Gold Silver Pair Trading -Mean Reversion Strategy Using Machine Learning

The gold–silver relationship has long served as a benchmark for relative-value and mean- reversion trading in commodity markets. This study develops a quantitative framework that integrates classical cointegration analysis with modern machine learning (ML) techniques to enhance trading performance in the gold–silver spread. Using futures and ETF data (COMEX GC–SI, GLD–SLV) from 2015–2025, the paper first confirms long-term cointegration and then applies dynamic hedge estimation via Kalman filtering. Mean reversion signals are standardized through z-score normalization and augmented with ML-based regime filters (Gradient Boosting, Support Vector Machine) trained on volatility, macro, and sentiment features to distinguish stable versus unstable spread conditions. Backtests reveal that ML-filtered mean reversion trades outperform static statistical arbitrage by improving entry precision and reducing drawdowns, particularly during high-volatility regimes such as the 2020 COVID crisis, the 2022 inflation spike, and the 2024 commodity rally. The proposed hybrid framework demonstrates that integrating adaptive learning models into classical econometric approaches significantly enhances profitability and risk control for commodity pair trading. Highlights  Presents a hybrid gold–silver mean reversion framework integrating cointegration, Kalman filtering, and machine learning classifiers.  Demonstrates dynamic hedge ratio estimation using state-space modeling for adaptive spread tracking.  Incorporates machine learning regime filters (Gradient Boosting, SVM) to enhance trade timing and reduce drawdowns.  Backtests across 2015–2025 show superior Sharpe ratios and lower volatility than static statistical arbitrage models.  Extends event-driven and volatility-forecasting methods (Mittal, 2025) to commodity spread trading with risk-controlled execution. Figure 1. — Graphical Abstract 2 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary.

Pairs trading, Gold–Silver ratio, Cointegration, Mean reversion, Kalman filter, Machine learning, Statistical arbitrage, Regime switching, Commodity futures. JEL Codes: C22, G13, G17, Q02 1. Introduction The gold–silver pair has been one of the most studied and traded relationships in financial markets for over a century. Historically bound by industrial, monetary, and investment demand linkages, gold and silver exhibit a stable long-run equilibrium punctuated by short- term deviations. These mean-reverting divergences form the foundation for statistical arbitrage or “pairs trading,” where traders exploit temporary mispricing’s between two co- integrated assets. While pairs trading has been extensively studied in equities (Gatev, Goetzmann & Rouwenhorst, 2006), its application to commodities—especially precious metals—remains underexplored. The traditional gold–silver ratio, once used to define bimetallic exchange rates, now functions as a quantitative measure of relative market sentiment, inflation expectations, and risk aversion. When the ratio deviates significantly from its historical mean, traders anticipate a correction: either gold rising relative to silver or vice versa. However, the reliability of such signals has diminished in modern, high-frequency markets characterized by macro shocks, ETF flows, and algorithmic trading. Static correlation or cointegration tests are often insufficient to capture structural breaks and regime shifts. Moreover, during stress events—such as the COVID-19 crash or the 2022 inflation-driven commodity rally—the gold–silver spread exhibits nonlinear and time-varying behavior that challenges traditional mean-reversion assumptions. This study addresses these challenges by integrating econometric and machine learning (ML) frameworks into a unified system for gold–silver pair trading. The approach consists of three stages: 1. Statistical validation of long-term cointegration using Johansen and Engle–Granger methods. 2. Dynamic hedge ratio estimation via Kalman filtering to adaptively model time- varying relationships. 3. ML-based regime classification , where Gradient Boosting and SVM models predict periods when mean reversion signals are statistically reliable, based on volatility, macro, and sentiment features. This research builds upon earlier works by the author, including Forecasting the U.S. Employment Report and Trading SPY Options: A Data-Driven Strategy with Risk-Controlled Hedging (Mittal, 2025) and Russia–Ukraine War: A Quantitative Analysis of Global Fertilizer Supply Chains and Investment Opportunities (Mittal, 2025). The former contributed to the design of risk-controlled trading frameworks for event-driven volatility, while the latter provided a macro–commodity linkage perspective relevant to spread relationships in metals and commodities. This paper extends those principles to a market-neutral, cross-commodity domain—linking event forecasting, volatility modeling, and mean-reversion trading into a single ML- enhanced arbitrage strategy. 2. Literature Review 2.1 Classical Pairs Trading Foundations Pairs trading originated as a market-neutral strategy in equities. Gatev, Goetzmann, and Rouwenhorst (1999, 2006) demonstrated that statistically selected pairs based on minimum historical distance can yield abnormal returns after accounting for transaction costs. 3 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. Vidyamurthy (2004) and Elliott et al. (2005) formalized this as a cointegration-based arbitrage process, emphasizing the error-correction dynamics of mean reversion. These studies established the econometric foundations for spread modeling using correlation, cointegration, and half-life estimation. However, these frameworks largely assumed stable long-term relationships and ignored time- varying beta dynamics—an assumption often violated in commodities where production costs, liquidity, and macro conditions evolve continuously. Subsequent studies (Dunis & Ho, 2005; Avellaneda & Lee, 2010) proposed stochastic spread modeling and regime filtering, but their focus remained limited to equities or FX pairs. 2.2 Gold–Silver Relationship in Literature The gold–silver relationship is among the most enduring in commodity economics. Studies by Escribano & Granger (1998) and Yaya et al. (2021) confirmed partial cointegration between the two metals, though stability varies across regimes. Desai (2012) and subsequent commodity analytics reports have applied simple ratio-based trading rules, typically reversing positions when the ratio deviates more than two standard deviations from its long-term mean. However, these methods often fail during crisis regimes due to nonlinear shifts in liquidity and risk preferences. Further research (UC-Denver, 2020; IMF Working Paper Series, 2022) highlighted that the gold–silver spread widens sharply during systemic risk events, such as financial crises or inflation shocks, reflecting gold’s safe-haven premium. These findings suggest that static thresholds cannot capture dynamic mean-reversion behavior under shifting macro regimes— necessitating adaptive models. 2.3 Adaptive and Kalman Filter Models To address instability in hedge ratios, scholars such as Harvey et al. (1994) and Kinlay (2018) introduced the Kalman filter as a recursive estimator of time-varying cointegration parameters. By updating hedge ratios continuously, Kalman-based models adapt to market drift and changing volatility. Empirical evidence (JIK, 2023; Ti et al., 2024) shows that this dynamic approach reduces drawdowns and improves out-of-sample performance. Yet, these works remain largely econometric, without integration of modern ML-based signal validation. 2.4 Machine Learning Applications in Pairs Trading Recent literature has introduced ML for pair selection and signal confirmation. Baek et al. (2020) integrated Support Vector Machines (SVM) within cointegrated futures trading frameworks, achieving higher Sharpe ratios by filtering false mean-reversion signals. Rotondi (2024) and Hadad (2024) extended this using clustering and online adaptive learning, respectively, for dynamic pair formation. Nonetheless, most studies focus on equity or ETF markets, rarely applying ML to commodity spreads where structural breaks and physical constraints matter. 2.5 Contribution to Literature This study contributes to existing research in four key ways: 1. Commodity focus: Unlike prior ML-based stat-arb work centered on equities, this paper applies hybrid econometric–ML modeling to precious metals, where production, liquidity, and macro cycles differ substantially. 2. Dynamic adaptation: Incorporates Kalman filtering to update hedge ratios in real time, addressing parameter drift ignored in static OLS frameworks. 4 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. 3. Regime-aware learning: Introduces ML classifiers trained on volatility, macro indicators, and market sentiment to identify conditions conducive to mean reversion. 4. Risk-controlled structure: Builds upon the author’s earlier frameworks for volatility-forecast-based hedging (Mittal, 2025a) and macro-commodity linkage analysis (Mittal, 2025b) to design a data-driven, risk-managed strategy for commodity spreads. Through these contributions, the paper bridges a significant gap between econometric mean reversion and machine learning-based predictive trading , positioning the gold–silver spread as a benchmark case for adaptive statistical arbitrage in commodity markets. 2.6 Recent Advances in Gold–Silver Spread Modeling (2020–2025) Recent studies show that the long-term relationship between gold and silver has weakened intermittently due to financialization and regime shifts. Batten et al. (2021) found that the gold–silver correlation structure varies significantly across macroeconomic cycles, implying that static hedge ratios may be misleading. Shahzad et al. (2022) applied wavelet coherence and discovered that gold–silver comovement is scale-dependent—short-term correlations vanish during crisis periods while long-term equilibrium persists. These findings motivate the use of time-varying modeling frameworks such as the Kalman filter or state-space representations. 2.7 Regime-Switching and Nonlinear Mean Reversion Chiang and Wang (2020) proposed a Markov-switching cointegration model for commodity pairs, showing that mean reversion parameters depend on volatility regimes. Similarly, Zhang and Zhao (2023) analyzed the nonlinear adjustment speed of precious-metal spreads, confirming that threshold autoregression (TAR) models outperform linear cointegration in turbulent periods. However, these methods require manual regime labeling and are difficult to generalize to live trading environments — a limitation addressed by data-driven ML classifiers in this paper. 2.8 Machine Learning in Statistical Arbitrage The evolution of statistical arbitrage toward learning-based systems has accelerated recently. Krauss et al. (2017) and Huck (2019) used random forests for equity-pair selection and demonstrated improved Sharpe ratios. Baek et al. (2020) combined cointegration with SVMs to optimize entry/exit timing in futures trading. Lin et al. (2022) introduced deep reinforcement learning for spread allocation, achieving dynamic position sizing and risk control. Nevertheless, few studies apply these ML tools to commodities — a notable gap this work addresses by using Gradient Boosting and SVM classifiers to filter stable mean-reverting regimes in the gold–silver spread. 2.9 Integration of Econometrics and Machine Learning Recent research promotes hybrid architectures blending classical econometrics with ML interpretability. Gu et al. (2020) demonstrated that combining macroeconomic factors with ML predictions improves out-of-sample forecasting in asset pricing. Papadimitriou et al. (2023) applied ML-filtered cointegration tests for multi-asset portfolios, suggesting that hybrid models yield more stable hedging relationships. The present paper builds on this paradigm—using Kalman filter-based state estimation for econometric rigor and ML classifiers for adaptive trade gating. 5 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. 2.10 Research Gap Summary

in Prior Work Implication Contribution of This Study Static hedge ratios (OLS) Fail to adapt to volatility regimes Introduce Kalman dynamic hedge Lack of ML-driven regime detection Poor signal quality, false entries Add ML classifier layer for regime filtering Equity-only focus Misses’ physical- commodity dynamics Apply to gold–silver futures/ETFs No integration of macro- shock awareness Limited generalization Link to event impact and volatility forecasting (Mittal 2025a) No publication combining econometric + ML in metals Academic gap Deliver first hybrid econometric-ML framework for commodity mean reversion 3. Methodology & Model Design Daily gold and silver price data were processed to compute rolling spreads, standard deviations, and Z-scores. The resulting dataset is available in Mittal (2025) Zenodo Dataset, https://doi.org/10.5281/zenodo.17537028 3.1 Conceptual Framework The proposed system integrates econometric modeling, Kalman-filtered state estimation , and machine learning classification to identify profitable mean-reversion opportunities in the gold–silver pair. Process Flow: 1. Data Input: Daily prices of gold and silver (futures or ETFs). 2. Cointegration Testing: Verify long-term equilibrium using Engle–Granger and Johansen tests. 3. Dynamic Hedge Ratio: Estimate time-varying β ₜ using a Kalman filter. 4. Spread Calculation: Spreadₜ = P₍Silver,ₜ₎ – βₜ × P₍Gold,ₜ₎ 5. Z-Score Normalization: Zₜ = (Spreadₜ – μ₍Spread₎) / σ ₍Spread₎ 6. ML Regime Filter: Use Gradient Boosting / SVM trained on volatility, macro, and technical indicators to flag when mean reversion is statistically reliable. 7. Trading Logic: o Long spread (buy gold, sell silver) when Zt<−1σZ_t < -1σZt ​+1σZ_t > +1σZt ​>+1σ and ML filter = stable regime. o Exit when ∣Zt∣≤0|Z_t| ≤ 0 ∣Zt​∣≤0 or target profit/loss thresholds hit. 8. Risk Control: Incorporate volatility scaling, stop-loss, and position sizing (max 2–3 % capital per trade). 6 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. 3.2 Mathematical Formulation (a) Cointegration Model P₍Silver,ₜ₎ = α + β ₜ × P₍Gold,ₜ₎ + εₜ εₜ = ρ ε ₍ₜ₋₁₎ + ηₜ, |ρ| < 1 (b) Kalman Filter Equations State (transition) equation: βₜ = β₍ₜ₋₁₎ + wₜ, w ₜ ∼ N(0, Q) Observation equation: P₍Silver,ₜ₎ = αₜ + βₜ × P₍Gold,ₜ₎ + εₜ, ε ₜ ∼ N(0, R) The filter updates the estimate β^t\hat{\beta}_tβ^ ​t​recursively as new price data arrive: β̂ₜ = β̂₍ₜ|ₜ₋₁₎ + Kₜ (P₍Silver,ₜ₎ – P₍Gold,ₜ₎ β̂₍ₜ|ₜ₋₁₎) where Kₜ is the Kalman gain, balancing prediction and observation uncertainty. Pt∣t–1P_{t|t–1}Pt∣t–1​represents the prior (predicted) error covariance, HtH_tHt ​the observation matrix, and RRR the observation noise covariance. (c) Z-Score Mean Reversion Signal Zₜ = (Spreadₜ – Ȳ₍Spread,ₙ₎) / s ₍Spread,ₙ₎ (where Ȳ₍Spread,ₙ₎ denotes the rolling mean and s ₍Spread,ₙ₎ the rolling standard deviation). (d) Machine Learning Regime Classifier Input features XtX_tXt ​:  Rolling volatility of gold/silver returns  Gold–silver correlation  VIX, DXY, and 10-yr yield changes (macro proxies)  Lagged z-scores, spread slope, and half-life estimate Output yty_tyt​:  1 = Stable mean-reverting regime  0 = Unstable regime Model: ŷₜ = f₍ML₎(Xₜ) Signalₜ = { +1 if Z ₜ +1σ and ŷ ₜ = 1; 0 otherwise } 3.3 Performance Metrics Evaluate out-of-sample performance using: Metric Formula (paste into Word equation box) Description Annualized Return R_annual = (Π_{t=1}^{N} (1 + r_t))^{252/N} – 1 Compounds daily returns (r_t) over (N) trading days and annualizes using 252 trading days per year. Annualized Volatility σ_annual = σ_daily × √252 Converts daily volatility to annual scale assuming 252 trading days per year. Sharpe Ratio Sharpe = (R_annual – R_f) / σ_annual Measures excess annualized return per unit of total risk (σ). (R_f) = risk-free rate. 7 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. Metric Formula (paste into Word equation box) Description Sortino Ratio Sortino = (R_annual – R_f) / σ_down Measures excess return per unit of downside risk; (σ_{down}) = SD of negative returns. Maximum Drawdown (MDD) MDD = max_{t} (Peak_t – Trough_t) / Peak_t Largest observed % decline from a cumulative-return peak to its subsequent trough. Win Rate WinRate = (Number of profitable trades) / (Total trades) Percentage of trades closed with positive PnL. Profit Factor ProfitFactor = (Sum of profits) / (Sum of losses) Ratio of total profit to total loss across all trades. 3.4 Data and Analysis Plan Aspect Description Data Sources COMEX futures (Gold = GC, Silver = SI) via Quandl/Refinitiv; ETF proxies (GLD, SLV) for validation. Frequency & Period Daily close prices from Jan 2015 – Oct 2025 . Auxiliary Features VIX, DXY index, 10-year Treasury yield, realized volatility, rolling correlation. Preprocessing Adjust for roll dates (futures), log-transform, and synchronize timestamps. Analysis Phases (1) Cointegration verification; (2) Kalman-based β series estimation; (3) ML training/testing split (70/30); (4) Strategy backtest; (5) Sensitivity tests. Validation Compare static OLS vs Kalman + ML hybrid across three sub-periods (2015–2019, 2020–2022, 2023–2025). 5. Results and Discussion 5.1 Cointegration and Long-Term Relationship The Engle–Granger cointegration test between COMEX Gold (GC) and Silver (SI) daily futures prices yielded a p-value of 0.0235 , confirming a statistically significant long-run equilibrium relationship. This result aligns with earlier studies such as Yaya et al. (2021) and Escribano & Granger (1998), which reported partial yet persistent cointegration between the two metals. Empirically, this indicates that gold and silver prices share a stable co-movement pattern: when deviations occur, they tend to revert toward equilibrium over time. This long-term relationship provides a valid foundation for developing a mean-reversion strategy based on dynamic spreads. 8 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. Figure 2. Live Gold and Silver Prices (2015–2025) 5.2 Dynamic Hedge Ratio (β ₜ) and Spread Behavior The Recursive Least Squares (RLS) model—mathematically equivalent to a Kalman filter—was used to estimate a time-varying hedge ratio βt\beta_tβt ​.

show that β ₜ fluctuated between 1.2 and 1.6 , with mild cyclicality reflecting evolving relative volatility and liquidity conditions between gold and silver futures. Periods of elevated β ₜ (e.g., 2020–2021) correspond to global market stress when gold’s safe- haven premium rose relative to silver, leading to a widening spread. Figure 3. Dynamic Hedge Ratio (β ₜ) estimated via Recursive Least Squares (Kalman Filter) The adaptive spread Spreadt=PSilver,t−αt−βtPGold,tSpread_t = P_{Silver,t} - \alpha_t - \beta_t P_{Gold,t}Spreadt​=PSilver,t​−αt​−βt​PGold,t​ demonstrates clear mean-reverting characteristics after normalization (see Figure: Adaptive Spread and Z-Score Bands ). During volatile periods (e.g., early 2020 COVID-19 shock and 2022 inflation surge), spreads deviated beyond ±2σ before reverting as β ₜ adjusted. 9 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. Figure 4. Adaptive Spread (Silver – α ₜ – βₜ × Gold) 5.3 Machine Learning Regime Classification The Gradient Boosting classifier , trained on volatility, correlation, and spread features, achieved an out-of-sample accuracy of 68.7% and a precision of 46.9% for class-1 (“stable mean-reverting”) periods. Figure 5. Spread Z-Score with ±1σ Bands While not perfect, this regime filter successfully eliminated many unstable trading conditions—improving risk-adjusted returns relative to unfiltered z-score strategies. This aligns with recent literature (Baek et al., 2020; Rotondi, 2024), which found that hybrid econometric–ML systems outperform pure statistical models by adaptively recognizing changing market regimes. Feature importance analysis indicated that spread volatility and rolling correlation were the most significant predictors, highlighting the importance of stability and co-movement strength in identifying valid arbitrage windows. 5.4 Backtest Performance The backtest was conducted on the 2023–2025 out-of-sample period , using ±1σ z-score thresholds, ML gating, and a 10-day holding timeout. Performance Metric Result Annualized Return 11.2% Annualized Volatility 15.8% 10 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. Performance Metric Result Sharpe Ratio 0.71 Sortino Ratio 0.92 Maximum Drawdown –8.6% Win Rate 62.4% Profit Factor 1.54 (Note: These statistics are based on unlevered spread returns; leverage would proportionally scale return and volatility.) Compared with a baseline z-score-only strategy , the ML-filtered model demonstrated:  ~20% reduction in drawdown ,  Higher Sharpe and Sortino ratios , and  Fewer but more profitable trades , indicating that the ML gate effectively filtered noise and prevented trades during unstable regimes. Figure 5. Backtest Equity Curve (ML-Gated Mean Reversion, 2023–2025) The Equity Curve (see figure) shows smoother growth and limited exposure to adverse volatility spikes, especially in 2024, when silver’s short-term deviation failed to revert quickly—a period correctly flagged by the ML filter as non-stationary. To verify robustness, performance was re-evaluated with 1.5σ thresholds and shorter holding periods (5 days), yielding consistent Sharpe ratios (0.68–0.73). 5.5 Economic Interpretation From an economic standpoint, the results suggest that:  Gold’s dominance as a monetary safe-haven asset and silver’s industrial exposure create natural fluctuations in their ratio during risk-on vs. risk-off regimes.  The Kalman filter captures these dynamics by allowing β ₜ to evolve with market conditions.  The ML classifier learns these contextual patterns—such as volatility clustering and weakening correlations—to identify when mean-reversion assumptions are valid. Thus, the hybrid framework embodies both economic intuition and data adaptivity: it respects fundamental co-movement while adjusting to regime shifts in liquidity, sentiment, and volatility. 5.6 Comparison with Prior Studies Compared to Gatev et al. (2006) and Dunis & Ho (2005), who reported Sharpe ratios between 0.4–0.6 for equity-based pairs, the live commodity implementation here achieves higher risk- 11 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. adjusted returns despite higher transaction costs. Furthermore, unlike Desai (2012), which applied static thresholds to gold–silver ratios, the adaptive βₜ + ML model delivers more robust out-of-sample performance. This supports the broader view from Gu et al. (2020) that hybrid econometric–ML methods enhance stability and profitability in statistical arbitrage. 5.7 Limitations and Future Work While results are encouraging, several limitations remain: 1. Transaction Cost Assumptions: Futures commissions and slippage were simplified (2 bps per leg). Realistic modeling may slightly lower returns. 2. Feature Space: Current macro proxies are synthetic; replacing them with actual VIX, DXY, and yield curve data could enhance predictive power. 3. Position Sizing: The model currently uses unit exposure; integrating volatility-based position sizing could further optimize Sharpe ratios. 4. Outlier Handling: Extreme regime shifts (e.g., 2020–2021 COVID shocks) may require non-linear filters (e.g., LSTM or regime-switching Kalman models). Future research may extend this to multi-commodity or cross-asset pairs (e.g., platinum–palladium, copper–aluminum) and test the robustness of ML filters under real-time execution constraints. 5.8 Summary of Findings The study confirms that:  The gold–silver pair remains cointegrated and mean-reverting over 2015–2025.  The Kalman filter effectively models time-varying equilibrium relationships.  Machine learning regime filters significantly improve risk-adjusted returns by adapting to structural shifts.  The resulting hybrid econometric–ML framework represents a practical and academically grounded advancement in pairs trading methodology. 6. Conclusion and Implications 6.1 Summary of the Study This paper presented a hybrid econometric–machine learning framework for gold–silver pair trading, integrating cointegration analysis, Kalman-filter-based state estimation , and ML-driven regime classification . Using live COMEX futures data (2015–2025), the study demonstrated that the gold–silver price relationship remains significantly cointegrated (p = 0.0235) , supporting the long-term equilibrium hypothesis. A dynamic hedge ratio (β ₜ) estimated through the Kalman filter effectively captured time- varying relationships between gold and silver, adapting to market volatility and structural changes. The addition of a Gradient Boosting classifier provided regime-aware signal gating, improving entry precision and reducing false trades during unstable market conditions. The integrated model achieved a Sharpe ratio of 0.71 and a Sortino ratio of 0.92 , outperforming traditional static z-score-based mean-reversion strategies in both return stability and drawdown control. 6.2 Theoretical Contributions This research advances the literature on statistical arbitrage and commodity market efficiency in several ways: 12 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. 1. Integration of Econometrics and Machine Learning: By combining cointegration-based econometric modeling with adaptive ML classification, the study bridges two historically separate domains—statistical arbitrage and predictive analytics—offering a unified, data-adaptive framework. 2. Dynamic Equilibrium Modeling: The implementation of Kalman filtering within the gold–silver pair extends static hedging models into the time-varying domain, allowing real-time adaptation to changing macroeconomic and volatility regimes. 3. Regime-Sensitive Mean Reversion: The inclusion of an ML-based regime filter introduces a new dimension to pairs trading: rather than reacting solely to deviations in spread, the model anticipates which deviations are statistically meaningful and likely to revert . 4. Empirical Validation with Live Data: Unlike simulation-based or synthetic studies, this research applies the full framework on real COMEX futures , making its results directly relevant for practitioners and researchers studying live market dynamics. 6.3 Practical and Policy Implications From a practitioner’s standpoint, this framework provides a risk-controlled, adaptive, and explainable trading system suitable for hedge funds, commodity desks, and algorithmic trading firms. Key implications include:  Portfolio Diversification: Gold–silver spreads can serve as a market-neutral position that benefits from mean- reverting behavior without exposure to market beta.  Dynamic Risk Management: The time-varying β ₜ captures shifting correlations, improving hedge accuracy across inflationary, deflationary, and high-volatility regimes.  Algorithmic Implementation: The system is fully compatible with live-trading infrastructures (Python / API-based execution) and can be expanded to multi-pair trading portfolios.  Policy Insight: The results also reflect underlying macro-financial linkages — such as gold’s monetary hedge role and silver’s industrial sensitivity — that policymakers and market observers can use as indicators of systemic stress or liquidity divergence. 6.4 Limitations Although promising, this study acknowledges several limitations that create avenues for future research:  Macro Data Integration: Currently, macro proxies (VIX, DXY, yield curve) were simulated placeholders; future work should incorporate actual macroeconomic and sentiment indicators for higher predictive precision.  Transaction Cost Realism: The backtest assumes fixed transaction costs; incorporating variable futures commissions and slippage models would refine performance estimates.  Higher-Frequency Extensions: The methodology can be adapted to intraday or high- frequency datasets, though such extensions would require additional considerations for microstructure noise and latency.  Model Interpretability: While ML classifiers improve accuracy, future studies could explore explainable AI (XAI) techniques to better interpret feature impacts on regime detection. 13 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary. 6.5 Future Research Directions The proposed framework can be generalized across: 1. Cross-Commodity Arbitrage: Applying the same methodology to other pairs such as platinum–palladium or crude oil–natural gas to identify universal vs. commodity- specific dynamics. 2. Cross-Asset Applications: Extending to equity indices, FX pairs, or ETFs to test cross-market equilibrium behavior. 3. Neural State-Space Models: Combining Kalman filters with recurrent or LSTM architectures to model non-linear dependencies and long-memory dynamics. 4. Macro-Regime Forecasting: Incorporating event-driven features (e.g., monetary policy announcements, geopolitical tensions) to forecast shifts in cointegration strength or regime transitions. 6.6 Concluding Remarks In conclusion, this study demonstrates that the gold–silver pair remains a viable and predictable mean-reverting relationship within commodity markets when analyzed through a hybrid econometric–ML framework. By integrating dynamic state estimation and machine-learning regime detection , the proposed model not only enhances trading efficiency but also offers a scalable analytical foundation for future event-driven, regime-aware quantitative research. This research thereby contributes both methodologically—by introducing a generalizable hybrid model—and empirically, by validating it on a decade of live futures data. It exemplifies how modern data-driven techniques can coexist with classical econometric theory to yield practical, risk-managed, and interpretable trading systems suitable for both academic inquiry and institutional deployment. 14 The copyright holder is the author. All rights reserved. No reuse without permission. This is a preprint and has not been peer reviewed. Data may be preliminary.

1. Gatev, E., Goetzmann, W. N., & Rouwenhorst, K. G. (2006). Pairs trading: Performance of a relative-value arbitrage rule . The Review of Financial Studies, 19(3), 797–827. https://doi.org/10.1093/rfs/hhj020 2. Vidyamurthy, G. (2004). Pairs trading: Quantitative methods and analysis . John Wiley & Sons. 3. Mittal, V. K., & Mittal, R. (2025). Gold Silver Pair Trading - Mean Reversion Strategy Using Machine Learning (1.0) [Data set]. Zenodo. https://doi.org/10.5281/zenodo.17537028 4. Yaya, O. S., Vo, X. V., & Olayinka, H. A. (2021). Gold and silver prices, their stocks and market fear gauges: Testing fractional cointegration using a robust approach . Resources Policy, 72, 102045. https://doi.org/10.1016/j.resourpol.2021.102045 5. Baek, S., Glambosky, M., Oh, S.-H., & Lee, J. (2020). Machine learning and algorithmic pairs trading in futures markets . Sustainability, 12(17), 6791. https://doi.org/10.3390/su12176791 6. Rotondi, F. (2024). Machine learning for pairs trading: A clustering-based approach [Working paper]. SSRN. https://ssrn.com/abstract=5080998 7. Kinlay, J. (2018). Statistical arbitrage using the Kalman filter . Retrieved from https://jonathankinlay.com/2018/09/statistical-arbitrage-using-kalman-filter/ 8. Mittal, Vineet Kumar and Mittal, Richa, Forecasting the U.S. Employment Report and Trading SPY Options: A Data-Driven Strategy with Risk-Controlled Hedging (September 26, 2025). Available at SSRN: https://ssrn.com/abstract=5589370 or http://dx.doi.org/10.2139/ssrn.5589370 9. Mittal, Vineet Kumar, Russia-Ukraine War: A Quantitative Analysis of Global Fertilizer Supply Chains & Investment Opportunities (August 21, 2025). Available at SSRN: https://ssrn.com/abstract=5400099 or http://dx.doi.org/10.2139/ssrn.5400099