of the MCCS and the ESPF.
Table 1. Expected annual inflation (survey responses) and realized annual CPI inflation rate

Case 1: (-0.1, 0.1)	Expected inflation (survey response)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	Marginal
Realized CPI inflation rate	(1)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
(2)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
(3)	0.11	0.38	1.72	12.15	8.37	4.39	0.93	2.12	30.16
(4)	0.13	0.47	1.70	12.24	11.01	7.14	1.87	2.65	37.21
(5)	0.15	0.36	0.80	5.08	6.68	8.49	4.85	1.89	28.30
(6)	0.03	0.04	0.06	0.20	0.52	1.54	1.65	0.29	4.34
(7)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Marginal	0.41	1.25	4.28	29.67	26.58	21.57	9.30	6.94	100.00
Case 2: (-0.3, 0.3)	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	Marginal
Realized CPI inflation rate	(1)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
(2)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
(3)	0.00	0.02	0.07	0.56	0.45	0.21	0.04	0.16	1.51
(4)	0.28	1.01	3.89	27.81	23.08	14.35	3.66	5.59	79.67
(5)	0.10	0.18	0.26	1.10	2.53	5.46	3.95	0.90	14.49
(6)	0.03	0.04	0.06	0.20	0.52	1.54	1.65	0.29	4.34
(7)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Marginal	0.41	1.25	4.28	29.67	26.58	21.57	9.30	6.94	100.00
Case 3: (-0.5, 0.5)	(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	Marginal
Realized CPI inflation rate	(1)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
(2)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
(3)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
(4)	0.28	1.04	3.99	28.52	23.86	15.11	3.99	5.83	82.63
(5)	0.10	0.17	0.23	0.95	2.20	4.92	3.65	0.82	13.04
(6)	0.03	0.04	0.06	0.20	0.52	1.54	1.65	0.29	4.34
(7)	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
Marginal	0.41	1.25	4.28	29.67	26.58	21.57	9.30	6.94	100.00
Table 2. Performance of individual household expectations relative to the mean of professional forecasts (Share of households that outperformed the mean of professional forecasts on an ex post basis)

Intervals for “remain the same” response	Comparison calculated bias	based on 1/	absolute size of	Comparison based on RMSE	2/
If | Et[πt,t+12] | < 0.1	33.1%	10.3%
If | Et[πt,t+12] | < 0.3	19.5%	12.3%
If | Et[πt,t+12] | < 0.5	29.2%	17.5%

Notes: 1. A household is counted if its average forecast error is smaller than the average forecast error of the mean of professional forecasts.

2. A household is counted if its RMSE ( $ \equiv $ squared average forecast error + standard deviation) is smaller than that of the mean of professional forecasts. Forecast errors are calculated on the basis of the selected inflation interval number, which ranges from 1 to 7.
Table 3. Forecasting power of the mean of household expectations and the mean professional forecasts $ ^{1/} $

Estimated Model : $ \pi_{t,t+12} = \beta_{1} \times E_{t}[\pi_{t,t+12}] + \beta_{2} \times N_{t}[\pi_{t,t+12}] + \beta_{3} \times \pi_{t-14,t-2} + \beta_{0} + \varepsilon_{t} $

Actual percentage number basis [1]	2/ [2]	[3]	[4]	[5]	[6]
Household mean (t)	(β1)	0.04	0.18	-0.10	0.21
( 0.16 )	( 0.15 )	( 0.20 )	( 0.21 )
Professional mean (t)	(β2)	1.24	**	1.29	**	1.32	***	1.24	***
( 0.56 )	)	( 0.51 )	)	( 0.37 )	( 0.33 )
Current inflation (t-2)	(β3)	-0.30	-0.29	-0.28	-0.26	-0.13	-0.28
( 0.29 )	)	( 0.30 )	)	( 0.27 )	)	( 0.21 )	( 0.23 )	)	( 0.20 )
Constant	(β0)	-0.05	0.09	-0.01	0.01	0.26	-0.09
( 0.24 )	)	( 0.24 )	)	( 0.17 )	)	( 0.30 )	( 0.33 )	)	( 0.22 )
Durbin-Watson d-statistic	0.24	0.23	0.23	0.56	0.47	0.58
Number of observations	48	48	48	48	48	48
Adj. R-squared	0.07	-0.01	0.09	0.19	-0.02	0.20
Root MSE	0.73	0.76	0.72	0.94	1.05	0.93

Notes: 1. All regressions were conducted using OLS. Numbers in parentheses are standard errors.

*** and ** indicate significance at the 1 percent and 5 percent level, respectively.

2. As actual percentage numbers are not available for household expectations, we used the median of range value to calculate the household mean.

3. All numbers, including realized inflation, the mean of professional forecasts, and current inflation, were converted to median of range values.
Table 4. The impact of the mean of professional forecasts on individual household expectations

Dependent variable	Independent variables	Number of observations	Adj. R-squared	Root MSE
Constant	Sum of coefficients on
Ei,t-4(πt-4,t+8)...Ei,t-1(πt-1,t+11)	Nt-4(πt-4,t+8)...Nt-1(πt-1,t+11)
Real number based regression	[1]	Ei,t(πt,t+12)	0.22(0.00)	0.80(0.00)	0.66(0.00)	158,602	0.42	1.83
Median value based regression	[2]	Ei,t(πt,t+12)	0.16(0.00)	0.80(0.000)	0.46(0.00)	158,602	0.42	1.84

Notes: All regressions were conducted using OLS. Numbers in parentheses are p-values for the exclusion tests.

$ N_{t-4}(\pi_{t-4,t+8})\ldots N_{t-1}(\pi_{t-1,t+11}) $ in [1] are on an actual percent number basis, while those in [2] are on a median of range basis.
Table 5. Estimating and testing the mean inflation expectations model (8)

Model estimated: $ E_{t}[\pi_{t,t+12}] = \alpha_{1-1} \times N_{t}[\pi_{t,t+12}] + \alpha_{1-2} \times \pi_{t-14,t-2} + \alpha_{2} \times E_{t-1}[\pi_{t-1,t+11}] + \alpha_{0} + \varepsilon_{t} $

Real number based regressions	Median value based regressions
[1]	[2]	[3]	[4]	[5]	[6]
Mean of professional forecasts (t)	(α1-1)	0.37	***	0.31	**	0.25	***	0.24	***
(0.14	)	(0.12	(0.08	)	(0.08
Current inflation (t-2)	(α1-2)	-0.33	***	-0.30	***	-0.16	**	-0.14	**
(0.08	)	(0.08	)	(0.07	)	0.06	)
Mean of household expectations (t-1)	(α2)	0.92	***	1.12	***	1.07	***	0.93	***	1.03	***	0.99	***
(0.04	)	(0.05	)	(0.05	)	(0.03	)	(0.04	)	(0.04	)
Constant	(α0)	0.06	-0.12	-0.13	0.01	0.64	**	0.52	**
0.07	(0.08	)	(0.08	)	(0.07	)	(0.25	)	(0.23	)
Test whether (F-statistic)	α1-1+α2=1	5.50	***	5.64	**
Test whether (F-statistic)	α1-2+α2=1	16.83	***	6.49	**
Test whether (F-statistic)	α1-1+α1-2+α2=1	0.41	0.98
Number of observations	58	58	58	58	58	58
Adj. R-squared	0.94	0.95	0.96	0.95	0.94	0.95
Root MSE	0.27	0.26	0.24	0.27	0.28	0.26

Notes: All regressions were conducted using OLS. Numbers in parentheses are standard errors.

*** and ** indicate significance at the 1 percent and 5 percent level, respectively.
Table 6. Micro data based regressions of inflation expectation dynamics

(Assuming that professional forecasts in t-1 are available when households form their expectations in t-1)

Panel A: Percentage number based regression
Dependent variable: Percentage point change in inflation expectations by individual households from t-1 to t.
[1]	[2]	[3]	[4]	[5]
Gap between professional forecast and household expectation in t-1	(β1-1)	0.702	***	1.589	***	0.800	***	1.122	***
( 0.002	)	( 0.012	)	( 0.003	)	( 0.013	)
Gap between current inflation and household expectation in t-1	(β1-2)	0.622	***	-0.858	***	0.755	***	-0.315	***
( 0.002	)	( 0.012	)	( 0.003	)	( 0.013	)
Change in the mean of professional forecasts (from t-1 to t)	(β2-1)	0.705	***	1.407	***	0.682	***	1.271	***
( 0.004	)	( 0.015	)	( 0.003	)	( 0.015	)
Change in current inflation rate (from t-1 to t)	(β2-2)	0.624	***	-0.681	***	0.598	***	-0.571	***
( 0.003	)	( 0.014	)	( 0.003	)	( 0.014	)
Constant	(β0)	0.569	***	0.693	***	0.563	***
( 0.005	)	( 0.005	)	( 0.006	)
Test of rational expectations 1/	β1-1=0 & β2-1=1	β1-2=0 & β2-2=1	β1-1=0 & β2-1=1 & β1-2=0 & β2-2=0	β1-1=0 & β2-1=1 & β1-2=0 & β0=0	β1-2=0 & β0=0 & β0=0	β2-2=1 & β0=0 & β0=0	β1-1=0 & β2-1=1 & β0=0 & β0=0 & β

Notes: All regressions were conducted using OLS. Numbers in parentheses are standard errors.

*** indicates significance at the 1 percent level.
Table 6. Micro data based regressions of inflation expectation dynamics (continued)

Panel B: Multiple choice based regression
Dependent variable: Change in the inflation expectations range by individual households from t-1 to t.
[1]	[2]	[3]	[4]	[5]	[6]
Gap between professional forecast and household expectation in t-1	(β1-1)	0.810	***	0.823	***	0.827	***	0.723	***
( 0.002	)	( 0.004	)	( 0.002	)	( 0.004	)
Gap between current inflation and household expectation in t-1	(β1-2)	0.554	***	-0.013	***	0.658	***	0.114	***
( 0.002	)	( 0.004	)	( 0.002	)	( 0.004	)
Change in the mean of professional forecasts (from t-1 to t)	(β2-1)	0.794	***	0.773	***	0.782	***	0.736	***
( 0.004	)	( 0.005	)	( 0.004	)	( 0.047	)
Change in current inflation rate (from t-1 to t)	(β2-2)	0.546	***	0.023	***	0.529	***	0.047	***
( 0.003	)	( 0.004	)	( 0.003	)	( 0.004	)
Constant	(β0)	0.114	***	0.293	***	0.159	***
( 0.002	)	( 0.003	)	( 0.003	)
Test of rational expectations 1/	β1-1=0 & β2-1=1	β1-2=0 & β2-2=1	β1-1=0 & β2-1=1 & β1-2=0 & β2-2=0	β1-1=0 & β2-1=1 & β0=0	β1-2=0 & β0=0	β1-1=0 & β2-1=1 & β1-2=0 & β0=0
(F-statistic)	1.2e+5	***	1.2e+5	***	61763	***	84735	***	93369	***	51294	***
Test of sticky information model	β1-1=1 & β2-1=1	β1-2=1 & β2-2=1	β1-1=1 & β2-1=1 & β1-2=0 & β2-2=0	β1-1=1 & β2-1=1 & β0=0	β1-2=1 & β0=0	β1-1=1 & β2-1=1 & β1-2=0 & β0=0
(F-statistic)	3046	***	18790	***	1561	***	3059	***	17811	***	2021	***
Number of observations	161,321	161,321	161,321	161,321	161,321	161,321
Adj. R-squared	0.404	0.264	0.405	0.415	0.318	0.418
Root MSE	0.821	0.913	0.821	0.814	0.879	0.812

Notes: All regressions were conducted using OLS. Numbers in parentheses are standard errors.

*** indicates significance at the 1 percent level.
Table 7. How well can the sticky information model explain disagreement in inflation expectations?

$$ E_{i,t}[\pi_{t,t+1}]=\sum\beta_{1,t}Year-Month-Dummy_{t}+\sum\beta_{2,t}Update Year-Month Dummy s(i,t)+\beta_{0}+\varepsilon_{i,t} $$

P-values for exclusion F tests
All	β1,s=0	All	β2,s=0	Number of observations	Adj. R-squared	Root MSE
[1]	F(58, 322737) = 1337.06	322,796	0.194	2.241
Prob > F =	0.0000
[2]	F(58, 322679) = 661.45	F(58, 322679) = 90.88	322,796	0.206	2.224
Prob > F =	0.0000	Prob > F =	0.0000

Notes: All regressions were conducted using OLS.

$ s(i,t) $ denotes the (past) period in which household i's inflation expectation in period t was updated. By definition, $ s(i,t) \leq t $ always holds.
Table 8. Probit model of inflation expectation updates

The dependent variable takes one if a household revised its inflation expectation in period t and takes zero if it did not.

[1]	[2]	[3]	[4]	[5]
log(no. of media articles)	0.007	**	0.004	-0.004	0.002	-0.003
( 0.003	)	( 0.003	)	( 0.004	)	( 0.004	)	( 0.004	)
$ \Delta $ log(no. of media articles)	0.039	***	0.039	***	0.061	***	0.070	***	0.057	***
( 0.005	)	( 0.005	)	( 0.005	)	( 0.005	)	( 0.005	)
E $ _{i,t} $ [ $ \pi_{t-1,t+11} $ ]-N $ _{t} $ [ $ \pi_{t-1,t+11} $ ]	0.009	***	0.003	***	0.004	***	0.007	***
( 0.000	)	( 0.001	)	( 0.001	)	( 0.001	)
Months since the last update (a)	-0.076	***	0.013	***
( 0.001	)	( 0.001	)
Average no. of months between updates (b)	-0.254	***
( 0.002	)
(a)÷(b)	0.037	***
( 0.001	)
Number of observations	267,269	267,269	224,379	224,379	219,092
Pseudo R-squared	0.0003	0.001	0.046	0.144	0.007

Notes: Reported coefficients are estimated marginal effects, that is, the change in the probability for a change in each independent variable.

Numbers in parentheses are standard errors.

*** and ** indicate significance at the 1 percent and 5 percent level, respectively.
Figure 1. Average inflation expectations and actual inflation by survey (from 2001 to 2009)
Figure 2. Distribution of the average no. of months between expectation updates

Note: To identify the average number of months between expectation updates for each individual household, we dropped observations of each household before its first update.
Table A.1. Basic statistics of the characteristics of the households surveyed

Total	FY2004	FY2005	FY2006	FY2007	FY2008
Sex
Male	80.9	82.3	81.8	81.1	79.9	78.1
Work status
No job	34.1	32.6	33.1	34.3	35.2	36.1
Farmer	1.3	1.2	1.3	1.2	1.3	1.2
Employee	42.7	43.4	42.9	42.8	42.1	41.8
Self-employed	17.4	18.5	18.0	17.5	17.0	15.1
Others	4.6	4.3	4.6	4.2	4.4	5.7
Age of household head
18 to 20	0.2	0.3	0.3	0.2	0.2	0.2
21 to 30	5.7	5.3	5.8	6.0	5.7	5.6
31 to 40	10.9	11.2	10.2	11.2	11.4	10.5
41 to 50	14.2	14.7	14.8	13.9	13.7	13.2
51 to 60	20.5	22.0	20.8	19.9	20.0	19.3
61 to 70	24.3	24.8	24.7	24.0	23.6	24.2
71 to 80	18.6	16.9	18.2	19.1	19.2	20.4
81 to 90	5.3	4.4	4.9	5.4	5.9	6.2
91 or above	0.3	0.3	0.3	0.4	0.5	0.4
Number of household members
1	26.9	25.0	24.9	24.8	29.0	33.0
2	26.8	26.9	27.2	28.1	26.2	24.9
3	17.9	18.6	18.8	18.2	16.7	16.7
4	15.6	16.3	15.9	15.4	15.8	14.5
5+	12.8	13.3	13.2	13.5	12.3	10.9
Number of working household members
0	25.3	23.5	24.4	25.1	26.3	28.5
1	41.0	41.7	41.2	40.5	40.5	40.7
2	23.6	24.5	23.8	24.1	23.3	21.6
3+	10.1	10.3	10.7	10.3	9.9	9.1
Household annual income
Less than 3 million yen	34.0	32.3	32.6	33.6	34.6	38.2
3 to 4 million yen	17.8	17.9	18.1	18.4	17.4	16.5
4 to 5.5 million yen	16.3	16.2	16.5	16.5	15.9	16.2
5.5 to 7.5 million yen	14.5	15.0	15.0	14.0	14.7	13.0
7.5 to 9.5 million yen	8.7	9.1	8.7	8.8	8.5	8.2
9.5 to 12 million yen	5.1	5.4	5.1	5.0	5.3	4.7
More than 12 million yen	3.7	4.0	4.0	3.7	3.5	3.1
Main income source
Salary	52.0	53.4	53.0	51.7	50.9	50.5
Business income	12.4	13.2	12.8	12.9	12.4	10.1
Pension	31.4	29.1	30.3	31.7	32.6	34.8
Other	4.1	4.4	3.9	3.6	4.1	4.7
House
Owner-occupied, detached	69.9	69.7	71.5	70.5	69.5	67.7
Owner-occupied, condominium	4.4	4.2	3.8	4.5	4.6	4.9
Publicly provided	2.8	2.1	2.1	2.4	2.8	5.0
Employer-provided	2.4	2.4	2.4	2.3	2.5	2.2
Private rental	20.6	21.6	20.2	20.2	20.6	20.2
Housing loan
Yes	21.7	22.1	22.0	21.7	21.4	20.7
Table A.2. Questions in the ESPF

1. Fiscal year based projection
(1)	Nominal GDP (percent change from the previous fiscal year)	(9) Consumer price index excluding fresh food (percent change from the previous fiscal year)
(2)	Real GDP (percent change from the previous fiscal year)	(10) Unemployment rate (percent)
(3)	Real private final consumption expenditure (percent change from the previous fiscal year)	(11) Euroyen TIBOR - 3 month (average during the period)
(4)	Real non-residential investment (percent change from the previous fiscal year)	(12) 10-year JGB yield (average during the period)
(5)	Export volumes of goods and services (percent change from the previous fiscal year)	(13) Stock prices - NIKKEI 225 (average during the period)
(6)	Import volumes of goods and services (percent change from the previous fiscal year)	(14) Money stock (percent change from the previous fiscal year)
(7)	Indices of industrial production (percent change from the previous fiscal year)	(15) Yen-dollar exchange rate (average during the period)
(8)	Current account balance (trillion yen)	(16) U.S. growth rate (percent change from the previous calendar year)
2. Quarterly based projection
(1)	Real GDP (seasonally adjusted annualized growth rate)	(3) Unemployment rate (percent)
(2)	Consumer price index excluding fresh food (percent change from the previous year)
3. Other questions
ESRI Discussion Paper Series No.231

# Time-varying Analysis of Dynamic Stochastic General Equilibrium Models Based on Sequential Monte Carlo Methods

by

Koiti Yano

February 2010

Economic and Social Research Institute

Cabinet Office

Tokyo, Japan
# Time-varying Analysis of Dynamic Stochastic General Equilibrium Models Based on Sequential Monte Carlo Methods

By Koiti Yano $ ^{*} $

## Abstract

This paper proposes a new method to estimate parameters, natural rates, and business cycles of dynamic stochastic general equilibrium models simultaneously and consistently. It is based on the Monte Carlo particle filter and a self-organizing state space model. In our method, we estimate the parameters and the natural rates using the time-varying-parameter approach, which is often used to infer invariant parameters practically. In most previous papers on DSGE models, structural parameters of them are assumed to be “deep (invariant).” However, our method analyzes how stable structural parameters are. Adopting the TVP approach creates the great advantage that the structural changes of parameters are detected naturally. Moreover, we estimate time-varying natural rates of macroeconomic data: real output and an equilibrium real interest rate. In empirical analysis, we estimate a new Keynesian DSGE model using the US data. The analysis shows that the average of the growth of natural output is 3.02 and the average of inflation target is 2.46 from 1985 to 2007. From the late Volcker era to the early Greenspan era, the reaction coefficient to inflation in the Taylor rule is increasing, and from the mid Greenspan era to the early Bernanke era it is stable around 4.0. These results indicate that the behavior of the Fed had changed to realize the stability of inflation from the late Volcker era to the early Greenspan era. An equilibrium real rate is negative from the early 2000s to the mid-2000s because the Fed cuts policy rates to prevent deflation.

JEL Classification Number: C11, C15, E00, E52, and D50

Key words : Dynamic Stochastic General Equilibrium Model, Bayesian Statistics, Monte Carlo particle filter, Self-organizing state space model, US economy
## 2 The Model

### 2.1 Households

In the economy, there is a continuum of households indexed by $ j \in (0, 1) $ . The households consume and provide labor. The utility of the household j is given by

$$ E_{0}^{j}\sum_{t=0}^{\infty}\beta^{t}\Big[\log(C_{j,t}-h C_{t-1})+\Upsilon\big(\frac{M_{j,t}}{P_{t}}\big)-\frac{\Psi_{L}}{1+\sigma_{L}}L_{j,t}^{1+\sigma_{L}}\Big], $$

where $ E_{0}^{j} $ is the expectation operator, conditional on household j's information at time 0, $ C_{j,t} $ is household j's consumption, $ C_{t-1} $ is past aggregate consumption, $ M_{j,t}/P_{t} $ is the household j's real money balances, $ L_{j,t} $ is household j's labor hours, t is a time index, and h, $ \chi $ , $ \Psi_{L} $ , and $ \eta $ are constants. The constraint condition of the household j is given by

$$ C_{j,t}+I_{j,t}+\frac{M_{j,t}}{P_{t}}+\frac{B_{j,t}}{P_{t}}\leq W_{t}L_{j,t}+\frac{M_{j,t-1}}{P_{t}}+r_{t}^{K}K_{j,t}+(1+i_{t-1})\frac{B_{j,t-1}}{P_{t}}+\Pi_{j,t}, $$

where $ I_{j,t} $ is investment by household j, $ B_{j,t} $ is household j's domestic bonds, $ W_{t} $ is the average real wage, $ i_{t} $ is the short-term nominal interest rate, $ K_{j,t} $ is household j's capital, $ r_{t}^{K} $ is the rental rate of $ K_{j,t} $ , and $ \Pi_{j,t} $ is the profit of the firm j. In addition to Eq. (2), we assume the households are subject to the no-Ponzi condition.

$$ \lim_{T\to\infty}E_{0}[(\prod_{t=0}^{T}\frac{1}{1+i_{t}})B_{j,T}]=0. $$

### 2.2 Capital Accumulation and Adjustment Cost

The time evolution of Capital, $ K_{j,t} $ is given by

$$ K_{j,t}=(1-\delta)K_{j,t-1}+\Big[1-s\big(\frac{I_{j,t}}{I_{j,t-1}}\big)\Big]I_{j,t}, $$

where $ \delta $ is the depreciation cost of capital, $ K_{j,t} $ , and $ s(\cdot) $ is an adjustment cost function. We restrict the function $ s(\cdot) $ to satisfy the following properties: $ s(1) = s'(1) = 0 $ and $ s''(1) = 1/\nu > 0 $ .

### 2.3 Final Good Sector

In the final good sector, a single final good is produced by a perfectly competitive, representative firm. The final good is produced using a continuum of intermediate good, $ Y_{j,t} $ , indexed by $ j \in (0,1) $ . The final good, $ Y_{t} $ , is produced using the aggregate technology.

$$ Y_{t}=\left[\int_{0}^{1}\left(Y_{j,t}\right)^{\frac{1}{1+\lambda_{p}}}d j\right]^{1+\lambda_{p}}, $$

where $ Y_{j,t} $ is the quantity of intermediate good j, $ \lambda_{p} $ is a constant. The demand curve for $ Y_{j,t} $ is given by

$$ Y_{j,t}=\left(\frac{P_{j,t}}{P_{t}}\right)^{-\frac{1+\lambda_{p}}{\lambda_{p}}}Y_{t}, $$
where $ P_{j,t} $ is the price of intermediate good j and $ P_{t} $ is the aggregate price of the final good. The aggregate price is given by

$$ P_{t}=\left[\int_{0}^{1}\left(P_{j,t}\right)^{-\frac{1}{\lambda_{p}}}d j\right]^{-\lambda_{p}}. $$

### 2.4 Intermediate Goods Firms

In the intermediate goods sector, monopolistic competitive domestic firms produce intermediate goods which is indexed by $ j \in (0,1) $ . The firm j's production function is given by

$$ Y_{j,t}=Z_{t}K_{j,t}^{\alpha}L_{j,t}^{1-\alpha}, $$

The aggregate technology level, $ Z_{t} $ , is given by

$$ \log Z_{t}=\left(1-\xi_{Z}\right)\log\bar{Z}+\xi_{Z}\log Z_{t-1}+\epsilon_{Z,t}, $$

where $ \epsilon_{Z,t}\sim N(0,\sigma_{Z,t}^{2}) $ and $ \bar{Z} $ and $ \xi_{Z} $ are constants. Solving the cost minimization of the firm j, the first order condition becomes

$$ \frac{W_{t}}{r_{t}^{K}}=\frac{1-\alpha}{\alpha}\frac{K_{j,t}}{L_{j,t}} $$

The firms j's real marginal cost is given by

$$ M C_{t}=\frac{1}{Z_{t}}\Big(\alpha^{-\alpha}(1-\alpha)^{-(1-\alpha)}W_{t}^{1-\alpha}\big(r_{t}^{K}\big)^{\alpha}\Big) $$

In the sticky prices model, proposed by Calvo (1983), a fraction $ 1 - \xi_{p} $ of all firms re-optimize their nominal prices while the remaining $ \xi_{p} $ fraction of all firms do not re-optimize their nominal prices. Following Christiano et al. (2005), firms that cannot re-optimize their price index to lagged inflation are as follows.

$$ P_{j,t}=\pi_{t-1}P_{j,t-1}, $$

where $ \pi_{t}=P_{t}/P_{t-1} $ . We call this price setting “lagged inflation indexation.” The firm j chooses $ P_{j,t} $ to maximize

$$ E_{t}\sum_{l=0}^{\infty}\left(\beta\xi_{p}\right)^{l}\Big[\frac{P_{j,t}}{P_{t+l}}X_{t l}-M C_{t+l}\Big]Y_{j,t+l}, $$

$$ subject to Y_{j,t}=\left(\frac{P_{j,t}}{P_{t}}\right)^{-\frac{1+\lambda_{p}}{\lambda_{p}}}Y_{t}, $$

where $ X_{tl} $ is

$$ X_{tl}=\left\{\begin{array}{l}\pi_{t}\times\pi_{t+1}\times\cdots\times\pi_{t+l-1}for l\geq1\\ 0for l=0.\end{array}\right. $$

The aggregate price index of sticky prices and inflation indexation is obtained by

$$ P_{t}=[(1-\xi_{p})(\tilde{P}_{t})^{\frac{1}{1-\lambda_{p}}}+\xi_{p}(\pi_{t-1}P_{t-1})^{\frac{1}{1-\lambda_{p}}}]^{1-\lambda_{p}}. $$