## Annex 2 — Data Sources, Interest Rate–Growth Differentials, and Fiscal Multipliers

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### Data sources, sample coverage, and variable definitions
- General sample: group of advanced economies as defined by the World Economic Outlook (WEO), total of 36 economies. Exact samples vary with analyses based on time coverage and data availability.
- Indicators extended backwards with additional sources (Annex Table 2.1.1):
  - Short-term policy rate: Bank for International Settlements; Global Data Scource; Haver; International Financial Statistics; and national sources.
  - Primary fiscal balance to GDP: Mauro and others (2015); and World Bank.
  - Gross public debt to GDP: Jordà and others (2019); Mauro and others (2015); IMF Historical Public Debt Database; and World Economic Outlook database.
  - Central bank assets to GDP: European Central Bank; Haver; and Ferguson and others (2015).
  - Nominal GDP: Jordà and others (2019); and World Economic Outlook database.
  - Real GDP: Mauro and others (2015); Global Data Source; Maddison Project database; and World Economic Outlook database.
  - Short-term government interest rate: Global Data Source; Jordà and others (2019); Organisation for Economic Co-operation and Development; and World Economic Outlook database.
  - Long-term government interest rate: Global Data Source; Jordà and others (2019); Organisation for Economic Co-operation and Development; and World Economic Outlook database.
  - Primary deficit: World Economic Outlook database.
  - Population by age: United Nations.
  - Interest cost of reserves: Federal Reserve.
  - Long-run total factor productivity: Bergeaud, Cette, and Lecat (2016).
  - Real public consumption: OECD Economic Outlook database.
  - Exchange rate classification (flex vs. fixed): Ilzetzki, Reinhart, and Rogoff (2019).
  - Systemic banking crisis classification: Laeven and Valencia (2018).
- Representative eras and sample windows:
  - Bretton Woods (1946–71); Post-Bretton Woods (1972–89); Globalization (1990–2007); GFC and post-GFC (2008–present).
  - Example samples by period listed (e.g., 1960–90 sample; 1991–2018 sample; 1871–2019 long-run r − g sample of 15–17 advanced economies).
- Forecast errors and vintages:
  - Forecast errors for r − g and debt dynamics: WEO vintages beginning in 1990.
  - Forecast errors for fiscal multipliers: OECD Economic Outlook vintages beginning in 1985.
- Construction notes:
  - Long-run short-term policy rate construction varies by country; WEO vintages used to reconstruct forecasted ten-year government bond spreads and five-year forecasts of policy rate for forward/backward decompositions (methodology in Annex 2.2).

### Stylized facts on interest rate–growth differentials (r − g)
- Sample: 17 advanced economies covering 1871–2018 for long-run r − g analysis (Annex Figure 2.2.1); 35 advanced economies for other series (Annex Figure 2.2.2).
- Key observations:
  - Advanced economies experienced negative interest rate–growth differentials a majority of the time between 1871 to 2018.
  - Median r − g fluctuated markedly before World War II.
  - Between World War II and throughout the 1970s, the median r − g was negative.
  - Through the 1980s median r − g varied near zero.
  - During the 1990s the median r − g was positive then declining, with overall trend mostly negative apart from a brief positive period around the Global Financial Crisis (GFC).

### Government budget constraint, decomposition framework, and notation
- Simplified one-period maturity identity:
  - B_t = D_t + B_{t−1}(1 + r_{t−1}).
  - In ratios: b_t − b_{t−1} = d_t + (r_{t−1} − g_t) b_{t−1}.
  - Interpretation: change in debt-to-GDP ratio equals primary deficit plus (r − g) times previous-period debt-to-GDP.
- Extended framework with long-term debt (exponentially decaying coupons, Hatchondo and Martinez 2009):
  - Government issues long-term debt with coupon c and principal repayment λ; yield-to-maturity r̅_t and bond premium τ_t^j defined.
  - Budget constraint with net debt and stock-flow adjustment η_t:
    - b_{t+1} = (b_t − θ_t s_t) e^{r̅_t − g_t} + η_t,
    - where θ_t and first-order approximations discussed (θ_t ≈ 1 if c = (1 − λ) r̅_t).
- Notation highlights:
  - g_t^j = (1/j)(log Y_{t+j} − log Y_t).
  - Bond premium τ_{t}^j = r_{t}^j − (1/j) ∑_{k=0}^{j−1} E_t r_{t+k}^{rf}.

### Backward- and forward-looking decomposition exercises, vintages, and assumptions
- Illustration data: IMF October 2015 WEO forecasts for 30 advanced economies for 2016–18; end point 2018 (latest final data on public debts and deficits available).
- Yield-curve construction assumptions:
  - Ten-year bond spread τ_t^{10} = r_t^{10} − (1/10) ∑_{k=0}^{4} E_t r_{t+k}^{rf} − (1/2) E_t r_{t+5}^{rf}.
  - Term spread grows linearly: τ_t^j = (1/j) τ_t^{10}. Robustness variant: τ_t^j = τ_t^{10} for all j.
  - For j > 10: r_t^j = (10/j) r_t^{10} + ((j − 10)/j)(E_t r_{t+5}^{rf} + τ_t^{10}).
  - λ set to λ = 1/8 to match average debt maturity (inverse of Macaulay duration).
- Backward-history example:
  - b̃_{2015} = b_{2015}
  - b̃_{t+1} = (b̃_t − θ_t s̃_t) e^{r̅_t − g_t} + η_t.
- Forward-looking inversion example:
  - b̃_{2024} = b_{2024}
  - b̃_t = θ_t s̃_t + (b̃_{t+1} − η_t) e^{−r̅_t + g̃_t}.

### Key quantitative results from backward decomposition (Table highlights)
- Table 2.2.1 reported cross-sectional distributions (vintages 2013, 2014, 2015). Exact entries:
  - Median impact on debt ratio
    - News about r − g: -2.94; -2.79; -1.47
    - News about primary deficits: -1.00; -0.05; -2.35
  - Mean impact on debt ratio
    - News about r − g: -5.70; -3.81; -2.20
    - News about primary deficits: -0.06; -0.20; -2.30
  - Cross-country standard deviation of impact on debt ratio
    - News about r − g: 8.27; 7.05; 3.27
    - News about primary deficits: 5.99; 5.35; 4.23
  - Relative contributions to cross-country debt ratio differences
    - News about r − g: 58%; 57%; 44%
    - News about primary deficits: 42%; 43%; 56%
  - Correlation of news about r − g and primary deficits
    - Point estimate: -0.14; 0.08; 0.11
    - p-value: 0.48; 0.67; 0.58
- Interpretation: median differences of r − g vs primary deficits typically small (around 1–3pp) relative to within-component cross-country standard deviations (around 3–8pp).

### Hypothetical forward experiment (Annex Figure 2.2.3)
- Thought experiment: a 100 basis point drop in the common component of r − g in 2019; persistence matches estimated common factor.
- Key outcomes:
  - If decline in r − g occurs through an increase in nominal growth: increase in borrowing capacity averages about 3 percentage points of GDP across advanced economies.
  - If decline in r − g occurs due to short-term policy rates: at the average 8-year maturity of debt, impact is small, averaging only a fraction of 1 percent of GDP.
  - Peak additional borrowing capacity about 3 percent as shown in the kernel-density experiment.

### Drivers, forecasting, and empirical estimation of r − g
- Variable used for r − g in forecasting: difference between the annual average short-term policy rate and the annual nominal growth rate.
- Theoretical decomposition: r_t − g_{t+1} = (E_t g^y_{t+1} − g^y_{t+1}) + (E_t π_{t+1} − π_{t+1}) + constant.
- Empirical specification (unbalanced sample of 15 countries):
  - (r_{i,t} − g_{i,t+1}) = α_i + δ_t + β_π π^*_{i,t+1} + β_g g^*_{i,t+1} + ε_{i,t}.
- Expectations constructed by era averages and five-year moving average smoothing across eras (Pre-WW1 through Global Financial Crisis and aftermath).
- Regression results (Annex Table 2.2.2) — selected exact reported values:
  - Inflation surprise coefficients: –1.022***; –0.368**; –0.297**; –0.367**; –0.364** (standard errors: (0.117)(0.179)(0.135)(0.179)(0.177)).
  - Growth surprise coefficients: –1.082***; –0.742***; –0.792***; –0.750***; –0.776*** (standard errors: (0.156)(0.198)(0.164)(0.202)(0.175)).
  - UIP debt gain: –0.098*** (standard error (0.036)).
  - UIP error: 0.120*** (standard error (0.036)) where included.
- Share of variation (exact values across columns): Growth surprise: 0.14; 0.16; 0.17; 0.16; 0.17. Inflation surprise: 0.28; 0.11; 0.09; 0.11; 0.11. Country Fixed Effects: 0.11; 0.09; 0.08; 0.09; 0.09. Time Fixed Effects: 0.19; 0.32; 0.32; 0.32; 0.32. Residuals: 0.28; 0.26; 0.26; 0.26; 0.25. Other variables: 0.00; 0.06; 0.09; 0.05; 0.07.
- Model diagnostics (exact reported values):
  - Observations: 2125; 1466; 1511; 1466; 1466
  - R^2: 0.848; 0.299; 0.288; 0.299; 0.3
  - Adjusted R^2: 0.835; 0.223; 0.213; 0.223; 0.224
  - Residual autocorrelation p-value: 0; 0; 0; 0; 0
  - Mean within-country residual persistence: 0.67; 0.35; 0.33; 0.33; 0.33 (standard errors (0.06)(0.07)(0.09)(0.09)(0.09))
  - Significance: *p<0.1; **p<0.05; ***p<0.01.
- Interpretation:
  - Time fixed effects explain around 20 percent of total variation and are persistent; forecast errors (inflation and growth surprises) explain around 40 percent and are transitory.
  - Country-specific controls largely insignificant; global drivers dominate in this sample.
  - UIP-related variables significant and consistent with financial repression effects lowering country-specific r − g.

### Forecasting the common international component and persistence
- Time fixed effects from specification (1) fitted with ARIMA selected by AIC: three autoregressive lags and two moving average terms, no unit root.
- Long-term persistence estimate: 0.87 (half-life of five years for a unit shock).

### International drivers and financial repression measures
- Explanatory variables and sources:
  - Long-run TFP: Bergeaud, Cette, and Lecat (2016).
  - Global share of middle-aged: UN 2019 Revision and Human Mortality Database.
  - EMDE GDP share: WEO spliced with Maddison Project 2018.
  - Opportunity cost of required reserves (US): cost_t = (Overnight rate_t − interest on reserves_t) × Required reserves_t / GDP_t (fraction of GDP); since 2009 this cost is zero for the Federal Reserve.
  - Financial repression proxy: Abiad, Detragiache, and Tressel (2008) index and UIP differential measures.
- Annex Figure 2.2.6 findings:
  - Correlation between (1 − global annual average Abiad et al. index) and the interest cost of unremunerated reserves: 0.87.
  - Timing: fast liberalization in the 1980s then easing in the 1990s; matches interest cost timing closely.
  - Cross-country regressions: country-level financial repression affects relative r − g but proxy not reliably significant for cross-country average.
- Annex Table 2.2.3 main takeaways:
  - TFP growth and global fraction 40–64 coefficients consistently significant.
  - EMDE GDP share explains a relatively large share of variance though less robust across specs.
  - Proxy for global financial repression not reliably significant.
  - Selected coefficient examples reported in table (exact values preserved in source).

### Fiscal multipliers — empirical strategy and baseline results
- Definition of government consumption shocks: real-time forecast errors FE_it = %ΔGC_it − E_{t−1}[%ΔGC_it]; winsorized at 1st and 99th percentiles; normalized to percent of GDP.
- Sample: 23 advanced OECD countries.
- Estimation: Auerbach and Gorodnichenko methodology with Ramey and Zubairy (2018) IV approach; Newey-West and clustered SEs; results robust to Driscoll-Kraay SEs.
- Baseline linear multiplier:
  - Baseline public consumption multiplier: about 1 throughout up to 4 years after the shock.
  - Multiplier different from zero at the 90 percent level in the first 3 periods.
  - First-stage F-statistics above Stock and Yogo (2005) rule of thumb of 10 and above Olea and Pflueger (2013) critical value.

### Robustness checks (Annex Table 2.3.1) — 1-year multiplier estimates (exact reported values)
- Column (1) baseline: 1.240** (standard error 0.600)
- Column (2) excluding 2008–2010 (No GFC): 0.950*** (0.365)
- Column (3) excluding the 1980s (No 80s): 1.360** (0.647)
- Column (4) Driscoll-Kraay SEs: 1.240** (0.613)
- Column (5) Fall vintages shocks: 1.640** (0.802)
- Observations across columns: 614, 545, 599, 614, 631.
- First-stage F-statistics: 54.719; 48.324; 50.260; 36.865; 38.306.
- Example R^2 reported: 0.631 (baseline).

### State-dependent multipliers (Annex Table 2.3.2) — selected one-year estimates (exact reported values)
- Unemployment below country-specific median: 1.000* (0.604)
- Unemployment above country-specific median: 1.410* (0.717)
- Unemployment below country-specific mean: 0.540 (0.689)
- Unemployment above country-specific mean: 1.710** (0.728)
- Expansion: 0.900 (0.726)
- Recession: 0.790 (0.934)
- Observations: 614; R2 examples: 0.634, 0.639, 0.666.
- First-stage F-statistics: 23.974; 31.467; 21.722.
- P-value of difference rows: 0.385; 0.026; 0.918 (across columns).
- Interpretation: multipliers above one during periods of slack when slack defined using country-specific mean; mixed evidence for other slack definitions.

### Multipliers under monetary accommodation and ELB (Annex Table 2.3.3)
- ELB indicator: short-term policy rate below 0.75 proxies the effective lower bound.
- Selected one-year multiplier estimates (exact reported values):
  - No ELB: 0.530 (0.579)
  - ELB: 2.590** (1.320)
  - Flexible exchange rates: 0.090 (0.315)
  - Fixed exchange rates: 2.050** (1.042)
  - Pre-GFC: −0.280 (0.687)
  - Post-GFC: 2.920* (1.604)
- Observations: 614. Example R2 values: 0.657; 0.650; 0.520.
- First-stage F-statistics: 22.754; 19.085; 24.991.
- P-value of difference rows: 0.062; 0.048; 0.004.
- Contextual notes:
  - Column (1) indicates a multiplier above two when monetary policy is constrained by ELB; significance sensitive to error structure and lag choices.
  - Excluding Japan: ELB effect equals 2.25.
  - Alternate specification with year fixed effects and baseline controls: multiplier at ELB around 3.
  - Fixed exchange rate regimes (monetary policy constrained) also show higher multipliers.

### Mechanisms, heterogeneity, and additional evidence
- Two channels for larger multipliers at ELB:
  - Interest rates do not rise to crowd out stimulus.
  - Increased inflation expectations at ELB reduce real rates permanently, supporting aggregate demand (theory suggests multipliers in the range of 2 to 5).
- Empirical suggestive evidence:
  - A 1 percent of GDP increase in government consumption increases one-year-ahead inflation expectations by 0.3 after one year; estimate not statistically significant.
- Debt and heterogeneity:
  - Multipliers are higher when household debt is large (consistent with Bernardini and Peersman (2017) and Klein (2017)).
  - Role of public debt on multipliers is mixed in literature; evidence varies.

*International Monetary Fund | April 2020 — IMF staff calculations.*

### Annex 2.1 Data Sources, Sample Coverage, and Variable Definitions

### Annex 2.1 Data Sources, Sample Coverage, and Variable Definitions

### Data sources and indicator coverage
- General sample: group of advanced economies as defined by the World Economic Outlook (WEO), total of 36 economies. Exact samples vary with analyses based on time coverage and data availability.
- WEO data are extended backwards for key indicators using additional sources listed below.
- Indicators extended backwards with respective additional sources (Annex Table 2.1.1):
  - Short-term policy rate: Bank for International Settlements; Global Data Scource; Haver; International Financial Statistics; and national sources.
  - Primary fiscal balance to GDP: Mauro and others (2015); and World Bank.
  - Gross public debt to GDP: Jordà and others (2019); Mauro and others (2015); IMF Historical Public Debt Database; and World Economic Outlook database.
  - Central bank assets to GDP: European Central Bank; Haver; and Ferguson and others (2015).
  - Nominal GDP: Jordà and others (2019); and World Economic Outlook database.
  - Real GDP: Mauro and others (2015); Global Data Source; Maddison Project database; and World Economic Outlook database.
  - Short-term government interest rate: Global Data Source; Jordà and others (2019); Organisation for Economic Co-operation and Development; and World Economic Outlook database.
  - Long-term government interest rate: Global Data Source; Jordà and others (2019); Organisation for Economic Co-operation and Development; and World Economic Outlook database.
  - Primary deficit: World Economic Outlook database.
  - Population by age: United Nations.
  - Interest cost of reserves: Federal Reserve.
  - Long-run total factor productivity: Bergeaud, Cette, and Lecat (2016).
  - Real public consumption: OECD Economic Outlook database.
  - Exchange rate classification (flex vs. fixed): Ilzetzki, Reinhart, and Rogoff (2019).
  - Systemic banking crisis classification: Laeven and Valencia (2018).

### Sample coverage and time windows (analytical exercises)
- Primary analytical sample: advanced economies. Annex Table 2.1.2 provides economy lists, time coverage, and which analytical/statistical exercises each economy enters.
- Representative eras and sample notes shown in Annex Table 2.1.2:
  - Bretton Woods (1946–71)
  - Post-Bretton Woods (1972–89)
  - Globalization (1990–2007)
  - GFC and post-GFC (2008–present)
- Example economy lists by era (as in the source):
  - 1960–90 sample includes: Australia; Austria; Belgium; Canada; Denmark; Finland; France; Germany; Greece; Iceland; Ireland; Italy; Japan; Luxembourg; Netherlands; New Zealand; Norway; Portugal; Spain; Sweden; Switzerland; United Kingdom; United States.
  - 1991–2018 sample includes: Australia; Austria; Belgium; Canada; Cyprus; Czech Republic; Denmark; Estonia; Finland; France; Germany; Greece; Iceland; Ireland; Israel; Italy; Japan; Korea; Latvia; Lithuania; Luxembourg; Malta; Netherlands; New Zealand; Norway; Portugal; Singapore; Slovak Republic; Slovenia; Spain; Sweden; Switzerland; Taiwan Provice of China; United Kingdom; United States.
  - 1871–2019 sample (long-run r − g analysis) includes: Belgium; Denmark; Finland; France; Germany; Italy; Japan; Netherlands; Norway; Portugal; Spain; Sweden; Switzerland; United Kingdom; United States.

### Forecast errors and vintages used
- Forecast errors for analyses of contributions to r − g and debt dynamics: calculated using forecasted annual data from World Economic Outlook database vintages beginning in 1990.
- Forecast errors for analysis of fiscal multipliers: calculated using forecasted annual data from OECD Economic Outlook database vintages beginning in 1985.

### Construction notes and data extensions
- Construction of long-run historical data for short-term monetary policy rate varies by country; vast majority of countries’ data come from national sources.
- WEO database vintages are used to reconstruct forecasted ten-year government bond spreads and sequences of five-year forecasts of the policy rate for forward/backward decomposition exercises (methodology described in Annex 2.2).

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### Annex 2.2 Interest Rate–Growth Differentials, Primary Deficits, and Contributions to Debt Dynamics

### Stylized facts on interest rate–growth differentials (r − g)
- Sample for long-run r − g analysis: 17 advanced economies covering 1871–2018 (Annex Figure 2.2.1).
- Key observations:
  - Advanced economies experienced negative interest rate–growth differentials a majority of the time between 1871 to 2018.
  - Median r − g fluctuated markedly before World War II.
  - Between World War II and throughout the 1970s, the median r − g was negative.
  - Through the 1980s median r − g varied near zero.
  - During the 1990s the median r − g was positive, then declining, with the overall trend mostly negative apart from a brief positive period around the Global Financial Crisis (GFC).
- Annex Figure 2.2.2: sample includes 35 advanced economies for interest rate–growth differentials (percentage points), showing median and individual country observations over time.

### Simplified government budget constraint (conceptual)
- Starting point: nominal debt B_t and deficit D_t. Under one-period maturity assumption:
  - B_t = D_t + B_{t−1}(1 + r_{t−1}).
- Denote nominal output Y_t and debt and deficit ratios b_t = B_t / Y_t and d_t = D_t / Y_t. Then:
  - b_t = d_t + b_{t−1} (1 + r_{t−1}) × (Y_{t−1} / Y_t).
  - With nominal growth g_t = Y_t / Y_{t−1} − 1 and first-order Taylor approximation:
    - (1 + r_{t−1})/(1 + g_t) ≈ 1 + r_{t−1} − g_t.
  - Rearranged simplified budget constraint:
    - b_t − b_{t−1} = d_t + (r_{t−1} − g_t) b_{t−1}.
- Interpretation: change in debt-to-GDP ratio equals primary deficit plus (r − g) times previous-period debt-to-GDP. As r − g falls, change in debt-to-GDP comes down, all else equal.

### Extended framework for decomposing unanticipated debt changes
- Notation highlights:
  - Growth: annualized nominal log output growth between t and t+j denoted g_t^j = (1/j)(log Y_{t+j} − log Y_t).
  - Interest rates: one-period policy rate r_t^{rf} (risk-free). Government debt yield curve at time t is {r_{t}^j}_{j≥1}, where j is maturity and r_{t}^j is the log yield on maturity-j zero-coupon bond priced as e^{−j r_{t}^j}.
  - Bond premium at horizon j: τ_{t}^j = r_{t}^j − (1/j) ∑_{k=0}^{j−1} E_t r_{t+k}^{rf}. E_t is expectation conditional on information at time t.
- Debt structure assumption: government issues only long-term debt with exponentially decaying coupons (Hatchondo and Martinez 2009). For each unit issued, government pays coupon c and principal repayment λ; remaining quantity becomes 1 − λ. A unit issued in period t yields coupon/principal streams (c + λ), (1 − λ)(c + λ), (1 − λ)^2(c + λ), ...
- Bond price q_t is the discounted price of that stream using the yield curve; yield-to-maturity r̅_t is the constant yield that prices the bond, with the implied inversion formula (equation (1) in the source).
- Government budget constraint with long-term debt (net of assets) and notation:
  - Let b_t denote debt-to-GDP at start of period t and s_t the primary surplus ratio during period t. Then:
    - b_{t+1} = (1 − λ) b_t e^{−g_t} + (1 / (1 + 1/q_t)) ((c + λ) b_t − s_t) e^{−g_t} (full expression in source).
  - Substituted simplification shown as:
    - b_{t+1} = (b_t − θ_t s_t) e^{r̅_t − g_t} + ... where θ_t = (1 − (1 − λ) e^{r̅_t} / (c + λ)).
  - First-order approximation yields θ_t ≈ 1 if c = (1 − λ) r̅_t.
- The government budget constraint used in exercises holds for net debt (financial liabilities minus financial assets); a stock-flow adjustment η_t is included:
  - b_{t+1} = (b_t − θ_t s_t) e^{r̅_t − g_t} + η_t, where η_t is computed as the residual from data.

### Sample and data for backward and forward decomposition exercises
- Illustration data source: IMF October 2015 WEO forecasts for 30 advanced economies for 2016–18. Vintage chosen because projections incorporate expected effects of large-scale asset purchase programs undertaken prior to that date. End point 2018 selected as latest final data on public debts and deficits available.
- Each forecast vintage contains contemporaneous measure of ten-year government bond spread and the sequence of five-year forecasts of the policy rate; these are treated as measures for r_t^{10} and E_t r_{t+k}^{rf} (k = 0,...,5) respectively.
- Yield-curve construction assumptions:
  - Ten-year bond spread τ_t^{10} computed as: τ_t^{10} = r_t^{10} − (1/10) ∑_{k=0}^{4} E_t r_{t+k}^{rf} − (1/2) E_t r_{t+5}^{rf} (equation (2) in the source).
  - Term spread grows linearly with horizon: τ_t^j = (1/j) τ_t^{10}. Robustness variant: τ_t^j = τ_t^{10} for all j.
  - For j > 10, expectations hypothesis supplemented by assumption that bond premium is constant: r_t^j = (10/j) r_t^{10} + ((j − 10)/j)(E_t r_{t+5}^{rf} + τ_t^{10}).
  - Yield-to-maturity r̅_t computed using equation (1) for given maturity λ. λ is set to λ = 1/8 to match average debt maturity of countries in sample (inverse of Macaulay duration).
- Backward-looking decomposition constructs alternative histories by replacing realized sequences with 2015 WEO forecasts for primary surplus ratios s̃_t, nominal growth g̃_t, and nominal interest rates (requires recomputing yield curve and yield-to-maturity at each point). Example backward-history rule:
  - b̃_{2015} = b_{2015}
  - b̃_{t+1} = (b̃_t − θ_t s̃_t) e^{r̅_t − g_t} + η_t
- Cross-country distributions for contributions of unanticipated changes in components to unanticipated change in debt path are shown in Figure 2.4 (main text reference).

### Correlations and summary statistics for post-2015 changes in public debt (Table 2.2.1 highlights)
- Table 2.2.1 summarizes cross-sectional distributions (average, cross-country standard deviation, correlations) of components’ effects on evolution of debt ratio 2015–2018 using WEO forecast vintages prior to 2015. Rightmost column in that table is source for numbers cited in main text.
- Key summary numbers (vintages 2013, 2014, 2015) — entries are presented exactly as in the source table:
  - Median impact on debt ratio
    - News about r − g: -2.94; -2.79; -1.47
    - News about primary deficits: -1.00; -0.05; -2.35
  - Mean impact on debt ratio
    - News about r − g: -5.70; -3.81; -2.20
    - News about primary deficits: -0.06; -0.20; -2.30
  - Cross-country standard deviation of impact on debt ratio
    - News about r − g: 8.27; 7.05; 3.27
    - News about primary deficits: 5.99; 5.35; 4.23
  - Relative contributions to cross-country debt ratio differences
    - News about r − g: 58%; 57%; 44%
    - News about primary deficits: 42%; 43%; 56%
  - Correlation of news about r − g and primary deficits
    - Point estimate: -0.14; 0.08; 0.11
    - p-value: 0.48; 0.67; 0.58
- Interpretation provided in source: difference in median impact on debt ratios of r − g compared to primary deficits is typically small (around 1–3pp) relative to cross-country standard deviation within each component (around 3–8pp). Thus average variation across components is a relatively small determinant of changes in the debt ratio relative to variation across countries within components.

### Forward-looking decomposition and hypothetical exercise
- Forward-looking decomposition holds fixed the 2024 WEO debt level and iterates back to 2019 by inverting the budget constraint. Example inversion for alternate future nominal growth sequence g̃_t:
  - b̃_{2024} = b_{2024}
  - b̃_t = θ_t s̃_t + (b̃_{t+1} − η_t) e^{−r̅_t + g̃_t}
- Hypothetical experiment reported in the annex (Annex Figure 2.2.3):
  - Estimates how much advanced economies could hypothetically increase borrowing while keeping debt stable at its 2024 projected level if interest rate–growth differentials were to drop by a further 100 basis points.
  - Key outcomes (as described in text):
    - If decline in r − g occurs through an increase in nominal growth, increase in borrowing capacity averages about 3 percentage points of GDP across advanced economies.
    - If decline in r − g occurs due to short-term policy rates, impact on government borrowing costs depends on debt maturity. At the average 8-year maturity of debt in advanced economies, impact is small, averaging only a fraction of 1 percent of GDP.

*Source: IMF staff compilation.*

### Annex  Table  2. 2. 1.   Summary  Statistics  for  Unexpected  Changes  of

### Annex  Table  2. 2. 1.   Summary  Statistics  for  Unexpected  Changes  of Fiscal Variables on Debt Ratio, 2015–18, by WEO Forecast Vintage

### Key findings on debt sensitivity and r − g
- If the shock is permanent, the impact of a given change in the interest-growth differential is much larger; sensitivity to debt maturity goes away when the decline in short-term interest rates is permanent (no difference between one-period and infinitely long-lived debt).
- Countries with larger debts are more sensitive to changes in interest rate–growth differentials, as r − g determines the growth rate of the debt ratio; a given change in r − g therefore leads to a larger change in borrowing capacity when the debt ratio is higher.
- Magnitude sensitivities computed are not a function of the sign of the shock and remain valid for increases in r − g (albeit with a negative sign); higher-debt countries are more exposed to increases in interest rate–growth differentials.
- If the interest rate–growth differential were to fall further, the average additional borrowing capacity consistent with debt stability would be about 3 percent at most, depending on how it would be financed. In general, the savings gained scales with the size of debt outstanding.

### Comparison to accounting decomposition (Annex Figure 2.2.4)
- The accounting decomposition explains movements in debt by realized components of the budget constraint: primary deficits, interest and growth rates, inflation, and a stock-flow adjustment.
- During the global financial crisis, debt levels rose sharply across advanced economies principally due to a sharp increase in primary deficits from 2009 onwards and a sharp decline in real growth and inflation in 2008–9.
- Primary deficits fell until 2015, slowing the rate at which the debt ratio grew.
- Since 2015, primary deficits and interest payments have been very close to pre-crisis levels, yet debt ratios are stable at much higher levels—implying that higher primary deficits during the crisis broadly offset gains from lower interest-growth differentials.
- Debt ratios have started to drift down since around 2016, mechanically due largely to higher nominal growth; however, fiscal consolidation (policy being broadly balanced rather than mildly expansionary as expected) underplays this role.

### Annex Figure 2.2.3 kernel-density thought experiment
- Thought experiment: unexpected 100 basis point drop in the common component of r − g across countries in 2019, evolving according to the statistical model and persistence matching the estimated common factor under alternative assumptions about debt maturity.
- Chart outcome: cross-country distribution of the increase in the debt ratio consistent with achieving the forecast debt levels in 2024 (as of the January 2020 WEO vintage) if r − g were to fall by 100 basis points in 2019.
- Mean shown in chart (percent of GDP) with financing through nominal growth or r on short-term debt, through r on long-term debt, and overall mean; peak additional borrowing capacity about 3 percent as noted above.

### Forecasting r − g — data and theoretical basis
- Interest rate–growth differential used: difference between the annual average short-term policy rate and the annual nominal growth rate.
- Rationale: strips out variation in risk premia and maps expected short-term policy rates to the yield curve; long decline in government interest rates since early 90s driven overwhelmingly by reductions in future expected policy rates, not term premia.
- Theoretical decomposition from Euler equation yields:
  - r_t − g_{t+1} = (E_t g^y_{t+1} − g^y_{t+1}) + (E_t π_{t+1} − π_{t+1}) + constant.
  - Thus, variation in realized interest-growth differential is driven by the sum of forecast errors on inflation and real growth.

### Empirical specification and estimation
- Estimated equation (unbalanced sample of 15 countries starting between 1871 (11 countries) and 1914 and ending in 2019):
  - (r_{i,t} − g_{i,t+1}) = α_i + δ_t + β_π π^*_{i,t+1} + β_g g^*_{i,t+1} + ε_{i,t},
  - where π^*_{i,t+1} and g^*_{i,t+1} are expectations, α and δ are country and time fixed effects.
- Expectations construction:
  - Step 1: averages over monetary eras: Pre-WW1 (1871–1913), WW1 (1914–1918), Interwar (1919–1938), WW2 (1939–1945), Bretton Woods (1946–1971), post-Bretton Woods (1972–1990), Global Financial Integration (1991–2007), Global Financial Crisis and aftermath (2008–2019).
  - Step 2: era averages filtered with equally-weighted five-year moving average to prevent sudden jumps.
- Time fixed effects capture components of r − g with predictable and cross-country predictive power; these time fixed effects are persistent and motivate focusing on global drivers.

### Regression results (Annex Table 2.2.2)
- Baseline points:
  - Coefficients on forecast errors statistically indistinguishable from unity in the first column when no additional explanatory variables are included; the simple Euler equation cannot be rejected in the text.
  - Inflation surprise coefficient in column (1): –1.022*** (standard error 0.117).
  - Growth surprise coefficient in column (1): –1.082*** (standard error 0.156).
- Selected coefficient pattern across columns (preserve exact reported values):
  - Inflation surprise: –1.022***; –0.368**; –0.297**; –0.367**; –0.364** (standard errors in parentheses: (0.117)(0.179)(0.135)(0.179)(0.177)).
  - Growth surprise: –1.082***; –0.742***; –0.792***; –0.750***; –0.776*** (standard errors in parentheses: (0.156)(0.198)(0.164)(0.202)(0.175)).
  - UIP debt gain: –0.098*** in relevant columns (standard error (0.036)).
  - UIP error: 0.120*** (standard error (0.036)) in the specification where included.
  - Other listed coefficients (Fraction 40–64, Dependency ratio, TFP growth, Labor productivity growth, NFA-GDP ratio) reported with exact values and standard errors as in table.
- Share of variation (exact values):
  - Growth surprise: 0.14; 0.16; 0.17; 0.16; 0.17
  - Inflation surprise: 0.28; 0.11; 0.09; 0.11; 0.11
  - Country Fixed Effects: 0.11; 0.09; 0.08; 0.09; 0.09
  - Time Fixed Effects: 0.19; 0.32; 0.32; 0.32; 0.32
  - Residuals: 0.28; 0.26; 0.26; 0.26; 0.25
  - Other variables: 0.00; 0.06; 0.09; 0.05; 0.07
- Model fit and diagnostics (exact reported values):
  - Observations: 2125; 1466; 1511; 1466; 1466
  - R^2: 0.848; 0.299; 0.288; 0.299; 0.3
  - Adjusted R^2: 0.835; 0.223; 0.213; 0.223; 0.224
  - Residual autocorrelation p-value: 0; 0; 0; 0; 0
  - Mean within-country residual persistence: 0.67; 0.35; 0.33; 0.33; 0.33 (standard errors (0.06)(0.07)(0.09)(0.09)(0.09))
  - Significance: *p<0.1; **p<0.05; ***p<0.01.
- Interpretation:
  - Time fixed effects explain around 20 percent of total variation in the baseline and represent the only predictable cross-country component with persistence; forecast errors (inflation and growth surprises) explain around 40 percent of variation and are an important channel but are transitory and unpredictable.
  - Country-specific controls are mostly insignificant, implying global trends are more important determinants of r − g than country-specific productivity or demographic trends in this sample.
  - UIP-related variables (UIP error and UIP debt gain) are significant and consistent with financial repression lowering country-specific r − g when domestic safe returns are low relative to foreign-currency alternatives.

### Forecasting the common international component of r − g
- Time fixed effects isolated from specification (1) are fitted with an ARIMA model; lag structure chosen by Akaike Information Criterion selects three autoregressive lags and two moving average terms and no unit root.
- Long-term persistence estimate: 0.87, suggesting a half-life of five years for a unit shock.

### Drivers of r − g — data sources and measures
- Explanatory variables and data sources:
  - Long-run total factor productivity (TFP) from long-run productivity database v2.3 (Bergeaud, Cette, and Lecat 2016); robustness checks use labor productivity measures.
  - Global share of middle-aged from UN’s 2019 Revision of World Population Prospects; advanced economy shares from Human Mortality Database aggregation.
  - Share of emerging market and developing economies in the global economy from WEO data spliced with Maddison Project 2018; robustness checks use cumulated current account deficits from the Jordà-Schularick-Taylor (2017) Macrohistory Database.
  - Opportunity cost of required reserves in the United States calculated as:
    - cost_t = (Overnight rate_t − interest on reserves_t) × Required reserves_t / GDP_t,
    - expressed as a fraction of GDP; since 2009 this cost has been zero because the Federal Reserve has paid interest on reserves equal to the federal funds rate.
  - Alternative natural measure of financial repression: the UIP differential (difference in bilateral rates of return after adjusting for nominal exchange rate depreciation), though average UIP differential is always zero and cannot be used as a global measure; cost of unremunerated reserves compared to the financial liberalization index of Abiad, Detragiache, and Tressel (2008) for global measure.

_International Monetary Fund | April 2020 — IMF staff calculations._

### Annex Figure 2.2.6.  Global Measures of Financial Repression

### Annex Figure 2.2.6.  Global Measures of Financial Repression

### Key findings on global financial repression and liberalization
- The Abiad, Detragiache, and Tressel (2008) index combines eight measures of financial liberalization and is shown as the average for the sample of 15 advanced economies used in Figure 2.2.1.
- Annex Figure 2.2.6 displays 1 minus the global annual average of this measure for the overlapping samples.
- The correlation between the Abiad, Detragiache, and Tressel index (1 minus the global annual average) and the interest cost of unremunerated reserves is 0.87.
- The timing of changes matches closely: increasingly fast liberalization in the 1980s before easing in the 1990s, which aligns with the timing of the interest cost of unremunerated reserves almost exactly.

### Cross-series interpretation and statistical notes
- The proxy of the global level of financial repression is not reliably significant in explaining the cross-country average of interest rate–growth differentials.
- Cross-country regressions suggest financial repression in a country can affect its relative interest rate–growth differential, but not the cross-country average.

*Source: Annex Figure 2.2.6 and accompanying text.*

### Annex Table 2.2.3.  International Drivers of Interest Rate–Growth Differentials

### Main empirical takeaways
- Coefficients on TFP growth and the global fraction of middle age are consistently statistically significant in explaining the common component of interest rate–growth differentials.
- The emerging market and developing economy (EMDE) GDP share is less reliably statistically significant in robustness checks, but its variation explains a relatively large share of the variance of the dependent variable.
- The proxy for global financial repression is not reliably significant.
- Note: full results reported in Annex Table 2.2.3; Newey-West standard errors in parentheses; no country fixed effects since this is a single time series (15 countries: 1950–2018).

### Selected coefficient highlights (as reported in table columns)
- TFP growth coefficients include: −0.660**, −0.879**, −0.331, −0.874***, −0.456**, −1.026***, −0.396** (standard errors reported in table).
- Global fraction 40–64 coefficients include: −0.889***, −0.883***, −0.265, −1.247***, −1.371*.
- Unremunerated reserves (GDP ratio): −1.484*** (standard error in table).
- EMDE GDP share: 0.290***, 0.257***, 0.143, 0.126***, 0.164*, −0.038, −0.053.
- Interest cost of reserves (GDP ratio) point estimates vary across specifications (example: 24.277*** in one column; other columns show large and mixed coefficients).
- Observations: 69 (67 in one specification). Adjusted R2 values reported, ranging up to 0.746 in the specification with a quadratic time trend.
- Statistical significance indicators: *p<0.1; **p<0.05; ***p<0.01.

*Source: Annex Table 2.2.3 and accompanying notes.*

### Annex 2.3 Fiscal Multipliers

### Empirical strategy and data
- Government consumption shocks are defined as real-time forecast errors: FE_it = %ΔGC_it − E_{t−1}[%ΔGC_it], where %ΔGC_it is percentage change in actual government consumption (measured in real time using data realized in year t+1).
- Forecast errors are winsorized excluding the bottom 1st and top 99th percentiles, normalized to percent of GDP using each country sample average of government consumption as a share of GDP.
- Sample: 23 advanced OECD countries.
- Baseline estimation follows Auerbach and Gorodnichenko methodology and uses the Ramey and Zubairy (2018) IV approach: shocks used as instruments to jointly estimate government consumption and real GDP responses. Newey-West and clustered standard errors are used; results robust to Driscoll-Kraay standard errors.

### Baseline linear multiplier results
- Baseline public consumption multiplier: about 1 throughout the estimation horizon (up to 4 years after the shock hits).
- Multiplier is different from zero at the 90 percent level in the first 3 periods (see main text Figure 2.9 referenced).
- First-stage F-statistics are above the Stock and Yogo (2005) rule of thumb of 10 and above the Olea and Pflueger (2013) critical value, indicating relevant instruments.

### Robustness (Annex Table 2.3.1)
- Reported 1-year multiplier estimates across robustness checks:
  - Column (1) baseline: 1.240** (standard error 0.600)
  - Column (2) excluding 2008–2010 (No GFC): 0.950*** (0.365)
  - Column (3) excluding the 1980s (No 80s): 1.360** (0.647)
  - Column (4) Driscoll-Kraay SEs: 1.240** (0.613)
  - Column (5) Fall vintages shocks: 1.640** (0.802)
- Observations across columns: 614, 545, 599, 614, 631.
- R2 reported (example: 0.631 for baseline). First-stage F-statistics: 54.719, 48.324, 50.260, 36.865, 38.306.

### State-dependent multipliers: slack and business cycle phases (Annex Table 2.3.2)
- Interaction specification uses indicator I_it for slack or recession states.
- Selected one-year multiplier estimates:
  - Unemployment below country-specific median: 1.000* (0.604)
  - Unemployment above country-specific median: 1.410* (0.717)
  - Unemployment below country-specific mean: 0.540 (0.689)
  - Unemployment above country-specific mean: 1.710** (0.728)
  - Expansion: 0.900 (0.726)
  - Recession: 0.790 (0.934)
- Observations: 614; R2 values: 0.634, 0.639, 0.666 (across specifications).
- First-stage F-statistics: 23.974, 31.467, 21.722.
- P-value of difference (coefficient difference t-test): 0.385, 0.026, 0.918 (across columns). Interpretation: multipliers are above one during periods of slack when slack is defined using the country-specific mean; evidence mixed when using other slack/recession definitions.

### Multipliers when monetary policy is accommodative (Annex Table 2.3.3)
- Indicator for ELB: short-term policy rate below 0.75 proxies the effective lower bound (ELB).
- Selected one-year multiplier estimates:
  - No ELB: 0.530 (0.579)
  - ELB: 2.590** (1.320)
  - Flexible exchange rates: 0.090 (0.315)
  - Fixed exchange rates: 2.050** (1.042)
  - Pre-GFC: −0.280 (0.687)
  - Post-GFC: 2.920* (1.604)
- Observations: 614 in reported columns. R2 values include 0.657, 0.650, 0.520. First-stage F-statistics: 22.754, 19.085, 24.991.
- P-value of difference rows: 0.062, 0.048, 0.004 (across reported comparisons).
- Interpretation and contextual notes:
  - Column (1) indicates a multiplier above two when monetary policy is constrained by ELB; statistical significance depends on error structure and lag inclusion.
  - Excluding Japan leaves ELB effect equal to 2.25.
  - Alternate specification with year fixed effects and baseline controls provides a multiplier at the ELB of around 3 (text note).
  - Fixed exchange rate regimes, where monetary policy is constrained, also show higher multipliers (consistent with prior literature).

### Mechanisms and additional evidence
- Two channels why fiscal policy may be more potent at the ELB:
  - Interest rates do not rise in response to fiscal stimulus, removing the normal crowding-out channel.
  - Increased inflation expectations at the ELB reduce real rates permanently, supporting aggregate demand (theoretical literature cites multipliers in the range of 2 to 5).
- Empirical suggestive evidence on inflation expectations:
  - A 1 percent increase in government consumption increases one-year-ahead inflation expectations by 0.3 after one year; inflation expectations move around zero in normal times. The point estimate is not statistically significant.
- Debt and multiplier heterogeneity:
  - Multipliers are higher when household debt is large (consistent with Bernardini and Peersman (2017) and Klein (2017)); this aligns with the theory that consumer deleveraging raises propensity to spend.
  - The role of public debt is mixed in the literature; some evidence suggests reduced effectiveness with public debt overhangs, while other studies report varying results.

*Source: Annex 2.3, Annex Figures and Tables, accompanying text.*

*Source: IMF staff calculations and Annex figures/tables from the provided document.*

### Annex Figure 2.3.3.  Response of One

### Annex Figure 2.3.3.  Response of One

### Figure description
- Variable plotted: Year Ahead Inflation Expectations (Units).
- Shock: public consumption shock equal to one percent of GDP.
- Comparisons:
  - Blue solid line: response under ELB.
  - Red line: response during no-ELB periods.
- Uncertainty: shaded area corresponds to 90 percent confidence interval.
- ELB = effective lower bound.

### Key findings and interpretation
- The figure isolates the effect of a one percent of GDP public consumption shock on year-ahead inflation expectations.
- The comparison differentiates responses when monetary policy is constrained by the ELB versus when it is not.
- Confidence intervals shown are 90 percent confidence intervals, indicating sampling uncertainty around the estimated responses.

### Sources and data
- Sources: Consensus Economics; Organisation for Economic Co-operation and Development; and IMF staff calculations.

*International Monetary Fund | Annex Figure 2.3.3.*

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_Source: https://www.imf.org/-/media/files/publications/weo/2020/april/english/annexch2.pdf_