## 9. Mean group estimates of the long-run effects of public debt and inflation on output growth for

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---

### Major themes and empirical setup
- Research question: whether a tipping point (threshold) for public indebtedness exists beyond which long-run economic growth drops, and more generally whether a build-up of public debt slows the economy in the long run.
- Sample: panel of N = 40 countries, period 1965–2010 (unbalanced panel, T_min = 30, N_min = 10).
- Key methodological concerns addressed:
  - Dynamics: ARDL and DL specifications, lags p in the range 0 to 3.
  - Heterogeneity: country-specific dynamics, Mean Group (MG) estimation.
  - Endogeneity / simultaneity between growth and debt: explicit modeling of feedbacks.
  - Cross-sectional error dependence: multi-factor error structure; cross-section augmentation using cross-section averages — CS-ARDL and CS-DL.
- Threshold variables considered:
  - g1(d_it; ξ) = I[d_it > ln(ξ)] (simple threshold).
  - g2(d_it; ξ) = I[d_it > ln(ξ)] * max(0; Δd_it) (interactive threshold capturing debt trajectory).

### Panel threshold tests — main findings
- Tests implemented: SupF / AveF (joint tests, r > 1) and SupT / AveT (single threshold, r = 1). Tests account for the nonstandard identification of ξ under the null.
- Cross-sectional dependence diagnostics:
  - CD tests (Pesaran) on residuals from ARDL and DL regressions are highly statistically significant for most specifications, indicating strong cross-sectional error dependence.
- Effect of accounting for common factors:
  - Without CS augmentation, tests frequently reject the null of no threshold, but Monte Carlo evidence shows large size distortions when common factors drive apparent threshold effects.
  - With CS augmentation (CS-ARDL, CS-DL), cross-sectional correlations decline substantially and tests avoid the spurious detection of thresholds.
- Empirical test outcomes (CS-ARDL and CS-DL):
  - Full sample (40 countries): weak evidence for a simple common debt-level threshold after CS-augmentation; the interactive threshold g2 (debt trajectory) remains statistically significant with estimated ξ in the range 40–60 percent (panel (c) of Table 4).
  - Advanced economies (19 countries): weaker joint evidence after CS-augmentation; some support for interactive threshold effects but threshold estimates vary widely (10% to 100% depending on lag order).
  - Developing economies (21 countries): after CS-augmentation, joint and interactive threshold tests generally fail to reject the null; less robust evidence of threshold effects.
- Interpretation: once unobserved common factors and spillovers are accounted for, there is no universally applicable simple debt-to-GDP threshold; debt trajectory (rising debt) is more important than level alone, especially when Δd_it > 0 and ξ > 50%.

### Long-run effects of debt (Mean Group estimates)
- Estimation approaches: ARDL and DL specifications and their cross-sectionally augmented versions (CS-ARDL, CS-DL); Mean Group (MG) estimates reported.
- Core empirical result: persistent increases in debt-to-GDP are associated with statistically significant negative long-run effects on GDP growth across specifications and country groups.
- Reported coefficient ranges:
  - ARDL / DL (without CS augmentation): long-run debt-growth coefficients b^LR_{Δd} generally in the range between -0.04 and -0.11 (statistically significant, often at the 1 percent level).
  - CS-ARDL / CS-DL (with CS augmentation): long-run debt-growth coefficients b^LR_{Δd} generally in the range between -0.03 and -0.15 (statistically significant in most cases; out of 168 reported coefficients in Table 8 only 9 are insignificant).
- Country-group patterns:
  - Advanced economies: negative long-run effects present; point estimates sometimes smaller in magnitude relative to developing economies but often statistically significant.
  - Developing economies: negative long-run effects present and often sizeable.
- Key interpretation: persistent rises in the debt-to-GDP ratio are associated with negative long-run growth effects; temporary, reversible debt increases need not harm long-run growth.

### Robustness to inclusion of inflation
- Motivation: in some developing economies deficit financing occurs via money creation leading to inflation rather than rising recorded debt.
- Findings when inflation (π_it) is included:
  - Debt long-run coefficients b^LR_{Δd} remain negative and statistically significant across ARDL, DL, CS-ARDL, and CS-DL.
  - Estimated ranges for inflation long-run effect b^LR_{π}:
    - ARDL / DL: b^LR_{π} lie between -0.04 and -0.08 in many specifications.
    - CS-ARDL / CS-DL: b^LR_{π} lie in the range -0.08 to -0.20 and are often statistically significant (1% level in most cases).
  - Cross-section augmentation reduces cross-sectional residual correlation (CD statistics decline) and yields stronger evidence that persistent inflation can be detrimental for growth, alongside persistent debt increases.

### Monte Carlo validation and methodological notes
- Monte Carlo experimental design: six different experimental designs (DGP1–DGP6) examined; performance of Sup and Ave tests of threshold effects ('1 = 0 and/or '2 = 0) evaluated; small-sample performance of pooled, fixed-effects, and filtered-pooled estimators assessed.
- Selected Monte Carlo numerical highlights (preserving reported values):
  - Table 1 (DGP1), Filtered Pooled estimator (N,T) = (40,100):
    - '1 Bias (x100) = 0.0172; RMSE (x100) = 0.0065.
    - ξ Bias (x100) = 0.000; RMSE (x100) = 0.001.
  - Table 1 (DGP1), Filtered Pooled estimator (N,T) = (100,100):
    - '1 Bias (x100) = 0.0054; RMSE (x100) = 0.0011.
    - ξ Bias (x100) = -0.020; RMSE (x100) = 0.000.
  - Table 2c (DGP1) rejection frequencies (x100) for '1 = -0:01 and (N,T) = (40,100):
    - SupT = 100.00; AveT = 99.95; T(0.2) = 100.00; T(0.5) = 77.85; T(0.9) = 98.05.
  - Table 2c (DGP1) size for '1 = 0 and (N,T) = (40,100):
    - SupT = 6.30; AveT = 5.00; T(0.2) = 6.10; T(0.5) = 5.60; T(0.9) = 5.35.
  - DGP2 filtered pooled estimator ('1 true value = -0:01), (N,T) = (40,100):
    - Bias (x100) = 0.0310; RMSE (x100) = 0.0164.
  - DGP4 filtered pooled estimator, '1 (true value = 0), (N,T) = (40,100):
    - '1 Bias (x100) = 0.0227; RMSE (x100) = 0.0123.
  - DGP6 (Table 11) rejection rates and biases for SupT / AveT with and without CS augmentation, (N,T) = (40,100):
    - Rejection rates without CS augmentation: SupT = 63.45; AveT = 79.60.
    - Rejection rates with CS augmentation: SupT = 65.55; AveT = 81.35.
    - Bias (x100) and RMSE (x100) for '1 without CS augmentation: Bias (x100) = 0.2344; RMSE (x100) = 0.2211.
    - Bias (x100) and RMSE (x100) for '1 with CS augmentation: Bias (x100) = 0.0000; RMSE (x100) = -0.0041 (other RMSE entries reported as 0.2258 and 0.1428).
- Monte Carlo conclusions:
  - Filtered pooled estimators (accounting for contemporaneous residual dependence and using reduced-form identification) have small bias and RMSE that decline with N and T.
  - SupT / AveT and SupF / AveF tests have correct size and good power when common factors are accounted for via cross-section augmentation; tests without CS augmentation can display large size distortions (e.g., rejection rates 63% to 93% in DGP6).
  - Small-sample performance is satisfactory for panels with N and T around 40–100 when cross-section augmentation is employed.
- Identification strategy: threshold effect in the output equation is identified under an exclusion restriction (threshold variable excluded from the debt equation) and linear simultaneity modeling e_it = λ_i ε_it + u_it; reduced-form ARDL allows consistent estimation of threshold parameters by grid search.

### Policy-relevant conclusions and implications
- No universal single debt-to-GDP threshold applicable across all countries after accounting for global factors and cross-country spillovers.
- Debt trajectory matters: rising debt-to-GDP (Δd_it > 0) beyond about 50–60 percent (in some specifications and for the full sample) is associated with statistically significant adverse long-run growth consequences.
- Persistent debt accumulation has robust negative long-run effects on growth even when accounting for feedbacks from growth to debt and for inflation; temporary fiscal deficits that are credibly reversible need not harm long-run growth.
- Inflation can also be harmful: persistent high inflation is associated with negative long-run growth effects, particularly when common factors are accounted for (CS-ARDL / CS-DL estimates indicate larger negative effects).
- Policy implication: Keynesian fiscal expansion can support short-run demand without adverse long-run growth effects provided fiscal plans credibly commit to and implement reversal of the debt increase (i.e., ensure debt trajectory is stable or declining over the long run).

*Source: _wp15197 - 9. Mean group estimates of the long-run effects of public debt and inflation on output growth for; Supplement and REFERENCES (PDF chapter, 1965–2010).*

### References .............................................................................................................

### _wp15197 - References .............................................................................................................

### Supplement overview
- Supplement sections and page references:
  - Supplement .............................................................................................................................. 41
  - A. Monte Carlo Results ..................................................................................................... 42
  - B. Additional Empirical Results........................................................................................ 55

### Figures listed
- Figure 1: Power functions of SupT and AveT tests for testing the null of 휑1=0 against the alternatives 휑1={−0.01,−0.009,...,0,0.001,...,0.01} in DGP1 ...................................................... 31
- Figure 2: Power functions for SupF and AveF tests for testing the null of 휑1=휑2=0 against the alternatives 휑1=0, 휑2={−0.01,−0.009,...,0,0.001,...,0.01} in the case of DGP4 ... 32

### Tables listed
- Table 1: MC findings for Bias(x100) and RMSE(x100) of the estimation of φ1 and τ in the baseline experiments without lags (DGP1) .................................................................................... 30
- Table 2: MC findings for Bias(x100) and RMSE(x100) for the estimation of φ1, 휑2, and 휏 in experiments with lagged dependent variable, feedback effects and two threshold indicators (DGP4) ............................................................................................................................. 30
- Table 3: MC findings for the estimation of φ1 and τ in DGP6 (experiments without threshold effects [휑1=휑2=0] and with unobserved common factors subject to threshold effects) ....... 32
- Table 4: Tests of debt-threshold effects for all countries, 1966-2010 ........................................... 33
- Table 5: Tests of debt-threshold effects for advanced economies, 1966-2010 .............................. 33
- Table 6: Tests of debt-threshold effects for developing economies, 1966-2010 ........................... 34
- Table 7: Tests of debt-threshold effects for all countries (robustness to the inclusion of inflation in the regressions), 1966-2010 ................................................................................................... 34
- Table 8: Mean group estimates of the long-run effects of public debt on output growth (1966-2010) ..................................................................................................................... 35

*Source: _wp15197 - References; Supplement; Figures and Tables listing.*

### 9. Mean group estimates of the long-run effects of public debt and inflation on output growth for

### 9. Mean group estimates of the long-run effects of public debt and inflation on output growth for

### Major themes and empirical setup
- Research question: whether a tipping point (threshold) for public indebtedness exists beyond which long-run economic growth drops, and more generally whether a build-up of public debt slows the economy in the long run.
- Sample: panel of N = 40 countries, period 1965–2010 (unbalanced panel, T_min = 30, N_min = 10).
- Key methodological concerns addressed:
  - Dynamics (ARDL and DL specifications, lags p in the range 0 to 3).
  - Heterogeneity (country-specific dynamics, Mean Group (MG) estimation).
  - Endogeneity / simultaneity between growth and debt (explicit modeling of feedbacks).
  - Cross-sectional error dependence (multi-factor error structure; cross-section augmentation using cross-section averages — CS-ARDL and CS-DL).
- Threshold variables considered:
  - g1(d_it; ξ) = I[d_it > ln(ξ)] (simple threshold).
  - g2(d_it; ξ) = I[d_it > ln(ξ)] * max(0; Δd_it) (interactive threshold capturing debt trajectory).

### Panel threshold tests — main findings
- Tests implemented: SupF / AveF (joint tests, r > 1) and SupT / AveT (single threshold, r = 1). Tests account for the nonstandard identification of ξ under the null.
- Cross-sectional dependence diagnostics:
  - CD tests (Pesaran) on residuals from ARDL and DL regressions are highly statistically significant for most specifications, indicating strong cross-sectional error dependence.
- Effect of accounting for common factors:
  - When regressions are NOT augmented with cross-section averages (no CS augmentation), tests frequently reject the null of no threshold — but Monte Carlo evidence shows large size distortions when common factors drive apparent threshold effects.
  - With CS augmentation (CS-ARDL, CS-DL), cross-sectional correlations decline substantially and tests avoid the spurious detection of thresholds.
- Empirical test outcomes (CS-ARDL and CS-DL):
  - For the full sample (40 countries): weak evidence for a simple common debt-level threshold after CS-augmentation; the interactive threshold g2 (debt trajectory) remains statistically significant with estimated ξ in the range 40–60 percent (panel (c) of Table 4).
  - For advanced economies (19 countries): weaker joint evidence after CS-augmentation; some support for interactive threshold effects but threshold estimates vary widely (10% to 100% depending on lag order).
  - For developing economies (21 countries): after CS-augmentation, joint and interactive threshold tests generally fail to reject the null; less robust evidence of threshold effects, possibly due to greater heterogeneity and smaller group size.
- Interpretation: once unobserved common factors and spillovers are accounted for, there is no universally applicable simple debt-to-GDP threshold; debt trajectory (rising debt) is more important than level alone, especially when Δd_it > 0 and ξ > 50%.

### Long-run effects of debt (Mean Group estimates)
- Estimation approaches: ARDL and DL specifications and their cross-sectionally augmented versions (CS-ARDL, CS-DL); Mean Group (MG) estimates reported.
- Core empirical result: persistent increases in debt-to-GDP are associated with statistically significant negative long-run effects on GDP growth across specifications and country groups.
- Reported coefficient ranges:
  - ARDL / DL (without CS augmentation): long-run debt-growth coefficients b^LR_{Δd} generally in the range between -0.04 and -0.11 (statistically significant, often at the 1 percent level).
  - CS-ARDL / CS-DL (with CS augmentation): long-run debt-growth coefficients b^LR_{Δd} generally in the range between -0.03 and -0.15 (statistically significant in most cases; out of 168 reported coefficients in Table 8 only 9 are insignificant).
- Country-group patterns:
  - Advanced economies: negative long-run effects present; point estimates sometimes smaller in magnitude relative to developing economies but remain often statistically significant.
  - Developing economies: negative long-run effects present and often sizeable.
- Key interpretation: if the debt-to-GDP ratio keeps rising persistently, it will have negative effects on economic growth in the long run; if debt-to-GDP rises temporarily and is reversed, long-run growth need not be harmed.

### Robustness to inclusion of inflation
- Motivation: in some developing economies deficit financing occurs via money creation leading to inflation rather than rising recorded debt.
- Findings when inflation (π_it) is included:
  - Debt long-run coefficients b^LR_{Δd} remain negative and statistically significant across ARDL, DL, CS-ARDL, and CS-DL.
  - Estimated ranges:
    - ARDL / DL estimates of inflation long-run effect b^LR_{π} lie between -0.04 and -0.08 in many specifications.
    - CS-ARDL / CS-DL estimates of b^LR_{π} lie in the range -0.08 to -0.20 and are often statistically significant (1% level in most cases).
  - Cross-section augmentation again reduces cross-sectional residual correlation (CD statistics decline) and yields stronger evidence that persistent inflation can be detrimental for growth, alongside persistent debt increases.

### Monte Carlo validation and methodological notes
- Monte Carlo experiments show:
  - Filtered pooled estimators (accounting for contemporaneous residual dependence and using the reduced-form identification) have small bias and RMSE that decline with N and T.
  - SupT / AveT and SupF / AveF tests have correct size and good power when common factors are accounted for via cross-section augmentation; tests without CS augmentation can display large size distortions (e.g., rejection rates 63% to 93% in DGP6).
  - Small-sample performance is satisfactory for panels with N and T around 40–100 when cross-section augmentation is employed.
- Identification strategy: threshold effect in the output equation is identified under an exclusion restriction (threshold variable excluded from the debt equation) and linear simultaneity modeling e_it = λ_i ε_it + u_it; reduced-form ARDL allows consistent estimation of threshold parameters by grid search.

### Policy-relevant conclusions and implications
- No universal single debt-to-GDP threshold applicable across all countries after accounting for global factors and cross-country spillovers.
- Debt trajectory matters: rising debt-to-GDP (Δd_it > 0) beyond about 50–60 percent (in some specifications and for the full sample) is associated with statistically significant adverse long-run growth consequences.
- Persistent debt accumulation has robust negative long-run effects on growth even when accounting for feedbacks from growth to debt and for inflation; temporary fiscal deficits that are credibly reversible need not harm long-run growth.
- Inflation can also be harmful: persistent high inflation is associated with negative long-run growth effects, particularly when common factors are accounted for (CS-ARDL / CS-DL estimates indicate larger negative effects).
- Policy implication: Keynesian fiscal expansion can support short-run demand without adverse long-run growth effects provided fiscal plans credibly commit to and implement reversal of the debt increase (i.e., ensure debt trajectory is stable or declining over the long run).

*Source: _wp15197 - 9. Mean group estimates of the long-run effects of public debt and inflation on output growth for (PDF chapter, 1965–2010).*

### REFERENCES

### _wp15197 - REFERENCES

### Key Bibliographic Citations
- Abbas, S. M. A., N. Belhocine, A. El-Ganainy, and M. Horton (2011). Historical Patterns and Dynamics of Public Debt: Evidence From a New Database. IMF Economic Review 59(4), 717–742.
- Abbas, S. M. A., L. Blattner, M. D. Broeck, A. El-Ganainy, and M. Hu (2014). Sovereign Debt Composition in Advanced Economies: A Historical Perspective. IMF Working Paper WP/14/162.
- Andrews, D. W. K. and W. Ploberger (1994). Optimal Tests when a Nuisance Parameter is Present Only Under the Alternative. Econometrica 62(6), pp. 1383–1414.
- Bai, J. (2009). Panel Data Models with Interactive Fixed Effects. Econometrica 77, 1229–1279.
- Checherita-Westphal, C. and P. Rother (2012). The impact of High Government Debt on Economic Growth and its Channels: An Empirical Investigation for the Euro Area. European Economic Review 56(7), 1392–1405.
- Chudik, A., K. Mohaddes, M. H. Pesaran, and M. Raissi (2013). Debt, Inflation and Growth: Robust Estimation of Long-Run Effects in Dynamic Panel Data Models. Federal Reserve Bank of Dallas, Globalization and Monetary Policy Institute Working Paper No. 162.
- Chudik, A., K. Mohaddes, M. H. Pesaran, and M. Raissi (2015). Long-Run Effects in Large Heterogeneous Panel Data Models with Cross-Sectionally Correlated Errors. Federal Reserve Bank of Dallas, Globalization and Monetary Policy Institute Working Paper No. 223.
- Chudik, A. and M. H. Pesaran (2015a). Common Correlated Effects Estimation of Heterogeneous Dynamic Panel Data Models with Weakly Exogenous Regressors. Journal of Econometrics forthcoming.
- Chudik, A. and M. H. Pesaran (2015b). Large Panel Data Models with Cross-Sectional Dependence: A Survey. In B. H. Baltagi (Ed.), The Oxford Handbook of Panel Data, pp. 3–45. Oxford University Press, New York.
- Davies, R. B. (1977). Hypothesis Testing when a Nuisance Parameter is Present only under the Alternative. Biometrika 64(2), 247–254.
- Davies, R. B. (1987). Hypothesis Testing when a Nuisance Parameter is Present Only Under the Alternatives. Biometrika 74(1), pp. 33–43.
- Eberhardt, M. and A. F. Presbitero (2015). Public Debt and Growth: Heterogeneity and Non-linearity. forthcoming in Journal of International Economics.
- Ghosh, A. R., J. I. Kim, E. G. Mendoza, J. D. Ostry, and M. S. Qureshi (2013). Fiscal Fatigue, Fiscal Space and Debt Sustainability in Advanced Economies. The Economic Journal 123(566), F4–F30.
- Hansen, B. E. (1996). Inference When a Nuisance Parameter Is Not Identified Under the Null Hypothesis. Econometrica 64(2), pp. 413–430.
- Hansen, B. E. (1999). Threshold Effects in Non-Dynamic Panels: Estimation, Testing, and Inference. Journal of Econometrics 93(2), 345–368.
- Krugman, P. (1988). Financing vs. Forgiving a Debt Overhang. Journal of Development Economics 29(3), 253–268.
- Moon, H. R. and M. Weidner (2014). Dynamic Linear Panel Regression Models with Interactive Fixed Effects. CeMMaP Working Paper CWP47/14.
- Panizza, U. and A. F. Presbitero (2013). Public Debt and Economic Growth in Advanced Economies: A Survey. Swiss Journal of Economics and Statistics 149(II), 175–204.
- Pesaran, M. H. (1997). The Role of Economic Theory in Modelling the Long Run. Economic Journal 107, 178–191.
- Pesaran, M. H. (2004). General Diagnostic Tests for Cross Section Dependence in Panels. IZA Discussion Paper No. 1240.
- Pesaran, M. H. (2015). Testing Weak Cross-Sectional Dependence in Large Panels. Econometric Reviews 34(6-10), 1089–1117.
- Pesaran, M. H. and Y. Shin (1999). An Autoregressive Distributed Lag Modelling Approach to Cointegration Analysis. In S. Strom (Ed.), Econometrics and Economic Theory in 20th Century: The Ragnar Frisch Centennial Symposium, Chapter 11, pp. 371–413. Cambridge: Cambridge University Press.
- Pesaran, M. H. and R. Smith (1995). Estimating Long-run Relationships from Dynamic Heterogeneous Panels. Journal of Econometrics 68(1), 79–113.
- Reinhart, C. M., V. R. Reinhart, and K. S. Rogoff (2012). Public Debt Overhangs: Advanced-Economy Episodes Since 1800. The Journal of Economic Perspectives 26(3), 69–86.
- Reinhart, C. M. and K. S. Rogoff (2010). Growth in a Time of Debt. American Economic Review 100(2), 573–78.
- Reinhart, C. M. and K. S. Rogoff (2011). From Financial Crash to Debt Crisis. American Economic Review 101(5), 1676–1706.
- Seo, M. H. and Y. Shin (2014). Dynamic Panels with Threshold Effect and Endogeneity. STICERD - Econometrics Paper Series /2014/577.
- Song, M. (2013). Asymptotic Theory for Dynamic Heterogeneous Panels with Cross-Sectional Dependence and Its Applications. Mimeo, January 2013.
- Woo, J. and M. S. Kumar (2015). Public Debt and Growth. Economica forthcoming.

### Supplement: "Is There a Debt-threshold Effect on Output Growth?"
- Purpose:
  - Provides supplemental results to the paper "Is There a Debt-threshold Effect on Output Growth?"
  - Presents a complete set of Monte Carlo findings (Part A) and additional empirical findings (Part B).

- Monte Carlo design and stated DGP implied probabilities and correlations:
  - DGP1 DGP2 DGP3 DGP4 DGP5 DGP6
  - ('1 = 0:01)('1 = 0:01)('1 = 0:01)('1 = 0)('1 = 0:01)('1 = 0)
  - ('2 = 0)('2 = 0)('2 = 0)('2 = 0:01)('2 = 0)('2 = 0)
  - P[g1(dit;) = 1] 37% 37% 37% - 38% 41%
  - P[g2(dit;) = 1] --- 23% --
  - Correlation between g1(dit;) and eit 39% 16% 17% - 14% 27%
  - Correlation between g2(dit;) and eit --- 27% -
  - Correlation between f1t and g1(dit;) ---- 0% -17%
  - Ej |1(i) j 0.90 0.90 0.90 0.90
  - R2 (output equation) 9% 50% 47% 51% 65% 50%

- Monte Carlo experimental coverage:
  - Six different experimental designs are investigated.
  - For each design, performance of Sup and Ave tests of the threshold effects ('1 = 0 and/or '2 = 0) is examined.
  - Small sample performance of pooled, fixed-effects, and filtered-pooled estimators for threshold coefficients ('1 and/or '2) and threshold level, , are evaluated.

### Selected Monte Carlo numerical highlights (preserving reported values)
- Table 1: Bias(x100) and RMSE(x100) for estimation of '1 (true value = -0:01) and  (true value = 0:80) in DGP1:
  - Filtered Pooled estimator (N,T) = (40,100): '1 Bias (x100) = 0.0172; RMSE (x100) = 0.0065.  Bias (x100) = 0.000; RMSE (x100) = 0.001.
  - Filtered Pooled estimator (N,T) = (100,100): '1 Bias (x100) = 0.0054; RMSE (x100) = 0.0011.  Bias (x100) = -0.020; RMSE (x100) = 0.000.
  - Notes: Filtered pooled estimators use qit = (1; dit)0 as filtering variables.

- Table 2c: Rejection frequencies (x100) of tests of '1 = 0 in DGP1 based on filtered pooled estimator:
  - For '1 = -0:01 and (N,T) = (40,100): SupT = 100.00; AveT = 99.95; T(0.2) = 100.00; T(0.5) = 77.85; T(0.9) = 98.05; other t-tests reported across the grid.
  - For '1 = 0 (size) and (N,T) = (40,100): SupT = 6.30; AveT = 5.00; T(0.2) = 6.10; T(0.5) = 5.60; T(0.9) = 5.35.
  - For '1 = 0 (size) and (N,T) = (100,100): SupT = 5.30; AveT = 5.50; T(0.2) = 5.40; T(0.5) = 6.35; T(0.9) = 6.00.

- DGP2 and DGP3 (Tables 3–6):
  - Filtered pooled estimator bias and RMSE for '1 (true value = -0:01) in DGP2:
    - (N,T) = (40,100): Bias (x100) = 0.0310; RMSE (x100) = 0.0164.
    - (N,T) = (100,100): Bias (x100) = 0.0241; RMSE (x100) = 0.0145.
    -  (true value = 0:80): (40,100) Bias (x100) = -0.05; RMSE (x100) = -0.01; RMSE entries include 1.060.40 formatting as in source.
  - Rejection rates (x100) for tests based on filtered pooled estimator in DGP2 and DGP3 are reported across a grid of '1 values and test statistics SupT, AveT, and t-tests at  = 0.2, 0.5, 0.9. Example:
    - DGP2, '1 = -0:01, (N,T) = (40,100): SupT = 100.00; AveT = 100.00; T(0.2) = 100.00; T(0.5) = 58.45; T(0.9) = 97.40.
    - DGP3, '1 = -0:01, (N,T) = (40,100): SupT = 100.00; AveT = 100.00; T(0.2) = 100.00; T(0.5) = 62.10; T(0.9) = 98.40.

- DGP4 (Tables 7–8):
  - Filtered pooled estimator bias and RMSE for '1 (true value = 0) and '2 (true value = -0:01):
    - '1, (N,T) = (40,100): Bias (x100) = 0.0227; RMSE (x100) = 0.0123.
    - '2, (N,T) = (40,100): Bias (x100) = -0.0070; RMSE (x100) = -0.0025; RMSE formatted with 0.1362 and 0.0864.
    -  (true value = 0:80): (40,100) Bias (x100) = -0.06; RMSE (x100) = -0.03; other entries as reported.
  - Rejection frequencies (x100) of tests of '1 = '2 = 0 in DGP4 using SupF and AveF:
    - Example: For '1 = 0 and '2 = -0:01, (N,T) = (40,100): SupF = 100.00; AveF = 100.00; F = 100.00; for '1 = 0 and '2 = 0 (size), (N,T) = (40,100): SupF = 5.90; AveF = 4.55; F = 5.85; size reported.

- DGP5 (Tables 9–10):
  - Filtered pooled estimator bias and RMSE for '1 (true value = -0:01) and  (true value = 0:80):
    - (N,T) = (40,100): '1 Bias (x100) = 0.0630; RMSE (x100) = -0.0109; RMSE entries include 0.3197 and 0.0931 formatting.
    - (N,T) = (100,100): '1 Bias (x100) = -0.0076; RMSE (x100) = -0.0163; RMSE entries include 0.1074 and 0.0583.
  - Rejection rates (x100) for SupT, AveT, and t-tests across the grid; examples include SupT values reaching near 100.00 for strong alternatives and reported size near conventional levels for '1 = 0.

- DGP6 (Table 11):
  - Rejection rates for SupT and AveT tests and bias(x100) and RMSE(x100) for estimates of '1 with and without cross-section (CS) augmentation:
    - Rejection rates without CS augmentation (N,T) = (40,100): SupT = 63.45; AveT = 79.60.
    - Rejection rates with CS augmentation (N,T) = (40,100): SupT = 65.55; AveT = 81.35.
    - Bias (x100) and RMSE (x100) for '1 without CS augmentation, (N,T) = (40,100): Bias (x100) = 0.2344; RMSE (x100) = 0.2211.
    - Bias (x100) and RMSE (x100) with CS augmentation, (N,T) = (40,100): Bias (x100) = 0.0000; RMSE (x100) = -0.0041; RMSE formatting appears with 0.2258 and 0.1428 for other entries.

### Part B: Additional Empirical Results (Tables 1–4)
- Tests of debt-threshold effects for advanced and developing economies (1966–2010) with robustness to inclusion of inflation:
  - Panel types and specifications reported: ARDL, DL, CS-ARDL, CS-DL with various lags and p values.
  - Reported threshold variables:
    - g1(dit; ) = I[dit > ln()]
    - g2(dit; ) = I[dit > ln()] × max(0; Δdit)
  - Example numeric entries (advanced economies, Table 1, panel a):
    - b values include 0.80, 0.60, 0.60, 0.80, 0.80, 0.80, 0.80, 0.10, 0.10, 0.10, 0.10, 0.10, 0.20.
    - SupF statistics reported include 20.21, z 19.82, z 20.5, z 18.74, z 24.44, z 20.2, z 22.09, z 4.52, 12.19, 10.03, 4.54, 5.02, 12.12.
    - AveF statistics reported include 11.7, z 11.57, z 10.53, z 11.47, z 13.11, z 10.38, z 9.85, z 2.14, 3.04, 4.39.
    - CD test statistics (Pesaran 2004) include 18.16, 14.41, 14.13, 24.01, 18.91, 15.70, 14.02, 4.40, 2.70, 10.77, 6.86, 3.44, 3.10.
  - Mean group estimates of long-run effects of public debt and inflation on output growth:
    - Advanced economies (Table 3, panel a) b̄_d estimates: -0.052 z, -0.050 z, -0.051 z, -0.113 z, -0.075 z, -0.089 z, -0.060 z, -0.065 z, -0.091 z, -0.078 z, -0.075 z, -0.071 z, -0.015 (with standard errors in parentheses preserved as in source).
    - b̄_π estimates include values such as -0.048 , 0.026, 0.037, -0.012, -0.025, etc., with statistical significance denotations  (10%), y (5%), z (1%) as reported.
    - Corresponding CD statistics and panels (b), (c), (d) reported similarly.
    - Developing economies (Table 4) report analogous mean group estimates with b̄_d and b̄_π values, CD statistics, and significance markers preserved exactly as in the tables.

- Notes on empirical tables:
  - Inflation (πit) and lagged values are included in ARDL and DL specifications; CS-ARDL and CS-DL include cross-sectional averages of inflation and lags.
  - Statistical significance markers:  at 10%, y at 5%, z at 1%.
  - CD denotes Pesaran (2004) cross-section dependence test statistic.

*Italic: Source — _wp15197 - REFERENCES (Supplement and Tables) from the provided PDF content.*

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