## _wp13248 ## Source details **Canonical URL:** [_wp13248](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13248.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13248.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13248.pdf.json) --- ### Introduction: policy context, objectives, and preview of findings - Policy context and motivation: - Widespread belief in a common debt threshold influenced policy debates in the United States and the United Kingdom; Reinhart & Rogoff suggested a debt-to-GDP threshold of around 90% beyond which economic growth is seriously affected. - Research objective and key questions: - Investigate nonlinearity in the debt-growth relationship distinguishing: - nonlinearity across countries (cross-country heterogeneity), and - within-country nonlinearity (country-specific threshold or tipping point). - Assess whether a common threshold across countries exists and whether country-specific tipping points can be identified. - Data and empirical scope: - Sample: total public debt data from 105 developing, emerging and developed economies. - Time horizon: 1972 to 2009. - Empirical focus: long-run (levels) relationship within a neoclassical growth model framework, accounting for short-run effects and unobserved global and local shocks. - Preview of findings (as stated in the Introduction): - Long-run debt coefficients differ across countries. - Tentative evidence that countries with higher average debt-to-GDP ratios are more likely to see a negative effect on their long-run growth performance. - No evidence found for a specific debt threshold common to all countries that triggers a systematic parameter shift for individual countries. ### Methodological framework and innovations - Four distinguishing features of the empirical approach: - Flexible dynamic framework separating long-run from short-run debt-growth relationships using an error correction model (ECM), cointegration testing and attention to endogeneity using recent panel time series methods. - Emphasis on potentially differing debt-growth relationships across economies in an a priori unspecified way (cross-section heterogeneity). - Explicit accounting for cross-section dependence through unobserved global shocks and local spillover effects with heterogeneous factor loadings. - Investigation of nonlinearities both across countries and within countries via: - heterogeneous dynamic ECMs for cross-country nonlinearity; and - two within-country approaches: (i) asymmetric dynamic model testing thresholds at 52%, 75% and 90% debt-to-GDP; (ii) static regression models with squared and cubed debt terms informed by order-of-summability, balance and co-summability tests. - ECM representation and recovery of long-run parameters: - ∆y_it = π_0i + π^EC_i y_i,t−1 + π^K_i cap_i,t−1 + π^D_i debt_i,t−1 + ... + "it - β^K_i = − π^K_i / π^EC_i; β^D_i = − π^D_i / π^EC_i. - Half-life (in years) computable as [ log(0.5) / log(1 + π^EC_i) ]. - Treatment of unobservables and cross-section dependence: - Unobservable common factors f_t included in long-run equations; cross-section averages (CA) used to proxy unobservables per Pesaran (2006) and Chudik & Pesaran (2013). - Inclusion of lags of cross-section averages (p = T^{1/3} rule of thumb) and CA of additional covariates to aid identification. - Weak exogeneity testing via ECM / Granger Representation: - For cointegrated pairs, ECM λ coefficients interpreted for causal direction: if λ_1i ≠ 0 then x → y; if λ_2i ≠ 0 then y → x; both non-zero imply joint determination. ### Data construction and sample - Final sample and coverage: - 3,485 observations from N = 105 countries (23 Low-Income, 30 Lower Middle-Income, 23 Upper Middle-Income, 29 High Income). - Average T = 33.2 years (range 21 to 38), covering 1972–2009. - Key variables and construction: - Main variables: GDP, capital stock, total public debt stock — all in logarithms of real US$ and expressed in per capita terms. - Capital stock constructed from gross fixed capital formation by perpetual inventory method with depreciation = 5% per year; cubic spline interpolation used for country series gaps < 3 years (53 observations in 19 country series). - Debt variable: total (external and domestic) general government debt in nominal terms (face value), raw data as share of GDP (Panizza update); converted to real debt stock and per capita debt series. - Cross-section averages proxies: trade openness (imports+exports/GDP, logs) and financial development (bank credit to bank deposits, logs). - Descriptive highlights (from Table A1): - Debt/GDP ratio: mean = 62.177, median = 51.728, sd = 49.618, min = 0.97, max = 470.610. - ∆y_it (GDP per capita growth rate, log differences): mean = 0.016, median = 0.020, sd = 0.048, min = -0.633, max = 0.321. - Stylized descriptive patterns: - Peaks in debt-to-GDP clustered in three years: 1985, 1994 and 2009 (these three years account for over one third of all peaks). - Figure 2 unconditional regression reported as: .011[0.54] − .005[1.12] log(debt/GDP)_max_i (absolute t-ratios in brackets). - Interquartile ranges and growth distributions show considerable heterogeneity across income groups and time. ### Nonlinearity, summability and inference challenges - Reasons to expect heterogeneity: - Country-specific production technology, debt composition differences, reasons for debt accumulation (consumption vs investment), capacity to tolerate high debt depend on country-specific characteristics and past crises. - Challenge of nonlinear transformations with integrated variables: - For xt ∼ I(1), nonlinear transforms like x2t may not have a well-defined order of integration; variance properties can depend on time t, invalidating standard I(d) framework. - Order of Summability, Balancedness and Co-Summability approach: - Order of summability S(δ) used as a summary measure of stochastic persistence for linear or non-linear processes (Berenguer-Rico & Gonzalo approach). - Country-specific OLS on rescaled partial sums yields ˆδ∗i = (ˆβ∗i − 1)/2 as order-of-summability estimate. - Balance test: compares S(δ_y) to S(δ_z) in country-specific regressions; null S(δ_y) = S(δ_z) implemented via difference of estimated β∗ coefficients divided by 2. - Co-summability: residuals from estimated regression should have summability order statistically close to zero (S(δˆeit) ≈ 0). - Deterministic components handled by repeated partial demeaning; panel inference via random-subsample strategy. - Practical implications: - Polynomial specifications of integrated variables require summability/balance/co-summability validation before inference; otherwise cointegration-based analysis may be invalid and produce spurious regressions. ### Empirical results — linear dynamic models (heterogeneous ECMs) - Linear dynamic model averages (Table 1; selected reported values across models [1]–[10]): - LRA (long-run average) across columns: -0.034, 0.000, -0.004, 0.035, 0.016, 0.050, 0.044, 0.027, 0.034, 0.031 (standard errors shown in brackets in source). - ALR (average of country-specific long-run coefficients) examples: -0.011, 0.036, 0.016, 0.046, 0.040, 0.049, 0.029, 0.053. - Error-correction coefficients on y_{i,t−1}: -0.108, -0.339, -0.487, -0.559, -0.655, -0.587, -0.656, -0.608, -0.674, -0.634 (all reported ∗∗∗). - Implied half-lives (years): 6.06, 1.67, 1.04, 0.85, 0.65, 0.78, 0.65, 0.74, 0.62, 0.69. - RMSE across models: 0.040, 0.036, 0.030, 0.025, 0.023, 0.022, 0.019, 0.020, 0.018, 0.020. - CD test (Pesaran 2004) across models: -0.89, 12.57, 18.15, 2.05, 2.04, 1.84, 2.60, 3.74, 3.07, 2.87. - Diagnostics and heterogeneity: - Pooled estimators exhibit strong evidence of nonstationary residuals and cross-section dependence; heterogeneous estimators (MG, CMG, CMG+) with CA reduce cross-section dependence materially. - Country-specific long-run debt coefficients display substantial cross-country variation; some averaged estimates positive, some negative, depending on model augmentation and specification. ### Empirical results — within-country asymmetric dynamic models (thresholds) - Asymmetric dynamic regressions implemented with thresholds at 52%, 75% and 90% debt-to-GDP; countries included only if at least 20% of observations in a regime: - 52% threshold: N = 55 countries. - 75% threshold: N = 45 countries. - 90% threshold: N = 30 countries. - Implementation details: - Debt decomposition into partial sums debt^+ and debt^− constructed relative to exogenously given debt-to-GDP thresholds. - Reporting rule: long-run debt parameters in low and high regimes shown only for countries with at least 20% of observations in a regime (30 countries for 90% threshold; 45 for 75%; 55 for 52%). - Key empirical findings (Section 4.4 summary): - Working hypothesis: shift to ‘high debt’ regime would have a negative, step-change impact on long-run growth. - Result: hypothesis not borne out — no evidence for any systematic change in the relationship between debt and growth when countries shift from a ‘low’ to ‘high’ debt regime; only around one in two countries experience an increase in the debt coefficient. - Average coefficient changes in each of the three cases are statistically insignificant (standard or robust means). ### Empirical results — summability, balancedness and co-summability diagnostics - Summability tests (Table 4): - Levels: variables reject S(0); examples (deterministics: constant): - debt_it: Median = 1.135, Mean = 1.168 (CI: Lower = 1.011, Upper = 1.325). - y_it: Median = 1.109, Mean = 1.096 (CI: Lower = 0.948, Upper = 1.243). - cap_it: Median = 1.286, Mean = 1.357 (CI: Lower = 1.217, Upper = 1.497). - First differences: growth rates broadly S(0); example means: - ∆y_it mean = 0.062; ∆debt_it mean = 0.221; ∆cap_it mean = 0.248. - Balance tests (Table 5): - Panel variants: A — unaugmented; B — augmented with CA; C — augmented with CA plus additional CA of openness variables. - Strong evidence that linear specification represents a balanced model: mean and median balance statistics close to zero in some augmented specifications. - Less evidence for nonlinear specifications: only median estimates and 95% confidence intervals for the model with linear, squared and cubed debt terms contain zero; some marginal rejections noted. - Co-summability tests (Table 6): - Specifications without cross-section averages are not co-summable; residual summability statistics far from zero. - Including cross-section averages brings statistics closer to zero; with additional CA the linear specification is co-summable focusing on the median statistic. - Nonlinear models appear co-summable in the final set but are less convincing given balance test results. - Three overarching diagnostic conclusions: - Strong evidence for significant persistence in the data investigated. - Accounting for cross-section correlation materially alters results; approaches assuming cross-section independence yield very different outcomes. - The linear model augmented with standard and additional cross-section averages provides the most convincing evidence of being both balanced and co-summable; nonlinear polynomial specifications are less convincing and may be misspecified. ### Empirical results — nonlinear static models and country heterogeneity - Static polynomial specifications (Table 7; heterogeneous parameter specifications MG, CMG, CMG+): - Averaged findings: - Linear specifications: negative relationship in MG; negligible relationship in CMG variants when accounting for CA. - Models with linear and squared debt terms: averaged coefficients indicate a concave relationship (consistent with a threshold story) in all three model types on average. - Models including cubed debt term: on average statistically insignificant in MG and CMG models, but not in CMG+. - Diagnostics: - Residuals from all models found to be stationary. - MG models suffer from severe residual cross-section dependence (CD test statistics > 20). - CMG models: CD test statistics around 3.5 to 4.5 (still rejecting cross-section independence). - CMG+ models: dramatically reduced cross-section dependence; argued to be largely free from strong-type cross-section dependence. - Heterogeneity results: - Counts of countries with statistically significant debt coefficients show both positive and negative country-specific slopes once accounting for cross-section correlation. - Polynomial averages mask substantial heterogeneity; country-level shapes (concave/convex/insignificant) vary across the sample. - Key interpretation: - No uniform nonlinear within-country relationship common to all countries; pooled nonlinear findings (e.g., a common 90% threshold) appear to be artifacts of pooled misspecification. ### Weak exogeneity, stationarity and cross-section dependence diagnostics - Weak exogeneity tests (Table 2): - Examples: MG Output (1 lag) GM-t = -2.54, p = 0.01, Avg λ̂_i = -0.928, t-stat = -21.98. - CMG with trend variants show similar evidence of cointegration and directional causality with CA augmentations reducing biases. - Panel stationarity tests (TA-III) and cross-section dependence (TA-IV): - Fisher tests: debt pc Fisher = 388.920, p = 0.00; cap pc Fisher = 703.690, p = 0.00 (deterministics: constant). - Pesaran CIPS: GDP pc Z-tbar = 3.86, p = 1.00; debt pc = 4.07, p = 1.00 (lags = 0, deterministics: constant). - Cross-section dependence in levels strong: CD statistics for y_it = 157.62, debt_it = 148.36, cap_it = 154.54 (p = 0.00). - First differences show smaller average correlations but CD tests remain significant. - CA/CCE augmentation reduces residual cross-section dependence substantially. ### Conclusions and policy-relevant implications - Main contributions: - Investigated long-run public debt–growth relationship with dynamic panel time series methods that account for cointegration, heterogeneity and cross-section dependence. - First panel study on debt and growth to comprehensively address parameter heterogeneity and unobserved common factors in this way; employed novel estimators and summability/co-summability diagnostics. - Key empirical conclusions: - Some evidence for systematic differences in the debt-growth relationship across countries, but no evidence for systematic within-country nonlinearities common to all countries. - Long-run debt coefficients tended to be lower in countries with higher average debt burden, although the average long-run debt coefficient across countries was positive. - Empirical tests favored linear specification over polynomial specifications; piecewise linear specifications with pre-specified thresholds (including canonical 90%) produced changes in debt coefficient at the threshold that were just as likely to be positive as negative. - The commonly cited 90% debt threshold is likely an outcome of empirical misspecification (pooled homogeneous-parameter models) rather than evidence of a universal within-country tipping point. - Policy implication: - The shape and form of the debt-growth relationship differs across countries; policy rules or thresholds appropriate for one country may be seriously misguided in another. *Source: _wp13248 (IMF Working Paper) — content units provided in supplied PDF excerpts.* ### 1. Introduction....................................................................................... 3 ### 1. Introduction ### Policy context and motivation - Widespread belief in a common debt threshold informing policy debates in the United States (including the debt ceiling debate and the government shutdown of October 2013) and the United Kingdom (Chancellor George Osborne’s speech, Manchester, September 30, 2013). - The view of a common “dangerous” debt threshold was strongly influenced by Reinhart & Rogoff, who suggested a debt-to-GDP threshold of around 90% beyond which economic growth is seriously affected (Reinhart & Rogoff, 2009, 2010a,b, 2011; Reinhart, Reinhart & Rogoff, 2012). - The study distinguishes itself from the primarily descriptive Reinhart & Rogoff (2010b) analysis and from related empirical studies (e.g., Kumar & Woo, 2010; Cecchetti, Mohanty & Zampolli, 2011; Checherita-Westphal & Rother, 2012). ### Research objective and key questions - Investigate nonlinearity in the debt-growth relationship with empirical strategies that distinguish: - a nonlinearity across countries (cross-country heterogeneity), from - a within-country nonlinearity (country-specific threshold or tipping point). - Assess whether a common threshold across countries exists, and whether country-specific tipping points can be identified to guide macroeconomic and fiscal policy. ### Data and empirical scope (overview) - Sample: total public debt data from 105 developing, emerging and developed economies. - Time horizon: 1972 to 2009. - Empirical focus: long-run (levels) relationship within a neoclassical growth model framework, while accounting for short-run effects and unobserved global and local shocks. ### Four distinguishing features of the empirical approach - Flexible dynamic framework separating long-run from short-run debt-growth relationships (standard error correction model, ECM), with cointegration testing and attention to endogeneity using recent panel time series methods. - Emphasis on potentially differing debt-growth relationships across economies in an a priori unspecified way (cross-section heterogeneity), motivated by theory, specification uncertainty, and data shortcomings. - Explicitly account for cross-section dependence through unobserved global shocks and local spillover effects that affect economies differently. - Investigate nonlinearities both across countries and within countries using: - heterogeneous dynamic ECMs to study cross-country nonlinearity and analyze patterns of short-run and long-run coefficients; and - two within-country approaches: (i) an asymmetric dynamic model testing a range of thresholds, including the 90% debt-to-GDP ratio, as potential tipping points; (ii) static regression models with squared and cubed debt terms, informed by tests for variable summability, balance and co-summability. ### Theoretical considerations and prior literature - Theoretical foundations for a negative and/or nonlinear relationship between debt and growth are tenuous; arguments include: - Negative long-run relationship under some models (Elmendorf & Mankiw, 1999) and more pronounced effects if high debt leads to uncertainty or expectations of future financial repression (Cochrane, 2011). - Alternative models where negative relationship disappears under wage rigidities and unemployment (Greiner, 2011). - Debt overhang motivation for nonlinearity in developing countries (Krugman, 1988; Sachs, 1989), but extension to advanced economies is difficult. - Potential tipping points of fiscal sustainability (Ghosh et al., 2013; Greenlaw et al., 2013) without established growth-framework models incorporating such tipping points. - Literature heterogeneity noted across four dimensions: (a) data used and country coverage; (b) modelling of nonlinearity/thresholds; (c) time horizon of results (short-run vs long-run, static vs dynamic, averaged vs annual data); (d) identification strategies (IV/2SLS, Arellano & Bond (1991) estimators). Existing studies typically address at most one or two of these features. ### Methodological innovations and alignment with prior work - Builds on and extends approaches arguing for country-specific thresholds and heterogeneity (Kraay & Nehru, 2006; Reinhart, Rogoff & Savastano, 2003; Reinhart & Rogoff, 2010c; Kourtellos, Stengos & Tan, 2014). - Adopts panel time series methods to account for cross-country correlations (Pedroni, 2007; Eberhardt, Helmers & Strauss, 2013; Eberhardt & Teal, 2014; Pesaran, 2006; Kapetanios, Pesaran & Yamagata, 2011; Chudik & Pesaran, 2013). - Transfers asymmetric cointegration framework (Shin, Yu & Greenwood-Nimmo, 2013) and summability/balance/co-summability analyses (Berenguer-Rico & Gonzalo, 2013a,b) from single time series to the panel setting. ### Preview of findings (as stated in the Introduction) - Long-run debt coefficients differ across countries. - Tentative evidence that countries with higher average debt-to-GDP ratios are more likely to see a negative effect on their long-run growth performance. - No evidence found for a specific debt threshold common to all countries that triggers a systematic parameter shift for individual countries. ### Structure of the paper - Section 2: how economic theory and data realities inform the empirical analysis. - Section 3: data description and econometric methods overview. - Section 4: empirical results and detailed analysis of heterogeneity and nonlinearity across and within countries. - Section 5: conclusion. *Source: _wp13248 - 1. Introduction (IMF working paper), pages provided in supplied content.* ### 2.1 Commonality and Heterogeneity ### 2.1 Commonality and Heterogeneity ### Descriptive evidence of commonality and heterogeneity - Histogram of years in which countries reach their debt-to-GDP ratio peak: peaks clustered in three years — 1985, 1994 and 2009 — with over one-third of countries peaking in those three years despite a data span of forty years. - Figure linking debt-to-GDP peaks to deviation of per capita GDP growth in ‘peak years’ (defined ad hoc as running from two years prior to two years after the debt-to-GDP maximum) from that of the full time horizon (excluding the five peak years): - Negative correlation suggested between maximum debt level and relative growth performance between peak debt and other years; this negative relationship is not statistically significant (linear regression result reported in the figure footnote). - Considerable cross-country heterogeneity: among countries with debt-to-GDP peaked in 1994 (blue squares), one country experienced growth about 2% above its growth rate in all other years, while another experienced a ‘peak years’ average growth rate 4% lower. - Dashed vertical line at debt-to-GDP ratio of 90%: a considerable number of countries had better growth performance in their peak debt years than at any other point since 1972, even above 90% debt-to-GDP. - Note: for peaks at the start (end) of the sample averages are limited to the peak year and the two years after (before). - Footnote: green diamonds indicating 2009 show that all countries in which debt peaked in that year (all High-Income countries bar GRD and LCA) had worse growth performance in 2007-2009 than in all other years (average growth rates, respectively). - Interquartile ranges (IQR) analysis for debt (grey shading, right axis — three debt peak years highlighted in black) and growth (black whiskers, left axis) across all countries and by income group (high, middle, low): - Except for 2009 in the High-Income Country sample, the three highlighted debt-peak years are not remarkable in terms of debt or growth distribution for many country groups. - Growth rate distribution exhibits an inverted U-shape over time in the full sample and all three sub-samples. - Distribution of debt shows no clearly discernable long-run pattern, apart from an initial decline in the early 1970s. - Descriptive conclusion: timing of debt peaks reveals some common effects across countries but substantial heterogeneity in whether those years are economically remarkable for individual countries. ### Cross-section dependence and identification problems in econometrics - Conventional panel empirical approaches typically assume cross-section independence (regression residuals show no systematic correlation across countries); this assumption is questionable given global shocks and spillovers. - Static model with a single covariate x and a single unobserved common factor f with heterogeneous factor loadings λ_i: - y_{i t} = β_{i} x_{i t} + u_{i t} - u_{i t} = λ_{i} f_{t} + ψ_{i} + "_{i t} (1) - x_{i t} = %_{i} f_{t} + π_{i} g_{t} + φ_{i} + e_{i t} (2) - Cross-sectional dependence arises because f_{t} (and possibly g_{t}) affect both y and x. - Substituting f_{t} from (2) into (1) yields: - y_{i t} = Ä β_{i} + λ_{i} %_{i}^{−1} ä︸︷︷︸ ζ_{i} x_{i t} + ψ_{i} − λ_{i} %_{i}^{−1} φ_{i}︸︷︷︸ η_{i} + "_{i t} − λ_{i} %_{i}^{−1} π_{i} g_{t} − λ_{i} %_{i}^{−1} e_{i t}︸︷︷︸ ς_{i t} (3) - = ζ_{i} x_{i t} + η_{i} + ς_{i t} - Note: in principle ζ_{i} 6= β_{i}. - If unobserved factors are ‘weak’ (local spillovers), bias may be limited; if ‘strong’ (affecting all countries), β is not identified. - Instrumentation complication: an observable z potentially usable as instrument may also be driven by f_{t}: - z_{i t} = ρ_{i} f_{t} + φ_{i} x_{i t} + θ_{i} + ε_{i t} (4) - Some observable z is correlated with x and thus potentially informative, but via f_{t} is correlated with unobservables in (1) and therefore invalid. - Consequences: - Principal component analysis intuition: a small number of unobserved common factors can drive macroeconomic variables. - Lack of robustness of IV results in cross-country growth literature may arise from neglected common factors. - Pooled estimators can produce heterogeneity bias (Pesaran & Smith, 1995); pooling introduces data dependencies in residuals if series are integrated, risking spurious regression (Kao, 1999; Phillips & Sul, 2003). - Existing research shows different results when moving away from full-sample homogeneous-parameter models to sub-sample analyses by geography, institutions, or income. ### Reasons to expect heterogeneity in the debt-growth relationship - Country-specific production technology differences (new growth literature; Temple, 1999) suggest differing debt-growth relationships. - Vulnerability to public debt depends on debt composition (Inter-American Development Bank, 2006); data mixes (general vs central government, denominations, terms) compromise comparability (Panizza & Presbitero, 2013). - The economic effect of public debt may depend on why debt was accumulated and whether it financed consumption or investment (and which activities). - Capacity to tolerate high debt depends on country-specific characteristics, past crises, macro and institutional frameworks (Reinhart, Rogoff & Savastano, 2003; Kraay & Nehru, 2006; Manasse & Roubini, 2009). - Heterogeneity is a natural extension to allow more modelling flexibility and enables empirical testing of the assumption via residual diagnostics. ### Preview of empirical strategy addressing heterogeneity and nonlinearity - Many empirical studies use squared debt terms or spline specifications within pooled models to capture heterogeneous impacts across debt levels (Cordella, Ricci & Ruiz-Arranz, 2010; Pattillo, Poirson & Ricci, 2011; Checherita-Westphal & Rother, 2012), but pooled common-parameter assumptions may mask cross-country heterogeneity. - Potential for apparent nonlinearity to arise from slope heterogeneity: Haque, Pesaran & Sharma (1999) warn that linearity may be rejected due to slope heterogeneity rather than true nonlinearity. - Descriptive fractional polynomial regressions (within-transformed per capita GDP vs debt-to-GDP in logs): - Pooled regression suggests a nonlinearity with a ‘threshold’ of 4.5 log points (equivalent to 90% debt-to-GDP), higher debt associated with lower per capita GDP (no causal claim). - Country-specific fractional polynomial regressions show the pooled nonlinearity is far from obvious when allowing country-specific relationships. - Data trimming notes: full sample fractional polynomial regressions exclude observations for debt-to-GDP ratio below 2 log points (<7.3% debt-to-GDP ratio), amounting to 85 observations (primarily ARE, CHN and LUX). Scatter plots and country-specific regressions further exclude observations where within-transformed income change year-on-year exceeds 40%, amounting to 281 observations (primarily BWA, IRL, KOR, THA and other fast-growing Middle-Income Countries). - Strategy for empirical analysis: - Begin with nonlinearity across countries using linear regression models that account for observed and unobserved heterogeneity. - Identification of long-run and short-run coefficients via the Pesaran (2006) common correlated effects (CCE) estimator, adjusting for dynamic setup per Chudik & Pesaran (2013). - Analyze relationship between estimated long-run coefficients and country-specific averages of debt levels, debt-to-GDP ratios, and peak debt-to-GDP ratios. - For country-level nonlinearities: apply methods recognizing that polynomial transformations of integrated variables raise issues for integration/cointegration analysis; use order of summability and co-summability tests (Berenguer-Rico & Gonzalo, 2013a,b) for pre-estimation validation in the presence of cross-section dependence. - Two approaches to country-level nonlinearity: - Nonlinear dynamic model of Shin, Yu & Greenwood-Nimmo (2013) with exogenously given thresholds at 52%, 75% and 90% debt-to-GDP to investigate heterogeneous growth regimes while accounting for cross-section dependence. - Polynomial terms of the debt stock in a static regression model, accounting for cross-section dependence, informed by (co-)summability analysis. *Source: 2.1 Commonality and Heterogeneity (excerpt).* ### 3.1 Data ### 3.1 Data ### Data construction and sources - Main variables: GDP, capital stock, total public debt stock — all in logarithms of real US$ and expressed in per capita terms, imposing constant returns to scale. - Capital stock constructed from gross fixed capital formation using the standard perpetual inventory method and assuming a common and constant 5% depreciation rate. - Debt variable: total (external and domestic) general government debt in nominal terms (face value), in the raw data expressed as a share of GDP; data for debt stock from an update to Panizza (2008). - Primary data sources: - World Bank World Development Indicators (WDI) database (GDP, capital, other series). - Update to Panizza (2008) for debt stock. - NYU Global Development Network Growth Database (trade openness: imports plus exports as a share of GDP). - Thorsten Beck and Asli Demirguc-Kunt’s Financial Structure Database, updated in 2010 (financial development: ratio of bank credit to bank deposits). - All variables are expressed in per capita terms (including debt stock). - A Data Appendix provides more details on construction and descriptive statistics; sample composition details are in a Technical Appendix. - Note: If trade openness and financial development are included in empirical analysis the resulting estimates are for the data ending in 2008. ### Empirical setup: key modeling assumptions - Specification starts from a static neoclassical production function augmented with a debt stock term; capital and debt coefficients βK_i and βD_i are allowed to differ across countries (heterogeneity). - Country-specific Total Factor Productivity (TFP) levels α_i and a set of common factors f_t with country-specific factor loadings λ_i are included to account for unobservable time-varying TFP. - Common factors can be ‘strong’ (global shocks) or ‘weak’ (local spillovers); unobservables may be nonstationary, implying integrated observable and unobservable processes. - Standard instrumentation in a pooled framework is considered invalid due to omnipresent unobserved factors and heterogeneous equilibrium relationships across countries. --- ### 3.2 Empirical Specification: Linear Dynamic Model ### Model representation - Baseline augmented production function (in notation of the source): - y_it = α_i + β^K_i cap_it + β^D_i debt_it + u_it, with u_it = λ′_i f_t + "it - Error Correction Model (ECM) representation provided to distinguish short-run from long-run behavior and to test for cointegration. - ECM form (notation preserved): - ∆y_it = α_i + ρ_i [ y_i,t−1 − β^K_i cap_i,t−1 − β^D_i debt_i,t−1 − λ′_i f_t−1 ] + γ^K_i ∆cap_it + γ^D_i ∆debt_it + γ^F′_i ∆f_t + "it - Reparameterized as: - ∆y_it = π_0i + π^EC_i y_i,t−1 + π^K_i cap_i,t−1 + π^D_i debt_i,t−1 + π^F′_i f_t−1 + π^k_i ∆cap_it + π^d_i ∆debt_it + π^f′_i ∆f_t + "it - Long-run parameters recoverable from level coefficients: - β^K_i = − π^K_i / π^EC_i - β^D_i = − π^D_i / π^EC_i - π^EC_i indicates speed of return to long-run equilibrium; half-life (in our data: in years) computable as [ log(0.5) / log(1 + π^EC_i) ]. ### Advantages of ECM approach (as stated) - (i) Distinguish short-run from long-run behaviour. - (ii) Investigate the error correction term and deduce the speed of adjustment to long-run equilibrium. - (iii) Test for cointegration via statistical significance of the error correction term. ### Treatment of unobservables and endogeneity - Unobservable common factors f_t included in the long-run equation: investigating cointegration between output, capital, debt and TFP. - Use of cross-section averages to replace unobservables and omitted elements of the cointegration relationship, following Banerjee & Carrion-i-Silvestre (2011). - Augmented ECM with cross-section averages (notation preserved) to address cross-section dependence. - Chudik & Pesaran (2013) concerns: small sample bias for moderate T; relax strict exogeneity and allow feedback between debt, capital and output. - Chudik & Pesaran (2013) recommendations for weakly exogenous regressors: - Include lags of cross-section averages in the ECM (a sufficient number of lagged cross-section averages; p = T^{1/3} can be employed as a rule of thumb). - Include cross-section averages of additional covariates (∆z_t−p) to help identify unobserved common factors. - Estimation approach: all models estimated by OLS; modelling features such as nonstationarity, cross-section correlation, heterogeneity and nonlinearity/asymmetry are captured by specification augmentation and additional regression terms. - Reliance on simulated critical values for inferential and diagnostic statistics. --- ### 3.3 Empirical Specification: Weak Exogeneity Testing ### Endogeneity concerns and identification - Two forms of endogeneity emphasized: - (1) Common factors driving both inputs and output (addressed via factor augmentation). - (2) Reverse causality: possibility that the empirical model (production function with debt) may instead estimate a disguised demand equation for debt or investment. - Adjustment of factor model to examine direction of causation (notation preserved): - y_it = β_i x_it + u_it, u_it = λ_i f_t + ψ_i + "it - x_it = %_i f_t + π_i g_t + ψ_i"it + φ_i + e_it - Standard instrumentation (z) considered problematic in macro panels; weak exogeneity testing in cointegrated nonstationary panels used as an alternative identification strategy. ### Weak exogeneity test via ECM / Granger Representation - For a cointegrated pair x and y, ECM system (notation preserved): - ∆y_it = c_1i + λ_1i ê_i,t−1 + Σ ψ_11ij ∆y_i,t−j + Σ ψ_12ij ∆x_i,t−j + "1i t - ∆x_it = c_2i + λ_2i ê_i,t−1 + Σ ψ_21ij ∆y_i,t−j + Σ ψ_22ij ∆x_i,t−j + "2i t - ê_i,t−1 = y − β̂_i x − d̂ constructed from estimated cointegrating relationship. - Interpretation: - If λ_1i ≠ 0 then x has a causal impact on y. - If λ_2i ≠ 0 then causal impact is reversed (y → x). - If both non-zero they determine each other jointly. --- ### 3.4 Empirical Specification: Asymmetric Dynamic Model ### Asymmetric long-run regression and decomposition - Asymmetric long-run model (notation preserved): - y_it = α_i + β^K_i cap_it + β^{D+}_i debt^+_it + β^{D−}_i debt^−_it + λ′_i f_t + "it - Debt decomposition: - debt_it = debt_i0 + debt^+_it + debt^−_it, with partial sums: - debt^+_it = Σ_{j=1}^t ∆debt^+_ij = Σ_{j=1}^t max(∆debt_ij, 0) - debt^−_it = Σ_{j=1}^t ∆debt^−_ij = Σ_{j=1}^t min(∆debt_ij, 0) - In practice partial sums are constructed for debt stock below and above exogenously determined debt-to-GDP ratio thresholds: 52% (sample median), 75% and 90% (the ‘canonical’ 90%). - Assignment to regimes determined by the debt-to-GDP ratio, while partial sums use the per capita debt stock variable (to permit comparison with literature using debt-to-GDP ratio). ### ECM of the asymmetric model (notation preserved) - ∆y_it = π_0i + π^EC_i y_i,t−1 + π^K_i cap_i,t−1 + π^{D+}_i debt^+_i,t−1 + π^{D−}_i debt^−_i,t−1 + π^F′_i f_t−1 + π^k_i ∆cap_it + π^{d+}_i ∆debt^+_it + π^{d−}_i ∆debt^−_it + π^f_i ∆f_t + "it ### Implementation issues and precautions - Dynamic asymmetry can be included in long-run (levels) terms, short-run (first differences), or both. - Cross-country heterogeneity allowed in all long-run and short-run parameters; unobserved time-varying heterogeneity addressed via augmentation with cross-section averages and further lags per Chudik & Pesaran (2013). - Problem when threshold is high: if only a very small number/share of observations for a country are above the threshold estimated coefficients may be very imprecise. - Reporting rule to guard against imprecise estimates: present estimated long-run debt parameters in low and high debt regimes only for countries where at least 20% of all time series observations are in one regime. - For the 90% debt/GDP threshold this amounts to a total of 30 countries. - For the 75% threshold this amounts to 45 countries. - For the 52% threshold this amounts to 55 countries. *Source: _wp13248 - 3.1 Data (PDF chapter/section).* ### 3.5 Empirical Specification: Order of Summability, Balancedness and Co-Summability ### 3.5 Empirical Specification: Order of Summability, Balancedness and Co-Summability ### Challenge of Nonlinear Transformations with Integrated Variables - Standard notion of integration is linear; for a nonstationary covariate xt ∼ I(1) and non-linear f(·), the order of integration of f(xt) (and thus yt = f(xt, θ) + ut) may not be well defined. - Illustration: for xt = xt−1 + εt with εt ∼ i.i.d.(0, σ2ε), V[xt − xt−1] = σ2ε ⇒ xt ∼ I(1). But for f(xt) = θ x2t: - V[x2t − x2t−1] = E[ε4t] + 4(t − 1)σ4ε − σ4ε ⇒ x2t ∼ I(?) - Finite variance characteristic is violated (function of time t), so order of integration for x2t cannot be stated within standard I(d) framework. - Consequence: cointegration-based empirical analysis of non-linear relationships with integrated inputs can be fundamentally problematic. ### Order of Summability Method (Berenguer-Rico & Gonzalo approach) - Introduces the order of summability S(δ) as a summary measure of stochastic persistence for linear or non-linear processes, not relying on linear structures. - Country-specific OLS estimation: - Estimate Y∗ik = β∗i log k + U∗ik for k = 1, . . . , T - With Y∗ik = Yik − Yi1, U∗ik = Uik − Ui1 and Yik = log(∑t=1k( yit − mt))2, where mt is the country-specific partial mean of yit. - For the ‘constant and linear trend’ case mt = (1/t)∑t j=1 yij − (2/t)∑t j=1( yij − (1/j)∑j`=1 yi` ). - OLS estimator: ˆβ∗i = ∑T k=1 Y∗ik log k / ∑T k=1 log2 k - Order of summability estimate: ˆδ∗i = (ˆβ∗i − 1)/2 - Interpretation: investigates the rate of convergence of a rescaled sum constructed from the variable series yit. - Inference: - Single time series: confidence intervals via estimation in subsamples. - Panel: because no natural cross-section ordering, take random draws of p Ncountries (full time series T within each country) and compute mean and median summability statistics across subsamples for inference. ### Relation between Summability and Integration - If xt is integrated of order d, I(d) with d ≥ 0, then xt is also summable of order d, S(d). - The reverse need not hold when non-linear transformations are involved, motivating summability analysis for polynomial specifications. ### Balance (Balancedness) Testing - Balance tests whether both sides of the empirical equation have the same order of summability: S(δy) = S(δz) for z = f(xt, θ) = θ f(xt). - Equivalent to testing null βni ≡ (βy i − βzi) = 0 in country-specific regression: - Y∗yik − Y∗zik = βni log k + (Uyik − Uzik) - Y∗yik defined as for LHS variable; Y∗zik is the partially demeaned sum of all RHS processes Yzik = log(∑t=1k(zit − mt))2. - Implementation notes: - In practice, sum all elements of z(RHS variables), partially demean, estimate orders of summability for y and z, compute their difference and divide by 2. - Inference via subsamples in single time series; in panel use random-subsample strategy to build confidence bands. - Balancedness is necessary but not sufficient for valid empirical specification; under the null the confidence interval includes zero. ### Co-Summability (Strong Co-Summability) - After estimating a balanced country-specific regression yit = ˆθ g(xit) + ˆeit, strong co-summability implies the order of summability of residuals S(δˆeit) is statistically close to zero. - Practical detail: residual series ˆeit sum to zero by least squares, so in practice the intercept is not subtracted when estimating summability of residuals. - Inference for co-summability follows same subsample principles as summability and balance tests. ### Practical Implementation and Deterministic Components - Deterministic components (intercept and trend) are handled by repeated partial demeaning of variable series as suggested in Berenguer-Rico & Gonzalo (2013b). - The approach assumes non-linearity in variables but linearity in parameters: - yt = g(xt, θ) + εt = θ g(xt) + εt - Econometric theory is being extended to nonlinearity in parameters, but restriction is consistent with common empirical implementations (debt thresholds or polynomial debt terms). ### Panel Extensions and Cross-Section Dependence - Panel versions of balance and co-summability tests include cross-section averages (CA) of all variables following Pesaran (2006) and Chudik & Pesaran (2013). - Two CA augmentation variants: - (i) include CA of all model variables, - (ii) include CA of all model variables plus CA of ‘other covariates’ (similar to dynamic heterogeneous panel estimations). - Motivation: account for cross-section dependence due to globalization, trade, and shared shocks. *Italic source attribution: IMF Working Paper section "3.5 Empirical Specification: Order of Summability, Balancedness and Co-Summability" from the provided PDF content.* ### 4.4 Results: Asymmetric Dynamic Models ### 4.4 Results: Asymmetric Dynamic Models ### Within-country threshold tests (asymmetric dynamic regressions) - Estimation accounts for unobserved common factors by inclusion of cross-section averages of all covariates and one further lag of the cross-section averages. - Three subsamples correspond to thresholds of 52%, 75% and 90% for the debt-to-GDP ratio; only countries with at least 20% of observations in one regime are included, amounting to 55, 45 and 30 countries for the three thresholds, respectively. - X-axis in plots: average debt burden over entire time horizon, expressed as average debt-to-GDP ratio (in logs) in left column and total debt stock per worker (in logs) in right column. - Y-axis in plots: estimated long-run debt coefficient, allowed to differ across regimes (and countries). - Working hypothesis: a shift to the ‘high debt’ regime would have a negative, step-change impact on long-run growth (expect most arrows to indicate a negative relationship). - Empirical result: hypothesis not borne out — no evidence for any systematic change in the relationship between debt and growth when countries shift from a ‘low’ to ‘high’ debt regime; only around one in two countries experience an increase in the debt coefficient. - Average coefficient changes in each of the three cases are statistically insignificant (standard or robust means). - Note: the simple count of countries experiencing increase/decrease does not take statistical significance into account. ### 4.5 Results: Summability, Balancedness and Co-Summability ### Summability tests (Table 4) - Models estimated with a constant term (left panel) and with constant and trend terms (right panel); the latter is the more natural choice given trending data. - All investigated variables reject summability of order 0, S(0), justifying concerns about time series properties and nonstationarity. - Summability testing for growth rates (lower panel): growth rates of per capita GDP and debt stock are broadly S(0), while capital stock growth rate appears to reject S(0). ### Balance tests (Table 5) - Panels: A — unaugmented ‘standard’; B — augmented in the common correlated effect fashion; C — further add cross-section averages from two ‘openness’ variables. - For balance, balance statistic should be close to zero; estimates and 95% confidence bands underlined where statistically rejected. - Specifications with constant and with constant plus trend provided; trend specification a priori more suitable. - Findings: - Strong evidence that linear specification represents a balanced model: mean and median balance statistics close to zero. - Less evidence for two nonlinear specifications: only median estimates and 95% confidence intervals for model with linear, squared and cubed debt terms contain zero. - Rejection of null of equal order of summability is marginal in specification with linear and squared debt terms in Panels B and C. ### Co-summability tests (Table 6) - Three blocks: standard panel co-summability (Panel A), includes cross-section averages of all model variables (Panel B), includes additionally cross-section averages of ‘other covariates’ (Panel C). - Specifications without cross-section averages are not co-summable; summability statistics for model residuals are some distance from zero. - Specifications with cross-section averages move closer to zero but still reject co-summability in the linear model. - Further including additional cross-section averages moves statistics even closer to zero; the linear specification is co-summable if focusing on the median statistic. - Nonlinear models in this final set also appear co-summable, but less convincing evidence that they are balanced — therefore co-summability for these models is uncertain. ### Three overarching conclusions from summability/balance/co-summability analysis - There is strong evidence for significant persistence in the data investigated, which may seriously impact estimation and inference. - Approaches assuming cross-section independence yield very different results from those that relax this assumption; accounting for cross-section correlation is important in macro panel datasets. - The only empirical model with fairly convincing evidence of being both balanced and co-summable is the linear model augmented with standard and additional cross-section averages. Nonlinear models are less convincing; some fail balance tests only marginally. - Implication: adoption of linear and squared debt terms in a flexible specification to model debt thresholds may represent a seriously misspecified empirical model and could lead to spurious regression results. ### 4.6 Results: Nonlinear Static Models ### Model specifications and diagnostics (Table 7) - Presented averaged debt coefficients from static production function models; all models are heterogeneous parameter specifications. - Investigated pooled model specifications (Pooled OLS, Fixed Effects, CCE Pooled) and found strong evidence of nonstationary residuals in these pooled models (potentially spurious). - Nine models correspond to same nine tested for balance and co-summability: MG, CMG, and CMG+ (CMG plus cross-section averages of two ‘openness’ variables). - Only strong evidence for the linear model in column[7] to represent a balanced and co-summable specification. - Average estimates: - Linear specifications: negative relationship in MG; no substantive relationship in two CMG models. - Models with linear and squared debt terms: averaged coefficients indicate a concave relationship in all three model types. - Nonlinear model including cubed debt term: on average statistically insignificant in MG and CMG models, but not in CMG+. - Residual diagnostics: - Residual series from all models found to be stationary. - MG models suffer from very serious residual cross-section dependence (CD test statistics in excess of 20). - CMG models: CD test statistics around 3.5 to 4.5 (thus still rejecting cross-section independence). - CMG+ models: dramatically reduced cross-section dependence, could be argued to be largely free from strong-type cross-section dependence. - Reported counts of countries with statistically significant debt coefficients show heterogeneity: in linear models similar numbers of positive and negative slope coefficients once accounting for cross-section correlation. - Models with linear and squared debt show more evidence for concave relations (consistent with Reinhart & Rogoff (2010b) debt threshold story), but this is not uniform across countries. - Models with three debt terms: averaged coefficients mask substantial heterogeneity across countries. - Overall: cannot provide support for notion that countries possess similar or identical nonlinearities in the debt-growth relationship over time once common-parameter assumptions are relaxed. ### 5 Concluding Remarks ### Main contributions of the paper - Investigated long-run relationship between public debt and long-run growth using dynamic empirical model with time series considerations to establish presence of a long-run equilibrium and address endogeneity. - Adopted empirical specifications allowing for heterogeneity in long-run relationships across countries and accounted for unobservable determinants via a flexible common factor model framework — first panel study on debt and growth to address parameter heterogeneity and cross-section dependence. - Employed novel empirical estimators and testing procedures to examine potential nonlinearity in the debt-growth relationship, distinguishing nonlinearity across countries from within-country nonlinearity. ### Key empirical findings - Some evidence for systematic differences in debt-growth relationship across countries, but no evidence for systematic within-country nonlinearities in the debt-growth relationship for all countries in sample. - Observed that long-run debt coefficients tended to be lower in countries with higher average debt burden, although average long-run debt coefficient across countries was positive. - Empirical tests favored a linear specification over polynomial specifications; piecewise linear specifications with pre-specified thresholds produced changes in debt coefficient at the threshold that were just as likely to be positive as negative. - Conclusion: the shape and form of the debt-growth relationship differs across countries; policies appropriate for one country may be seriously misguided in another. - The commonly found 90% debt threshold is likely an outcome of empirical misspecification — a pooled instead of heterogeneous model — and misinterpretation assuming pooled model estimates imply a common within-country nonlinearity. *Source: _wp13248 - 4.4 Results: Asymmetric Dynamic Models* ### chapter 8, pp. 555–677. ### _wp13248 - chapter 8, pp. 555–677 ### Data and sample - Final sample: 3,485 observations from N = 105 countries (23 Low-Income, 30 Lower Middle-Income, 23 Upper Middle-Income, 29 High Income), average T = 33.2 years (range 21 to 38), covering 1972–2009. - Debt series: total debt (domestic + external) in face value terms as a percentage of GDP (Panizza update); converted to real debt stock and per capita debt series. - Key data construction choices: - Real GDP (2000 US$), population, gross fixed capital formation (investment/GDP) from WDI. - Capital stock constructed by perpetual inventory method with depreciation = 5% per year; cubic spline interpolation used for country series gaps < 3 years (53 observations in 19 country series). - Cross-section averages (CA) used for proxies: trade openness (exports+imports/GDP, Bill Easterly) and financial development (bank credit to bank deposits, Beck & Demirguc-Kunt). Both raw variables in logs. - Descriptive highlights (from Table A1): - Debt/GDP ratio: mean = 62.177, median = 51.728, sd = 49.618, min = 0.97, max = 470.610. - ∆y_it (GDP per capita growth rate, logs differences): mean = 0.016, median = 0.020, sd = 0.048, min = -0.633, max = 0.321. ### Stylized patterns and figures - Distribution of peak debt/GDP years: three years 1985, 1994 and 2009 account for over one third of all debt/GDP peaks in the 105-country sample. - Figure 2 unconditional result (exact reported regression): - .011[0.54] − .005[1.12] log(debt/GDP)_max_i (absolute t-ratios in brackets). - Interquartile ranges and coverage: data coverage varies across income groups; High Income coverage rises from ~70% (early 1970s) to >90% (later), Middle Income from ~60% (1970s) to >90% (1990s onward), Low Income coverage improves after 1982 to ~90% mid-1980s onward, with some drop after 2007. ### Methodology and model families - Main empirical framework: heterogeneous error-correction models (ECM) estimating the first difference of log real GDP per worker (∆y_it) with lagged levels to capture long-run relationships and country heterogeneity. - Estimators and models reported (Table 1 and subsequent tables): - 2FE (two-way fixed effects), CCEP, MG, CMG variants (CMG = Chudik & Pesaran / Pesaran multifactor heterogeneous panel estimators). - CMG implementations augment regressions with cross-section averages (CA) of additional lags and/or other variables (open = trade/GDP, findev = financial development), and with country-specific trends in some specifications. - Asymmetric dynamic models adapted from Shin, Yu & Greenwood-Nimmo (2013) allow LR and SR asymmetry and regime splits at debt/GDP thresholds (52%, 75%, 90%). - Co-summability and order-of-summability diagnostics computed for residuals and variables (Tables 4–6). ### Empirical findings — linear dynamic model averages (Table 1) - Reported robust mean long-run average (LRA) and related statistics across models [columns correspond to models 1–10 as in Table 1]: - LRA (columns [1]–[10]): -0.034, 0.000, -0.004, 0.035, 0.016, 0.050, 0.044, 0.027, 0.034, 0.031 - Standard errors for pooled model [1] shown as [0.023], for [2] [0.018], for [3]–[10] [0.011]–[0.017] as indicated. - Significance markers: e.g., column[4] LRA = 0.035 [0.013] ∗∗∗; column[6] LRA = 0.050 [0.013] ∗∗∗. - ALR (average of country-specific long-run coefficients) (selected values): columns show -0.011, 0.036, 0.016, 0.046, 0.040, 0.049, 0.029, 0.053 (with standard errors reported in brackets). - SR (short-run coefficients) (selected): 0.000, 0.001, -0.015, 0.007, -0.001, 0.009, 0.009, 0.010, 0.003, 0.004 (with standard errors). - ALR threshold (cross-country implied threshold where debt impact becomes negative): reported as 96.3%, 90.2%, 73.4%, 22.5%, 0.3%, 126.7%, 28.4%, 18.5% across models [1]–[9]. - Error-correction (EC) coefficient on y_{i,t−1} (columns [1]–[10]): -0.108, -0.339, -0.487, -0.559, -0.655, -0.587, -0.656, -0.608, -0.674, -0.634 (all with [0.014]–[0.038] standard errors and ∗∗∗ significance). - Implied half-life (years) corresponding to EC coefficients: 6.06, 1.67, 1.04, 0.85, 0.65, 0.78, 0.65, 0.74, 0.62, 0.69. - Model fit and diagnostics (selected): - RMSE (columns [1]–[10]): 0.040, 0.036, 0.030, 0.025, 0.023, 0.022, 0.019, 0.020, 0.018, 0.020. - CD test (Pesaran 2004) values across models: -0.89, 12.57, 18.15, 2.05, 2.04, 1.84, 2.60, 3.74, 3.07, 2.87. - Observations for each model: typically 3,485 (some models slightly fewer due to augmentations). ### Heterogeneity, asymmetry and threshold results (Table 3 and related) - Asymmetric dynamic models splitting sample at debt/GDP thresholds 52%, 75%, 90%: - 52% threshold (models [1]–[6], N = 55 countries where >=20% obs in a regime): - ALR debt > 52% GDP (selected): -0.038, -0.004, -0.049, -0.014, 0.008, 0.028 (with reported [0.024]–[0.027] SEs; some entries marked ∗∗). - ALR debt < 52% GDP (selected): 0.002, -0.009, -0.015, 0.014, 0.012, 0.020. - Lagged dependent y_{i,t−1} coefficients (columns [1]–[6]): -0.566, -0.711, -0.767, -0.692, -0.732, -0.754 (all ∗∗∗). - RMSE (columns [1]–[6]): 0.029, 0.022, 0.020, 0.023, 0.018, 0.017. - 75% threshold (N = 45 countries): - ALR debt > 75% GDP (selected): -0.053, -0.032, -0.040, -0.023, 0.013, 0.013. - ALR debt < 75% GDP (selected): -0.046, 0.002, -0.014, -0.003, 0.018, -0.004. - RMSE ranges: 0.031 to 0.018 across models. - 90% threshold (N = 30 countries): - ALR debt > 90% GDP (selected): -0.001, 0.003, -0.021, -0.010, 0.069, 0.037. - ALR debt < 90% GDP (selected): -0.005, 0.054, 0.001, 0.084, 0.049, 0.120 (one entry marked ∗∗). - RMSE ranges: 0.034 to 0.020. - Interpretation from asymmetric models: average long-run debt coefficients vary across regimes and threshold choices; both negative and positive average long-run coefficients appear depending on threshold and model augmentation. ### Co-summability, summability and balance diagnostics (Tables 4–6) - Estimated order of summability (δ̂* ) for levels (Table 4, N = 105) — mean and median results (deterministics: constant) for key variables (exact reported values): - debt_it: Lower CI band = 1.011; Mean = 1.168; Upper CI band = 1.325; Median = 1.135. - debt2_it: Lower CI band = 1.027; Mean = 1.180; Upper CI band = 1.334; Median = 1.156. - debt3_it: Lower CI band = 1.045; Mean = 1.205; Upper CI band = 1.364; Median = 1.174. - y_it: Lower CI band = 0.948; Mean = 1.096; Upper CI band = 1.243; Median = 1.109. - cap_it: Lower CI band = 1.217; Mean = 1.357; Upper CI band = 1.497; Median = 1.286. - First-difference summability panel (Table 4) (deterministics: constant) — means: - ∆y_it mean = 0.062; ∆debt_it mean = 0.221; ∆cap_it mean = 0.248. - Median ∆y_it = 0.000; median ∆debt_it = 0.182; median ∆cap_it = 0.213. - Balance statistics (δ̂_y − δ̂_g) across model variants (Table 5) — sample highlights: - Panel A (standard specification, deterministics constant): Mean = -0.147 (columns [1]–[3] variations), with Lower CI band = -0.316 and Upper CI band = 0.023 in one reported set; Median = -0.191 in one reported set. - Panels B and C (with CA and with additional CA) show mean and median balances that shift toward positive values when deterministics include constant & trend (e.g., Panel A deterministics constant & trend mean = 0.082, 0.386, 0.493 in columns [1]–[3]). - Underlined mean or median balance statistics indicate evidence against the hypothesis of a balanced regression model (δ̂_y − δ̂_g = 0). - Co-summability statistics (Table 6) — summary (N = 105 country-specific residuals): - Standard specification (no CA): mean = 0.929, 0.907, 1.188 in some columns; Lower CI band values and Upper CI band values reported (e.g., Lower CI band 0.792, Upper 1.065). - Specification with CA: mean = 0.270, 0.210, 0.222 with Lower CI band 0.109, 0.030, -0.012 and Upper CI band 0.431, 0.390, 0.456. - Specification with Additional CA (columns [7]–[9]): mean = 0.206, 0.136, 0.143 with Lower CI bands and Upper CI bands reported; median values reported as well. - Underlined mean or median co-summability statistics indicate evidence against co-summability (δ̂" = 0). Only specifications with convincing evidence from balance testing in Table 5 are printed in black (co-summability conditional on balance). ### Static polynomial specifications and country heterogeneity (Table 7) - Static linear and nonlinear country-by-country summaries (robust means across countries): - Reported averaged debt coefficients (examples from columns [1]–[9]): - debt_it coefficients across specifications include values such as -0.059, 0.286, 1.989, 0.000, 0.338, 1.307 (with associated reported SEs and significance markers where applicable). - debt2_it: reported values include -0.024, -0.330, -0.027, -0.178 (with SEs), debt3_it entries like 0.019, 0.009 in some columns. - Diagnostics: RMSE and CD Test vary by specification (e.g., RMSE reported 0.056 down to 0.031 in various specifications; CD Test values reported e.g., 25.69, 25.85, 22.97, 4.48, 4.06, 3.48). - Reporting of country counts with statistically significant positive/negative debt coefficients and counts of convex/concave polynomial relationships (explicit counts reported in table). - Some specifications flagged as “Bal & Co-Sum” indicate those specifications found to be balanced and co-summable in prior tests (Tables 5 and 6). ### Diagnostics on exogeneity, stationarity, and cross-section dependence (selected) - Weak exogeneity testing (Table 2): Panel A (without CA) and Panel B (with CA) present GM-t group-mean t-statistics and Avg λ̂_i with t-stats for output, capital, and debt stock equations across model variants (MG, CMG, CMG with trend, CMG with added lags/CA/covariates). Examples: - MG[3] Output (1 lag): GM-t = -2.54, p = 0.01, Avg λ̂_i = -0.928, t-stat = -21.98. - CMG with trend[5] Output1: GM-t = -2.56, p = 0.01, Avg λ̂_i = -1.000, t-stat = -24.87 (without CA); with CA GM-t = -2.44, p = 0.01, Avg λ̂_i = -0.956, t-stat = -20.92. - Panel stationarity testing (TA-III) (selected exact entries): - Maddala and Wu (1999) Fisher test (deterministics: constant), lags = 0: GDP pc Fisher = 222.590, p = 0.263; debt pc Fisher = 388.920, p = 0.00; cap pc Fisher = 703.690, p = 0.00. - Pesaran (2007) CIPS test (deterministics: constant), lags = 0: GDP pc Z-tbar = 3.86, p = 1.00; debt pc = 4.07, p = 1.00; cap pc = 2.67, p = 1.00. - Cross-section dependence (TA-IV): - Levels: avg ρ for y_it = 0.39, debt_it = 0.36, cap_it = 0.38; average absolute correlations avg|ρ| = 0.66, 0.52, 0.75 respectively; CD statistic values: 157.62, 148.36, 154.54 (p-values 0.00). - First differences: avg ρ small (0.07); CD test still significant (29.93, 28.03, 18.32; p = 0.00). - Heterogeneous AR(2) residuals vs AR(2) with CCE: AR(2) residuals show avg ρ ~ 0.09–0.13 and CD highly significant; AR(2) with CCE residuals reduce avg ρ to ≈0 and CD statistics much smaller (not always significant), illustrating the effectiveness of CA/CCE augmentation. ### Key analytical conclusions (as reported in tables and figures) - Country heterogeneity matters: heterogeneous estimators (MG, CMG) produce a distribution of country-specific long-run debt coefficients (ALR) and reveal substantial cross-country variation in debt-growth relationships. - Nonlinearity and asymmetry: polynomial and regime-split specifications (debt, debt^2, debt^3; thresholds at 52%, 75%, 90%) produce differing average long-run coefficients above and below thresholds, with evidence that the sign and magnitude of the debt-growth link depend on regime choice and model augmentation. - Balance and co-summability: testing indicates that conclusions about long-run debt effects depend on whether the production-function regression is balanced and whether residuals are co-summable; only some specifications pass balance and co-summability diagnostics. - Cross-sectional common factors: inclusion of cross-section averages (CA) materially affects estimation, reduces residual cross-section dependence in many specifications, and alters estimated coefficients and diagnostic outcomes. *Source: _wp13248 - chapter 8, pp. 555–677 (source PDF filename: _wp13248 - chapter 8, pp. 555–677).* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13248.pdf_