## wp1882 - 1. Or, equivalently, that a country can only service a lower debt given its surpluses.

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### Historical patterns of interest-growth differentials in advanced economies
- Point estimates and long-run averages:
  - Since 1880, the average annual interest-growth differential in six advanced economies has been -1.7 percent.
  - Since 1960, the average annual interest-growth differential has been -0.8 percent.
- Medium-run (decadal) variation:
  - Decadal average differentials varied from around minus five percentage points per year in the 1960s and 70s to nearly two percentage points in the 1980s and 90s, before dropping back below zero more recently.
- Short-term volatility:
  - Short-term volatility is very high; the differential frequently changes by several percentage points per year.
  - For the USA (1880-2016), using the short-term rate, the average interest-growth differential was -1.9 percent during 1880-2016, and -1.6 percent since 1950.
  - In the USA the differential moves by over 2.5 percentage points in more than one-fifth of years.
  - A 2.5 percentage point fall in the annual growth rate causes the debt-GDP ratio to increase mechanically by 2.5 percentage points.

### Consequences for maximum sustainable debt levels
- Three data aspects with contrasting implications:
  - Low average differentials imply maximum sustainable debt levels are likely to be high.
  - Persistent medium-run movements imply maximum sustainable debt levels may vary substantially over time.
  - High short-term volatility induces unpredictable variation in the debt-GDP ratio, increasing default risk and pulling down maximum sustainable debt levels.
- Critical role of the long-run differential:
  - If the long-run interest-growth differential is negative, governments can issue arbitrarily large debt without becoming insolvent so long as deficits are bounded.
  - Because of this, testing statistically whether the long-term interest-growth differential may be positive is essential.

### Statistical testing and estimated bounds on the long-run differential
- Two statistical approaches employed:
  - Likelihood ratio approach based on combinations of VAR coefficients applied to two datasets (five advanced economies): one annual since 1880 and one quarterly since 1956.
    - For both datasets, VAR point estimates are negative.
    - The associated likelihood ratio test cannot reject the possibility that the long-run differential is positive.
    - Typical upper bound from this method: in the order of zero to one percent per year.
  - Spectral approach developed by Müller and Watson (2016).
    - Produces similar negative point estimates for the long-run differential.
    - Generates larger confidence intervals and is more sensitive to outliers.
    - For countries with long, uninterrupted data, provides upper bounds around one to two percent per year.
- Joint implication:
  - Both methods suggest small and positive long-run interest-growth differentials are unlikely, but still consistent with the data.

### Model linking interest-growth processes to debt limits
- Model features:
  - Based on Ghosh, Kim, Mendoza, Ostry, and Qureshi (2013) with extensions:
    - Correlated aggregate shocks to the nominal risk-free rate and nominal GDP growth.
    - Idiosyncratic shocks to governments’ primary balances.
    - Surplus-GDP ratio is a bounded function of the aggregate state, including outstanding debts.
  - Deficits funded by issuing noncontingent defaultable bonds to risk-neutral investors.
  - Investors price government debt at a premium over the risk-free rate reflecting default risk; a sufficiently high debt level causes loss of market access (debt limit).
- Definition of debt limit:
  - The debt level at which government loses access to debt markets because investors refuse to lend at any finite interest rate.

### Simple motivating model and analytic results (Proposition 1)
- Setup:
  - Constant nominal growth rate 1 + g and risk-free nominal interest rate 1 + r.
  - Expected primary surplus se_t ∈ [s_, s̄]; realized surplus st = se_t + ut with ut mean-zero and support [−ū, ū].
  - One-period zero coupon nominal bonds with price qt and face value bt; budget constraint qt bt + st = b_{t-1}(1 + g).
  - Investors infinitely risk averse: qt = 1/(1 + r) if probability of default is zero; qt = 0 otherwise.
- Proposition 1 results:
  - If r > g then there exists b̄, the maximum sustainable debt, given by:
    - b̄ = (s̄ − ū)(1 + g)(1 + r) / (r − g)
  - If r < g then there exists no such b̄; the government never defaults, no matter the debt level.
- Implications:
  - Debt limit is enormously sensitive to r − g, especially when r − g is close to zero.
  - The sign of r − g is critical for the existence of a sustainable debt limit: negative r − g implies no finite debt limit so long as deficits are bounded.
  - More general stochastic specifications retain these qualitative properties (e.g., if r and g stochastic, a negative long-run expectation of r − g leads debt to diverge with probability one).

### Model environment, recursive equilibrium, and key propositions
- Environment and government problem:
  - Nominal risk-free interest rates and nominal GDP growth rates are state-dependent: R_t = R(x_t), G_t = G(x_t); x_t follows an N−state ergodic Markov process with transition matrix M.
  - Surplus rule: s_t = s(d_{t-1}, x_t) + e_t, e_t ∼ F(·), E e_t = 0; s(d,x) is weakly increasing in d and bounded; F(·) has finite support [−\bar{ε}, \bar{ε}].
  - Budget constraint (period-t GDP units): d_{t-1} / G(x_t) q_{t-1} = s_t + d_t; government prevented from accumulating assets: d_t ≥ 0.
  - Investors risk-neutral; in default they reclaim share φ; debt price q_t satisfies q_t = (1 / R(x_t)) [ p_t φ q_t + (1 − p_t) ] with p_t = probability of default next period.
- Recursive equilibrium solution strategy:
  - Period-t debt price as function of belief p: \hat{q}(x,p) = (1 − p) / [ R(x) − p φ ].
  - Default probability for government with debt d: Z(x,d,p,D) = ∑_{x'} π(x'|x) F( H(x,x',d,p,D(x')) ), with H defined in the text.
  - Rational beliefs: p^*(x,d,D) is the smallest fixed point of p = Z(x,d,p,D) (p = 1 always a fixed point).
  - Default boundary \hat{D}(x) defined via existence/non-existence of p^* < 1 for debt levels.
  - Equilibrium satisfies price equation, fixed-point conditions for p^*, default-boundary conditions, equality of current and future thresholds D(x) = \hat{D}(x), and debt path divergence condition lim_{T→∞} P(d_T > d) = 1 ∀ d > 0 ⇒ d_0 > D(x_0).
- Key theoretical results:
  - Proposition 3:
    - There exists an equilibrium with D(x) = ∞ ∀ x if and only if E log(G_t / R_t) > 1.
    - Interpretation: debt limits are infinite iff the long-run nominal growth rate exceeds the risk-free interest rate on average.
  - Proposition 4 (tangency condition):
    - At equilibrium: ∂/∂p Z( x, \hat{D}(x), p^*(x, \hat{D}(x)), \hat{D} ) = 1.
    - Role: reduces equilibrium computation to solving 2N simultaneous equations (N tangency conditions and N fixed-point requirements for an N-state model).

### Empirical approach: estimating processes for R(·) and G(·) and main findings
- Objectives:
  1. Estimate stochastic processes for R(·) and G(·) to feed into the model and predict debt limits.
  2. Test whether the long-run interest-growth differential is positive or negative.
- Data and methods:
  - VARs estimated for nominal interest and nominal GDP growth in seven advanced economies using two datasets:
    1. Jordà, Schularick, and Taylor (2017) annual data for USA, UK, France, Germany covering 1880-2015 (with missing intervals for France and Germany).
       - Observations: USA 134 (1880-2015), UK 134 (1880-2015), France 114 (1880-1913, 1922-1938, 1951-2015), Germany 128 (1880-1921, 1925-1944, 1950-2015).
    2. Quarterly data (average overnight central bank rates and national accounts growth) covering a subset of G7 (excludes Italy and Japan; Germany starts in 1976; others from 1956 or 1961 for Canada).
  - Nominal rates used to reflect nominal debt dynamics; focus on policy interest rates with robustness checks using long-term bond and effective rates (similar findings).
  - Complementary spectral method of Müller and Watson (2016) applied to annual data 1880-2015.
- Stylized VAR evidence (annual 1-lag VAR, 1880-2015; all rates annualized):
  - Regularities:
    1. Interest rates are highly persistent; past nominal growth has little predictive power for interest rates.
    2. Nominal GDP growth is highly volatile (annual standard deviations of ~6 percentage points common) and less persistent than interest rates.
    3. Innovations to interest and growth covary positively; correlation ~0.3 for the USA and similar in other countries.
  - Long-run differentials from simple VARs are negative, between −1.6 and −2.1 for all countries in the illustrative table.
- VAR-based inference (long-lag VARs and likelihood ratio tests):
  - Longer-lag VARs change point estimates slightly but implied long-run means remain negative for all countries.
  - For the US, the standard error of the interest-growth differential is about 1.4; a simple Wald test of the hypothesis that the long-run interest-growth differential is less than zero has a p-value of 0.06.
  - Selected LR Means (annual data) reported:
    - USA: Growth LR Mean 5.96; Int. rate LR Mean 3.84.
    - United Kingdom: Growth LR Mean 5.49; Int. rate LR Mean 3.87.
    - France: Growth LR Mean 5.67; Int. rate LR Mean 4.02.
    - Germany: Growth LR Mean 5.96; Int. rate LR Mean 4.04.
  - Cross-country conservative 5% critical values from VAR tests:
    - UK, France, Germany: around 0 to 0.4 percentage points per year.
    - Canada: nearer 0.8 percentage points per year (partly due to shorter sample).
- Spectral (Müller-Watson) results (annual 1880-2015):
  - I(0) spectral estimates (most restrictive) produce narrower confidence sets; Bayes and Bayes superset produce wider upper bounds sensitive to outliers (e.g., WWII-era nominal growth).
  - Selected I(0) results (Midpoint and 95% c.v. for average interest-growth differential over next 100 years):
    - USA I(0): Mid. -1.90; 95% c.v. 0.90.
    - UK I(0): Mid. -1.50; 95% c.v. 1.10.
    - France I(0): Mid. -2.00; 95% c.v. 0.90.
    - Germany I(0): Mid. -3.40; 95% c.v. 2.50.
  - ML fractional integration estimates: USA -0.10; UK 0.00; France 0.30; Germany 0.00.
- Main empirical conclusions:
  - Point estimates of the long-run interest-growth differential are negative across countries, periods, and estimation methods.
  - Confidence intervals typically include small positive numbers; for long, unbroken datasets and few extreme events (UK, USA, France), the largest plausible long-run interest-growth differential is between 0 and 2 percent per year.
  - Policy guidance: conservative sustainable-debt estimates should (a) feature long-run differentials in the range 0 to 2 percent per year, and (b) clearly explain sensitivity to the assumed long-run differential.

### Model application and sensitivity analyses
- UK illustrative exercise:
  - Conservative historical debt limits generated for the UK move little in response to variation in the interest-growth differential, increasing around 5 percent of GDP following the 2008-9 financial crises.
- Sensitivity to long-run differential uncertainty:
  - Solving the model with a long-run differential at the 2.5% critical value for the VAR-based estimates (instead of the 5% value) leads to maximum sustainable debt-GDP ratios about 5pp lower.
  - The increase in fiscal space due to large declines in interest rates since 2008 could be entirely offset by a fractionally more conservative assumption on the long-run interest-growth differential.

### Estimating the surplus function (UK, annual data 1880-2015)
- Functional form and parameterization:
  - s(x′,d) = α + β(d − γ)^2 I_{d<γ} + δ(R(x′) − 1) + η(G(x′) − 1)
  - Parameter restrictions to satisfy empirical requirements: α, γ > 0 and β < −α/γ^2.
- Estimated parameters (generalized least squares, UK data):
  - α = 3.36
  - β = -0.006763
  - γ = 3.60
  - δ = 0.99
  - η = -0.83
- Empirical implications from estimates:
  - α > 0 implies surpluses are positive for high enough debt levels.
  - β < 0 implies surpluses are low when debt is low.
  - δ > 0 implies surpluses are positively conditionally correlated with nominal rates.
  - η < 0 implies surpluses are negatively conditionally correlated with growth rates.
  - The fiscal policy rule implied is counter-cyclical: lower growth induces higher surplus ratios.
  - δ + η is positive, so the surplus function responds positively to the interest-growth differential beyond dynamic debt responses.
- Role in model:
  - Surplus responsiveness to worsening debt dynamics mitigates increases in debt in high r − g states; boundedness captures fiscal fatigue.

### Robustness and discretization validations (summary)
- Alternative interest-rate measures:
  - Long-term interest-rate and effective interest-rate measures tested; point estimates of long-run interest-growth differential remain universally negative; upper confidence limits usually small and positive.
  - Example long-term spreads (1880-2013): USA 0.55, UK 0.90, France 1.16, Germany 1.28.
  - Example effective interest rates (1880-2013): USA -0.88, UK 0.35, France -0.46, Germany 0.12.
- Spectral method on quarterly data:
  - Quarterly series (1956Q1-2016Q4) show apparent non-stationarity; spectral confidence sets grow large and estimator-sensitive; likely driven by short sample and declining differentials since early 1980s.
- Discretization of joint R–G process:
  - Nodes constructed along rays using Gauss-Hermite quadrature; transition probabilities chosen to match conditional moments.
  - Validation: simulated Markov chain (100,000 periods) yields VAR estimates closely matching historical VAR coefficients and long-run means (example: Data Growth 5.02 vs Simulation Growth 5.01; Data Int. rate 5.18 vs Simulation Int. rate 5.17).
  - Discretization reproduces growth-rate volatility and range well; handles major episodes with caveats (e.g., World War II financial-repression episodes).

*Source: wp1882 - 1. Or, equivalently, that a country can only service a lower debt given its surpluses.*

### 1. Or, equivalently, that a country can only service a lower debt given its surpluses.

### wp1882 - 1. Or, equivalently, that a country can only service a lower debt given its surpluses.

### Historical patterns of interest-growth differentials in advanced economies
- Point estimates and long-run averages:
  - Since 1880, the average annual interest-growth differential in six advanced economies has been -1.7 percent.
  - Since 1960, the average annual interest-growth differential has been -0.8 percent.
- Medium-run (decadal) variation:
  - Decadal average differentials varied from around minus five percentage points per year in the 1960s and 70s to nearly two percentage points in the 1980s and 90s, before dropping back below zero more recently.
- Short-term volatility:
  - Short-term volatility is very high; the differential frequently changes by several percentage points per year.
  - For the USA (1880-2016), using the short-term rate, the average interest-growth differential was -1.9 percent during 1880-2016, and -1.6 percent since 1950.
  - In the USA the differential moves by over 2.5 percentage points in more than one-fifth of years.
  - A 2.5 percentage point fall in the annual growth rate causes the debt-GDP ratio to increase mechanically by 2.5 percentage points.

### Consequences for maximum sustainable debt levels
- Three data aspects with contrasting implications:
  - Low average differentials imply maximum sustainable debt levels are likely to be high.
  - Persistent medium-run movements imply maximum sustainable debt levels may vary substantially over time.
  - High short-term volatility induces unpredictable variation in the debt-GDP ratio, increasing default risk and pulling down maximum sustainable debt levels.
- Critical role of the long-run differential:
  - If the long-run interest-growth differential is negative, governments can issue arbitrarily large debt without becoming insolvent so long as deficits are bounded.
  - Because of this, testing statistically whether the long-term interest-growth differential may be positive is essential.

### Statistical testing and estimated bounds on the long-run differential
- Two statistical approaches employed:
  - Likelihood ratio approach based on combinations of VAR coefficients applied to two datasets (five advanced economies): one annual since 1880 and one quarterly since 1956.
    - For both datasets, VAR point estimates are negative.
    - The associated likelihood ratio test cannot reject the possibility that the long-run differential is positive.
    - Typical upper bound from this method: in the order of zero to one percent per year.
  - Spectral approach developed by Müller and Watson (2016).
    - Produces similar negative point estimates for the long-run differential.
    - Generates larger confidence intervals and is more sensitive to outliers.
    - For countries with long, uninterrupted data, provides upper bounds around one to two percent per year.
- Joint implication:
  - Both methods suggest small and positive long-run interest-growth differentials are unlikely, but still consistent with the data.

### Model linking interest-growth processes to debt limits
- Model features:
  - Based on Ghosh, Kim, Mendoza, Ostry, and Qureshi (2013) with extensions:
    - Correlated aggregate shocks to the nominal risk-free rate and nominal GDP growth.
    - Idiosyncratic shocks to governments’ primary balances.
    - Surplus-GDP ratio is a bounded function of the aggregate state, including outstanding debts.
  - Deficits funded by issuing noncontingent defaultable bonds to risk-neutral investors.
  - Investors price government debt at a premium over the risk-free rate reflecting default risk; a sufficiently high debt level causes loss of market access (debt limit).
- Definition of debt limit:
  - The debt level at which government loses access to debt markets because investors refuse to lend at any finite interest rate.

### Simple motivating model and analytic results (Proposition 1)
- Setup:
  - Constant nominal growth rate 1 + g and risk-free nominal interest rate 1 + r.
  - Expected primary surplus se_t ∈ [s_, s̄]; realized surplus st = se_t + ut with ut mean-zero and support [−ū, ū].
  - One-period zero coupon nominal bonds with price qt and face value bt; budget constraint qt bt + st = b_{t-1}(1 + g).
  - Investors infinitely risk averse: qt = 1/(1 + r) if probability of default is zero; qt = 0 otherwise.
- Proposition 1 results:
  - If r > g then there exists b̄, the maximum sustainable debt, given by:
    - b̄ = (s̄ − ū)(1 + g)(1 + r) / (r − g)
  - If r < g then there exists no such b̄; the government never defaults, no matter the debt level.
- Implications:
  - Debt limit is enormously sensitive to r − g, especially when r − g is close to zero.
  - The sign of r − g is critical for the existence of a sustainable debt limit: negative r − g implies no finite debt limit so long as deficits are bounded.
  - More general stochastic specifications retain these qualitative properties (e.g., if r and g stochastic, a negative long-run expectation of r − g leads debt to diverge with probability one).

### Model application and sensitivity analyses
- UK illustrative exercise:
  - Conservative historical debt limits generated for the UK move little in response to variation in the interest-growth differential, increasing around 5 percent of GDP following the 2008-9 financial crises.
- Sensitivity to long-run differential uncertainty:
  - Solving the model with a long-run differential at the 2.5% critical value for the VAR-based estimates (instead of the 5% value) leads to maximum sustainable debt-GDP ratios about 5pp lower.
  - The increase in fiscal space due to large declines in interest rates since 2008 could be entirely offset by a fractionally more conservative assumption on the long-run interest-growth differential.

### Relation to literatures and scope
- Linkages:
  - Capacity-to-repay models: extends nonlinear fiscal reaction function approach to allow stochastic interest and growth rates.
  - Empirical literature on interest-growth differentials: differs by formally testing the sign of the long-run differential rather than only reporting point estimates.
  - Sovereign default models: shares stochastic drivers with strategic default literature; correlation between growth and risk-free interest rates is key determinant of sustainable debt levels.
- Scope and limits:
  - The paper treats behavior of the interest-growth differential as a statistical phenomenon and does not provide economic explanations for past movements.
  - The analysis focuses on whether higher debt limits are possible given observed dynamics, not whether they are desirable.

*Source: wp1882 - 1. Or, equivalently, that a country can only service a lower debt given its surpluses.*

### 3.1    Environment

### 3.1    Environment

### Model setup
- Government treats nominal risk-free interest rates and nominal GDP growth rates as exogenous and state-dependent: R_t = R(x_t) and G_t = G(x_t).
- Aggregate state x_t is drawn each period from an N−state discrete Markov process with transition matrix M; x_t is assumed ergodic.
- Government surplus rule:
  - s_t = s(d_{t-1}, x_t) + e_t,  e_t ∼ F(·),  E e_t = 0
  - s(d,x) is weakly increasing in d and bounded; F(·) has finite support [−\bar{ε}, \bar{ε}].
  - Bounded s(d,x) captures limited ability to raise surpluses (referred to as “fiscal fatigue”).
- Government issues non-contingent one-period bonds; end-of-period debt-to-GDP ratio is d_t (market-value).
- Government budget constraint (expressed in period-t GDP units):
  - d_{t-1} / G(x_t) q_{t-1} = s_t + d_t
  - Government is prevented from accumulating assets: d_t ≥ 0.
- Investors are risk-neutral; opportunity cost = risk-free rate R_t.
  - In default investors reclaim a share φ of outstanding liabilities.
  - If investors expect default next period with probability p_t, debt price satisfies:
    - q_t = (1 / R(x_t)) [ p_t φ q_t + (1 − p_t) ]
  - Important modeling note: recovery in default is proportional to the preceding-period price (prevents debt price being bounded below by φ).

### Recursive equilibrium (solution strategy)
- Equilibrium objects are state-dependent: debt prices consistent with investors’ beliefs, debt thresholds (maximum debt issuable at those prices), and investors’ beliefs p_t equal to actual default probabilities implied by thresholds.
- Recursive formulation variables: x (aggregate state) and d (market value of outstanding debt).
- Period-t debt price as function of investor belief p:
  - \hat{q}(x,p) = (1 / R(x)) [ p φ \hat{q}(x,p) + (1 − p) ] ⇒ \hat{q}(x,p) = (1 − p) / [ R(x) − p φ ]
  - Note: when p = 1, \hat{q}(x,p) = 0.
- Given next-period debt price and default rule D(·), the probability of default in period t for government with debt d is:
  - Z(x,d,p,D) = ∑_{x'} π(x'|x) F( H(x,x',d,p,D(x')) )
  - H(x,x',d,p,D(x')) = d / [ G(x') \hat{q}(x,p) ] − D(x') − s(x',d)
  - H(·) is the minimum surplus shock required to push next-period debt above the default threshold D(x').
- Rational beliefs requirement: p must equal actual default probability Z(·).
  - Define p^*(x,d,D) as the smallest fixed point of p = Z(x,d,p,D). p^* always exists because p = 1 is always a fixed point.
- Default boundary \hat{D}(·) defined by:
  - ∃ p^*(x,d,D) < 1 ∀ x, ∀ d ≤ \hat{D}(x)
  - @ p^*(x,d,D) < 1 ∀ x, ∀ d > \hat{D}(x)
- Recursive equilibrium defined by functions D(x'), \hat{q}(x,p), p^*(x,d,D), \hat{D}(x) satisfying:
  1. \hat{q}(x,p) satisfies the price equation for all (x,p).
  2. For all (x,d), p^*(x,d,D) is the smallest p satisfying p = Z(x,d,p,D).
  3. Given p^*(x,d,D), \hat{D}(x) satisfies the default-boundary conditions.
  4. Current and future default thresholds equal: D(x) = \hat{D}(x) ∀ x.
  5. If {d_t}_{t=0}^∞ is the sequence of debt levels generated, then lim_{T→∞} P(d_T > d) = 1 ∀ d > 0 ⇒ d_0 > D(x_0).

### Properties of equilibrium (key theoretical results)
- Proposition 3:
  - There exists an equilibrium with D(x) = ∞ ∀ x if and only if E log(G_t / R_t) > 1.
  - Interpretation: debt limits are infinite iff the long-run nominal growth rate exceeds the risk-free interest rate on average (E log(G_t / R_t) > 1). This generalizes earlier simple examples where the risk-free differential governs existence of a debt limit.
  - Intuition: with D(x) = ∞ and zero risk premium, debt path depends only on the risk-free interest-growth differential; if positive, debt diverges w.p.1, violating finite initial condition requirement.
- Proposition 4 (tangency condition):
  - At equilibrium:
    - ∂/∂p Z( x, \hat{D}(x), p^*(x, \hat{D}(x)), \hat{D} ) = 1
  - Role: reduces computation of equilibrium to solving simultaneous equations. With N states, N tangency conditions and N fixed-point requirements yield 2N equations for 2N unknowns (default probabilities and thresholds).
  - Modeling trade-off: assuming one-period debt simplifies solution but overstates exposure to interest-growth differential volatility relative to long-maturity debt.

### Empirical approach: estimating processes for R(·) and G(·)
- Objectives:
  1. Estimate a stochastic process for R(·) and G(·) to feed into the model and predict debt limits.
  2. Test whether the long-run interest-growth differential is positive or negative (critical for finite debt limits).
- Methods:
  - Estimate VARs for nominal interest and nominal GDP growth in seven advanced economies using two datasets: annual (1880-2015) and quarterly (from 1956 for many countries).
  - Use VAR coefficients and variance-covariance matrices to form confidence intervals around long-run interest-growth differential.
  - Complementary robustness checks:
    - Spectral method of Müller and Watson (2016) to estimate confidence sets for long-run averages robust to varied data-generating processes (applied to annual data).
    - Reference to Mehrotra (2017) using probit regression to infer probability future interest rates are positive.
- Main empirical summary and findings:
  - Point estimates of the long-run average interest-growth differential are negative for all countries, periods, and estimation methods reported.
  - Confidence intervals typically include small positive numbers; VAR-based estimates often cannot reject at the 5% level that the long-run differential is greater than zero (both using annual data from 1880 and quarterly data from 1956).
  - Spectral method (annual data) produces confidence intervals centered on negative values but these confidence sets are wider due to method generality.
- Data description:
  - Two datasets:
    1. Jordà, Schularick, and Taylor (2017) annual data for USA, UK, France, Germany covering 1880-2015 (with specified missing intervals for France and Germany).
       - Observations: USA 134 (1880-2015), UK 134 (1880-2015), France 114 (1880-1913, 1922-1938, 1951-2015), Germany 128 (1880-1921, 1925-1944, 1950-2015).
    2. Quarterly data (average overnight central bank rates and national accounts growth) covering a subset of G7 (excludes Italy and Japan; Germany starts in 1976; others from 1956 or 1961 for Canada).
  - Nominal rates are used because debt is typically nominal and nominal shocks matter for debt-GDP dynamics.
  - Focus on policy interest rates for conceptual consistency (risk-free rate), predictability concerns of effective rates, and data availability; robustness checks using long-term bond and effective rates show similar findings (reported in Appendix B.2).
- Stylized VAR evidence (annual 1-lag VAR, 1880-2015; all rates annualized):
  - Key regularities across countries:
    1. Interest rates are highly persistent; past nominal growth has little predictive power for interest rates.
    2. Nominal GDP growth is highly volatile (annual standard deviations of ~6 percentage points common) and less persistent than interest rates.
    3. Innovations to interest and growth covary positively; correlation ~0.3 for the USA and similar in other countries.
  - Extract from 1-lag VAR table (annual data 1880-2015) — long-run interest-growth differentials from these simple VARs are negative, between −1.6 and −2.1 for all countries in the table (imprecisely estimated).

*Source: IMF Working Paper (section 3.1 Environment).*

### 23. See, for example, the coefficient of 0.03 on lagged growth for the US interest rate regression

### wp1882 - 23. See, for example, the coefficient of 0.03 on lagged growth for the US interest rate regression

### VAR-based inference on the long run interest-growth differential
- Motivation and approach:
  - Estimate long-lagged VARs (four-lag VARs) for risk-free interest and nominal growth rates to test whether the long-run interest-growth differential is negative.
  - Use a likelihood ratio test to find the largest long-run interest-growth differential consistent with the data (the critical value θ0 not rejected at a given significance level).
  - Longer-lag VARs change point estimates slightly but yield negative differences in implied long-run means for all countries; innovations to interest and growth remain positively correlated and growth is much noisier.
- Key empirical diagnostics and examples:
  - Example simple-statistic: coefficient of 0.03 on lagged growth for the US interest rate regression.
  - For the US, the standard error of the interest-growth differential is about 1.4. A simple Wald test of the hypothesis that the long-run interest-growth differential is less than zero has a p-value of 0.06.
  - Tables 3 and 4 report multi-lag VAR estimates (annual data 1880-2015; annualized quarterly data 1956Q1-2016Q4). Selected LR Means from Table 3 (annual data) include:
    - USA: Growth LR Mean 5.96; Int. rate LR Mean 3.84.
    - United Kingdom: Growth LR Mean 5.49; Int. rate LR Mean 3.87.
    - France: Growth LR Mean 5.67; Int. rate LR Mean 4.02.
    - Germany: Growth LR Mean 5.96; Int. rate LR Mean 4.04.
  - For quarterly post-war US data, using the unconditional likelihood ratio test the 5% critical value crosses at -0.06 (Figure 5a). Using quarterly post-war data, one cannot reject that the interest-growth differential is positive; the 5% critical value is 0.06.
- Tests compared:
  - Preferred: unconditional likelihood ratio test (re-estimates full model under constraint; allows other parameters to adjust).
  - Wald test: simpler but only a local approximation; can be inaccurate for large deviations from the null because it holds other parameters fixed.
  - Conditional likelihood tests (use only information about changes) are computationally simpler but can be misleading in short or highly persistent samples; conditional statistics can level out and fail to reject very large positive θ0.
- Cross-country summary from VAR tests:
  - Figures 6 and 7 show tests for UK, France, Germany, and Canada; results are similar to the US.
  - At the 5% confidence level, conservative estimates of the long-run average interest-growth differential vary:
    - UK, France, Germany: around 0 to 0.4 percentage points per year.
    - Canada: nearer 0.8 percentage points per year (partly due to shorter sample).

### Spectral inference on the long run interest-growth differential (Müller-Watson method)
- Motivation and approach:
  - Use spectral decomposition to compute standard errors of the future mean from low-frequency movements; robust to intermittent regime changes, fractional integration, and other low-frequency features hard to capture with VARs.
  - The variance of the limiting distribution of the forecasted average is expressed in terms of spectral weights near zero (low-frequency movements).
  - Method provides a cross-check on stationarity and partial integration assumptions embedded in VAR analysis.
- Implementations reported:
  - Table 5 reports midpoint and 95% upper critical values for three Müller-Watson spectral estimates for the average interest-growth differential over the next 100 years using annual data 1880-2015:
    - I(0), Bayes, and Bayes superset specifications are presented, plus the ML fractional integration estimates.
  - Selected values from Table 5 (Midpoint and 95% c.v.):
    - USA I(0): Mid. -1.90; 95% c.v. 0.90.
    - UK I(0): Mid. -1.50; 95% c.v. 1.10.
    - France I(0): Mid. -2.00; 95% c.v. 0.90.
    - Germany I(0): Mid. -3.40; 95% c.v. 2.50.
    - Bayes and Bayes superset upper bounds are larger (example: USA Bayes superset 95% c.v. 3.50).
    - Memo: 4-lag VAR results for comparison (Table 5):
      - USA VAR point estimate -2.12; VAR 95% critical value -0.07.
      - UK VAR point estimate -1.63; VAR 95% critical value -0.03.
      - France VAR point estimate -1.65; VAR 95% critical value 0.26.
      - Germany VAR point estimate -1.92; VAR 95% critical value 0.27.
    - ML fractional integration estimates: USA -0.10; UK 0.00; France 0.30; Germany 0.00.
- Interpretation of spectral results:
  - I(0) estimates (flat spectral density) are most restrictive and produce narrower confidence sets; Bayes and Bayes superset allow more flexible spectral densities and produce wider upper bounds that can be driven by outliers (e.g., extreme nominal growth fluctuations around World War II for France and Germany).
  - Maximum likelihood estimates of fractional integration are close to zero and always less than 0.5 in absolute magnitude, suggesting likelihood favors near-I(0) models; differences between Bayesian and I(0) estimates largely reflect the flat prior over spectral families versus flexible priors.
  - The author expresses a preference for the I(0) spectral estimates over Bayesian ones given parsimony and data-consistent degree of integration.

### Main empirical findings and implications for debt limits
- Robust point estimates:
  - Point estimates of the long-run interest-growth differential are negative across countries, periods, and estimation methods.
  - This implies a challenge to models of debt sustainability: if long-run interest-growth differentials are negative, debt limits may not be finite.
- Uncertainty and conservative bounds:
  - Upper bounds of confidence sets for the long-run average can be positive.
  - For countries with long, unbroken datasets and few extreme events (UK, USA, France), both VAR-based and I(0) spectral estimates agree that the largest plausible long-run interest-growth differential is somewhere between 0 and 2 percent per year.
- Policy-relevant guidance (author's recommendations):
  - Conservative estimates of sustainable debt levels should:
    - a) feature long-run differentials that are somewhere in the range of 0 to 2 percent per year, and
    - b) explain clearly the sensitivity of results to the assumed long-run differential.
- Transition to application:
  - The statistical processes for interest and growth differentials estimated here are used in the model developed in section 3 to estimate debt limits, with the UK used as an example in the following section.

*Source: wp1882 - 23. See, for example, the coefficient of 0.03 on lagged growth for the US interest rate regression (excerpt).*

### 5.1    Estimating the surplus function

### 5.1    Estimating the surplus function

### Functional form and required properties
- The average surplus function s(x′,d) is parameterized as a bounded quadratic function of debt:
  - s(x′,d) = α + β(d − γ)^2 I_{d<γ} + δ(R(x′) − 1) + η(G(x′) − 1) (equation (10))
  - Constants: α, β, γ, δ, η.
- Four empirical requirements any plausible average surplus function must satisfy:
  - 1) Average balances must be negative when debt is low so that average debt levels are positive.
  - 2) The government must be able to raise surpluses high enough (for at least some level of debt) to pay the (average) risk-free interest so that debt limits are positive.
  - 3) Surpluses must be bounded so that debt limits are finite.
  - 4) The correlation of the surplus ratio with the state must reflect cyclicality of surpluses.
- Parameter restrictions ensuring the four requirements:
  - α, γ > 0 and β < −α/γ^2.
- Interpretation:
  - The formulation embodies “fiscal fatigue”: governments can raise surpluses to pay large debts but ability is eventually limited; the maximum expected surplus when debt is high is α.
  - The relationship is treated as a descriptive correlation (not causal) capturing usual correlations of surpluses with interest, growth, and debt; the functional form is fit directly to the data without additional controls.

### Estimated parameters (UK, annual data 1880-2015)
- Parameters estimated using generalized least squares on UK data (Mauro et al. (2015)).
- Table 6 estimates:
  - α = 3.36
  - β = -0.006763
  - γ = 3.60
  - δ = 0.99
  - η = -0.83

### Empirical implications from the estimates
- Key empirical points highlighted by the estimation:
  - α > 0 implies surpluses are positive for high enough debt levels.
  - β < 0 implies surpluses are low when debt is low.
  - δ > 0 implies surpluses are positively conditionally correlated with nominal rates.
  - η < 0 implies surpluses are negatively conditionally correlated with growth rates.
  - The fiscal policy rule implied is counter-cyclical: lower growth induces higher surplus ratios (all else equal).
  - The sum δ + η is positive, so the surplus function responds positively to the interest-growth differential, over and above the dynamic response via movements in the debt stock.

### Role of the surplus function in the model
- The surplus function is an important determinant of model-predicted debt limits:
  - Surpluses that increase when debt dynamics deteriorate mitigate increases in debt in states with high interest-growth differentials.
- Appendix C (in source) contains further discussion of surplus-function estimation (not reproduced here).

*Source: IMF Working Paper — section "5.1    Estimating the surplus function" (parameters estimated on UK annual data 1880-2015).*

### References

### wp1882 - References

### A. Proofs — theoretical results and conditions
- Proposition 1 (motivating model)
  - If r > g then there exists b̄, the maximum sustainable debt, given by:
    - b̄ = (s̄ − ū)(1 + g)(1 + r) / (r − g)
  - If r < g then there exists no such b̄; the government never defaults, no matter the debt level.
  - Key equality used to define b̄:
    - 1/(1 + r) b∗ + s̄ − ū = 1/(1 + g) b∗

- Proposition 3 (debt limits with negative average interest-growth differentials)
  - There exists an equilibrium with D(x) = ∞ ∀x if and only if E log(Gt/Rt) > 1.
  - Iterated budget relation:
    - dT = d0 Q1T − ∑_{r=1}^T s_r Q_tr
    - Q1T = ∏_{s=1}^T G_s R_{s−1}^{−1} (in equilibrium with p_t = 0, q_t = 1/R_t)
  - As T → ∞, Q1T → (e^{E log(G_s/R_s)})^T with probability 1.
  - Case implications:
    - If E log(G_s/R_s) > 1 then Q1T diverges → ∞ and deficits bounded ⇒ d0 will not diverge ⇒ D(x) = ∞ is an equilibrium.
    - If E log(G_s/R_s) < 1 then Q1T → 0 and debt can become unbounded unless d_{t−1} ≤ ∑_{r=t}^∞ Q_{r}^t s_r, implying a debt limit exists.

- Proposition 4 (tangency condition at equilibrium)
  - At an equilibrium, the tangency condition holds:
    - ∂/∂p Z(x, D(x), p*(x, D(x)), ˆD) = 1
  - Proof sketch:
    - If derivative ≠ 1 a contradiction is constructed by showing existence of additional fixed points or violation of maximality of D(x).
    - Uses monotonicity ∂/∂d Z(x,d,p,ˆD) > 0 and continuity properties of Z(·).

### B. Further empirical work — robustness, alternative measures, and stationarity issues
- B.1 Restricted VAR estimation
  - VAR estimates were re-estimated with long-run means restricted to be at the 5% critical value for each country.
  - Finding: dynamics coefficients are little changed relative to unrestricted estimates; change mainly in intercept terms.
  - Conclusion: dynamics of interest and growth rates are statistically well pinned down, but relative levels are not.

- B.2 Alternative interest rate measures
  - Two alternative interest rate measures tested: long-term government debt rate and effective interest rate on the stock of government debt (data drawn from Mauro et al. (2015)).
  - Table 8 (average spread over risk-free rate) reports country/time-block spreads (selected entries):
    - Long-term interest rates (1880-2013): USA 0.55, UK 0.90, France 1.16, Germany 1.28
    - Effective interest rates (1880-2013): USA -0.88, UK 0.35, France -0.46, Germany 0.12
    - Example subperiod: 2000-2013 long-term spreads: USA 1.91, UK 1.43, France 1.73, Germany 1.37
  - Figure 15 results (annual 1880-2015): estimates using long-term rates are higher than risk-free rates; point estimates of long-run interest-growth differential are universally negative; upper limits of confidence sets usually small and positive.

- B.3 Spectral method on quarterly data — stationarity concerns
  - Spectral estimates for quarterly data (1956Q1-2016Q4, annualized) diverge from VAR-based estimates due to apparent non-stationarity in the sample period.
  - Table 9 reports midpoint and 95% upper critical values of Mueller-Watson prediction ranges for average interest-growth differential of next 100 years. Selected entries:
    - I(0) midpoints and 95% c.v.: USA midpoint 3.40, 95% c.v. 5.20; UK midpoint 4.90, 95% c.v. 6.80; France midpoint 4.00, 95% c.v. 6.00; Germany midpoint 4.40, 95% c.v. 6.50
    - ML fractional integration estimates: USA 1.00, UK 1.00, France 1.00, Germany 0.80
  - Interpretation:
    - Many countries show estimated degree of integration equal to unity (non-stationary), so confidence sets grow without bound and are estimator-sensitive.
    - Conclusion: quarterly dataset is likely too short; apparent non-stationarity probably driven by sample period features (declining differentials since early 1980s) rather than fundamental long-run behavior. Longer annual series implies stationarity for most countries (exception noted: Canada).

### C. Estimating the surplus function — functional forms and fit
- Two estimated surplus functions displayed:
  - Fifth-order polynomial (black curve).
  - Bounded quadratic functional form (blue curve; featured in the text).
- Both specifications include linear terms in interest and growth rates (held at sample means when plotting).
- Qualitative observation:
  - Fifth-order polynomial sharply increases for low lagged debt levels and is broadly flat for high debt levels, except it diverges sharply outside the estimation range; bounded quadratic avoids sharp divergence.
- Figure 16 also displays debt density alongside surplus function plots.

### D. Discretizing the growth-interest rate process — method and validation
- Discretization approach overview:
  - Construct a matrix of two-dimensional nodes (interest rate R(x) and growth rate G(x)) and a transition matrix M.
  - Choose R and D (number of distances and directions) to generate R D + 1 discrete nodes placed along D equally-spaced rays using Gauss-Hermite quadrature for the conditional density along each ray.
  - Generating formula for node with index (r,d):
    - x(r,d) = μ + √(2 r) A [ cos(d 2π/(D+1)), sin(d 2π/(D+1)) ]'
    - Where μ = Long run mean, Σ = Long run variance, A A' = Σ.
  - Remaining node: r = d = 0 (the long-run mean).
  - Transition probabilities per row chosen by minimizing weighted errors on conditional mean, variance, and skew to avoid negative probabilities.

- Discretization parameters and visualization
  - Example discretization shown with D = 20, R = 3 in Figure 17; point sizes proportional to ergodic frequency.
  - Positive correlation between interest and growth nodes evident.

- Validation by simulation
  - Simulate the fitted Markov chain for 100,000 periods, then estimate a VAR from the simulation.
  - Table 10 compares VAR from UK historical data ("Data") and VAR from simulated discretized approximation ("Simulation") (one-lag restricted VAR). Selected entries:
    - Coefficients — Constant: Data Growth 1.43, Int. rate 0.27; Simulation Growth 1.81, Int. rate 0.85
    - Long-run means: Data Growth 5.02, Int. rate 5.18; Simulation Growth 5.01, Int. rate 5.17
    - Growth(-1): Data 0.67, 0.05; Simulation 0.64, 0.03
    - Innovation covariance: Growth Data 22.05, Simulation 24.02; Int. rate Data 1.43, Simulation 1.52
  - Conclusion: coefficients are almost identical — discretized process well-represents the VAR estimated from the data.

- Historical series reconstruction
  - To produce figures, each historical period is mapped to the nearest discretized node.
  - Figure 18 compares UK data (black) and discretized series (blue) for nominal growth rate and risk-free rate.
  - Discretization handles growth-rate volatility and range of both series well; struggles during episodes like World War II where correlation structure changed (financial repression).

*Italic: Source — wp1882 - References (PDF content provided).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2018/wp1882.pdf_