## 2.1    Sources

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---

### Data sources and coverage
- United States:
  - Quantity, coupon rate, frequency of coupon payments, and maturity date of each marketable US debt security outstanding at the end of each month (January, 1869– present) from the US Treasury’s Monthly Statement of the Public Debt (MSPD).
  - Prior to July, 1974 data furnished in the month end Daily Statement of the United States Treasury.
  - Issuances cross-checked with the US Treasury’s database of treasury auction results (1982–2020).
  - Appendix A.2 comparison: post-1950 dataset agrees exactly with CRSP; CRSP coverage incomplete pre-1950 and almost non-existent prior to 1931. Authors’ coverage universal back to 1869.
- United Kingdom:
  - Gilts and Consols prior to 2018: marketable debt data from Ellison and Scott (2020).
  - Gilts outstanding for 2019–2020: UK Debt Management Office’s Gilts in Issue reports.
  - Treasury Bills compiled as weekly amount by maturity sold during auction since 1877; sources include The Economist Historical Archive, Financial Times Historical Archive, Bank of England Bulletins and Annual Reports, and UK DMO’s Treasury Bill Tender Results (various coverage windows listed).
  - Where tender allotments unavailable, tender offers used instead.

### Period used for headline results and rationale
- Headline results focus on the period 1920–2020 despite data gathered from 1869 onwards; rationale: large changes in financial markets post-World War I motivate focusing on 1920–2020.

### Data representation and visual summaries
- Figures 1 and 2:
  - Decade-start snapshots of entire outstanding set of government debt assets: each bar = individual security; x-axis = months to maturity; y-axis = face value as a share of GDP; bars colored by security type; vertical line = average maturity.
  - Consols treated with truncated x-axis for presentation; y-axis not harmonized across panels.
- Figure 3: number of debt assets outstanding, 1920–2020; vast increase for US and UK; US issuances in circulation more than doubled in past 15 years (driven almost entirely by new notes); UK securities peaked mid-1980s.
- Figure 4: average debt maturity, 1920–2020; UK average maturity far longer than US; UK very long in 1930s and 1940s.
- Figures 5 and 6 (payment densities):
  - Empirical payment densities constructed by converting each bond into its stream of future payments and aggregating across all outstanding assets at month end (coupons capped at a maturity of 1000 months for plotting).
  - Each dot = promised payment by due date (months to payment) and fraction of total payments (payment density); vertical line = average months to payment.

### Empirical patterns and notable observations
- Complexity and variety:
  - Vast increase in number of debt securities and complexity for both US and UK.
- Maturity differences:
  - Average maturity of UK government debt far longer than that of the US; especially long in 1930s–1940s.
- Distributional changes:
  - US debt distribution acquired much fatter tails: increasing share of very short and very long bonds.
  - UK maturity structure changes dominated by rise and fall of the consol until ~1980.
- Limitations of simple summaries:
  - One-time snapshots leave questions about representativeness and higher moments; choice of summary parameter (e.g., average maturity) can omit meaningful information.

### Constructing densities and aggregation framework (notation and identities)
- Discrete time notation (t = 1,2,...):
  - Government total obligations at start of period t coming due in period t + s: b^s_t.
  - Net issuance of maturity-s claims in period t: i^s_t.
  - Maturity-s debt stock evolution: b^s_{t+1} = b^{s+1}_t + i^s_t, s ≥ 1.
  - Aggregate totals: b_t = sum_{s=1}^∞ b^s_t, i_t = sum_{s=1}^∞ i^s_t.
  - Aggregate debt evolution: b_{t+1} = b_t − b^1_t + i_t.
  - Debt and issuance ratios (densities):
    - f_t(s) = b^s_t / b_t
    - g_t(s) = i^s_t / i_t
  - f_t(s) sums to one and is theoretical equivalent of empirical debt distribution y^s_t.
- Evolution equation linking old debt, issuance, and new debt (Appendix A.3):
  - f_{t+1}(s) = (1−w_t) g_t(s) + w_t ( f_t(s+1) / (1−f_t(1)) ) ∀ s  (equation (4))
  - w_t ∈ [0,1] is a decreasing function of aggregate issuance relative to outstanding debt, i_t / b_t.
  - Issuance i_t expression: i_t = d_t + b^1_{t−1} / ∑_{s=1}^{∞} g_t(s) q^t_s, where d_t is overall deficit and q^t_s price of s-period zero-coupon bond.

### Conversion to payment densities (practical mapping)
- Each bond’s promised stream converted to common framework before aggregation; Appendix A.3 describes details.
- Coupons capped at maturity of 1000 months for plotting exercise.
- Two-stage construction:
  - Individual security payment density y_{s it} constructed from C_{its} (coupon at t+s) and P_{it} (principal at terminal payment).
  - Aggregate portfolio density y_{s t} = ∑_{i=1}^{n_t} ω_{it} y_{s it}, with ω_{it} = (P_i + C_i) / ∑_{n} (P_i + C_i). ∑_{s=1}^{∞} y_{s t} = 1.

### Treatment of special instruments and yield-curve construction
- Floating-rate instruments treated with country-specific approaches; examples:
  - US: Svensson (1994) parametric model on daily treasury yield curve rates to develop monthly-horizon yield curve up to 30 years.
  - UK: Nelson-Siegel (1987) parametric model to develop monthly-horizon yield curve up to 40 years.
  - Peru: insufficient yield curve data; step-up coupons treated with known future coupons; other floating-rate coupons assumed fixed at reference.
  - Poland: use LIBOR/EURIBOR sources and Svensson or Nelson-Siegel as appropriate.
- Perpetual bonds (consols) treated as finite schedule of coupon payments set to 100 years from any given time for accounting.
- Callable notes/bonds issued before 1985 treated in baseline as not called; alternative "called-at-first-availability" exercise reported in robustness.

### Key conceptual point
- Evolution of maturity structure depends only on net issuance of claims, irrespective of yield curve or deficits; issuance maps into deficits via prices of future claims but that mapping is not focal here.

### Theoretical implication preview
- Under light assumptions (basic accounting and governments can freely trade in a fixed set of asset markets), only one parametric model both consistent and more parsimonious than listing all future obligations: government trades a fixed set of exponential-coupon bonds.
- Any other setting either imposes arbitrary issuance restrictions or requires as many parameters as the longest horizon of obligations.
- Formal uniqueness claims and proofs provided in Appendix B; main text uses illustrative examples.

---

### Appendix A.3.  We further impose thatf

### Constraints and evolution identities
- f_t(s) ≥ 0 ∀ s; f_t(s) is a probability mass function.
- g_t(s) need not be non-negative for all s because maturity-structure changes (e.g., debt swaps) can make g_t(s) negative for some s.
- Equation (4) restated:
  - f_{t+1}(s) = (1−w_t) g_t(s) + w_t ( f_t(s+1) / (1−f_t(1)) ) ∀ s
- w_t = 1 − f_t(1) / (1 − f_t(1) + i_t / b_t ) ∈ [0,1].

### Representation of debt by parametric family
- If f_t(s) = f(s | ω_t) and f_{t+1}(s) = f(s | ω_{t+1}) for parameter vector ω_t, then imposing functional form on f also imposes restrictions on issuance g_t(s).
- Aim: identify functional forms where resulting g_t(s) allows government free access to a variety of debt markets.

### Exponential-coupon bonds (no restriction on issuance)
- M outstanding bonds indexed by i with decay rates θ_i; each period fraction θ_i of outstanding balance comes due.
- Share of bond i in obligations: ω_{i,t}; f_t(s) = ∑_{i=1}^{M} ω_{i,t} θ_i (1−θ_i)^{s−1}.
- If government has access to each of the M bond markets, issuance density g_t(s) = ∑_{i=1}^{M} σ_{i,t} θ_i (1−θ_i)^{s−1}, where σ_{i,t} share of issuance in market i.
- Evolution of weights:
  - ω_{t+1,i} = (1−w_t) σ_{i,t} + w_t ( ω_{i,t} (1−θ_i) / (1− ∑_{j=1}^{M} ω_{j,t} θ_j) ).
- Key property: residual maturity of an exponential-coupon bond is exponential-coupon; therefore any issuance among same set yields f_{t+1}(·) in same family with no arbitrary issuance constraints.

### Bullet bond with coupon (restrictions on issuance)
- Single coupon-paying bond for T periods with terminal payment 1 yields outstanding debt density y_{c,T}(s) with discrete pieces (as specified).
- Government must trade in two markets: buy back old bonds in entirety and issue new debt of duration T.
- Issuance g_t(s) = (1−σ_t) y_{c,T}(s) + σ_t y_{c,T−1}(s).
- For equation (4) to hold requires σ_t = −w_t / (1−w_t), implying government buys back whole outstanding stock every period — an arbitrary and impractical restriction.
- Conclusion: except for full set of bullet markets at all horizons, single fixed-T bullet parameterizations impose arbitrary issuance restrictions; exponential-coupon setup is the only parsimonious parameterization without such restrictions.

### Empirical evidence: summarizing changes in US and UK debt maturity
- Sample limited to post-1920.
- Headline estimates obtained by fitting portfolios of exponentially-decaying-coupon bonds; comparison to constant-coupon portfolios.
- Maximum empirical maturity S set to 1200.

#### Methodology (exponential-coupon mixture MLE)
- Decay rate of bond in market i: θ_i; share in time-t density: ω_{i,t}.
- Likelihood maximized:
  - l(θ, ω | y) = ∑_{t=1}^{T} ∑_{s=1}^{S} y^t_s log( ∑_{i=1}^{M} ω_{i t} θ_i (1−θ_i)^{s−1} )  (equation (5))
- Constraint: ∑_{i=1}^{M} ω_{i t} = 1 ∀ t.
- Parameters to estimate: M + T×(M−1).

#### Headline results (exponentially decaying coupons, 1920–2020) — selected exact entries
- United States (Table 1 excerpts):
  - M=1: θ_1 = 0.012*** (0.0003); ̄ω_1 = 0.744*** (0.0146)
  - M=2: θ_1 = 0.009*** (0.0003), θ_2 = 0.129*** (0.0185); ̄ω_1 = 0.605*** (0.0207), ̄ω_2 = 0.256*** (0.0146)
  - M=3: θ_1 = 0.008*** (0.0004), θ_2 = 0.032*** (0.0052), θ_3 = 0.237*** (0.0344); ̄ω_1 = 0.601*** (0.0154), ̄ω_2 = 0.243*** (0.0227), ̄ω_3 = 0.152*** (0.0147)
  - M=4: θ_1 = 0.008*** (0.0003), θ_2 = 0.030*** (0.0044), θ_3 = 0.009*** (0.0009), θ_4 = 0.334*** (0.1056); ̄ω_1 = 0.466*** (0.0191), ̄ω_2 = 0.235*** (0.0179), ̄ω_3 = 0.109*** (0.0201), ̄ω_4 = 0.055*** (0.0181)
  - T = 12, S = 1200, Params. 11, Log lik = −6607.68 (M=1) → Log lik = −6388.83 (M=4); AIC: 13217.36 (M=1) → 20057.67 (M=4)
  - Example interpretation for US M = 3: θ_1 = 0.008, θ_2 = 0.032, θ_3 = 0.237 (implied intuitive durations noted in text).
- United Kingdom (Table 1 excerpts):
  - M=1: θ_1 = 0.004*** (0.0001); ̄ω_1 = 0.437*** (0.0138)
  - M=2: θ_1 = 0.003*** (0.0001), θ_2 = 0.008*** (0.0005); ̄ω_1 = 0.457*** (0.0193), ̄ω_2 = 0.220*** (0.0290)
  - M=3: θ_1 = 0.002*** (0.0001), θ_2 = 0.006*** (0.0005), θ_3 = 0.009*** (0.0006); ̄ω_1 = 0.437*** (0.0138), ̄ω_2 = 0.201*** (0.0358), ̄ω_3 = 0.304*** (0.0316)
  - M=4: θ_1 = 0.002*** (0.0001), θ_4 = 0.446*** (0.0685); ̄ω_1 = 0.466*** (0.0191), ̄ω_2 = 0.534*** (0.0191), ̄ω_3 = 0.343*** (0.0335), ̄ω_4 = 0.038*** (0.0065)
  - T = 12, S = 1200, Params. 11, Log lik = −7919.62 (M=1) → Log lik = −7744.89 (M=4); AIC: 15841.25 (M=1) → 22769.78 (M=4)
- Decadal variation (Figure 7 summary):
  - US: substitution patterns across very long, short, and medium maturities across sample; long-maturity bonds regained prominence in later decades.
  - UK: long tradition of ultra-long debt (consols) reflected in θ estimates; ultra-long debt before ~1950 (θ_1 = 0.002 average duration > 40 years) → long debt in 1970s–80s (θ_3 = 0.009 ≈ 9 years) → slightly longer debt in 2010s (average duration ≈ 16 years).

### Model selection and fit metrics
- AIC minimized at M = 1 for both countries due to large penalty from adding parameters (each increment in M increases free parameters by ~T ≈ 1200).
- Robustness: recompute AIC dropping long-end observations up to cutoff maturity S; M = 1 still minimizes AIC even at very short horizons.
- Alternative goodness-of-fit (err(s)):
  - For US, substantial improvement in fit when M increases from 1 to 2; two-bond model for US is hard to improve on.
  - For UK, benefits of extra assets less evident; notable short-maturity improvement only when M = 4.
- Conclusion: modeller’s choice in number of assets; small-number exponential-coupon models are supported.

### Comparison to constant-coupon (finite-duration) approaches
- Constant-coupon market density f(s | μ_i) specified as uniform over integer buckets (exact formula preserved in source).
- Constant-coupon densities cannot be rationalized by unrestricted market access unless full set of constant-coupon markets exist at every horizon.
- Empirical fit:
  - For US, constant-coupon approach fits far worse; constant-coupon M=4 worse than exponentially-decaying-coupon M=1.
  - Constant-coupon parameterization struggles to match sharp decay at short maturities and can alter qualitative descriptions of maturity changes.
- Conclusion: exponential-coupon parameterization theoretically coherent and empirically superior for examined datasets.

### Extensions to select emerging markets (data and findings summarized)
- Four emerging market economies with sufficiently rich long-run public debt data: Peru, Poland, Egypt, Nigeria.
- Data coverage and sources:
  - Peru: daily reports from Ministry of Economy and Finance (September 2006–December 2020); extended back to 1999 using Bloomberg.
  - Poland: bond transactions database from Ministry of Finance (May 1991–April 2020).
  - Egypt and Nigeria: central bank and ministry/DMO sources back to 2002; Egypt extended to 1995 using Bloomberg.
- Coverage limitations: dataset captures only marketable debt; excludes bilateral and non-marketable state-held debt; focus restricted to central government debt.
- Empirical highlights (selected exact figures):
  - Peru (M = 3): shortest decay rate θ3 ≈ 0.9 percent per month; longest-maturity decay ≈ 0.6 percent per month; weight on longest bond ω3 ≈ 5 percent on average (US and UK equivalents: 15 and 34 percent).
  - Poland, Egypt, Nigeria: require shorter bonds; longest-maturity decay rates “around 2-3 percent for M = 1 or 1.5-2 percent for larger M”; shortest bonds decay above 20 percent per month.
  - Poland and Egypt: average weight of the longest bonds ω1 around 55 to 75 percent. Nigeria: shorter debt accounts for more than half of debt.
  - Dynamic example (Poland, M = 3): initial transition from very short-term debt (θ3 = 0.52) to longer bonds post-Iron Curtain, steady lengthening through mid-1990s to mid-2000s, stable long-duration structure from 2005 onwards.
- Goodness-of-fit: log likelihoods and AICs reported in tables; exponential-coupon favored over constant-coupon in fit metrics.

### Robustness, real obligations, and special exercises
- Sample-period sensitivity: Tables 5–7 repeat MLE for 1960–2020, 1980–2020, and 1869–2020; qualitative θ_i similar in later samples; pre-1920 θ_i quite different.
- Real obligations: real debt densities constructed by deflating nominal payments using expected inflation term structure ˆΠ_{m,t} = ∏_{h=1}^m (1 + E_t π_{t+h})^{1/12}; 30-year CPI inflation expectations imputed monthly beginning January 1920 for US and UK.
  - Empirical note: decay rates of fitted assets a little higher in real-obligations parametrization; US one-bond decay rate around 0.002 higher than headline case, consistent with average inflation.
- Callable bonds exercise (called-at-first-availability):
  - Assuming call at first availability slightly raises decay rates over 100-year sample; average weights relatively unchanged.
  - MLE under callable assumption reported (Tables 9 and 10) with exact parameter estimates preserved in source.

### Methodological implications and suggested extensions
- Modeling government balance sheets as small number of exponentially-decaying bonds fits data better than constant-coupon approach and affects fiscal policy interpretation.
- Suggested extensions:
  - Use estimated θ1,...,θM from well-covered countries to impute maturity distributions for countries with limited data by estimating weights ω1,t,...,ωM,t to match observed moments.
  - Strengthen model selection by estimating statistical process for weights ω_t, possibly via filtering to reduce parameter space and allow richer models to compete.

_Italic: Source — wpiea2021101-print-pdf - 2.1    Sources (IMF working paper PDF)._

### 2.1    Sources

### 2.1    Sources

### Data sources and coverage
- United States:
  - Data on quantity, coupon rate, frequency of coupon payments, and maturity date of each marketable US debt security outstanding at the end of each month (January, 1869– present) come from the US Treasury’s Monthly Statement of the Public Debt (MSPD).
  - Prior to July, 1974 these data were furnished in the month end Daily Statement of the United States Treasury.
  - Issuances are cross-checked with the US Treasury’s database of treasury auction results (1982–2020).
  - Appendix A.2 comparison: post-1950 the dataset agrees exactly with CRSP; CRSP coverage is incomplete pre-1950 and almost non-existent prior to 1931. The authors’ coverage is universal back to 1869 and freely available to other researchers.
- United Kingdom:
  - Gilts (coupon-bearing debts of more than 12 months maturity) and Consols (debts with no maturity) prior to 2018: marketable debt data from Ellison and Scott (2020).
  - Gilts outstanding for 2019–2020: UK Debt Management Office’s (DMO) Gilts in Issue reports.
  - Treasury Bills (zero-coupon debts of fewer than or equal to 12 months maturity): compiled as a weekly amount by maturity sold during auction (competitive bidding) since UK treasury bills were first offered in 1877.
  - Treasury bill dataset sources include The Economist Historical Archive (1877–1946, 1950–68), Financial Times Historical Archive (1973–86, 1997–98), Bank of England Quarterly Bulletins and Annual Reports (1946–50, 1968–73, 1986–99), and UK DMO’s Treasury Bill Tender Results (2000–20).
  - Where data are not available in the form of tender allotments (i.e. the amount sold), tender offers (i.e. the amount offered for sale) are used instead.

### Period used for headline results and rationale
- Although data are gathered from 1869 onwards, large changes in financial markets post-World War I motivate focusing on the period 1920–2020 for headline results and for the remainder of the section.

### Data representation and visual summaries
- Figures 1 and 2:
  - Present decade-start snapshots of the entire outstanding set of government debt assets; each bar represents an individual debt security with months to maturity on the x-axis and face value as a share of GDP on the y-axis.
  - Bars are colored by security type. Vertical line in each figure indicates average maturity.
  - For presentation, despite infinite maturity of consols, the x-axis is truncated; the y-axis is not harmonized across panels to abstract from changes in total issuance and focus on distribution of outstanding obligations.
- Figure 3:
  - Shows the number of debt assets outstanding, 1920–2020, and documents a vast increase in the number of securities for both the US and UK.
  - For the US, the number of issuances in circulation has more than doubled in the past 15 years (driven almost entirely by new notes).
  - UK securities peaked in the mid-1980s, with growth in the 2000s due to index-linked gilts and bills; notable spikes in number of bills outstanding correspond to deviations from a one-issue-per-week baseline (e.g., weekly issuance of six-month bills from May 1989 to August 1993).
- Figure 4:
  - Shows average debt maturity, 1920–2020.
  - Average maturity of UK government debt is far longer than that of the US, and was very long in the 1930s and 1940s.
- Figures 5 and 6 (payment densities):
  - Construct empirical payment densities by converting each bond into its stream of future payments and aggregating across all outstanding assets at the end of each calendar month (capping coupons at a maturity of 1000 months).
  - Each dot in these figures represents a promised payment positioned by due date (x-axis, months to payment) and fraction of total payments (y-axis, payment density). Vertical line indicates average months to payment.
  - Payment densities display the future distribution of promised obligations and convert security types into a common unit: promises of future payments.

### Empirical patterns and notable observations
- Complexity and variety:
  - There has been a vast increase in the number of debt securities and complexity of government debts for both the US and UK (increased variety of horizontal bars in Figures 1 and 2; explicitly shown in Figure 3).
- Maturity differences:
  - The average maturity of UK government debt is far longer than that of the US; the UK maturity was especially long in the 1930s and 1940s (see Figure 4).
- Distributional changes:
  - US government debt distribution has acquired much fatter tails, with an increasing share of very short and very long bonds.
  - UK maturity structure changes have been largely dominated by the rise and fall of the consol, at least until around 1980.
- Limitations of simple summaries:
  - One-time snapshots (Figures 1 and 2) leave questions unanswered about representativeness, higher moments of distributions, and which features map into stylized models of debt maturity choice.
  - Choice of summary parameter (e.g., average maturity) can omit meaningful information; the paper assesses alternative parameterizations.

### Constructing densities and aggregation framework
- Conversion to payment densities:
  - Each bond’s promised stream of payments is converted into a common framework before aggregation across outstanding assets to form the monthly distribution of future payments.
  - Appendix A.3 describes details of this exercise.
  - Coupons are capped at a maturity of 1000 months for the plotting exercise.
- Notational definitions (discrete time, t = 1,2,...):
  - Government total obligations at start of period t coming due in period t + s denoted b^s_t.
  - Net issuance of maturity-s claims in period t is i^s_t.
  - Maturity-s debt stock evolution: b^s_{t+1} = b^{s+1}_t + i^s_t, s ≥ 1.
  - Aggregate totals: b_t = sum_{s=1}^∞ b^s_t, i_t = sum_{s=1}^∞ i^s_t.
  - Aggregate debt evolution: b_{t+1} = b_t − b^1_t + i_t. Equivalently, total issuance i_t equals change in borrowing, b_{t+1} − b_t, plus maturing debt b^1_t.
  - Debt and issuance ratios (densities):
    - f_t(s) = b^s_t / b_t
    - g_t(s) = i^s_t / i_t
  - f_t(s) sums to one and is the theoretical equivalent of the empirical debt distribution y^s_t defined earlier.
- Key conceptual point:
  - The evolution of the maturity structure depends only on the net issuance of claims, irrespective of the yield curve or deficits; issuance maps into deficits via prices of future claims, but that mapping is not the focal concern here.

### Theoretical implication preview
- Under very light assumptions (basic accounting and that governments can freely trade in a fixed set of asset markets), there is only one parametric model of the debt maturity structure that is both consistent with these assumptions and more parsimonious than listing all future obligations: the setting where the government can trade a fixed set of exponential-coupon bonds.
- Any other setting either imposes arbitrary issuance restrictions or requires as many parameters as the longest horizon of obligations.
- Formal uniqueness claims and proofs are provided in Appendix B; the paper proceeds with illustrative examples in the main text.

*Source: wpiea2021101-print-pdf - 2.1    Sources*

### Appendix A.3.  We further impose thatf

### Appendix A.3.  We further impose thatf

### Constraints on debt and issuance densities
- f_t(s)≥0 ∀ s, consistent with the data and meaning that f_t(s) is also a probability mass function.
- Requiring g_t(s) to meet a similar non-negativity constraint would be overly restrictive because many situations where the government changes the maturity structure of its debt, such as a debt swap, will necessarily require that g_t(s) be negative for some s.

### Evolution equation linking old debt, issuance, and new debt
- Equations (1)–(3) imply:
  - f_{t+1}(s) = (1−w_t) g_t(s) + w_t ( f_t(s+1) / (1−f_t(1)) ) ∀ s  (equation (4))
- w_t∈[0,1] is a decreasing function of aggregate issuance relative to outstanding debt, i_t / b_t.
- Issuance i_t is gross financing needs divided by the issuance-weighted average price of debt: i_t = d_t + b^1_{t−1} / ∑_{s=1}^{∞} g_t(s) q^t_s, where d_t is the overall deficit in period t and q^t_s the price of the s-period zero-coupon bond.

### Representation of debt by parametric family
- If new and old debt can be represented by the same family of distributions, debt in each period can be described by a parameter vector ω_t with f_t(s) = f(s | ω_t) and f_{t+1}(s) = f(s | ω_{t+1}).
- Imposing a functional form on f_t(·) and f_{t+1}(·) also imposes restrictions on issuance g_t(s); the aim is to understand when resulting g_t(s) allows the government free access to a variety of debt markets.

### Exponential-coupon bonds (no restriction on issuance)
- Setup:
  - M outstanding bonds indexed by i, bond i has decay rate θ_i, each period a fraction θ_i of outstanding balance i comes due.
  - Share of bond i in total time-t obligations is ω_{i,t}. Then ω_t = (ω_{1,t},...,ω_{M,t}).
  - f_t(s) = f(s | ω_t) = ∑_{i=1}^{M} ω_{i,t} θ_i (1−θ_i)^{s−1}
- If government has access to each of the M bond markets with amortization rates θ_1,...,θ_M, issuance density:
  - g_t(s) = ∑_{i=1}^{M} σ_{i,t} θ_i (1−θ_i)^{s−1}, where σ_{i,t} is the share of issuance in market i.
- For any aggregate issuance w_t∈[0,1] and bond-specific issuance shares σ_{i,t}, distribution for new debt is in the same family:
  - ω_{t+1,i} = (1−w_t) σ_{i,t} + w_t ( ω_{i,t} (1−θ_i) / (1− ∑_{j=1}^{M} ω_{j,t} θ_j) )
- Key property: an exponential-coupon bond’s residual maturity is itself exponential-coupon; therefore any choice of issuance among the same set of exponential-coupon securities yields f_{t+1}(·) in the same family with no arbitrary constraints on issuance.

### Bullet bond with a coupon (restrictions on issuance)
- Single bond paying coupon c for T periods with terminal payment 1; outstanding debt density:
  - f_t(s) = y_{c,T}(s) ≡
    - c/(1 + cT) for s < T
    - 1 + c/(1 + cT) for s = T
    - 0 for s > T
- Government must trade in two markets:
  - Buy back old bonds in their entirety (market for maturity T−1).
  - Issue new debt of duration T.
- Issuance:
  - g_t(s) = (1−σ_t) y_{c,T}(s) + σ_t y_{c,T−1}(s)
- For equation (4) to hold, constraint σ_t = −w_t / (1−w_t) is required, implying the government buys back the whole outstanding debt stock every period. This reveals an arbitrary restriction on issuance stemming from assuming the debt stock always a bond of maturity T.
- Conclusion: except for the special case where bullet bonds exist at all horizons up to T (i.e., a full set of bullet markets), most convenient parameterizations (like a single fixed-T bullet bond) impose arbitrary and economically unjustified restrictions on issuance. The exponential-coupon setup is the only parsimonious parameterization that does not do so.

### Empirical evidence: summarizing changes in US and UK debt maturity
- Sample limited to post-1920 for both countries.
- Headline estimates obtained by fitting portfolios of bonds with exponentially-decaying coupons; comparison also made to portfolios of constant-coupon bonds.
- Maximum empirical maturity S set to 1200.

#### Methodology (exponential-coupon mixture MLE)
- Decay rate of bond in market i: θ_i, share in time-t density: ω_{i,t}.
- Observations indexed by t=1,...,T (months).
- Likelihood (mixture distribution) maximized:
  - l(θ, ω | y) = ∑_{t=1}^{T} ∑_{s=1}^{S} y^t_s log( ∑_{i=1}^{M} ω_{i t} θ_i (1−θ_i)^{s−1} )  (equation (5))
- Constraint: ∑_{i=1}^{M} ω_{i t} = 1 ∀ t = 1,...,T.
- Parameters to estimate: M + T×(M−1).

#### Headline results (exponentially decaying coupons, 1920–2020)
- Table 1 reports MLE estimates for United States and United Kingdom for M = 1,2,3,4.
- Example interpretation:
  - For M = 3 (United States) best-fit decay parameters are 0.8%, 3.2% and 23.7% (θ_1 = 0.008, θ_2 = 0.032, θ_3 = 0.237).
  - Intuitive durations implied (from note): 1/0.008/12 ≈ 10.4 years, 1/0.032/12 ≈ 2.6 years, 1/0.237 ≈ 4.2 months.
- Average weights (sample averages over > 1200 periods; standard errors by delta method):
  - US, M=3 average weights roughly a 60-25-15 split across long, medium, short markets.
- Decadal variation (Figure 7 summary):
  - US: first half of sample, substitution of very long-maturity obligations first for short debt (1930s–40s) and later medium-dated securities; by 1970s most government debt was in the middle of term structure (around half); since then long-maturity bonds have again accounted for majority of future promises with short-end reliance declining, particularly in last decade.
  - UK: tradition of issuing very long debt (consols) reflected in θ estimates. For M=3, the shortest bond approximating UK debt is approximately as long as the longest bond used in US case (θ_UK_3 = 0.009 vs θ_US_1 = 0.008). Story: ultra-long debt before ~1950 (θ_1 = 0.002 average duration > 40 years) → long debt in 1970s–80s (θ_3 = 0.009 ≈ 9 years) → slightly longer debt in 2010s (average duration ≈ 16 years).
- Selected exact parameter estimates (from Table 1; significance levels: 5%*, 1%**, 0.1%***):
  - United States:
    - M=1: θ_1 = 0.012*** (0.0003); ̄ω_1 = 0.744*** (0.0146)
    - M=2: θ_1 = 0.009*** (0.0003), θ_2 = 0.129*** (0.0185); ̄ω_1 = 0.605*** (0.0207), ̄ω_2 = 0.256*** (0.0146)
    - M=3: θ_1 = 0.008*** (0.0004), θ_2 = 0.032*** (0.0052), θ_3 = 0.237*** (0.0344); ̄ω_1 = 0.601*** (0.0154), ̄ω_2 = 0.243*** (0.0227), ̄ω_3 = 0.152*** (0.0147)
    - M=4: θ_1 = 0.008*** (0.0003), θ_2 = 0.030*** (0.0044), θ_3 = 0.009*** (0.0009), θ_4 = 0.334*** (0.1056); ̄ω_1 = 0.466*** (0.0191), ̄ω_2 = 0.235*** (0.0179), ̄ω_3 = 0.109*** (0.0201), ̄ω_4 = 0.055*** (0.0181)
    - T = 12, S = 1200, Params. 11, Log lik = −6607.68 (M=1) → Log lik = −6388.83 (M=4)
    - AIC: 13217.36 (M=1) → 20057.67 (M=4)
  - United Kingdom:
    - M=1: θ_1 = 0.004*** (0.0001); ̄ω_1 = 0.437*** (0.0138)
    - M=2: θ_1 = 0.003*** (0.0001), θ_2 = 0.008*** (0.0005); ̄ω_1 = 0.457*** (0.0193), ̄ω_2 = 0.220*** (0.0290)
    - M=3: θ_1 = 0.002*** (0.0001), θ_2 = 0.006*** (0.0005), θ_3 = 0.009*** (0.0006); ̄ω_1 = 0.437*** (0.0138), ̄ω_2 = 0.201*** (0.0358), ̄ω_3 = 0.304*** (0.0316)
    - M=4: θ_1 = 0.002*** (0.0001), θ_4 = 0.446*** (0.0685); ̄ω_1 = 0.466*** (0.0191), ̄ω_2 = 0.534*** (0.0191), ̄ω_3 = 0.343*** (0.0335), ̄ω_4 = 0.038*** (0.0065)
    - T = 12, S = 1200, Params. 11, Log lik = −7919.62 (M=1) → Log lik = −7744.89 (M=4)
    - AIC: 15841.25 (M=1) → 22769.78 (M=4)

### Model selection
- Akaike Information Criterion (AIC) minimized at M = 1 for both countries due to large penalty from adding parameters (each increment in M increases free parameters by ~T ≈ 1200).
- Result implies statistically the debt maturity structure is indistinguishable from being constant over time under AIC, though economically unsatisfying.
- Robustness check: recompute AIC dropping long-end observations up to a cutoff maturity S; M = 1 still minimizes AIC even when recalculated at very short horizons (Figure 8).
- Alternative goodness-of-fit: absolute average error up to horizons:
  - err(s) = (1 / sT) ∑_{t=1}^{T} ∑_{u=1}^{s} | f_t(u) − ˆf(u, θ, ω_t) |
  - For US, substantial improvement in fit when M increases from 1 to 2; two-bond model for US is hard to improve on.
  - For UK, benefits of extra assets less evident; notable improvement at short maturities arises only when M = 4.
- Interpretation: number of assets to describe maturity structure is a modeller’s choice; no strong statistical evidence for a minimum M; results back models using a small number of exponential-coupon bonds.

### Comparison to alternate approaches (constant-coupon bonds)
- Alternative framework: M finite-duration constant-coupon bonds with market i duration μ_i; density for each market is uniform adjusted for non-integer durations:
  - f(s | μ_i) = 
    - 1/μ_i for s < ⌊μ_i⌋
    - 1 − ⌊μ_i⌋ / μ_i for ⌊μ_i⌋ ≤ s < ⌊μ_i⌋ + 1
    - 0 otherwise
- This density cannot be rationalized by unrestricted market access unless μ_i = i ∀ i (full set of constant-coupon markets).
- Estimation results (Table 2, 1920–2020) — μ_i in months; average duration of bond i is μ_i / 2:
  - United States, M=3 fitted bonds average durations ≈ 1, 4, and 14 years with average weights ~30-30-40 percent (exact μ estimates: μ_1 = 22.3*** months, μ_2 = 100.4*** months, μ_3 = 339.4*** months in one reported specification).
  - Constant-coupon approach fits far worse (inferior likelihoods); for US, constant-coupon M=4 worse than exponentially-decaying-coupon M=1.
- Reasons for poorer fit of constant-coupon model:
  - Government payment schedules are naturally downward-sloping; constant-coupon (histogram-like) struggles to match sharp decay at short maturities.
  - Poorer fit alters qualitative descriptions of maturity changes: for US, the exponential-coupon model shows long-duration bonds regained prominence in second half of 20th century; constant-coupon depiction loses that pattern and suggests a rise in short/medium-term shares instead.
- Conclusion: exponential-coupon parameterization is both theoretically coherent (rationalizable by unrestricted market access) and empirically superior to the constant-coupon benchmark for the datasets examined.

### Extension to select emerging markets
- Emerging markets pose data challenges; only four emerging market economies had sufficiently rich long-run public debt data for inclusion: Peru, Poland, Egypt, and Nigeria.
- Shorter histories and less-complete public records make characterization harder compared to US and UK.

*Italic: Source — wpiea2021101-print-pdf - Appendix A.3.  We further impose thatf (IMF working paper PDF).*

### 5.1    Data

### 5.1    Data

### Data sources and sample coverage
- Peruvian bond-level data: daily reports of public debt outstanding published by the Ministry of Economy and Finance of Peru for September 2006 to December 2020; extended back to 1999 using Bloomberg complete history of issuance.
- Polish bond-level data: detailed database of bond transactions from the Ministry of Finance of Poland covering May 1991 to April 2020, including detailed sale and buy-back data to describe marketable debts outstanding at any given time.
- Egypt and Nigeria: combination of debt issuance data from each country’s central bank and Egypt’s Ministry of Finance and Nigeria’s Debt Management Office, with data going back to 2002; Egypt extended back to 1995 using Bloomberg.
- Note on Bloomberg: “Bloomberg is not a sufficient source for reconstructing the full portfolio of public debt obligations.” For Peru official data from 2006 onwards corroborate summed issuance from Bloomberg; for Egypt Bloomberg is comprehensive prior to 2002.

### Coverage limitations and focus
- Dataset captures only marketable debt, consistent with prior work on the US and the UK.
- Excluded obligations: bilateral debts with other sovereigns and non-marketable debt held by other state entities.
- Focus restricted to central government debt; subnational government debt typically excluded.
- As a result, the sample is typically only a subset of emerging market debt and the total face value in the sample diverges from commonly-cited aggregate measures. Differences are most notable in recent years for Egypt and Nigeria.
- Authors assert comprehensiveness for the subset of marketable debts, which are the only obligations for which changes in maturity structure are relevant.

### Currency and coupon treatment in payment-density construction
- Terms of each debt security are used to construct the empirical payment density, following the methodology summarized by Table 4 and its discussion.
- Specific adjustments for foreign-currency and floating-rate instruments:
  - Foreign-currency denominated securities are converted to the respective home currency assuming that future exchange rate movements are perfectly foreseen.
  - Foreign-currency denominated floating-rate bonds of Poland are indexed to one of either the 3-month LIBOR, 6-month LIBOR, or 3-month EURIBOR.
  - For four Polish Brady bonds with “step-up” coupon rate terms that are not fully known, a fixed rate of 4.75% is used (described as “roughly the average of the future ‘step-ups’”).
  - Peru yield curve data are insufficient to project floating-rate coupons, so future coupon payments are assumed perfectly foreseen.

### Empirical results (summary from 5.2 and tables)
- Headline summaries are reported in Tables 3 and 11 in Appendix D for Peru and Poland, and Egypt and Nigeria, respectively.
- Characterization of maturity structures:
  - Peru:
    - Debt maturity structure differs on average and variation driven by long-dated bonds.
    - Short end similarity: decay rate for θ3 of 0.9 percent per month in the M = 3 case, similar to the UK.
    - Long end similarity: longest-maturity debt decays at 0.6 percent per month, closer to the US equivalent of 0.8 percent than the UK’s 0.2 percent.
    - In the M = 3 case, weight on the longest bond, ω3, is on average around 5 percent of Peruvian government debt; equivalent numbers for the US and UK are 15 and 34 percent respectively.
    - Peru has used floating-rate bonds at various points and issued a 100-year bond in November 2020.
  - Poland, Egypt, Nigeria:
    - Require shorter bonds to describe maturity structures.
    - Longest-maturity bond markets have decay rates “around 2-3 percent for M = 1 or 1.5-2 percent for larger M.”
    - Shortest bonds decay at rates above 20 percent per month, consistent with issuance concentrated in very short maturities.
    - For Poland and Egypt the average weight of the longest bonds in the government’s debt portfolio, ω1, is around 55 to 75 percent.
    - For Nigeria, shorter debt accounts for more than half of the debt.
- Dynamic evolution example (Poland, M = 3):
  - Monthly weights show a swift initial transition from very short-term debt relying on an asset with duration of around two months (θ3 = 0.52) to slightly longer bonds as markets were re-established after the fall of the Iron Curtain, then steady lengthening through the mid-1990s to mid-2000s, and a relatively stable long-duration debt structure from 2005 onwards.
  - Constant-coupon models fail to capture the rapid lengthening in the early 1990s and produce misleading high-frequency fluctuations after 2005; goodness-of-fit measures (log likelihood) favor the exponential-coupon specification over the constant-coupon parameterization (see Tables 12 and 13 in Appendix D).

### Quantitative estimation highlights (excerpt from Table 3)
- Peru, 1999-2020 and Poland, 1991-2020 exponential-decaying coupon MLE estimates (selected entries):
  - Bond decay parameters (examples):
    - Peru θ1 = 0.007*** (M = 1), θ1 = 0.006*** (M = 2,3,4) with reported standard errors in parentheses.
    - Poland θ1 = 0.021*** (M = 1), θ1 = 0.017*** (M = 2,3), θ1 = 0.016*** (M = 4).
  - Average weights (examples):
    - Peru ̄ω1 = 0.583*** (M = 1), 0.568*** (M = 2), 0.365*** (M = 3).
    - Poland ̄ω1 = 0.768*** (M = 1), 0.756*** (M = 2), 0.754*** (M = 3).
  - Sample sizes and fit statistics:
    - T and S reported (e.g., T 264 for Peru M variants; T 348 for Poland M variants).
    - Log likelihood values shown (e.g., Peru M = 1 log lik = -1588.40; Poland M = 1 log lik = -1685.19).
    - AIC values reported (examples: Peru M = 1 AIC = 3178.81; Poland M = 1 AIC = 3372.37).
  - Significance levels indicated as: 5%*, 1%**, 0.1%***.

### Methodological implications and extensions
- Modeling government balance sheets as a small number of exponentially-decaying bonds fits data better than a constant-coupon approach and yields different economic interpretations relevant for fiscal policy.
- Potential extensions suggested:
  - Use estimated θ1,...,θM from well-covered countries to impute maturity distributions for countries with limited data by estimating weights ω1,t,...,ωM,t to match observed moments (e.g., time series for “short” or “long” debt).
  - Strengthen model selection by estimating a statistical process for the weights ωt, likely requiring a filtering approach since weights are not directly observed; this would reduce parameter space and allow richer models to compete with simpler models.

*Source: wpiea2021101-print-pdf - 5.1    Data*

### 1.  Regress each horizonh’s RPI inflation expectation at timeton lags of RPI infla-

### 1.  Regress each horizonh’s RPI inflation expectation at timeton lags of RPI infla-

### Imputing RPI inflation expectations (regression procedure)
- Regress each horizon h’s RPI inflation expectation at time t on lags of RPI inflation of orders up to 120 months where:
  - E_t π_{t+h} = α + ∑_{j=1}^{p} β_j π_{t−j} + ε_{t+h}  ∀ p ∈ {1,2,3,...,120}
  - ρ = ∑_{j=1}^{p} β_j
- For each horizon h, select the optimal lag order using the choice of p that minimizes the Bayesian information criterion (BIC).
- The parameters of the chosen optimal models are summarized in Figure 14 and are compared to those of US CPI forecasts following a similar procedure.
- Notable empirical observation: there is much more movement in the estimated RPI parameters from short- to long-horizon.
- Construct “imputed” values for each forecast horizon h from monthly vintages prior to 1998 by summing the estimated constant and each lag of RPI weighted with lag weights from the optimal model.
- Figure 15 summarizes the resulting UK RPI inflation imputed forecasts with selected forecasts taken every five years and compared to actual RPI inflation out-turns.
- Data source: RPI inflation data are sourced from the Global Financial Data (GFD) database.

### Key statistics and references in the imputation section
- Lag orders considered: up to 120 months.
- Vintage coverage for imputed forecasts: monthly vintages prior to 1998.
- Comparative reference: US CPI forecasts constructed using a similar procedure (details and imputed CPI forecasts saved for Appendix C.2).
- Figures referenced: Figure 14 (Imputed inflation forecast parameter estimates) and Figure 15 (United Kingdom RPI inflation expectations, 1981–2020).

### Floating-rate coupon-bearing instruments and yield-curve construction
- Floating-rate instruments:
  - These instruments do not have a fixed coupon and instead pay varying amounts of interest that rise and fall based on current relative yields.
  - US: the US Treasury began issuing floating rate notes (FRNs) in January 2014 which have a duration of two years.
  - UK: the UK has issued floating rate gilts (FRGs) at various points with the last remaining FRG redeemed in July 2001.
  - Peru and Poland make more extensive use of floating rate bonds.
- Methodological challenge: computing appropriate yield curves to compute monthly forward rates at any horizon is country-specific.
- Country-specific yield-curve methodologies:
  - United States:
    - Data: Daily treasury yield curve rates published by the US Treasury (yields at 1-, 2-, 3-, and 6-month horizons and 1-, 2-, 3-, 5-, 7-, 10-, 20-, and 30-year horizons).
    - Method: model the yield curve with the Svensson (1994) parametric model to develop a complete monthly-horizon yield curve up to 30 years.
  - United Kingdom:
    - Data: Bank of England yield curves provide monthly-horizon rates up to 60 months and additional semi-annual rates out to 40 years; dataset often missing yields in the first 1-11 months.
    - Method: model the complete monthly-horizon yield curve up to 40 years using the Nelson and Siegel (1987) parametric model (Nelson-Siegel adapts better than Svensson to lack of short-end data).
  - Peru:
    - Yield curve data insufficient.
    - Majority of floating rate bonds are step-up coupon bonds with known future coupons accounted for accordingly.
    - For additional floating rate bonds, take the coupon at the time of the reference data and let it stay fixed for projection purposes.
  - Poland:
    - Floating rate international bonds indexed to short-term reference rates including 3- or 6-month LIBOR or EURIBOR.
    - Data sources: LIBOR data from FRED; EURIBOR data from the ECB (available up to 12 months).
    - Method: use the Svensson model to estimate a yield curve out several years for forward rates beyond 1 year; for floating rate domestic bonds, derive term structure using yields on transactions and fit a monthly-horizon yield curve using the Nelson-Siegel model.

### Treatment of special instrument types
- Perpetual bonds (consols):
  - Treated as if government must account for a finite schedule of coupon payments set to 100 years from any given time in which the bond is outstanding (rather than true perpetuity).
- Callable notes and bonds:
  - For certain notes and bonds issued before 1985 where the US Treasury reserved the right to stop paying interest before maturity, these are treated as if the government assumes ex-ante that they will pay principal at maturity and not redeem prior to that date.
  - Current status: there are currently no notes or bonds in circulation eligible to be called.
- Certificates of indebtedness:
  - Predecessor of the US treasury bill issued in parallel with treasury bills until the late 1960s (phased out thereafter).
  - Nominal coupon-bearing instruments that pay a coupon at maturity and sometimes a coupon after 6 months.

### Comparison to CRSP Monthly US Treasury Database
- CRSP monthly data begin December, 1925, but do not comprehensively cover amounts outstanding of individual securities until around 1950.
- Example: In January, 1930, CRSP only has quantity outstanding data for the 3% Panama Canal Loan of 1961 and the 4 1 4% Bonds of 1947-52, notably missing data for the Fourth Liberty Loan (which made up nearly 40% of all government debt at the time).
- No securities with maturity prior to June, 1946 have quantity data in CRSP.
- Divergence: total marketable debt outstanding in CRSP begins to diverge from the headline amount (provided by the Federal Reserve Bank of Dallas) beyond 1950 (illustrated in Figure 17).
- Contribution: the authors’ data are comprehensive going back to January, 1869 and mirror available headline data back to January, 1942.

### Converting outstanding debts into streams of future payments (payment mapping)
- For almost all outstanding securities, the stream of future payments is a function of:
  - Number of units outstanding B_{it}, coupon rate r_i, coupon frequency F_i, and maturity date (residual maturity M_{it}).
- Two types of future payouts from debt of type i:
  - C_{its} (coupon): intermediate payments in period t+s for s = 1,...,M_{it}; coupon payments may be zero in some periods if payable at quarterly or semi-annual frequency.
  - P_{it} (principal): the terminal payment due in period t+M_{it}.
- Floating-rate and real bonds depend on additional variables:
  - Π_{st} = ∏_{h=1}^{s} (1 + E_t π_{t+h})^{1/12} is cumulative gross expected inflation between period t and t+s; E_t π_{t+s} is the expected inflation at time t for inflation in period t+s.
  - f_{t,T} = ((1+z_T)^T / (1+z_t)^t)^{1/(T−t)} − 1 is the forward rate at time t for a bond maturing in period T.
  - z_t is the spot rate of a bond maturing in t months.
- Table 4 summarizes stream-of-payment formulae by instrument type (examples preserved from the source):
  - Zero-coupon (US name: Bills; UK name: Bills): P_{it} = B_{it}, C_{its} = 0.
  - Nominal coupon-bearing (US name: Bonds; UK name: Gilts): P_{it} = 0 or B_{it}, C_{its} = 0 or B_{it} r_i / F_i.
  - Perpetual bonds (Consol): P_{it} = 0, C_{its} = 0 or B_{it} r_i / F_i.
  - Real coupon-bearing (TIPS / Index-linked): P_{it} = Π_{M_{it}} B_{it}, C_{its} = 0 or Π_{st} B_{it} r_i / F_i.
  - Floating coupon-bearing (FRNs / FRGs): P_{it} = B_{it}, C_{its} = 0 or B_{it} f_{t−1,T} / F_i.
- Using these formulae, principal and coupon payments specific to each individual debt asset outstanding at the end of each month are calculated.

### Constructing aggregate debt maturity density
- Index assets by i and time by t; future payouts are mapped to a density of payments over s = 1,...,M_{it}.
- First stage (individual security payment density):
  - Define y_{s it} as the fraction of payment obligations for security i outstanding at time t that are due s months after t:
    - y_{s it} = C_{its} / (P_{it} + C_{it}) + [C_{it} * (M_i − s mod 12) / F_i] = C_{its} + P_{it} / (P_{it} + C_{it}) if s = M_{it}
    - Where C_{it} = ∑_{s=1}^{M_i} C_{its}; ∑_{s=1}^{M_{it}} y_{s it} = 1.
- Second stage (aggregate portfolio density):
  - Aggregate across n_t securities outstanding at time t:
    - y_{s t} = ∑_{i=1}^{n_t} ω_{it} y_{s it}
    - ω_{it} = (P_i + C_i) / ∑_{n} (P_i + C_i) is the fraction of future payments due to bond i.
    - ∑_{s=1}^{∞} y_{s t} = 1; {y_{s t}}_{s=1}^{∞} is the empirical maturity density at time t (nominal).
- Interpretation: the empirical maturity density represents the fraction of the zero-coupon equivalent which must be paid in each subsequent period; closely related to the zero coupon equivalent measure of total debt.

### Mathematical details of the conceptual framework (key propositions and conditions)
- Dynamics and weighting (Proposition 1):
  - The relationship between f_t, g_t and f_{t+1} can be expressed as a weighted average:
    - f_{t+1}(s) = w_t ( f_t(s+1) / (1−f_t(1)) ) + (1−w_t) g_t(s) ∀ s
  - Given f_t, g_t any f_{t+1} defined by the equation is a probability mass function for any w_t ∈ [ w̄_t, 1 ], where:
    - w̄_t = min_{s:g_t(s)<0} ( −g_t(s) / ( f_t(s+1)/(1−f_t(1)) − g_t(s) ) )
  - Interpretation: w_t is a decreasing function of aggregate issuance relative to outstanding debt, w_t = 1 − f_t(1) / (1 − f_t(1) + i_t / b_t ) ∈ [0,1].
- Parametric debt densities:
  - Assume f_t(s) = f(s|ω_t) ∀ t,s with parameters ω_t ∈ Ω.
  - Issuance implied by parametric evolution:
    - g_t(s) = g(s|ω_t, ω_{t+1}, w_t) := f(s|ω_{t+1}) / (1−w_t) − ( w_t / (1−w_t) ) ( f(s+1|ω_t) / (1−f(1|ω_t)) )
- Market structure and unrestricted market access (Definition 1):
  - Suppose there are N debt markets with payment densities h_j(s); issuance shares σ_{j t} sum to unity.
  - Unrestricted market access requires that for any ω_t ∈ Ω, any σ_t ∈ S, and any w_t ∈ [ w̄, 1 ] there exists ω_{t+1} ∈ Ω such that:
    - f(s|ω_{t+1}) = w_t ( f(s+1|ω_t) / (1−f(1|ω_t)) ) + (1−w_t) ( ∑_{j=1}^{N} σ_{j t} h_j(s) ) ∀ s
  - Implication: frameworks that constrain σ_t beyond σ_t ∈ S impose additional limits not justified by market incompleteness alone.
- Mixture distributions and market rationalization:
  - If debt distributions converge while issuance continues (∃ T > t such that w_T < 1), outstanding obligations will eventually be a mixture distribution (Proposition 2).
  - For a mixture density f(s|ω_t) = ∑_{i} ω_{i,t} f_i(s), unrestricted market access requires existence of an M×M matrix W such that:
    - W f(s) = f(s+1) ∀ s
    - W 1 = 1 − f(1)
  - Theorem 1 (preview): the only solutions to the above conditions are Mexponentially-decaying-coupon bonds or a full set of zero-coupon bonds at every horizon between 1 and some maximum horizon T (statement truncated in source).

*Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021101-print-pdf.pdf*

### 1.  For alls,f(s) =

### 1.  For alls,f(s) = ˆf(s), with the elements of ˆf satisfying

### Primitive densities and market mapping
- For i = 1,...,M there exist λ1,...,λM such that:
  - ˆf_i(s) = λ_i^{s−1}(1−λ_i) for all s
- If T > 0 then for i = M+1,...,M+T:
  - ˆf_i(s) = { 1 if s = i−M; 0 if s ≠ i−M }
- The rows of P sum to one, and can be expressed as:
  - P = [ Q1 Q2 0_{T×(M−T)} ] where Q1 is M×M, Q2 is T×T and both are invertible.
- Interpretation:
  - The primitive densities ˆf come in two types:
    - i ≤ M: exponentially-decaying-coupon bonds (fraction 1−λ_i pays out each period).
    - i > M: one-period bonds (each paying only at a single horizon between 1 and T).
  - The matrix P converts from primitive densities ˆf to market densities f.
  - Because Q2 is invertible, any non-exponential-coupon asset must be able to replicate a full set of zero-coupon bonds between horizons 1 and T.
  - Using zero-coupon bonds to capture an empirical debt density with maximum maturity T requires T assets.

### Key formal notes
- Note that T ≤ M does not restrict analysis: an exponential-coupon bond with λ_i = 0 is a one-period zero-coupon bond; one can always add T−M redundant exponential-coupon bonds with λ_i = 0.
- If ˆf included fewer than T distinct zero-coupon bonds then Q2 would be rank deficient.

### Equations referenced (exact)
- W = A(Λ + Ψ)A^{−1} where A is a change of basis, Λ is square and diagonal, Ψ has zeros except possibly ones on the first off-diagonal.
- Ψ = { ψ_{ij} } = { 1 if i = j−1, i < T; 0 otherwise }.
- f(s+1) = (Λ + Ψ) f(s) (intuition when A = I)
- If Ψ = 0 then Λ f(s) = f(s) and f_i(s+1) = λ_i f_i(s) ∀ i,s, so f_i(s) are exponentially-decaying-coupon bonds with parameter λ_i.
- If Λ = 0 and T = M then f(1) is the unit vector (0,0,...,1)′ and Ψ shifts up entries so f_i(s) = 1 if s = i and 0 otherwise (i-period zero-coupon bond).

### Structural conclusion
- Solutions to equations (11) and (12) are linear combinations of primitive densities ˆf.
- Because Q1 and Q2 are select rows and columns of A, they inherit invertibility from A.

---

### Proof sketch: Jordan decomposition and mapping to primitive densities
- Decompose W into Jordan canonical form: W = A(Λ + Ψ)A^{−1}.
- Define ˜A = diag(A1). Let Q = ˜A^{−1} A (row-normalized A). Q is invertible.
- Rewrite f(s+1) = W f(s) ⇒ Q^{−1} f(s+1) = (Λ + Ψ) Q^{−1} f(s).
- Define Φ = [ Λ 0 ; 0 Ψ ] and R = [ I_M I_M ] so that Λ + Ψ = R Φ R′.
- Let ˆf(s) = R Q^{−1} f(s). Then:
  - ˆf(s+1) = Φ ˆf(s)
- Because Ψ is block-diagonal, upper M rows of ˆf satisfy:
  - ˆf_i(s+1) = λ_i ˆf_i(s), implying ˆf_i(s) = λ_i^{s−1}(1−λ_i).
- Two cases complete the proof:
  - Case I: T = M yields one-period bond in last row and a chain generating zero-coupon one-period bonds; set Q1 = Q2 = 1/2 Q and P = [ Q1 Q2 ] so f(s) = P ˆf(s).
  - Case II: T < M requires redefining ˆf dropping last M−T entries and defining Q_T and P = [ Q1 Q2 0 ] with Q1 = 1/2 Q and Q2 = 1/2 Q_T; rowsums of P are one and Q1, Q2 invertible.

---

### Propositions and important identities (exact forms preserved)
- w_t = (1 − f_t(1)) / (1 − f_t(1) + i_t / b_t) ∈ [0,1]
- Definition 2 (Non-redundancy): Ω is non-redundant if for all ω_{t+1} ∈ Ω there exist ω_t ∈ Ω, σ_t ∈ { x ∈ R^N | ∑_{N} = 1 }, and w_t ∈ [ ̄w, 1 ] with ̄w defined by equation (10), such that equation (9) holds.
- Proposition 4 (Necessary conditions for unrestricted market access). If f(s|ω) can be rationalized by unrestricted access to asset markets defined by h_1,...,h_N, and Ω is non-redundant, then:
  1. Truncation: ∀ ω ∈ Ω, ∃ ̃ω ∈ Ω such that:
     - f(s| ̃ω) = f(s+1|ω) / (1 − f(1|ω)) ∀ s
  2. Parametric mixture: ∀ ω, ω′ ∈ Ω, ∀ α ∈ R ∃ ̃ω ∈ Ω such that:
     - f(s| ̃ω) = α f(s|ω′) + (1−α) f(s|ω) ∀ s
- Recursive substitution result (equation (14)):
  - f_{t+1}(s) = Q_t^t f_0(s+1+t) + ∑_{s=0}^t Q_s^t (1 − w_{t−s}) g_{t−s}(s+s)
  - Q_s^t = { 1 if s = 0; ∏_{r=0}^{s−1} ( w_{t−r} / (1 − f_{t−r}(1)) ) otherwise }
- For mixture issuance g_t(s) = ∑_{j=1}^N σ_{j,t} h_j(s), in the limit f_{t+1}(s) is a mixture distribution with basis densities h_j(s).

- From Proposition 3: the truncation condition in vector notation is:
  - ̃ω′ f(s) = ω′ f(s+1) / (1 − ω′ f(1)) and stacking yields W f(s) = f(s+1) ∀ s with W 1 = 1 − f(1).

- Checking densities: J = ∑_{s=1}^∞ f(s) = (I − W)^{−1} f(1) and since f(1) = (I − W) 1, J = 1.

---

### Robustness exercises: sample-period sensitivity (exact sample periods and table notes)
- Tables 5-7 repeat analysis for samples since 1960, since 1980, and back to 1869.
- Qualitative finding: In later samples the estimated θ_i are very similar to headline results though with slightly different average weights; prior to 1920 the estimated θ_i are quite different.
- Table 5: MLE Estimates, Exponentially decaying coupon 1960-2020 (selected exact entries shown)
  - United States, M = 1: θ_1 = 0.015*** (0.0005), ̄ω_1 = 0.688*** (0.0225)
  - United States, M = 2: θ_1 = 0.010*** (0.0006), θ_2 = 0.139*** (0.0237), ̄ω_1 = 0.496*** (0.0329), ̄ω_2 = 0.312*** (0.0225)
  - United Kingdom, M = 1: θ_1 = 0.006*** (0.0002), ̄ω_1 = 0.243*** (0.0316)
  - T = 732, S = 1200, Params. = 1734 1467 2200 (table shows parameter counts by model), Log lik = -3813.51 (US M=1), AIC = 7629.01 (US M=1)
  - Significance levels: 5%*, 1%**, 0.1%***
- Table 6: MLE Estimates, Exponentially decaying coupon 1980-2020 (selected exact entries)
  - United States, M = 1: θ_1 = 0.014*** (0.0006), ̄ω_1 = 0.725*** (0.0273)
  - T = 492, S = 1200, Params. = 1494 987 1480 (table shows parameter counts), Log lik = -2605.22 (US M=1), AIC = 5212.44 (US M=1)
  - Significance levels: 5%*, 1%**, 0.1%***
- Table 7: MLE Estimates, 1869-2020 (selected exact entries)
  - United States, M = 1: θ_1 = 0.004*** (0.0001), ̄ω_1 = 0.336*** (0.0042)
  - T = 1817, S = 1200, Params. = 1181 936 3754 5118 2136 415461 (table shows parameter counts; preserved as printed), Log lik = -11654.62 (US M=1), AIC = 23311.24 (US M=1)
  - Significance levels: 5%*, 1%**, 0.1%***

### Robustness: real obligations analysis (exact formulas and notes)
- Real debt densities constructed by deflating nominal payment obligations using the expected inflation term structure:
  - ˆB_{dollars,t}^{m} = B_{dollars}^{m}_t / ˆΠ_{m,t}
  - ˆΠ_{m,t} = ∏_{h=1}^m (1 + E_t π_{t+h})^{1/12}
- 30-year CPI inflation expectations for the US and UK each month beginning January 1920 are imputed from Federal Reserve Bank of Cleveland’s projections (imputation procedure identical to that used for imputing UK RPI).
- Empirical note: decay rates of fitted assets are a little higher in real-obligations parametrization; the real versions of the debt maturity distributions decay at the rate of expected inflation. Example statement: the US one-bond decay rate is around 0.002 higher than in the headline case, consistent with an average inflation rate.

*Source: wpiea2021101-print-pdf - 1.  For alls,f(s) = (IMF working paper content provided).*

### 2.4 percent (as decay rates are monthly).

### wpiea2021101-print-pdf - 2.4 percent (as decay rates are monthly).

### Data sources and frequency
- CPI inflation data sourced from the Global Financial Data (GFD) database.
- Monthly frequency beginning January 1914 for the UK and January 1875 for the US.
- "2.4 percent (as decay rates are monthly)."

### MLE Estimates, 1920-2020 (Real units) — United States and United Kingdom (Table 8)
- Model specifications: M = 1, 2, 3, 4 for each country.
- United States — Bond decay parameters (θ) and average weights ( ̄ω):
  - θ1: 0.014*** (0.0004); 0.010*** (0.0004); 0.008*** (0.0004); 0.008*** (0.0004)
  - θ2: 0.114*** (0.0164); 0.030*** (0.0040); 0.029*** (0.0033); 0.011*** (0.0008)
  - θ3: 0.231*** (0.0302); 0.182*** (0.0339); 0.422*** (0.0619); 0.016** (0.0071)
  - θ4: 0.343*** (0.1165); 0.444*** (0.0636)
  - ̄ω1: 0.679*** (0.0157); 0.497*** (0.0211); 0.491*** (0.0153); 0.545*** (0.0191)
  - ̄ω2: 0.321*** (0.0157); 0.322*** (0.0244); 0.317*** (0.0187); 0.455*** (0.0191)
  - ̄ω3: 0.181*** (0.0156); 0.135*** (0.0214); 0.050*** (0.0073); 0.135*** (0.0195)
  - ̄ω4: 0.057*** (0.0191); 0.047*** (0.0072)
  - T: 1205 1205 1205 1205
  - S: 1200 1200 1200 1200
  - Params.: 11 20 72 41 33 36 19 11 20 72 41 33 36 19 (as presented)
  - Log lik: -6357.39; -6130.78; -6096.08; -6093.71
  - AIC: 12716.78; 14675.56; 17018.15; 19425.43
- United Kingdom — Bond decay parameters (θ) and average weights ( ̄ω):
  - θ1: 0.004*** (0.0001); 0.002*** (0.0001); 0.002*** (0.0001); 0.002*** (0.0002)
  - θ2: 0.009*** (0.0007); 0.007* (0.0041)
  - θ3: (only one value shown) 0.422*** (?) and 0.016** (0.0071) — as presented
  - θ4: 0.343*** (0.1165); 0.444*** (0.0636) — as shown in table alignment
  - ̄ω1: 0.545*** (0.0182); 0.545*** (0.0182); 0.518*** (0.0206)
  - ̄ω2: 0.404*** (0.0176); 0.300*** (0.0245)
  - ̄ω3: 0.135*** (0.0214); 0.050*** (0.0073); 0.135*** (0.0195)
  - ̄ω4: 0.057*** (0.0191); 0.047*** (0.0072)
  - T: 1205 1205 1205 1205
  - S: 1200 1200 1200 1200
  - Params.: 11 20 72 41 33 36 19 11 20 72 41 33 36 19 (as presented)
  - Log lik: -7935.18; -7784.86; -7727.08; -7720.22
  - AIC: 15872.37; 17983.73; 20280.15; 22678.45
- Significance levels: 5%*, 1%**, 0.1%***

### Imputed inflation expectations (Figure 18)
- Time coverage: 1920–2020.
- Series shown: CPI Inflation, Imputed inflation expectations, Inflation expectations.
- Vertical percent scales: United States range roughly -10 to 20 Percent; United Kingdom range roughly -25 to 25 Percent (as plotted).
- Decadal time axis labels: 1910 1930 1950 1970 1990 2010 2030 2050 (as in figure).

### Called-at-first-availability analysis (Appendix C.3)
- Historical context: Certain historical US Treasury bonds included call options allowing repayment of principal and stopping interest before maturity; these were prolific in the first half of the 100-year sample.
- Baseline assumption prior: callable bonds assumed not called.
- Alternative exercise: assume the call option is taken at date of first availability (called-at-first-availability).
- Mechanical effect: shortens the payment schedule of every bond that includes a call option.
- Observed impact:
  - Slightly higher decay rates over the 100-year sample across all model specifications.
  - Respective average weights relatively unchanged.
- Tables reflecting this exercise: Table 9 and Table 10 (United States, callable assumption).

### MLE Estimates, Exponentially decaying coupon, Callable assumption — United States (Table 9)
- Bond decay parameters θ and average weights ̄ω for M = 1, 2, 3, 4:
  - θ1: 0.015*** (0.0004); 0.012*** (0.0004); 0.010*** (0.0005); 0.010*** (0.0004)
  - θ2: 0.187*** (0.0229); 0.042*** (0.0088); 0.038*** (0.0059)
  - θ3: 0.279*** (0.0441); 0.230*** (0.0299)
  - θ4: 1.000*** (0.1834)
  - ̄ω1: 0.770*** (0.0145); 0.653*** (0.0218); 0.647*** (0.0122)
  - ̄ω2: 0.230*** (0.0145); 0.191*** (0.0240); 0.181*** (0.0136)
  - ̄ω3: 0.156*** (0.0150); 0.159*** (0.0092)
  - ̄ω4: 0.014** (0.0049)
  - T: 12 12 12 12 12 12 12 12
  - S: 1200 1200 1200 1200
  - Params.: 11 21 42 42 73 64 0 (as presented)
  - Log lik: -6306.87; -6132.37; -6118.06; -6112.13
  - AIC: 12615.73; 14692.74; 17090.11; 19504.25
- Significance levels: 5%*, 1%**, 0.1%***

### MLE Estimates, Constant coupon, Callable assumption — United States (Table 10)
- Bond maturities μ and average weights ̄ω for M = 1, 2, 3, 4:
  - μ1: 306.1*** (0.17); 54.4*** (0.21); 10.4*** (0.15); 11.4*** (0.16)
  - μ2: 347.3*** (0.41); 101.4*** (0.34); 83.2*** (0.46)
  - μ3: 314.3*** (0.53); 107.0*** (0.92)
  - μ4: 295.5*** (0.62)
  - ̄ω1: 0.541*** (0.02); 0.240*** (0.01); 0.244*** (0.01)
  - ̄ω2: 0.459*** (0.02); 0.445*** (0.02); 0.283*** (0.03)
  - ̄ω3: 0.316*** (0.02); 0.171*** (0.03)
  - ̄ω4: 0.303*** (0.02)
  - T: 12 12 12 12 12 12 12 12
  - S: 1200 1200 1200 1200
  - Params.: 11 21 42 42 73 64 0 (as presented)
  - Log lik: -7504.30; -6608.50; -6628.86; -6746.34
  - AIC: 15010.59; 15644.99; 18111.71; 20772.69
- Significance levels: 5%*, 1%**, 0.1%***

### Extra figures and tables for emerging markets (Section D)
- Figure 19: Face value of total and marketable debt outstanding, emerging markets — panels for:
  - Poland (1996–2020 axis), Nigeria (2004–2020 axis), Peru (2000–2020 axis), Egypt (2004–2020 axis).
  - Series: Marketable debt (data) and Gross debt (headline).
- Figure 20: Peru government debt obligations by half-decade, 2000–2020.
  - Security categories: Bill, Bond, Floating−rate bond, Floating−rate international, Inflation−linked bond, International.
  - Vertical line indicates average maturity.
- Figure 21: Egypt government debt obligations by half-decade, 2000–2020.
  - Security categories: Bill, Bond, International bill, International bond, Zero−coupon bond.
  - Vertical line indicates average maturity.
- Figure 22: Nigeria government debt obligations by half-decade, 2005–2020.
  - Security categories: Bill, Bond, International.
  - Vertical line indicates average maturity.

### MLE Estimates, Exponentially decaying coupon — Egypt and Nigeria (Table 11)
- Egypt, 1995-2020; Nigeria, 2002-2020; M = 1, 2, 3, 4.
- Bond decay parameters θ:
  - Egypt θ1: 0.032*** (0.0018); 0.020*** (0.0017); 0.020*** (0.0016); 0.020*** (0.0016)
  - Egypt θ2: 0.218*** (0.0261); 0.159*** (0.0244); 0.157*** (0.0229); 0.204*** (0.0248)
  - Egypt θ3: 0.328*** (0.0510); 0.288** (0.1041); 0.240*** (0.0403); 0.162*** (0.0271)
  - Egypt θ4: 0.359*** (0.1189); 0.343*** (0.0592)
  - Nigeria θ1: 0.029*** (0.0019); 0.014*** (0.0016); 0.012*** (0.0016); 0.012*** (0.0015)
  - Nigeria θ2: 0.204*** (0.0248); 0.054** (0.0190); 0.045*** (0.0110)
  - Nigeria θ3: 0.162*** (0.0271) (as presented)
- Average weights ̄ω:
  - Egypt ̄ω1: 0.594*** (0.0334); 0.566*** (0.0146); 0.565*** (0.0087); 0.462*** (0.0352)
  - Egypt ̄ω2: 0.406*** (0.0334); 0.281*** (0.0160); 0.274*** (0.0040); 0.538*** (0.0352)
  - Egypt ̄ω3: 0.153*** (0.0210); 0.080** (0.0355); 0.433*** (0.0460); 0.300*** (0.0387)
  - Egypt ̄ω4: 0.081** (0.0380); 0.204*** (0.0374)
  - Nigeria ̄ω1: 0.346*** (0.0336); 0.343*** (0.0139)
  - Nigeria ̄ω2: 0.220*** (0.0490); 0.153*** (0.0165)
  - Nigeria ̄ω3: 0.300*** (0.0387)
  - Nigeria ̄ω4: 0.081** (0.0380); 0.204*** (0.0374)
- Sample sizes and metrics:
  - T: 308 308 308 308 (Egypt); 228 228 228 228 (Nigeria)
  - S: 1200 1200 1200 1200
  - Params.: 13 106 199 281 230 459 688 (as presented)
  - Log lik (Egypt): -1359.45; -1289.11; -1283.71; -1283.48
  - Log lik (Nigeria): -1034.35; -926.68; -917.37; -911.71
  - AIC (Egypt): 2720.91; 3198.21; 3805.43; 4422.97
  - AIC (Nigeria): 2070.70; 2313.37; 2752.73; 3199.42
- Significance levels: 5%*, 1%**, 0.1%***

### MLE Estimates, Constant coupon — Peru and Poland (Table 12)
- Peru, 1999-2020; Poland, 1991-2020; M = 1, 2, 3, 4.
- Bond maturities μ:
  - Peru μ1: 394.1*** (0.37); 186.0*** (1.36); 101.2** (8.56); 98.0*** (7.33)
  - Peru μ2: 367.2*** (0.73); 182.0*** (0.90); 103.0*** (13.77); 168.3*** (0.64)
  - Peru μ3: 345.5*** (1.15); 181.1*** (1.03); 172.8*** (1.75); 119.8*** (1.09)
  - Peru μ4: 345.5.2*** (1.13); 174.9*** (1.85)
  - Poland μ1: 188.1*** (0.22); 62.1*** (0.31); 58.2*** (0.38); 11.6*** (0.31)
  - Poland μ2: 118.8*** (0.41); 62.2*** (0.70)
  - Poland μ3: 119.8*** (1.09)
- Average weights ̄ω:
  - Peru ̄ω1: 0.449*** (0.05); 0.053* (0.03); 0.039* (0.03); 0.629*** (0.03)
  - Peru ̄ω2: 0.551** (0.05); 0.377*** (0.06); 0.021 (0.02); 0.371*** (0.03)
  - Peru ̄ω3: 0.570*** (0.05); 0.366*** (0.06); 0.142*** (0.03); 0.311*** (0.04)
  - Peru ̄ω4: 0.574*** (0.05); 0.136*** (0.03)
  - Poland ̄ω1: 0.539*** (0.03); 0.273*** (0.02)
  - Poland ̄ω2: 0.319*** (0.04); 0.280*** (0.04)
  - Poland ̄ω3: 0.311*** (0.04)
  - Poland ̄ω4: 0.574*** (0.05); 0.136*** (0.03)
- T: 264 264 264 264 (Peru); 348 348 348 348 (Poland)
- S: 12 12 12 12 12 12 1200 1200 (as presented)
- Params.: 12 665 31 796 13 506 99 1048 (as presented)
- Log lik (Peru): -1687.89; -1673.96; -1708.09; -1746.23
- Log lik (Poland): -2029.95; -1945.63; -1907.25; -1791.79
- AIC (Peru): 3377.79; 3879.92; 4478.17; 5084.46
- AIC (Poland): 4061.91; 4591.27; 5212.50; 5679.59
- Significance levels: 5%*, 1%**, 0.1%***

### MLE Estimates, Constant coupon — Egypt and Nigeria (Table 13)
- Egypt, 1995-2020; Nigeria, 2002-2020; M = 1, 2, 3, 4.
- Bond maturities μ:
  - Egypt μ1: 102.1*** (0.22); 79.1*** (0.37); 10.3*** (0.19); 11.4*** (0.19)
  - Egypt μ2: 118.8*** (1.25); 83.7*** (0.72); 79.8*** (0.93); 162.6*** (1.29)
  - Egypt μ3: 136.2*** (0.83); 97.7*** (1.96); 166.7*** (1.60); 98.6*** (3.52)
  - Egypt μ4: 134.2*** (0.95); 159.9*** (2.00)
  - Nigeria μ1: 146.1*** (0.29); 40.1*** (0.21); 11.3*** (0.20); 11.3*** (0.20)
  - Nigeria μ2: 162.6*** (1.29); 64.1*** (0.62); 58.1*** (0.79)
  - Nigeria μ3: 98.6*** (3.52)
  - Nigeria μ4: 134.2*** (0.95); 159.9*** (2.00)
- Average weights ̄ω:
  - Egypt ̄ω1: 0.796*** (0.04); 0.460*** (0.02); 0.475*** (0.02); 0.732*** (0.03)
  - Egypt ̄ω2: 0.204*** (0.04); 0.397*** (0.03); 0.307*** (0.04); 0.268*** (0.03)
  - Egypt ̄ω3: 0.143*** (0.03); 0.101*** (0.03); 0.224*** (0.03); 0.094** (0.04)
  - Egypt ̄ω4: 0.116*** (0.03); 0.169*** (0.03)
  - Nigeria ̄ω1: 0.732*** (0.03); 0.571*** (0.03); 0.571*** (0.03)
  - Nigeria ̄ω2: 0.268*** (0.03); 0.205*** (0.03); 0.166*** (0.03)
  - Nigeria ̄ω3: 0.224*** (0.03); 0.094** (0.04)
  - Nigeria ̄ω4: 0.116*** (0.03); 0.169*** (0.03)
- T: 308 308 308 308 (Egypt); 228 228 228 228 (Nigeria)
- S: 1200 1200 1200 1200
- Params.: 13 106 199 281 230 459 688 (as presented)
- Log lik (Egypt): -1759.31; -1567.78; -1388.94; -1386.75
- Log lik (Nigeria): -1508.20; -1300.86; -1184.40; -1206.81
- AIC (Egypt): 3520.62; 3755.57; 4015.88; 4629.50
- AIC (Nigeria): 3018.40; 3061.71; 3286.79; 3789.62
- Significance levels: 5%*, 1%**, 0.1%***

*Content extracted from wpiea2021101-print-pdf - 2.4 percent (as decay rates are monthly).

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021101-print-pdf.pdf_