## wpiea2023085 — 1870. To analyze some key facts about the estimatedr (excerpt)

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---

### Data and sample
- Demographic data combines three sources: the World Bank’s World Development Indicators, which is extended with data from the Human Mortality Database and the International Historical Statistics (Mitchell 2007).
- Income inequality data comes from the World Inequality Database (www.wid.world).
- Data on capital openness comes from Quinn (2003).
- All remaining data comes from the IMF’s World Economic Outlook database.
- Sample composition: annual data on 16 advanced economies — Australia, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Netherlands, Norway, Spain, Sweden, Switzerland, United Kingdom, and the United States.
- Sample properties:
  - Unbalanced; starts as early as 1878 and ends in 2019.
  - Years excluded to avoid outliers: 1913 to 1921 and 1939 to 1947.

### r* estimates: phases, uncertainty, and cross-country patterns
- Three identified phases for r*:
  - A relatively stable, or slightly declining, series up until WWII.
  - An increase in r* after WWII until the 1970s.
  - A decline to all-time lows thereafter.
- Two notable characteristics:
  - The average r* exhibits the same downward trend starting around 1960 as documented in previous literature.
  - The range of r* across countries is substantially narrower in the latter half of the sample, which could reflect increasing global integration.
- Estimation uncertainty:
  - Standard errors can be as large as 1.9% percentage points.
  - Appendix Figure A.2 shows the average r* estimate with a one-standard-error range.
- Visualization notes:
  - Figure 1 plots time series of average and median r* estimates across countries alongside the interquartile range.

### Output gap estimates
- Output gap is a key output of the estimation procedure and is imposed to be stationary around zero.
- Appendix Figure A.3 plots:
  - Time series of the average output gap across countries.
  - Output gap of the United States along with the NBER recession phases.
- Large observed drops:
  - Biggest drop around the Great Depression to around 5 percent of potential GDP (individual countries observed output gaps greater than 10 percent of potential GDP).
  - Other large drops observed around the oil... (text truncated in source excerpt).

### Sensitivity checks and estimation uncertainty
- Baseline r∗ estimates:
  - Not particularly sensitive to changes in assumptions for cross-country average time series; individual-country differences can be larger.
  - Level of r∗ estimates sensitive to assumed initial values and error term variances; time series patterns are robust and often differ only by a constant across sensitivity tests.
- One-sided (real-time) filter estimates are more volatile than two-sided estimates, but overall time series patterns are similar (Appendix Figure A.5).
- High estimation uncertainty:
  - The uncertainty around the level of r∗ limits its use for gauging the precise appropriate level of real interest rates and the implied monetary stance.
  - Regressions that control for the level of r∗ (e.g., through country fixed effects) can still yield robust results.
- Reported sensitivity tests (definitions preserved):
  - S1: increase error term variances by 50 percent
  - S2: decrease error term variances by 50 percent
  - S3: force error term variances to fall by 50 percent after 1990
  - S4: force Phillips curve to flatten by 50 percent
  - S5: change the initial value of r* by -50 percent
  - S6: change the initial value of r* by 50 percent
  - S7: estimate only from 1950 onwards

### Comparison to other r* measures and observed real rates
- Comparison findings:
  - On the common sample period, the paper’s r∗ tracks Holston, Laubach and Williams (2017) closely, with some level differences for the United States and Canada and a more pronounced drop since 2000 for the United Kingdom.
  - Compared to Rachel and Summers (2019): their OECD aggregate r∗ declines from about 3.5 percent to about 0.4 percent by 2017; the paper’s r∗ shows a very similar decline.
- Observed real interest rate (r) versus structural r∗:
  - Observed r is much more volatile than structural r∗.
  - Correlation coefficient between r and r∗ is 6.8 percent; excluding outliers (removing observations outside the [-5,5] interval for r and r∗) increases the correlation to 18.3 percent.
- Construction of observed r:
  - Real rate constructed as nominal yield on long-term government bonds (10y when available) minus 10-year trailing moving average of inflation.
  - Alternative inflation-expectations measures were experimented with, but trailing-data measures exhibited more intuitive properties.

### Stylized bivariate patterns for determinants of r∗
- Variables analyzed (available for all 16 economies over long periods): old-age dependency ratio, life expectancy, population growth, relative price of capital, TFP growth, trend in real GDP growth, public debt to GDP, inequality (income share of top 10 percent), capital account openness (Quinn 2003).
- Long-run bivariate patterns:
  - Old-age dependency ratio and life expectancy: steady upward trend; negatively correlate with the decline in neutral rates since the 1970s.
  - Population growth and trend real GDP growth: tight positive relationship with neutral rates since at least the beginning of the 20th century.
  - Public debt to GDP: positive correlation with r∗ until WWI and clearly negative correlation thereafter.
  - Decline in relative price of capital and decline in TFP growth coincide with the decline in neutral rates.
  - Neutral rates began falling before inequality started increasing in the sample countries.
  - Capital account openness displays a negative correlation with neutral rates for most of the sample.
- After residualizing for year fixed effects:
  - Old-age dependency ratio remains negatively related to r∗.
  - Life expectancy is negatively related to r∗ but becomes statistically insignificant once year effects are considered.
  - Faster population growth is positively and statistically significantly associated with r∗.
  - TFP growth is statistically significant and positively associated with r∗ after residualizing.
  - Real GDP trend growth: tight positive and statistically significant relationship with r∗.
  - Public debt to GDP: negative relationship with r∗ but not statistically significant in bivariate plots.
  - Inequality: associated with lower neutral interest rates in bivariate analysis, but not statistically significant once common factors are controlled for.
  - Capital account openness: mildly negative relationship with r∗, not statistically significant in bivariate analysis.

### Multivariate regression analysis (panel regressions) — specification and key results
- Estimated specification: r∗i,t = αi + τt + βx Xi,t−1 + εi,t, where αi are country fixed effects, τt are year fixed effects, and Xi,t−1 are lagged determinants. Coefficients β describe associations, not causal effects.
- Key regression results (all regressions include country and time fixed effects; standard errors clustered at the country level in parentheses):
  - Column (1) — Period 1878–2020, Observations 1,834, R2 0.713:
    - Old-age dependency ratio: -0.251*** (0.042)
  - Column (2) — Period 1956–2020, Observations 974, R2 0.867:
    - Old-age dependency ratio: -0.224*** (0.068)
    - Life expectancy: -0.055 (0.113)
    - Population growth: 0.431*** (0.142)
    - TFP growth: 0.113*** (0.033)
    - Relative price of capital: -0.024 (0.091)
  - Column (3) — Period 1878–2020, Observations 1,831, R2 0.776:
    - Old-age dependency ratio: -0.189*** (0.047)
    - Real GDP trend growth: 0.454*** (0.086)
  - Column (4) — Period 1878–2020, Observations 1,741, R2 0.817:
    - Old-age dependency ratio: -0.103** (0.046)
    - Real GDP trend growth: 0.376*** (0.063)
    - Public debt to GDP: -0.017** (0.007)
  - Column (5) — Period 1989–2020, Observations 1,000, R2 0.892:
    - Old-age dependency ratio: -0.148*** (0.027)
    - Real GDP trend growth: 0.458*** (0.087)
    - Public debt to GDP: -0.011*** (0.002)
    - Inequality: -2.702 (1.985)
  - Column (6) — Period 1878–2020, Observations 1,744, R2 0.825:
    - Old-age dependency ratio: -0.127*** (0.034)
    - Real GDP trend growth: 0.391*** (0.061)
    - Public debt to GDP: -0.016** (0.007)
    - Capital account openness: 0.013*** (0.004)
  - Significance notation: ***p <0.01, **p <0.05, *p <0.1.
- Regression insights:
  - Demographics (old-age dependency ratio) and long-run growth (real GDP trend growth, population growth, TFP growth) are important correlates of neutral interest rates.
  - Public debt shows a negative association with r∗ in multivariate regressions.
  - Capital account openness appears significant in column (6) with a positive coefficient, but results vary across specifications.
  - Inequality is not robustly significant in multivariate regressions once country and year effects are included.

### On proxying r∗ with observed long-term real interest rates
- Using observed long-term real interest rates as a proxy for r∗ can be misleading.
- Regressions using observed real rates as the dependent variable are generally less consistent with theory and more unstable than those using estimated r∗.
- Results using observed real rates:
  - Do not show robust evidence that population aging (old-age dependency ratio) has a negative effect.
  - Other coefficients can be counterintuitive (e.g., negative associations with relative price of capital and TFP growth in some specifications).
- Conclusion: evidence supports using model-based measures of r∗ rather than simply proxying with observed long-term real rates.

### Robustness, time-varying patterns, and extensions
- Exclusion exercise:
  - Appendix Table A.3 shows results are robust in the exclusion of any country (example shown excluding Australia: Old-age dependency ratio -0.128*** (0.034); Real GDP trend growth 0.399*** (0.061); Public debt to GDP -0.015** (0.007); Capital account openness 0.015*** (0.003); Observations 1,623; R2 0.828).
- Rolling regressions (40-year moving window; Figure 8):
  - Old-age dependency ratio: positive pre-WWI, moved into negative territory after that; statistically insignificant for most of the sample, with identification driven by the aging acceleration of the 90s.
  - Real GDP trend growth: always positive and generally statistically significant; magnitude varies from about 0.1 to 0.6.
  - Public debt to GDP: coefficient hovers between -0.005 to -0.018 and is statistically significant most of the time.
  - Capital account openness: statistically insignificant for much of the sample, except during major liberalizations.
- Current account and post-1973 interactions (Appendix Table A.4, column (4) — Period 1978-2020, Observations 1,700, R2 0.832):
  - Old-age dependency ratio: -0.118*** (0.034)
  - Real GDP trend growth: 0.404*** (0.062)
  - Public debt to GDP: -0.016** (0.007)
  - Capital account openness: 0.012*** (0.003)
  - Openness×Post73: -0.003 (0.010)
  - Current account balance: 0.040 (0.048)
  - CA×Post73: -0.268* (0.140)
  - Openness×CA: -0.001 (0.001)
  - Openness×CA×Post73: 0.004** (0.002)
  - Interpretation: in the recent period of globalization, a more positive current account is associated with higher r∗; interactions suggest stronger association of current account imbalances and r∗ in more open countries.
- Robustness checks (Table 3 overview):
  - Alternative estimation approaches (pooled OLS, only country fixed effects, baseline with country and year fixed effects, WLS, double clustered SEs, Driscoll-Kraay SEs, dynamic specification with lagged dependent variable) produce broadly similar conclusions.
  - Dynamic specification shows high persistence in r∗: lag dependent variable 0.883*** (0.034), which reduces the significance of slowly evolving covariates.

### Contributions to historical changes in r∗ (decomposition)
- Periods analyzed: i) pre-WWI (1885–1913), ii) inter wars (1919–1923), iii) post-WWII until 60s (1946–1969), iv) 70s and 80s (1970–1989), v) post-inflation targeting (1990–2020).
- Long-run magnitudes:
  - For the median country, decline since the 1960s peak is 4.5 percentage points, bottoming out at 0.5 percent in 2019.
  - Japan experienced the largest declines; fall of r∗ in Japan is about a third of the average decline excluding Japan.
  - Other large post-60s declines: Belgium, France, Italy, the Netherlands, Spain. Smallest declines: Canada, Denmark, United States.
- Decomposition method:
  - Country-specific contribution to changes in r∗ between t and t′: C_x i,t,t′ = β̂_x (x_i,t′ − x_i,t), using β̂_x from column (6) of Table 1 and 5-year averages around t and t′.
- Decomposition findings by period (cross-country averages):
  - Pre-WWI: explanatory variables account for little of the change; residual negative and as large as one percentage point.
  - Inter-war: decline in r∗ largely accounted for by lower trend GDP growth and, to a lesser extent, increased age dependency ratio.
  - Post-WWII until 60s: declining public debt stocks and capital account liberalization contributed to increases in r∗; partially offset by increasing old-age dependency ratio.
  - 70s and 80s: falling r∗ associated with slowing GDP growth, population aging, and debt accumulation; capital account opening and common factors provided positive contributions; residual as large as 2 percentage points implies unobserved factors.
  - Post-IT (post-1990): acceleration of demographic aging in the 90s is the main factor behind the fall in r∗; debt accumulation and slowing trend GDP growth also contributed.

### Conclusions and policy interpretation
- Main empirical contributions:
  - Comparable r∗ estimates for 16 advanced economies over 1878–2019 using the Laubach-Williams estimation approach.
  - Three distinct historical phases: 1870s–WWII (stable/slightly declining), post-WWII–1960s (increasing), since 1960s (steady decline).
  - Median decline since the 1960s peak: 4.5 percentage points; median r∗ bottomed at 0.5 percent in 2019.
- Determinants and interpretation:
  - Population aging and increases in life expectancy associated with lower r∗.
  - Production-factor proxies (population growth, TFP growth, real GDP trend growth) correlate positively with r∗.
  - Higher public debt-to-GDP ratios associated with lower r∗ (association, not claimed causal).
  - More open capital accounts associated with higher r∗ in some specifications.
  - Differences from some prior studies attributed to using semi-structural estimation rather than observed real rates as proxies for r∗.
- Policy implication:
  - Broad macroeconomic policy regimes and long-run structural factors (demographics, public debt dynamics, capital account openness) are important to explain secular shifts in the neutral interest rate.

### Appendix A — key reported quantitative outputs and descriptive statistics (exact values)
- Table A.1: Selected imposed values for error term variances (country rows shown exactly in source; example entries preserved in source).
- Table A.2: Descriptive statistics (exact values)
  - r*: Obs 1,857 Mean 3.1 SD 2.2 Min -2.1 Max 17.7
  - Old-age dependency ratio: Obs 1,983 Mean 15.8 SD 7.1 Min 3.0 Max 48.0
  - Life expectancy at birth: Obs 1,952 Mean 64.5 SD 14.0 Min 29.5 Max 84.4
  - Population growth: Obs 1,913 Mean 0.9 SD 0.6 Min -2.1 Max 4.2
  - Relative price of capital: Obs 1,097 Mean 2.2 SD 1.9 Min 0.9 Max 17.4
  - TFP growth: Obs 1,017 Mean 1.0 SD 1.8 Min -6.9 Max 10.4
  - Real GDP trend growth: Obs 1,978 Mean 3.1 SD 2.0 Min -5.8 Max 12.9
  - Public debt to GDP: Obs 1,859 Mean 53.0 SD 38.6 Min 1.9 Max 239.6
  - Inequality: Obs 1,114 Mean 0.3 SD 0.0 Min 0.2 Max 0.6
  - Capital account openness: Obs 1,996 Mean 79.6 SD 26.4 Min 0.0 Max 100.0
- Figure A.2: Weighted mean of r* constructed using PPP GDP weights; 1SE range is PPP GDP-weighted mean of standard errors across countries.
- Figure A.6: Correlation between r* and observed rates: 6.8 percent; excluding outliers brings correlation to 18.3 percent.

*Source: wpiea2023085 — 1870. To analyze some key facts about the estimatedr (excerpt).*

### 1870. To analyze some key facts about the estimatedr

### 1870. To analyze some key facts about the estimatedr

### Data and sample
- Demographic data combines three sources: the World Bank’s World Development Indicators, which is extended with data from the Human Mortality Database and the International Historical Statistics (Mitchell 2007).
- Income inequality data comes from the World Inequality Database (www.wid.world).
- Data on capital openness comes from Quinn (2003).
- All remaining data comes from the IMF’s World Economic Outlook database.
- The sample consists of annual data on 16 advanced economies: Australia, Belgium, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Japan, Netherlands, Norway, Spain, Sweden, Switzerland, United Kingdom, and the United States.
- The sample is unbalanced and starts as early as 1878 and ends in 2019.
- To avoid outliers driving results, years around the two World Wars are excluded: specifically 1913 to 1921 and 1939 to 1947.

### r* estimates
- Figure 1 plots time series of average and median r* estimates across countries alongside the interquartile range.
- Three phases for r* are identified:
  - A relatively stable, or slightly declining, series at the beginning of the sample up until WWII.
  - An increase in r* after WWII until the 1970s.
  - A decline to all-time lows thereafter.
- Two notable characteristics:
  - The average r* exhibits the same downward trend starting around 1960 as documented in previous literature.
  - The range of r* across countries is substantially narrower in the latter half of the sample, which could reflect ever-increasing integration of the global economy.
- A finding common with related literature is relatively wide standard errors, as large as 1.9% percentage points.
- Appendix Figure A.2 shows the average r* estimate with a one-standard-error range.

### Output gap estimates
- A key output of the estimation procedure is the estimate of the output gap.
- Appendix Figure A.3 plots the time series of the average output gap across countries, as well as the output gap of the United States along with the NBER recession phases.
- Under the imposed assumption in the estimation, the output gap is stationary around zero.
- The biggest drop in the average output gap is observed around the Great Depression to around 5 percent of potential GDP, even though individual countries observed output gaps greater than 10 percent of potential GDP.
- Other large drops in output gaps are observed around the oil... 

*Source: wpiea2023085 - 1870. To analyze some key facts about the estimatedr (excerpt).*

### Appendix Table A.2 reports the summary statistics of all variables.

### Appendix Table A.2 reports the summary statistics of all variables

### Sensitivity checks and estimation uncertainty
- Baseline r∗ estimates are not particularly sensitive to changes in assumptions for cross-country average time series; for individual countries differences can be larger.
- Level of r∗ estimates is sensitive to assumed initial values and error term variances; time series patterns are robust and often differ only by a constant across sensitivity tests.
- All estimates exhibit sizeable standard errors; estimation uncertainty is high (illustrated in Appendix Figure A.2).
- One-sided (real-time) filter estimates are more volatile than two-sided estimates, but overall time series patterns are similar (Appendix Figure A.5).
- The uncertainty around the level of r∗ limits its use for gauging the precise appropriate level of real interest rates and the implied monetary stance; regressions that control for the level of r∗ (e.g., through country fixed effects) can still yield robust results.

### Comparison to other r∗ measures and observed real rates
- On the common sample period, the shape of the paper’s r∗ tracks Holston, Laubach and Williams (2017) (HLW) closely, with some differences (e.g., a more pronounced drop since 2000 for the United Kingdom).
- There are level differences for the United States and Canada relative to HLW; the paper’s estimates are smoother than HLW’s, likely due to different data frequency (annual vs quarterly).
- Comparison to Rachel and Summers (2019): their OECD aggregate r∗ declines from about 3.5 percent to about 0.4 percent by 2017; the paper’s r∗ shows a very similar decline.
- Observed real interest rate (r) is much more volatile than structural r∗. The correlation coefficient between r and r∗ is 6.8 percent; excluding outliers (removing observations outside the [-5,5] interval for r and r∗) increases the correlation to 18.3 percent.
- Constructing r as: nominal yield on long-term government bonds (10y when available) minus 10-year trailing moving average of inflation is the primary approach used; alternative inflation-expectations measures based on leading data were experimented with but trailing-data measures exhibited more intuitive properties.

### Stylized facts about determinants of r∗ (bivariate patterns)
- Variables analyzed (available for all 16 advanced economies over long periods): old-age dependency ratio, life expectancy, population growth, relative price of capital, TFP growth, trend in real GDP growth, public debt to GDP, inequality (income share of top 10 percent), capital account openness (Quinn 2003 index).
- Long-run patterns:
  - Old-age dependency ratio and life expectancy display a steady upward trend and negatively correlate with the decline in neutral rates since the 1970s.
  - Population growth and trend real GDP growth show a tight positive relationship with neutral rates since at least the beginning of the 20th century.
  - Public debt to GDP shows a positive correlation with r∗ until WWI and a clearly negative correlation thereafter.
  - Decline in relative price of capital and decline in TFP growth coincide with the decline in neutral rates.
  - Neutral rates began falling before inequality started increasing in the sample countries.
  - Capital account openness displays a negative correlation with neutral rates for most of the sample.
- After residualizing variables with respect to year fixed effects (to remove common time trends):
  - Old-age dependency ratio remains negatively related to r∗.
  - Life expectancy is negatively related to r∗ but becomes statistically insignificant once year effects are considered.
  - Faster population growth is positively and statistically significantly associated with r∗.
  - TFP growth is statistically significant and positively associated with r∗ after residualizing.
  - Real GDP trend growth shows a tight positive and statistically significant relationship with r∗.
  - Public debt to GDP has a negative relationship with r∗ but is not statistically significant in bivariate plots.
  - Inequality is associated with lower neutral interest rates in bivariate analysis, but the relationship is not statistically significant once common factors are controlled for.
  - Capital account openness shows a mildly negative relationship with r∗, not statistically significant in bivariate analysis.

### Multivariate regression analysis (panel regressions)
- Estimated specification: r∗i,t = αi + τt + βx Xi,t−1 + εi,t, where αi are country fixed effects, τt are year fixed effects, and Xi,t−1 are lagged determinants. Coefficients β describe associations, not causal effects.
- Key regression results (Table 1; all regressions include country and time fixed effects; standard errors clustered at the country level in parentheses):

  - Column (1) — Period 1878–2020, Observations 1,834, R2 0.713:
    - Old-age dependency ratio: -0.251*** (0.042)

  - Column (2) — Period 1956–2020, Observations 974, R2 0.867:
    - Old-age dependency ratio: -0.224*** (0.068)
    - Life expectancy: -0.055 (0.113)
    - Population growth: 0.431*** (0.142)
    - TFP growth: 0.113*** (0.033)
    - Relative price of capital: -0.024 (0.091)

  - Column (3) — Period 1878–2020, Observations 1,831, R2 0.776:
    - Old-age dependency ratio: -0.189*** (0.047)
    - Real GDP trend growth: 0.454*** (0.086)

  - Column (4) — Period 1878–2020, Observations 1,741, R2 0.817:
    - Old-age dependency ratio: -0.103** (0.046)
    - Real GDP trend growth: 0.376*** (0.063)
    - Public debt to GDP: -0.017** (0.007)

  - Column (5) — Period 1989–2020, Observations 1,000, R2 0.892:
    - Old-age dependency ratio: -0.148*** (0.027)
    - Real GDP trend growth: 0.458*** (0.087)
    - Public debt to GDP: -0.011*** (0.002)
    - Inequality: -2.702 (1.985)

  - Column (6) — Period 1878–2020, Observations 1,744, R2 0.825:
    - Old-age dependency ratio: -0.127*** (0.034)
    - Real GDP trend growth: 0.391*** (0.061)
    - Public debt to GDP: -0.016** (0.007)
    - Capital account openness: 0.013*** (0.004)

  - Significance notation: ***p <0.01, **p <0.05, *p <0.1.

- Regression insights:
  - Demographic trends (old-age dependency ratio) and long-run growth (real GDP trend growth, population growth, TFP growth) are important and closely related to neutral interest rates.
  - Public debt shows a negative association with r∗ in multivariate regressions.
  - Capital account openness appears significant in column (6) with a positive coefficient, but results vary across specifications.
  - Inequality is not robustly significant in multivariate regressions once country and year effects are included.

### On proxying r∗ with observed long-term real interest rates
- Using observed long-term real interest rates as a proxy for r∗ can be misleading.
- Regressions using observed real rates as the dependent variable are generally less consistent with theory and more unstable than those using estimated r∗.
- Results using observed real rates do not show robust evidence that population aging (old-age dependency ratio) has a negative effect; other coefficients can be counterintuitive (e.g., negative associations with relative price of capital and TFP growth in some specifications).
- Evidence supports using model-based measures of r∗ rather than simply proxying with observed long-term real rates.

*Source: Appendix Table A.2 and accompanying text in the provided IMF content unit.*

### Appendix Table A.3 shows that our results are robust in the exclusion of any country.

### wpiea2023085 - Appendix Table A.3 shows that our results are robust in the exclusion of any country.

### Multivariate regressions (Table 2 overview)
- The paper reports estimates of regressions of observed long-term real interest rates on lagged determinants, with country and year fixed effects; standard errors clustered at the country level.
- Selected coefficient estimates from Table 2 columns (1)–(6):
  - Old-age dependency ratio: 0.054, 0.053*, 0.033, 0.031, 0.052, 0.058 (standard errors in parentheses reported for each)
  - Life expectancy: -0.137**, -0.290 (standard errors reported)
  - Population growth: -0.443 (0.310)
  - TFP growth: -0.090* (0.050)
  - Relative price of capital: -0.344*** (0.111)
  - Real GDP trend growth: 0.103, 0.107, 0.230**, 0.093 (standard errors reported)
  - Public debt to GDP: 0.000, 0.002, -0.000 (standard errors reported)
  - Inequality: -5.448** (2.492)
  - Capital account openness: -0.013 (0.009)
- Sample periods and fit:
  - Periods: 1878–2020, 1956–2020, 1878–2020, 1878–2020, 1989–2020, 1878–2020 (by column)
  - Observations: 1,834; 974; 1,831; 1,741; 1,000; 1,741 (by column)
  - R2 values: 0.561, 0.508, 0.527, 0.530, 0.668, 0.536 (by column)
- Note on construction: Real interest rate = nominal yield on long term bond (10y when available) minus 10 year trailing moving average of inflation.

### Time-varying coefficients (rolling regressions; Figure 8)
- Rolling regressions use a 40-year moving window; coefficients reported at the last year of the window with 90 percent confidence intervals.
- Key time-varying patterns:
  - Old-age dependency ratio: positive in the pre-WWI period, moved into negative territory after that; statistically insignificant for most of the sample, with identification driven by the aging acceleration of the 90s.
  - Real GDP trend growth: always positive and generally statistically significant; magnitude varies from about 0.1 to 0.6.
  - Public debt to GDP: coefficient hovers between -0.005 to -0.018 and is statistically significant most of the time.
  - Capital account openness index: statistically insignificant for a large part of the sample, except during the period of major capital account liberalizations.

### Current account, post-1973 interactions, and capital openness
- Appendix Table A.4 extends regressions by including current account balance (% of GDP).
  - The coefficient on the current account balance is insignificant in the baseline.
  - Interacting current account with a post-1973 dummy: current account × post-1973 is positive and significant at a 10 percent level — in the recent period of globalization, a more positive current account is associated with higher r∗.
  - Interpretation: consistent with the global savings glut hypothesis linking large capital inflows (current account deficits) to low r∗ in recipient countries.
  - Interaction of current account and capital account openness after 1973: positively related to r∗, suggesting stronger association of current account imbalances and r∗ in more open countries.
- Conclusion from these tests: capital account openness index and current account balance capture different features; coefficient on capital account openness remains nearly unaffected when adding current account.

### Robustness checks (Section 4.2.1 and Table 3)
- Alternative estimation approaches reported in Table 3:
  - Column (1) Pooled OLS: Old-age dependency ratio -0.083** (0.030); Real GDP trend growth 0.576*** (0.117); Public debt to GDP -0.006 (0.004); Capital account openness -0.002 (0.005).
  - Column (2) Only country fixed effects: Old-age dependency ratio -0.083*** (0.022); Real GDP trend growth 0.498*** (0.047); Public debt to GDP -0.017*** (0.006); Capital account openness 0.002 (0.004).
  - Column (3) Baseline (country and year fixed effects): Old-age dependency ratio -0.127*** (0.034); Real GDP trend growth 0.389*** (0.061); Public debt to GDP -0.016** (0.007); Capital account openness 0.013*** (0.004).
  - Column (4) Weighted least squares (WLS with weights = inverse squared standard errors): results close to baseline.
  - Column (5) Double clustered SEs (country and year): similar results.
  - Column (6) Driscoll-Kraay SEs: similar results.
  - Column (7) Dynamic specification with lagged dependent variable: Lag dependent variable 0.883*** (0.034); inclusion leads to Old-age dependency ratio -0.014 (0.012) and indicates high persistence in r∗.
- Observations and fit in Table 3:
  - Observations: 1,741 in columns (1)–(6); column (7) reports 656 (lagged specification).
  - R2 values: 0.463, 0.786, 0.825, 0.854, 0.825, 0.825, 0.970 (by column).
- Robustness conclusions:
  - Time-invariant factors and common factors across countries are important for identification.
  - Public debt to GDP becomes significant when purging time-invariant factors, with magnitude close to baseline.
  - Results robust to WLS weighting and alternative standard error computations.
  - High persistence in r∗ (lag ≈ 0.883***) reduces significance of variables with steady trends (e.g., old-age dependency ratio) in dynamic specifications.

### Contributions to changes in r∗ (Section 4.3, Figures 9–10)
- Periods analyzed: i) pre-WWI (1885–1913), ii) inter wars (1919–1923), iii) post-WWII until 60s (1946–1969), iv) 70s and 80s (1970–1989), v) post-inflation targeting (1990–2020).
- Long-run patterns in r∗:
  - For the median country, decline since the 1960s peak is 4.5 percentage points, bottoming out at 0.5 percent in 2019.
  - Cross-country co-movements are evident; Japan experienced the largest declines (fall of r∗ in Japan is about a third of the average decline excluding Japan).
  - Other large post-60s declines: Belgium, France, Italy, the Netherlands, Spain. Smallest declines: Canada, Denmark, United States.
- Decomposition method:
  - Country-specific contribution to changes in r∗ between t and t′: C_x i,t,t′ = β̂_x (x_i,t′ − x_i,t), using β̂_x from column (6) of Table 1 and 5-year averages around t and t′.
- Decomposition findings (cross-country averages by period):
  - Pre-WWI: unable to attribute a large part of changes in r∗ to the explanatory variables; residual negative and as large as one percentage point.
  - Inter-war: decline in r∗ largely accounted for by lower trend GDP growth and, to a much smaller degree, by increased age dependency ratio.
  - Post-WWII until 60s: declining public debt stocks and liberalization of the capital account contributed to increases in r∗; these were partially offset by increasing old-age dependency ratio.
  - 70s and 80s: falling r∗ associated with slowing GDP growth, population aging, and debt accumulation; capital account opening and factors common to all countries provided positive contributions; residual as large as 2 percentage points implies other unobserved factors associated with the decline.
  - Post-IT (post-1990): acceleration of demographic aging in the 90s is the main factor behind the fall in r∗; debt accumulation and slowing trend GDP growth also contributed.

### Conclusions (Section 5)
- The study provides comparable r∗ estimates for 16 advanced economies over 1878–2019 using the Laubach-Williams estimation approach.
- Three distinct phases in r∗ over the last 150 years:
  - 1870s to WWII: relatively stable or slightly declining r∗.
  - Post-WWII to 1960s: r∗ increasing.
  - Since the 1960s: r∗ has declined steadily (median decline 4.5 percentage points, bottoming at 0.5 percent in 2019).
- Determinants and interpretation:
  - Population aging and increases in life expectancy (old-age dependency ratio, life expectancy) associated with lower r∗.
  - Production-factor proxies (population growth, TFP growth, real GDP trend growth) correlate positively with r∗.
  - Higher public debt-to-GDP ratios associated with lower r∗ (association, not claimed causal).
  - More open capital accounts associated with higher r∗.
  - Findings differ from some prior studies that found traditional determinants play little role; attributed here to the semi-structural estimation approach rather than using observed real rates as r∗ proxies.
- Policy implication: broad macroeconomic policy regimes and long-run structural factors (demographics, public debt dynamics, capital account openness) are important to explain secular shifts in the neutral interest rate.

*Source: wpiea2023085 - Appendix Table A.3 shows that our results are robust in the exclusion of any country.*

### References

### wpiea2023085 - References

### References cited
- Arena, Marco; Gabriel Di Bella; Alfredo Cuevas; Borja Gracia; Vina Nguyen; and Alex Pienkowski.2020. “It is only natural: Europe’s low interest rates.” International Monetary Fund, IMF Working Paper No. 2020/116.
- Armelius, Hanna; Martin Solberger; and Erik Spänberg.2018. “Is the Swedish neutral interest rate affected by international developments.” Sveriges Riksbank Economic Review, 1: 22–37.
- Attanasio, Orazio P, and Guglielmo Weber.2010. “Consumption and saving: models of intertemporal allocation and their implications for public policy.” Journal of Economic Literature, 48(3): 693–751.
- Auclert, Adrien, and Matthew Rognlie.2018. “Inequality and aggregate demand.” National Bureau of Economic Research, Inc NBER Working Papers 24280.
- Bauer, Michael D., and Glenn D. Rudebusch.2020. “Interest rates under falling stars.” American Economic Review, 110(5): 1316–54.
- Bernanke, Ben S.2005. “The global saving glut and the US current account deficit.” Remarks at the Homer Jones Lecture, St. Louis, MO.
- Borio, Claudio; Piti Disyatat; Mikael Juselius; and Phurichai Rungcharoenkitkul.2017. “Why so low for so long? A long-term view of real interest rates.” Bank for International Settlements BIS Working Papers 685.
- Brand, Claus; Marcin Bielecki; and Adrian Penalver.2018. “The natural rate of interest: estimates, drivers, and challenges to monetary policy.” ECB Occasional Paper, (217).
- Carvalho, Carlos; Andrea Ferrero; Fernanda Nechio; et al.2017. “Demographic transition and low us interest rates.” FRBSF Economic Letter, 27.
- Cesa-Bianchi, Ambrogio; Richard Harrison; and Rana Sajedi.2022. “Decomposing the drivers of Global R.” Bank of England working papers 990.
- Del Negro, Marco; Domenico Giannone; Marc P. Giannoni; and Andrea Tambalotti.2017. “Safety, liquidity, and the natural rate of interest.” Brookings Papers on Economic Activity, 48(1 (Spring): 235–316.
- Del Negro, Marco; Domenico Giannone; Marc P. Giannoni; and Andrea Tambalotti.2019. “Global trends in interest rates.” Journal of International Economics, 118(C): 248–262.
- Driscoll, John C, and Aart C Kraay.1998. “Consistent covariance matrix estimation with spatially dependent panel data.” Review of economics and statistics, 80(4): 549–560.
- Fujiwara, Shigeaki; Yuto Iwasaki; Ichiro Muto; Kenji Nishizaki; and Nao Sudo.2016. “Developments in the natural rate of interest in Japan.” Bank of Japan Review.
- Hamilton, James D.; Ethan S. Harris; Jan Hatzius; and Kenneth D. West.2016. “The equilibrium real funds rate: Past, present, and future.” IMF Economic Review, 64(4): 660–707.
- Holston, Kathryn; Thomas Laubach; and John C Williams.2017. “Measuring the natural rate of interest: International trends and determinants.” Journal of International Economics, 108: S59–S75.
- Johannsen, Benjamin K., and Elmar Mertens.2016. “The expected real interest rate in the long run : Time series evidence with the effective lower bound.” Board of Governors of the Federal Reserve System (U.S.) FEDS Notes 2016-02-09.
- Johannsen, Benjamin K., and Elmar Mertens.2021. “A Time-series model of interest rates with the effective lower bound.” Journal of Money, Credit and Banking, 53(5): 1005–1046.
- Jordà, Òscar; Moritz Schularick; and Alan M Taylor.2017. “Macrofinancial history and the new business cycle facts.” NBER macroeconomics annual, 31(1): 213–263.
- Kiley, Michael T.2020. “What can the data tell us about the equilibrium real interest rate?” International Journal of Central Banking, 16(3): 181–209.
- Laubach, Thomas, and John C Williams.2003. “Measuring the natural rate of interest.” Review of Economics and Statistics, 85(4): 1063–1070.
- Lian, Weicheng; Natalija Novta; Evgenia Pugacheva; Yannick Timmer; and Petia Topalova.2020. “The price of capital goods: a driver of investment under threat.” IMF Economic Review, 68(3): 509–549.
- Linde, Jesper; Josef Platzer; and Robin Tietz.2022. “Natural versus neutral rate of interest: Parsing disagreement about future short-term interest rates.” VoxEU.org, July 26.
- Lunsford, Kurt G., and Kenneth D. West.2019. “Some evidence on secular drivers of US safe real rates.” American Economic Journal: Macroeconomics, 11(4): 113–39.
- Mian, Atif; Ludwig Straub; and Amir Sufi.2021. “Indebted demand.” Quarterly Journal of Economics, 136(4): 2243–2307.
- Mian, Atif R; Ludwig Straub; and Amir Sufi.2020. “The saving glut of the rich.” NBER Working Paper No. 26941.
- Mishkin, Frederic S.2007. “Inflation dynamics.” International Finance, 10(3): 317–334.
- Mitchell, Brian R.2007. International historical statistics 1750-2005: Americas. Springer.
- Quinn, Dennis P.2003. “Capital account liberalization and financial globalization, 1890–1999: a synoptic view.” International Journal of Finance & Economics, 8(3): 189–204.
- Rachel, Lukasz, and Lawrence Summers.2019. “On secular stagnation in the industrialized world.” NBER Working Papers 26198.
- Reinhart, Carmen M., and M. Belen Sbrancia.2015. “The liquidation of government debt.” Economic Policy, 30(82): 291–333.
- Roberts, John M.2004. “Monetary policy and inflation dynamics.” Available at SSRN 633222.
- Rogoff, Kenneth S; Barbara Rossi; and Paul Schmelzing.2022. “Long-run trends in long-maturity real rates 1311-2021.” National Bureau of Economic Research.
- Sajedi, Rana, and Gregory Thwaites.2016. “Why are real interest rates so low? The role of the relative price of investment goods.” IMF Economic Review, 64(4): 635–659.
- Straub, Ludwig.2019. “Consumption, savings, and the distribution of permanent income.” Unpublished manuscript, Harvard University.

### Appendix A — Additional results: key quantitative outputs and specifications

- Table A.1: Values imposed for error term variances (selected country rows shown exactly as in source)
  - Australia: 0.330    0.125    0.625    0.015    0.023    0.045    0.000    HLW GB*
  - Belgium: 0.200    0.077    0.950    0.005    0.050    0.025    0.018    HLW Euro
  - Canada: 0.330    0.250    0.625    0.015    0.023    0.045    0.000    HLW CA*
  - Denmark: 0.200    0.077    0.950    0.005    0.050    0.025    0.009    HLW Euro
  - Finland: 0.330    0.125    0.625    0.015    0.023    0.045    0.017    HLW GB*
  - France: 0.200    0.077    0.950    0.005    0.050    0.025    0.011    HLW Euro
  - Germany: 0.630    0.200    0.625    0.035    0.033    0.055    0.015    HLW Euro*
  - Italy: 0.200    0.231    1.900    0.005    0.050    0.025    0.002    HLW Euro*
  - Japan: 0.723    0.185    2.005    0.023    0.410    0.031    0.094    FIMNS 2016
  - Netherlands: 0.200    0.077    0.950    0.005    0.050    0.025    0.002    HLW Euro
  - Norway: 0.330    0.125    0.625    0.015    0.023    0.045    0.017    HLW GB*
  - Spain: 0.150    0.044    1.002    0.013    0.054    0.083    0.053    HLW Euro
  - Sweden: 0.330    0.125    0.625    0.015    0.023    0.045    0.017    HLW GB*
  - Switzerland: 0.200    0.154    0.950    0.005    0.050    0.025    0.001    HLW Euro*
  - United Kingdom: 0.330    0.125    0.625    0.015    0.023    0.045    0.001    HLW GB*
  - United States: 0.330    0.375    0.625    0.015    0.023    0.045    0.008    HLW US*
  - Notes: The table reports the values of the error term variances used in the estimation of the Holston, Laubach and Williams (2017) model: λ_g = σ_ε,g / σ_ε,y* and λ_z = σ_ε,z / σ_ε,˜y × a_r / √2, where a_r is the initial value of the coefficient on the real rate gap in the IS curve. The sources are shown in the last column. HLW refers to Holston, Laubach and Williams (2017). FIMNS 2016 refers to Fujiwara et al. (2016). The asterisk denotes that we started at the values from the source, but manually adjusted the values in order to get convergence of the algorithm.

- Table A.2: Descriptive statistics (exact values)
  - r*: Obs 1,857 Mean 3.1 SD 2.2 Min -2.1 Max 17.7
  - Old-age dependency ratio: Obs 1,983 Mean 15.8 SD 7.1 Min 3.0 Max 48.0
  - Life expectancy at birth: Obs 1,952 Mean 64.5 SD 14.0 Min 29.5 Max 84.4
  - Population growth: Obs 1,913 Mean 0.9 SD 0.6 Min -2.1 Max 4.2
  - Relative price of capital: Obs 1,097 Mean 2.2 SD 1.9 Min 0.9 Max 17.4
  - TFP growth: Obs 1,017 Mean 1.0 SD 1.8 Min -6.9 Max 10.4
  - Real GDP trend growth: Obs 1,978 Mean 3.1 SD 2.0 Min -5.8 Max 12.9
  - Public debt to GDP: Obs 1,859 Mean 53.0 SD 38.6 Min 1.9 Max 239.6
  - Inequality: Obs 1,114 Mean 0.3 SD 0.0 Min 0.2 Max 0.6
  - Capital account openness: Obs 1,996 Mean 79.6 SD 26.4 Min 0.0 Max 100.0
  - Notes: The Old-age dependency ratio is defined as the share of the population of age 65 and older relative to the share of the population of age 15 to 64. Inequality refers to the share of pre-tax national income held by top 10 percent. Capital account openness is an index from Quinn (2003).

- Table A.3: Multivariate regressions of r* excluding one country at a time (column (1) — excluding Australia; exact estimates)
  - Old-age dependency ratio: -0.128*** (0.034)
  - Real GDP trend growth: 0.399*** (0.061)
  - Public debt to GDP: -0.015** (0.007)
  - Capital account openness: 0.015*** (0.003)
  - Observations: 1,623
  - R2: 0.828
  - Notes: All regressions include country and time fixed effects. Standard errors clustered at the country level are reported in parentheses. ***p <0.01, **p <0.05, *p <0.1.

- Table A.4: Multivariate regressions of r* including current account (column (4) — exact estimates)
  - Old-age dependency ratio: -0.118*** (0.034)
  - Real GDP trend growth: 0.404*** (0.062)
  - Public debt to GDP: -0.016** (0.007)
  - Capital account openness: 0.012*** (0.003)
  - Openness×Post73: -0.003 (0.010)
  - Current account balance: 0.040 (0.048)
  - CA×Post73: -0.268* (0.140)
  - Openness×CA: -0.001 (0.001)
  - Openness×CA×Post73: 0.004** (0.002)
  - Period: 1978-2020
  - Observations: 1,700
  - R2: 0.832
  - Notes: All regressions include country and time fixed effects. Standard errors clustered at the country level are reported in parentheses. Openness refers to capital account openness. CA refers to current account balance. Post73 refers to a dummy variable that is one if year is after 1973. ***p <0.01, **p <0.05, *p <0.1.

### Figures and additional diagnostic summaries (exact reported metrics)
- Figure A.2 (Uncertainty around r* estimates)
  - Weighted mean of r* constructed using PPP GDP weights. The 1SE range is calculated as the PPP GDP-weighted mean of the standard errors of the r* estimates across all countries.
  - War periods excluded from estimation: 1913–1921 and 1939–1947.
- Figure A.6 (r* versus observed short-term and long-term real rates)
  - The correlation coefficient between the two measures is 6.8 percent; excluding outliers (i.e., removing observations outside the [-5,5] interval for r and r*) brings the correlation to 18.3 percent.
- Sensitivity checks described for r* estimates (S1–S7) with definitions exactly as reported:
  - S1: increase error term variances by 50 percent
  - S2: decrease error term variances by 50 percent
  - S3: force error term variances to fall by 50 percent after 1990
  - S4: force Phillips curve to flatten by 50 percent
  - S5: change the initial value of r* by -50 percent
  - S6: change the initial value of r* by 50 percent
  - S7: estimate only from 1950 onwards

*Content derived from "References" and Appendix A of wpiea2023085 (source PDF: wpiea2023085 - References).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023085.pdf_