Dear Reviewer 3,
We are grateful for your detailed, expert evaluation. Your comments identified substantial gaps in the methodological transparency of the manuscript, and we have addressed each one thoroughly. In doing so, we discovered a problem in the original dataset that required rebuilding the entire empirical analysis from scratch. We disclose this proactively below before the point-by-point responses.
During preparation of this revision, we conducted a systematic audit of the original dataset and identified critical anomalies inconsistent with known market behavior. Specifically, the S&P 500–NASDAQ 100 daily return correlation was near zero (known documented range: ρ > 0.90), and the Bitcoin–Ethereum correlation was near zero and negative (known range: ρ ∈ [0.60, 0.85]). These values indicate the original dataset did not reflect genuine market dynamics.
We have therefore rebuilt the entire empirical analysis from verified, publicly available real-world data:
Return series (10 assets + VIX): Yahoo Finance via the yfinance Python library. All tickers listed in revised §3.2; daily data deposited as CSV files.
Market concentration (HHI, CR10): SEC EDGAR NPORT-P quarterly filings, SPDR S&P 500 ETF Trust (CIK: 0000884394). 19 filings, January 2020–October 2024, <pctVal> fields extracted. Python script and raw data deposited.
Sample: 1,215 trading days, 2020-01-03 to 2024-10-30.
All eight analysis scripts (Python 3.11) are deposited on Zenodo (DOI: 10.5281/zenodo.21706058), making every number in this response independently reproducible without proprietary software. The structural design of the paper — the concentration-fragility hypothesis, the four-method framework, and portfolio implications — is unchanged.
The revised manuscript replaces the parametric MS-VAR overlay — which was not reproducible without EViews and rested on the compromised original dataset — with two transparent and fully documented regime-identification approaches:
For each of the 10 return series, we estimate a GJR-GARCH(1,1) model with Student-t errors using the arch Python library. The model captures the asymmetric leverage effect that standard GARCH misses:
S&P 500 parameter estimates (full table for all 10 series in new Appendix Table A2):
| Parameter | Estimate | Std. Error | p-value |
|---|---|---|---|
| μ (mean) | 0.0621% | 0.0156 | <0.001 |
| ω (long-run variance) | 0.0012 | 0.0004 | 0.002 |
| α (ARCH) | 0.0836 | 0.0211 | <0.001 |
| γ (asymmetry / leverage) | 0.0814 | 0.0283 | 0.004 |
| β (GARCH) | 0.8563 | 0.0197 | <0.001 |
| ν (Student-t d.f.) | 7.38 | 1.24 | <0.001 |
GARCH persistence: α + β = 0.856 (below unity, confirming covariance stationarity). Peak VaR₁% = 23.76% on 2020-03-17 (COVID-19); minimum = 0.24% during the 2023–2024 bull market.
Using real quarterly HHI values from SEC EDGAR (forward-filled to daily frequency), we classify each trading day into one of three concentration regimes using empirical tertile cutoffs:
| Regime | HHI Threshold | Trading Days |
|---|---|---|
| LOW (dispersed market) | ≤ 0.01347 | 415 |
| MEDIUM (transitional) | 0.01347 – 0.01481 | 441 |
| HIGH (concentrated market) | > 0.01481 | 359 |
LOW HHI predominates in 2020–2021 (post-COVID recovery); HIGH HHI in 2023–2024 (AI/mega-cap surge). This approach is more transparent than a parametric MS-VAR because regimes are identified from observable HHI data rather than latent state probabilities.
V3 adds full VECM documentation in expanded §3.3.1 and new Appendix Table A1. All results are computed on the real 1,215-observation dataset using statsmodels.tsa.vector_ar.vecm.
| Series | ADF Statistic | p-value | Integration Order |
|---|---|---|---|
| VaR₁% (level) | −6.21 | <0.001 | I(0) |
| VIX (level) | −4.87 | 0.001 | I(0) |
| HHI (level) | −2.41 | 0.138 | I(1) |
| ΔHHI | −18.34 | <0.001 | I(0) after differencing |
| CR10 (level) | −2.08 | 0.253 | I(1) |
| ΔCR10 | −16.91 | <0.001 | I(0) after differencing |
| Criterion | Optimal lag k |
|---|---|
| AIC | 2 |
| BIC | 1 ← adopted (parsimony) |
| HQC | 1 |
| H₀ | Trace Statistic | Critical Value (5%) | Decision |
|---|---|---|---|
| r = 0 | 81.00 | 47.85 | Reject — at least 1 CI vector |
| r ≤ 1 | 32.14 | 29.79 | Marginal → adopt r = 1 |
| r ≤ 2 | 11.23 | 15.49 | Fail to reject |
Interpretation: HHI and CR10 operate through opposite channels. HHI during the 2023–2024 AI-stock era corresponded with lower VaR because concentration was driven by deeply liquid mega-caps. CR10 independently raises long-run fragility through single-name and sector-cluster risk.
| Equation | α̂ | Interpretation |
|---|---|---|
| VaR₁% | −0.0678 | 6.78% of disequilibrium corrected per day (≈ half-life 10 days) |
| HHI | +0.0012 | Near-zero — HHI is weakly exogenous (concentration drives fragility, not vice versa) |
| Regressor | Coefficient | HAC SE | t-statistic | p-value |
|---|---|---|---|---|
| Intercept | −0.00841 | 0.00318 | −2.64 | 0.008** |
| HHI | −2.555 | 1.280 | −1.996 | 0.046** |
| VIX | +0.00236 | 0.000128 | +18.4 | <0.001*** |
| CR10 | +0.217 | 0.0934 | +2.32 | 0.020** |
Adjusted R² = 0.756. Breusch-Pagan: BP = 352 (p < 0.001) — HAC standard errors are warranted.
| Test | Statistic | p-value | Decision |
|---|---|---|---|
| Ljung-Box Q(5) | 2,519 | <0.001 | Serial correlation (GARCH persistence by construction) |
| Breusch-Pagan | 352 | <0.001 | Heteroskedastic — HAC SE applied |
| Jarque-Bera | 6,844 | <0.001 | Non-normal; CLT valid at n = 1,215 |
V3 adds a fully documented MFI section (§3.3.2) and new Appendix Table A3. All five inputs are standardized to μ = 0, σ = 1 using sklearn StandardScaler before PCA.
VIX — CBOE Volatility Index (Yahoo Finance ^VIX)
RVol_SP500 — S&P 500 realized volatility: 20-day rolling annualized standard deviation
GARCH_VaR — S&P 500 GJR-GARCH(1,1) conditional VaR₁% (Script 02)
RVol_HY — HY corporate bond realized volatility: 20-day rolling annualized
RVol_BTC — Bitcoin realized volatility: 20-day rolling annualized
| Component | Eigenvalue | % Variance Explained | Cumulative % |
|---|---|---|---|
| PC1 (MFI) | 3.7594 | 75.13% | 75.13% |
| PC2 | 0.7367 | 14.73% | 89.87% |
| PC3 | 0.3083 | 6.17% | 96.04% |
| PC4 | 0.1388 | 2.78% | 98.81% |
| PC5 | 0.0590 | 1.18% | 100.00% |
We retain PC1 as the MFI. Justification: (a) Kaiser criterion (eigenvalue > 1): only PC1 qualifies; (b) scree plot elbow at PC2; (c) PC1 explains 75.13% of total variance.
| Input Variable | Loading on PC1 |
|---|---|
| RVol_SP500 | +0.500 |
| GARCH_VaR | +0.481 |
| VIX | +0.469 |
| RVol_HY | +0.455 |
| RVol_BTC | +0.301 |
All five loadings are positive: PC1 is unambiguously a scalar financial stress index.
| Indicator | AUC | Sensitivity | Specificity | F₁ |
|---|---|---|---|---|
| MFI (proposed) | 0.988 | 0.967 | 0.952 | 0.889 |
| VIX only (benchmark) | 0.974 | — | — | — |
| Improvement | +0.014 |
Robustness: leave-one-variable-out jackknife changes AUC by no more than 0.009; 90th percentile threshold: AUC = 0.981; 99th percentile: AUC = 0.993.
The GARCH specification is now fully documented in §3.3.1 and Appendix Table A2 (see also R3.1 for model equations and S&P 500 parameter estimates). Key additions in V3:
Model: GJR-GARCH(1,1) with Student-t errors, estimated by maximum likelihood via the arch library. The asymmetric specification is used because standard GARCH cannot capture the leverage effect documented for equity returns.
ARCH-LM pre-test: All 10 series reject H₀ of no conditional heteroskedasticity at the 1% level (LM(5) statistics: 41.2–314.7), confirming GARCH is appropriate.
VaR₁% time series: Peak = 23.76% on 2020-03-17 (COVID-19). Mean = 2.14%. Full daily series deposited as garch_var_daily.csv.
Appendix Table A2: ω, α, γ, β, ν, and α+β persistence for all 10 series (S&P 500, NASDAQ-100, Russell 2000, 10Y Treasury, IG Corp, HY Corp, Gold, Crude Oil, Bitcoin, Ethereum).
V3 provides full DY documentation in §3.3.3 and new Appendix Table A4. Method: Generalized Forecast Error Variance Decomposition (GFEVD) of Pesaran & Shin (1998), as applied by Diebold & Yilmaz (2012). The GFEVD is order-invariant — critical when no a priori causal ordering of the 10 asset classes exists.
Row normalization: θ̄ij(H) = 100 × θ̃ij(H) / Σj θ̃ij(H). All rows sum to 100 (net-flow sum = 0 ± machine epsilon, verified). Parameters: VAR lag p = 8 (AIC); horizon H = 10; volatility proxy: squared returns × 10,000; rolling window: 250 days.
| SP500 | NASD | RUT | 10Y | IG | HY | Gold | Oil | BTC | ETH | FROM | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| S&P 500 | 27.5 | 19.1 | 17.5 | 1.1 | 4.9 | 5.5 | 1.9 | 7.5 | 9.6 | 5.4 | 72.5 |
| NASDAQ | 26.1 | 26.6 | 14.6 | 1.0 | 4.5 | 5.3 | 2.0 | 6.1 | 8.6 | 5.2 | 73.4 |
| Russell 2000 | 24.3 | 14.0 | 25.7 | 1.5 | 3.2 | 6.7 | 1.5 | 8.9 | 9.0 | 5.3 | 74.3 |
| 10Y Treasury | 10.6 | 6.5 | 7.7 | 47.6 | 6.0 | 4.8 | 2.4 | 5.8 | 5.9 | 2.7 | 52.4 |
| IG Corp | 19.7 | 11.2 | 13.9 | 2.1 | 18.4 | 8.9 | 2.5 | 4.2 | 12.8 | 6.3 | 81.6 |
| HY Corp | 15.0 | 8.9 | 13.1 | 1.4 | 9.5 | 32.3 | 2.2 | 9.7 | 5.1 | 2.9 | 67.8 |
| Gold | 4.6 | 4.2 | 2.6 | 2.9 | 6.2 | 1.6 | 67.8 | 0.9 | 5.1 | 4.0 | 32.2 |
| Crude Oil | 7.6 | 4.4 | 6.0 | 0.5 | 6.5 | 19.4 | 2.4 | 49.3 | 3.1 | 0.9 | 50.7 |
| Bitcoin | 10.1 | 5.9 | 9.3 | 0.5 | 2.4 | 4.5 | 1.9 | 7.5 | 40.7 | 17.4 | 59.3 |
| Ethereum | 7.2 | 5.1 | 6.2 | 0.4 | 1.4 | 2.9 | 1.6 | 3.7 | 21.4 | 50.2 | 49.8 |
| TO others | 125.1 | 79.2 | 90.9 | 11.2 | 44.5 | 59.8 | 18.4 | 54.4 | 80.4 | 50.2 | TCI = 61.4% |
| NET | +52.6 | +5.8 | +16.6 | −41.1 | −37.2 | −8.0 | −13.8 | +3.7 | +21.1 | +0.3 |
TCI = 61.40% — substantial cross-asset connectedness covering the COVID-19 shock, Russia–Ukraine escalation, and the most aggressive Fed rate-hiking cycle in four decades.
Net senders: S&P 500 (+52.6), Russell 2000 (+16.6), Bitcoin (+21.1).
Net receivers: 10Y Treasury (−41.1), IG Corporate (−37.2).
Rolling TCI peak: 79.87% on 2022-06-13 — the exact date of S&P 500 bear-market confirmation and Bitcoin’s collapse below $23,000.
V3 replaces the undocumented Table 5 with a fully transparent regime-conditional optimization (§3.3.4, Appendix Table A5). Asset universe: S&P 500, NASDAQ-100, Russell 2000, HY Corporate, Gold. Optimizer: scipy.optimize.minimize (SLSQP; tolerance = 10⁻¹²; long-only). Risk-free rate rf = 4% p.a.
| Strategy | LOW SR | LOW MDD | MEDIUM SR | MEDIUM MDD | HIGH SR | HIGH MDD |
|---|---|---|---|---|---|---|
| EW | 0.445 | −26.1% | 0.445 | −26.1% | 0.445 | −26.1% |
| MV | 0.300 | −16.4% | 0.085 | −19.2% | 0.044 | −19.8% |
| MSR | 0.814 | −21.1% | 0.080 | −41.9% | 0.744 | −18.9% |
| RP | 0.527 | −20.6% | 0.458 | −22.0% | 0.452 | −22.4% |
(1) Risk Parity is the most regime-robust strategy: SR declines only from 0.527 to 0.452 (range = 0.075).
(2) MSR shifts in high-concentration regimes toward broad equity + defensive gold (29.7% S&P + 57.1% Gold; SR = 0.744).
(3) MV deteriorates sharply with concentration (SR 0.300 → 0.044), reflecting elevated equity correlations in high-HHI mega-cap markets.
Cautionary note added to §3.3.4: “Optimization inputs are in-sample regime-conditional moments. Results illustrate portfolio implications and should not be interpreted as investment recommendations or out-of-sample forecasts.”
All model equations in V3 have been reformatted in standard mathematical notation. The eight key equations now in the manuscript:
The Conclusions section (§5) has been substantially revised. Key language changes:
HHI–fragility moderated: Old: “Our results confirm that market concentration is a primary driver of systemic fragility.”
New: “Our results suggest that concentration operates through two distinct channels with opposite effects: overall HHI is negatively associated with VaR₁% in periods dominated by liquid mega-caps, while CR10 is positively associated with VaR₁%. This nuanced finding calls for disaggregated monitoring of concentration metrics.”
MFI qualified as coincident, not leading: New: “The MFI achieves AUC = 0.988 in classifying high-VaR days in-sample. The index is contemporaneous rather than leading — it signals stress concurrently rather than anticipating it.”
DY spillover contextualized: New: “The TCI of 61.4% reflects elevated linkages during a period containing three major shocks. Whether this level is structurally elevated or crisis-episode-driven cannot be determined from a single five-year window.”
Global qualification added: “All empirical findings are conditional on the study period (January 2020 – October 2024). Generalization beyond this window requires out-of-sample validation.”
| Section | Change Type | Description |
|---|---|---|
| Data | Complete replacement | All empirical data rebuilt from Yahoo Finance + SEC EDGAR; 8 Python scripts deposited on Zenodo (DOI: 10.5281/zenodo.21706058) |
| §3.2 Concentration | Expansion | HHI/CR10 from 19 real SEC EDGAR NPORT-P filings; forward-filled to daily frequency |
| §3.3.1 GARCH | New | GJR-GARCH(1,1) Student-t; ARCH-LM pre-tests; VaR₁% construction documented; all 10 series in Appendix A2 |
| §3.3.1 VECM | Expansion | ADF table, Johansen results, β vector, α coefficients, OLS-HAC table, diagnostic table added |
| §3.3.1 MS-VAR | Replaced | Replaced with transparent HHI-regime tertile conditioning using observable SEC EDGAR data |
| §3.3.2 MFI/PCA | Expansion | 5 real inputs documented; eigenvalue table; PC1 loadings; AUC = 0.988 vs. VIX benchmark (AUC = 0.974) |
| §3.3.3 DY Spillover | Complete replacement | Real 10-series GFEVD table; TCI = 61.40%; rolling TCI with peak 79.87% on 2022-06-13 |
| §3.3.4 Portfolio | Complete replacement | 4 strategies × 3 regimes; real weights and performance; Risk Parity identified as most regime-robust |
| All equations | Reformatting | 8 equations in standard mathematical notation (R3.7) |
| §5 Conclusions | Moderation | Language qualified throughout; two-channel finding; out-of-sample caveats added (R3.8) |
| Appendices A1–A5 | New | Full VECM (A1), GARCH params (A2), PCA loadings (A3), DY block table (A4), Portfolio full results (A5) |
| Zenodo deposit | Updated | 8 Python scripts + all CSV raw/results files; complete pipeline reproducible without proprietary software |
We believe the revised manuscript addresses every point raised by Reviewer 3 and represents a substantially more rigorous and transparent contribution. We thank the reviewer for the scrutiny that motivated this reconstruction.
Sincerely,
Jorge Aníbal Restrepo-Morales
Corresponding Author
Tecnológico de Antioquia — Institución Universitaria, Medellín, Colombia
gifatdea@gmail.comOn behalf of: Moreno Rodríguez RY · Zea Restrepo F · Giraldo Betancur EA
Zenodo V3: doi.org/10.5281/zenodo.21706058