## 3. Alternative Priors of c and αr

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---

### Context and research questions
- Japan’s nominal short-term interest rate has been at the zero lower bound (ZLB) since the late 1990s; the 10-year rate reached the ZLB in early 2016 when the negative interest rate policy was introduced.
- Core inflation remains well below the Bank of Japan (BoJ)’s inflation target of two percent, and inflation expectations are anchored near zero.
- Key questions addressed:
  - How large is the gap between the real interest rate and the natural rate of interest in Japan?
  - How much impact has demographic change had on Japan’s natural rate of interest?

### Conceptual definitions and mechanisms
- Real interest rate (ex ante) = nominal interest rate − expected inflation.
- Natural rate of interest (r*) = real rate consistent with output at potential and constant inflation (Laubach and Williams, 2003). It is the benchmark for monetary stance: policy is expansionary if real rate < r*.
- From household intertemporal optimization: r = 1/σ g_c + ϑ (equation reproduced as given), motivating a linear relationship between r* and trend per-capita potential output growth.
- Working-age population definition used: those aged between 15 and 64 years old (United Nations’ World Population Prospects).

### Empirical approach and model extension
- Two broad estimation approaches for r*: statistical (trend of real rate) and structural (equilibrium real rate from macro relationships). This paper uses a semi-structural (state-space) approach extending Laubach and Williams (2003) and Pescatori and Turunen (2016).
- Main model innovations:
  - Decompose potential output using a production-function approach:
    - Y*_t = A*_t (L*_t)^α (K*_t)^(1−α).
  - Potential labor input decomposed into four factors: working-age population (F_t), potential labor force participation (P*_t), potential employment rate (E*_t), potential total working hours per worker (H*_t).
  - Trend growth rates of production factors follow random walks with contemporaneous and lagged working-age population growth as exogenous influences, yielding a trend potential output growth process g_t affected by Δf_t and Δf_{t−1}.
  - Natural rate linked to trend per-capita potential output growth:
    - r*_t = c (g_t − Δn_t) + z_t, where Δn_t is total population growth and z_t captures other determinants.
  - The z_t process includes observable exogenous determinants:
    - excess global savings S_t (proxied by current account surplus of emerging and developing economies expressed in terms of U.S. GDP),
    - a measure of Japan risk premium E_t (proxied by equity risk premium and bond market premium),
    - policy uncertainty P_t (news-based measure).
  - Robustness check: replace excess global savings with Japan’s excess savings (current account surplus of Japan).

### State-space core system
- IS curve (output gap x_t related to interest rate gaps) and backward-looking Phillips curve for core inflation π_t are retained from Pescatori and Turunen (2016).
- The true output gap x_t is related to IMF WEO output gap x_t^WEO via:
  - x_t^WEO = x_t + μ_t + ε^w_t, with μ_t = ρ μ_{t−1} + ε^μ_t. WEO output gap treated as an imprecise signal to aid identification.

### Priors, data, and estimation setup
- Prior for persistence parameter 휌휌: Beta 0.95 (SD 0.01).
- Priors for all standard deviations: Inv. Gamma (as listed in Table 1).
- Selected prior values (Density / Mean / SD):
  - 푐푐: Gamma / 0.70 / 0.27
  - 훼훼1: Gamma / 1.01 / 0.10
  - 훼훼1+훼훼2: Beta / 0.92 / 0.05
  - 훼훼푟푟: Gamma / 0.06 / 0.04
  - 훽훽1: Beta / 0.45 / 0.10
  - 훽훽2: Beta / 0.37 / 0.10
  - 훽훽푥푥: Beta / 0.05 / 0.01
  - 훽훽푚푚: dogmatic / 0.01
  - 훿훿1: Gamma / 1 / 0.25
  - 훿훿1+훿훿2: Beta / 0.99 / 0.0025
  - 훾훾0: Gamma / 0.014 / 0.01
  - 훾훾1: Normal / 0.00 / 0.01
  - 훿훿푠푠, 훿훿푒푒, 훿훿푝푝: Normal / 0.10 / 0.25
  - 휎휎(휀휀푦푦): Inv. Gamma / 0.02 / Inf.
  - 휎휎(휀휀푥푥): Inv. Gamma / 0.97 / Inf.
  - 휎휎(휀휀휋휋): Inv. Gamma / 1.22 / Inf.
  - 휆휆푔푔 = 휎휎(휀휀푔푔)/휎휎(휀휀푦푦): Inv. Gamma / 0.45 / Inf.
  - 휆휆푧푧 = 휎휎(휀휀푧푧)/휎휎(휀휀푥푥): Inv. Gamma / 0.019 / Inf.
  - 휎휎푤푤, 휎휎휇휇: Inv. Gamma / 0.50 / Inf.
- Data sample: quarterly 1990Q1 to 2018Q1 on real GDP, core inflation, import relative price inflation, and short-term real interest rate.
- Key data sources: IMF WEO, Ministry of Internal Affairs and Communications (core inflation), Consensus Economics (short-term inflation expectations), UN World Population Prospects.
- Determinants for the 푧푧-process: excess global savings (current account surplus of emerging and developing economies in percent of U.S. GDP), equity risk premium (IMF Global Financial Stability Report, October 2018), bond market premium (spread between Japanese and U.S. 10-year government bonds), and Japan EPU (Arbatli et al., 2017).
- Estimation method: Bayesian MCMC (Random Walk Metropolis–Hastings). Mode, mean, median, and posterior percentiles from two chains of 150,000 draws (initial 50,000 discarded; retain ~1 every 4 draws). Convergence achieved in all estimations.

### Baseline posterior estimates (selected posterior statistics)
- 푐푐: Mode 0.566 / Median 0.643 / 10th 0.372 / 90th 1.021
- 훼훼1: Mode 1.045 / Median 1.034 / 10th 0.945 / 90th 1.126
- 훼훼1+훼훼2: Mode 0.927 / Median 0.909 / 10th 0.856 / 90th 0.951
- 훼훼푟푟: Mode 0.016 / Median 0.027 / 10th 0.009 / 90th 0.062
- 훽훽1: Mode 0.430 / Median 0.424 / 10th 0.342 / 90th 0.508
- 훽훽2: Mode 0.388 / Median 0.389 / 10th 0.287 / 90th 0.491
- 훽훽푥푥: Mode 0.047 / Median 0.051 / 10th 0.039 / 90th 0.065
- 훿훿1: Mode 0.919 / Median 0.962 / 10th 0.685 / 90th 1.305
- 훿훿1+훿훿2: Mode 0.991 / Median 0.990 / 10th 0.986 / 90th 0.993
- 훾훾0: Mode 0.023 / Median 0.012 / 10th 0.004 / 90th 0.027
- 훾훾1: Mode 0.009 / Median 0.000 / 10th -0.013 / 90th 0.013
- 훿훿푠푠: Mode 0.221 / Median 0.210 / 10th -0.118 / 90th 0.526
- 훿훿푒푒: Mode 0.156 / Median 0.106 / 10th -0.204 / 90th 0.437 (equity-risk-premium specification: Mode 0.236 / Median 0.235 / 10th -0.079 / 90th 0.555)
- 훿훿푝푝: Mode 0.225 / Median 0.206 / 10th -0.115 / 90th 0.532
- 휎휎(휀휀푦푦): Mode 0.009 / Median 0.014 / 10th 0.007 / 90th 0.031
- 휎휎(휀휀푥푥): Mode 0.951 / Median 0.944 / 10th 0.866 / 90th 1.036
- 휎휎(휀휀휋휋): Mode 0.900 / Median 0.904 / 10th 0.831 / 90th 0.989
- 휆휆푔푔 = 휎휎(휀휀푔푔)/휎휎(휀휀푦푦): Mode 0.221 / Median 8.485 / 10th 5.764 / 90th 13.090 (bond-premium variant: Median 10.492 / 10th 7.693 / 90th 16.979)
- 휆휆푧푧 = 휎휎(휀휀푧푧)/휎휎(휀휀푥푥): Mode 0.009 / Median 0.012 / 10th 0.007 / 90th 0.025
- 휎휎푤푤: Mode 0.365 / Median 0.293 / 10th 0.261 / 90th 0.332
- 휎휎휇휇: Mode 0.281 / Median 0.276 / 10th 0.248 / 90th 0.307
- 휌휌 (posterior): Mode 0.972 / Median 0.963 / 10th 0.950 / 90th 0.973

### Key baseline findings and quantitative results
- Natural rate trend:
  - Baseline shows a declining trend in Japan's natural rate since the 1990s.
  - Bayesian medians from both equity-risk-premium and bond-market-premium estimations suggest Japan’s natural rate in Q1 2018 was likely to be negative, but close to zero.
  - Median estimates suggest the natural rate in Japan has likely fallen into negative territory since the Asian Financial Crisis (AFC).
  - Range of estimates widens during and after the Global Financial Crisis (GFC), indicating greater uncertainty.
- Monetary policy stance and interest rate gap:
  - QQE by BoJ since April 2013 has pushed down real interest rates; short-term real interest rate on declining trend since 2013.
  - Short-term real interest rate below median natural rate since 2017 → negative interest rate gap (accommodative stance), but gap has been less than one percentage point and shrunk significantly in Q1 2018.
  - Interest rate gap is an incomplete and imprecise indicator of policy stance given uncertainty in natural rate estimation.
- Demographics:
  - Demographic change has had a significantly negative impact on the natural rate, with the impact increasing in recent years.
  - Since the global financial crisis (GFC), the decline in working-age population growth has contributed about −0.3 percentage points on average to the estimated negative natural rate — which is about −0.7 percent during this period.
  - The negative impact of working-age population growth on r* is increasing over time:
    - less than −0.2 percentage points in Q1 2010,
    - about −0.4 percentage points in Q1 2018.
  - Negative demographic impact increased steadily since around 2005 when working-age population growth declined faster than total population growth.
  - Negative demographic impact stabilized at around -0.5 percent in recent years (due to slight recovery in working-age population growth), but projected to increase significantly after the early 2020s as working-age population growth is projected to decline faster than total population growth.
  - Japan’s natural rate has likely fallen into negative territory since the late 1990s.
- Trend potential growth and other shocks:
  - Other shocks to trend potential growth (e.g., technology) are important drivers of the natural rate.
  - Trend potential growth (excluding demographics) was high in early 1990s, declined significantly in 1990s (pushing natural rate into negative territory), with partial recoveries pre- and post-GFC but stabilized at low levels (about one third of 1990 level) in recent years.
- z-determinant (excess global savings, equity risk premium, policy uncertainty):
  - Equity risk premium rebounded to historical highs after the GFC, exerting substantial downward pressure on the natural rate.
  - Japan’s equity risk premium remained relatively high compared to historical levels and compared to the U.S., sustaining a negative impact on the natural rate in Japan.
  - Evidence for the global savings glut hypothesis during the period between AFC and GFC: excess global savings (emerging market current account surpluses) contributed to downward pressure on Japan’s natural rate, particularly between AFC and GFC. After the GFC, the negative impact of excess global savings diminished and even reversed in recent years.

### Robustness checks and alternative specifications
- Alternative priors for 푐푐 and 훼훼푟푟:
  - Pescatori and Turunen (2016): 푐푐 Gamma / 0.81 / 0.05; 훼훼푟푟 Gamma / 0.08 / 0.05
  - Holston et al. (2017): 푐푐 Gamma / 1 / 0.05; 훼훼푟푟 Gamma / 0.07 / 0.02
  - Results: median natural rate slightly higher than baseline (larger interest rate gaps), but demographic impact rising in recent years and still explains about half of the negative natural rate in Q1 2018 (around -0.8 percent). Qualitatively similar to baseline.
- Japan-specific excess savings:
  - Replaced global excess savings with Japan current account surplus (percent of GDP).
  - Estimated demographic impact comparable to baseline; Japan’s excess savings had a negative impact on domestic natural rate in recent years.
- No z-controls (푧푧 follows random walk):
  - Replaced 푧푧-process with random walk z_t = z_{t-1} + ε_z,t.
  - Results: median natural rate slightly higher and more persistent; demographic contribution accounts for a larger share of drivers; implied larger interest rate gap (closing more recently).
- Using 10-year interest rate instead of short-term:
  - Replaced short-term real rate with 10-year real rate (10-year JGB yield minus inflation expectations).
  - Results: median natural rate based on 10-year interest rate declined since 1990s and fell into negative territory in recent years; median fell below zero only since around 2012 (vs. baseline negative since AFC).
  - Size of demographic contribution similar to baseline; z-determinant has a much smaller negative contribution to the 10-year-based natural rate than to the short-term-based natural rate.
  - Interpretation: more structural shocks (including demographics) relatively more important for the 10-year-based natural rate; relevance to BoJ Yield Curve Control where 10-year target is emphasized.

### Conclusions and policy implications
- Main conclusions:
  - Demographic change has contributed to the decline in Japan’s natural rate, with this contribution increasing over time—supporting the secular stagnation interpretation for Japan.
  - Other shocks to trend potential growth (e.g., technology/productivity) and z-determinants (excess savings, equity risk premium, policy uncertainty) also played important roles.
  - Bayesian median estimates suggest Japan’s natural rate likely fell below zero since the AFC; projected further declines in working-age population growth imply the negative demographic impact may increase by 2040.
  - Results are qualitatively robust across different priors and interest rate choices, but estimates have large error bands and should be interpreted with caution.
- Policy implications and recommendations:
  - Boost trend potential growth to offset negative demographic impacts and lift the natural rate to widen the interest rate gap.
  - Examples of policy actions include accelerating labor market reforms and other structural reforms (including more active migration policies).
  - Given limits to monetary policy when the natural rate is negative and falling, structural measures to raise potential growth are critical to restore policy space.

*Source: wpiea2019031 - Author’s calculations and IMF staff estimates as presented in the provided content.*

### References ________________________________________________________________28

### References ________________________________________________________________28

### Figures
- 1. Natural Rate of Interest ____________________________________________________18
- 2. Determinants of the Natural Rate of Interest  ___________________________________20
- 3. Natural Rate of Interest with Alternative Priors _________________________________22
- 4. Natural Rate of Interest without z - Controls ____________________________________23
- 5. Natural Rate of Interest without z - Controls ____________________________________25
- 6. Natural Rate of Interest Based on 10-Year Interest Rate __________________________26

### Tables
- 1. Priors __________________________________________________________________15
- 2. Estimation Results: Posteriors  ______________________________________________17

*Source PDF filename: wpiea2019031 - References ________________________________________________________________28*

### 3. Alternative Priors of c and αr _______________________________________________24

### 3. Alternative Priors of c and αr

### Context and research questions
- Japan’s nominal short-term interest rate has been at the zero lower bound (ZLB) since the late 1990s; the 10-year rate reached the ZLB in early 2016 when the negative interest rate policy was introduced.
- Core inflation remains well below the Bank of Japan (BoJ)’s inflation target of two percent, and inflation expectations are anchored near zero.
- Key questions addressed:
  - How large is the gap between the real interest rate and the natural rate of interest in Japan?
  - How much impact has demographic change had on Japan’s natural rate of interest?

### Conceptual definitions and mechanisms
- Real interest rate (ex ante) = nominal interest rate − expected inflation.
- Natural rate of interest (r*) = real rate consistent with output at potential and constant inflation (Laubach and Williams, 2003). It is the benchmark for monetary stance: policy is expansionary if real rate < r*.
- From household intertemporal optimization: r = 1/σ g_c + ϑ (equation reproduced as given), motivating a linear relationship between r* and trend per-capita potential output growth.
- Working-age population definition used: those aged between 15 and 64 years old (United Nations’ World Population Prospects).

### Empirical approach and model extension
- Two broad estimation approaches for r*: statistical (trend of real rate) and structural (equilibrium real rate from macro relationships). This paper uses a semi-structural (state-space) approach extending Laubach and Williams (2003) and Pescatori and Turunen (2016).
- Main model innovations:
  - Decompose potential output using a production-function approach:
    - Y*_t = A*_t (L*_t)^α (K*_t)^(1−α)  (equation (1) and log-differenced form in (2)).
  - Potential labor input decomposed into four factors:
    - working-age population (F_t), potential labor force participation (P*_t), potential employment rate (E*_t), potential total working hours per worker (H*_t). (Equation (3) and (4).)
  - Trend growth rates of production factors follow random walks with contemporaneous and lagged working-age population growth as exogenous influences (equations (13)–(17)), which yields a trend potential output growth process g_t affected by Δf_t and Δf_{t−1} (equation (20)).
  - Natural rate linked to trend per-capita potential output growth:
    - r*_t = c (g_t − Δn_t) + z_t  (equation (29)), where Δn_t is total population growth and z_t captures other determinants.
  - The z_t process includes observable exogenous determinants (equation (30)):
    - excess global savings S_t (proxied by current account surplus of emerging and developing economies expressed in terms of U.S. GDP),
    - a measure of Japan risk premium E_t (proxied by equity risk premium and bond market premium),
    - policy uncertainty P_t (news-based measure).
  - Robustness check: replace excess global savings with Japan’s excess savings (current account surplus of Japan).

### State-space core system
- IS curve (output gap x_t related to interest rate gaps) and backward-looking Phillips curve for core inflation π_t are retained from Pescatori and Turunen (2016) with equations reproduced as (22)–(24).
- The true output gap x_t is related to IMF WEO output gap x_t^WEO via:
  - x_t^WEO = x_t + μ_t + ε^w_t, with μ_t = ρ μ_{t−1} + ε^μ_t (equations (25)–(26)). WEO output gap treated as an imprecise signal to aid identification.

### Bayesian estimation strategy and priors
- Estimation uses Bayesian methods following Pescatori and Turunen (2016) to combine priors with the likelihood and to improve identification of trend potential growth and initial values.
- Priors:
  - Common parameters largely follow distributions used in Pescatori and Turunen (2016).
  - The new parameters capturing working-age population growth effects, γ_0 and γ_1 (derivable from θ parameters and α), are given priors:
    - γ_0 assumed positive and its prior follows a Gamma distribution.
    - γ_1 follows a Normal distribution.
  - Parameter c is weakly identified in equation (29); prior mean and standard deviation of c (and of α_r) use estimates from Fujiwara et al. (2016).
  - Priors on z_t process parameters (δ_s, δ_e, δ_p) are set relatively loose and on unbounded support.
  - Loose priors are set for σ_w and σ_μ and a high prior mean for ρ to reflect the precision of WEO gap signal.

### Main empirical findings (preserved numeric values)
- Demographic change has had a significantly negative impact on Japan’s natural rate of interest in recent years.
- Since the global financial crisis (GFC), the decline in working-age population growth has contributed about −0.3 percentage points on average to the estimated negative natural rate — which is about −0.7 percent during this period.
- The negative impact of working-age population growth on r* is increasing over time:
  - less than −0.2 percentage points in Q1 2010,
  - about −0.4 percentage points in Q1 2018.
- Japan’s natural rate has likely fallen into negative territory since the late 1990s.

### Comparisons and consistency with literature
- Findings broadly align with existing studies that show a downward trend in Japan’s natural rate since the 1990s, though some estimates in the literature are higher than those in this paper (e.g., Imakubo et al., 2015; Fujiwara et al., 2016).
- Okazaki and Sudo (2018) (DSGE with demographics and financial frictions) found a similar negative and increasing impact of working-age population changes on r* since the early 2000s, with similar magnitudes to this paper.
- Empirical literature (Aiyar et al., 2016; Liu and Westelius, 2017) supports the mechanism that demographic change reduces potential growth via declines in labor input and TFP growth.

### Model implications and interpretation
- The extended semi-structural model explicitly treats demographic contributions as part of trend potential growth and estimates their separate contribution to r*.
- The sign and magnitude expectations:
  - γ_0 expected positive because working-age population growth is expected to raise trend potential growth.
  - c is important but weakly identified; priors from Fujiwara et al. (2016) are used to stabilize estimation.
- The model captures both direct effects of working-age population growth on potential output growth (through α Δf_t in equation (12)) and indirect effects via trend growth rates of production factors (equations (13)–(17)).

*Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019031.pdf*

### 0.95 for the persistence parameter 휌휌. The priors for all standard deviations are inverse

### wpiea2019031 - 0.95 for the persistence parameter 휌휌. The priors for all standard deviations are inverse

### Priors, data, and estimation setup
- Prior for persistence parameter 휌휌: Beta 0.95 (SD 0.01).
- Priors for all standard deviations: Inv. Gamma (as listed in Table 1).
- Selected prior values from Table 1 (Density / Mean / SD):
  - 푐푐: Gamma / 0.70 / 0.27
  - 훼훼1: Gamma / 1.01 / 0.10
  - 훼훼1+훼훼2: Beta / 0.92 / 0.05
  - 훼훼푟푟: Gamma / 0.06 / 0.04
  - 훽훽1: Beta / 0.45 / 0.10
  - 훽훽2: Beta / 0.37 / 0.10
  - 훽훽푥푥: Beta / 0.05 / 0.01
  - 훽훽푚푚: dogmatic / 0.01
  - 훿훿1: Gamma / 1 / 0.25
  - 훿훿1+훿훿2: Beta / 0.99 / 0.0025
  - 훾훾0: Gamma / 0.014 / 0.01
  - 훾훾1: Normal / 0.00 / 0.01
  - 훿훿푠푠, 훿훿푒푒, 훿훿푝푝: Normal / 0.10 / 0.25
  - 휎휎(휀휀푦푦): Inv. Gamma / 0.02 / Inf.
  - 휎휎(휀휀푥푥): Inv. Gamma / 0.97 / Inf.
  - 휎휎(휀휀휋휋): Inv. Gamma / 1.22 / Inf.
  - 휆휆푔푔 = 휎휎(휀휀푔푔)/휎휎(휀휀푦푦): Inv. Gamma / 0.45 / Inf.
  - 휆휆푧푧 = 휎휎(휀휀푧푧)/휎휎(휀휀푥푥): Inv. Gamma / 0.019 / Inf.
  - 휎휎푤푤, 휎휎휇휇: Inv. Gamma / 0.50 / Inf.
- Data sample: quarterly 1990Q1 to 2018Q1 on real GDP, core inflation, import relative price inflation, and short-term real interest rate.
- Key data sources: IMF WEO, Ministry of Internal Affairs and Communications (core inflation), Consensus Economics (short-term inflation expectations), UN World Population Prospects.
- Determinants for the 푧푧-process: excess global savings (current account surplus of emerging and developing economies in percent of U.S. GDP), equity risk premium (IMF Global Financial Stability Report, October 2018), bond market premium (spread between Japanese and U.S. 10-year government bonds), and Japan EPU (Arbatli et al., 2017).
- Estimation method: Bayesian MCMC (Random Walk Metropolis–Hastings). Note: Mode, mean, median, and posterior percentiles from two chains of 150,000 draws (initial 50,000 discarded; retain ~1 every 4 draws). Convergence achieved in all estimations.

### Baseline posterior estimates (selected posterior statistics from Table 2)
- Posterior estimates (Equity risk premium / Bond market premium columns presented similarly). Selected posterior Mode / Median / 10th / 90th percentiles:
  - 푐푐: Mode 0.566 / Median 0.643 / 10th 0.372 / 90th 1.021
  - 훼훼1: Mode 1.045 / Median 1.034 / 10th 0.945 / 90th 1.126
  - 훼훼1+훼훼2: Mode 0.927 / Median 0.909 / 10th 0.856 / 90th 0.951
  - 훼훼푟푟: Mode 0.016 / Median 0.027 / 10th 0.009 / 90th 0.062
  - 훽훽1: Mode 0.430 / Median 0.424 / 10th 0.342 / 90th 0.508
  - 훽훽2: Mode 0.388 / Median 0.389 / 10th 0.287 / 90th 0.491
  - 훽훽푥푥: Mode 0.047 / Median 0.051 / 10th 0.039 / 90th 0.065
  - 훿훿1: Mode 0.919 / Median 0.962 / 10th 0.685 / 90th 1.305
  - 훿훿1+훿훿2: Mode 0.991 / Median 0.990 / 10th 0.986 / 90th 0.993
  - 훾훾0: Mode 0.023 / Median 0.012 / 10th 0.004 / 90th 0.027
  - 훾훾1: Mode 0.009 / Median 0.000 / 10th -0.013 / 90th 0.013
  - 훿훿푠푠: Mode 0.221 / Median 0.210 / 10th -0.118 / 90th 0.526
  - 훿훿푒푒: Mode 0.156 / Median 0.106 / 10th -0.204 / 90th 0.437 (equity-risk-premium specification: Mode 0.236 / Median 0.235 / 10th -0.079 / 90th 0.555)
  - 훿훿푝푝: Mode 0.225 / Median 0.206 / 10th -0.115 / 90th 0.532
  - 휎휎(휀휀푦푦): Mode 0.009 / Median 0.014 / 10th 0.007 / 90th 0.031
  - 휎휎(휀휀푥푥): Mode 0.951 / Median 0.944 / 10th 0.866 / 90th 1.036
  - 휎휎(휀휀휋휋): Mode 0.900 / Median 0.904 / 10th 0.831 / 90th 0.989
  - 휆휆푔푔 = 휎휎(휀휀푔푔)/휎휎(휀휀푦푦): Mode 0.221 / Median 8.485 / 10th 5.764 / 90th 13.090 (bond-premium variant: Median 10.492 / 10th 7.693 / 90th 16.979)
  - 휆휆푧푧 = 휎휎(휀휀푧푧)/휎휎(휀휀푥푥): Mode 0.009 / Median 0.012 / 10th 0.007 / 90th 0.025
  - 휎휎푤푤: Mode 0.365 / Median 0.293 / 10th 0.261 / 90th 0.332
  - 휎휎휇휇: Mode 0.281 / Median 0.276 / 10th 0.248 / 90th 0.307
  - 휌휌 (posterior): Mode 0.972 / Median 0.963 / 10th 0.950 / 90th 0.973

### Key baseline findings and quantitative results
- Natural rate trend:
  - Baseline shows a declining trend in Japan's natural rate since the 1990s.
  - Bayesian medians from both equity-risk-premium and bond-market-premium estimations suggest Japan’s natural rate in Q1 2018 was likely to be negative, but close to zero.
  - Median estimates suggest the natural rate in Japan has likely fallen into negative territory since the Asian Financial Crisis (AFC).
  - Range of estimates widens during and after the Global Financial Crisis (GFC), indicating greater uncertainty.
- Monetary policy stance and interest rate gap:
  - QQE by BoJ since April 2013 has pushed down real interest rates; short-term real interest rate on declining trend since 2013.
  - Short-term real interest rate below median natural rate since 2017 → negative interest rate gap (accommodative stance), but gap has been less than one percentage point and shrunk significantly in Q1 2018.
  - Interest rate gap is an incomplete and imprecise indicator of policy stance given uncertainty in natural rate estimation.
- Demographics:
  - Demographic change has had a significantly negative impact on the natural rate, with the impact increasing in recent years.
  - Negative demographic impact increased steadily since around 2005 when working-age population growth declined faster than total population growth.
  - Negative demographic impact stabilized at around -0.5 percent in recent years (due to slight recovery in working-age population growth), but projected to increase significantly after the early 2020s as working-age population growth is projected to decline faster than total population growth.
  - Demographic channel modeled through trend potential growth; other demographic channels (fertility, longevity, preferences for safe assets) are not captured by baseline channel.
- Trend potential growth and other shocks:
  - Other shocks to trend potential growth (e.g., technology) are important drivers of the natural rate.
  - Trend potential growth (excluding demographics) was high in early 1990s, declined significantly in 1990s (pushing natural rate into negative territory), with partial recoveries pre- and post-GFC but stabilized at low levels (about one third of 1990 level) in recent years.
- z-determinant (excess global savings, equity risk premium, policy uncertainty):
  - Equity risk premium rebounded to historical highs after the GFC, exerting substantial downward pressure on the natural rate.
  - Japan’s equity risk premium remained relatively high compared to historical levels and compared to the U.S., sustaining a negative impact on the natural rate in Japan.
  - Evidence for the global savings glut hypothesis during the period between AFC and GFC: excess global savings (emerging market current account surpluses) contributed to downward pressure on Japan’s natural rate, particularly between AFC and GFC. After the GFC, the negative impact of excess global savings diminished and even reversed in recent years.

### Robustness checks and alternative specifications
- Alternative priors:
  - Used alternative priors for 푐푐 and 훼훼푟푟 based on Pescatori and Turunen (2016) and Holston et al. (2017) (Table 3):
    - Pescatori and Turunen (2016): 푐푐 Gamma / 0.81 / 0.05; 훼훼푟푟 Gamma / 0.08 / 0.05
    - Holston et al. (2017): 푐푐 Gamma / 1 / 0.05; 훼훼푟푟 Gamma / 0.07 / 0.02
  - Results: median natural rate slightly higher than baseline (larger interest rate gaps), but demographic impact rising in recent years and still explains about half of the negative natural rate in Q1 2018 (around -0.8 percent). Qualitatively similar to baseline.
- Japan-specific excess savings:
  - Replaced global excess savings with Japan current account surplus (percent of GDP) to proxy Japan’s excess savings.
  - Estimated demographic impact comparable to baseline; Japan’s excess savings had a negative impact on domestic natural rate in recent years (consistent with longevity and old-age dependency influences).
- No z-controls (푧푧 follows random walk):
  - Replaced 푧푧-process with random walk z_t = z_{t-1} + ε_z,t.
  - Results: median natural rate slightly higher and more persistent; demographic contribution accounts for a larger share of drivers; implied larger interest rate gap (closing more recently).
- Using 10-year interest rate instead of short-term:
  - Replaced short-term real rate with 10-year real rate (10-year JGB yield minus inflation expectations).
  - Results: median natural rate based on 10-year interest rate declined since 1990s and fell into negative territory in recent years; median fell below zero only since around 2012 (vs. baseline negative since AFC).
  - Size of demographic contribution similar to baseline; z-determinant has a much smaller negative contribution to the 10-year-based natural rate than to the short-term-based natural rate.
  - Interpretation: more structural shocks (including demographics) relatively more important for the 10-year-based natural rate; relevance to BoJ Yield Curve Control where 10-year target is emphasized.

### Conclusions and policy implications
- Main conclusions:
  - Demographic change has contributed to the decline in Japan’s natural rate, with this contribution increasing over time—supporting the secular stagnation interpretation for Japan.
  - Other shocks to trend potential growth (e.g., technology/productivity) and z-determinants (excess savings, equity risk premium, policy uncertainty) also played important roles.
  - Bayesian median estimates suggest Japan’s natural rate likely fell below zero since the AFC; projected further declines in working-age population growth imply the negative demographic impact may increase by 2040.
  - Results are qualitatively robust across different priors and interest rate choices, but estimates have large error bands and should be interpreted with caution.
- Policy implications and recommendations:
  - Boost trend potential growth to offset negative demographic impacts and lift the natural rate to widen the interest rate gap.
  - Examples of policy actions include accelerating labor market reforms and other structural reforms (including more active migration policies).
  - Given limits to monetary policy when the natural rate is negative and falling, structural measures to raise potential growth are critical to restore policy space.

*Source: Author’s calculations and IMF staff estimates as presented in the provided content.*

### REFERENCES

### REFERENCES

### Natural rate of interest and monetary policy
- Barsky, R., Justiniano, A., and L. Melosi, 2014, “The Natural Rate of Interest and Its Usefulness for Monetary Policy,” American Economic Review, Vol. 104, No. 5, pp. 37-43.  
- Bernanke, B. S., 2005, “The Global Savings Glut and the U.S. Current Account Deficit,” Sandridge Lecture, Virginia Association of Economists, Richmond.  
- Bernanke, B. S., 2017, “Some Reflections on Japanese Monetary Policy,” Brookings Institute.  
- Caballero, R. J., Farhi, E., and P. O. Gourinchas, 2017, “The Safe Assets Shortage Conundrum,” Journal of Economic Perspectives, Vol. 31, No. 3, pp. 29-46.  
- Curdia, V., Ferrero, A., Ng, G. C., and A. Tambalotti, 2015, “Has U.S. Monetary Policy Tracked the Efficient Interest Rate?” Journal of Monetary Economics, Vol. 70, pp. 72-83.  
- Hakkio, C. S., and A. L. Smith, 2017, “Bond Premiums and the Natural Real Rate of Interest,” Federal Reserve Bank of Kansas City, KCFED Economic Review, First Quarter, pp. 5-38.  
- Holston, K., Laubach, T., and J. C. Williams, 2017, “Measuring the Natural Rate of Interest: International Trends and Determinants,” Journal of International Economics, Vol. 108, pp. 59-75.  
- Laubach, T., and J. C. Williams, 2003, “Measuring the Natural Rate of Interest,” Review of Economics and Statistics, Vol. 85, pp. 1063-1070.  
- Lewis, K. F., and F. Vazquez-Grande, 2017, “Measuring the Natural Rate of Interest: Alternative Specifications,” Board of Governors of the Federal Reserve System, Finance and Economics Discussion Series No. 2017-059.  
- Lubik, T. A., and C. Matthes, 2015, “Calculating the Natural Rate of Interest: A Comparison of Two Alternative Approaches,” Economic Brief, Federal Reserve Bank of Richmond, No. 15-10.  
- Pescatori, A., and J. Turunen, 2016, “Lower for Longer: Neutral Rate in the U.S.,” IMF Economic Review, Vol. 64, pp. 708-731.  
- Summers, L. H., 2014, “U.S. Economic Prospects: Secular Stagnation, Hysteresis, and the Zero Lower Bound,” Business Economics, Vol. 49, No. 2, pp. 65-73.  

### Demographics, productivity, and real interest rates
- Aiyar, S., Ebeke, C., and X. Shao, 2016, “The Impact of Workforce Aging on European Productivity,” IMF Working Paper No. 16/238, Washington D.C.: International Monetary Fund.  
- Carvalho, C., Ferrero, A., and F. Nechio, 2016, “Demographics and Real Interest Rates: Inspecting the Mechanism,” Working Paper 2016-05, Federal Reserve Bank of San Francisco.  
- Feyrer, J., 2007, “Demographics and Productivity,” The Review of Economics and Statistics, Vol. 89, Issue 1, pp. 100-109.  
- Ikeda D., and M. Saito, 2014, “The Effects of Demographic Changes on the Real Interest Rate in Japan,” Japan and the World Economy, Vol. 32, pp. 37-48.  
- Liu, Y., and N. Westelius, 2017, “The Impact of Demographics on Productivity and Inflation in Japan,” Journal of International Commerce, Economics and Policy, Vol. 8, No. 2.  
- Sudo, N., and Y. Takizuka, 2018, “Population Aging and the Real Interest Rate in the Last and Next 50 Years –A Tale Told by an Overlapping Generations Model,” Bank of Japan Working Paper Series No. 18-E-1.  

### Japan-focused studies and Bank of Japan working papers
- Arbatli, E. C., Davis, S. J., Ito, A., Miake, N., and I. Saito, 2017, “Policy Uncertainty in Japan,” IMF Working Paper No. 17/128, Washington D.C.: International Monetary Fund.  
- Colacelli, M., and E. Fernandez-Corugedo, 2018, “Macroeconomic Effects of Japan’s Demographics: Can Structural Reforms Reverse Them?” IMF Working Paper No. 18/248, Washington D.C.: International Monetary Fund.  
- Fujiwara, S., Iwasaki, Y., Muto, I., Nishizaki, K., and N. Sudo, 2016, “Development in the Natural Rate of Interest in Japan,” Supplementary Paper Series for the “Comprehensive Assessment” (2), Bank of Japan Review, 2016-E-12.  
- Hogen, Y., and R. Okuma, 2018, “The Anchoring of Inflation Expectations in Japan: A Learning-Approach Perspective,” Bank of Japan Working Paper Series No. 18-E-8.  
- Imakubo, K., Kojima, H., and J. Nakajima, 2015, “The Natural Yield Curve: Its Concept and Measurement,” Bank of Japan Working Paper Series No. 15-E-5.  
- Kawamoto, T., Ozaki, T., Kato, N., and K. Maehashi, 2017, “Methodology for Estimating Output Gap and Potential Growth Rate: An Update,” BoJ Reports & Research Papers, Research and Statistics Department, Bank of Japan.  
- Okazaki, Y., and N. Sudo, 2018, “Natural Rate of Interest in Japan –Measuring its Size and Identifying Drivers Based on a DSGE Model,” Bank of Japan Working Paper Series No. 18-E-6.  

### Financial frictions, credit, and global influences
- Coeurdacier, N., Guibaud, S., and K. Jin, 2015, “Credit Constraints and Growth in a Global Economy,” American Economic Review, Vol. 105, No. 9, pp. 2828-2881.  
- Daníelsson, Á., Helgason, Ó. S., and S., Thórarinsson, 2016, “Estimating the Natural Interest Rate for Iceland: An Exploratory Study,” Working Paper No. 74, Central Bank of Iceland.  
- Fries, S., Mesonnier, J., Mouabbi, S., and J. Renne, 2016, “National Natural Rates of Interest and the Single Monetary Policy in the Euro Area”, Banque de France, Working Paper No. 611.  
- Hogen, Y., and R. Okuma, 2018, “The Anchoring of Inflation Expectations in Japan: A Learning-Approach Perspective,” Bank of Japan Working Paper Series No. 18-E-8.  

### IMF and other institutional reports
- International Monetary Fund (IMF), 2017, G-20 Report on Strong, Sustainable, and Balanced Growth, IMF Annual Meetings, October, Washington D.C.: International Monetary Fund.  
- International Monetary Fund, 2018, Global Financial Stability Report, October, Washington D.C.: International Monetary Fund.  

*Source: wpiea2019031 - REFERENCES*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019031.pdf_