Regime-Conditional Distributional Comparison of Trading Strategies: A GAMLSS/ZAGA Framework Applied to the S&P 500

Abstract: Conventional comparisons of algorithmic trading strategies reduce each performance metric to a single number over the full backtest horizon, thereby discarding information about how performance varies with market conditions. This paper proposes a distributional framework that addresses this shortcoming. A walk-forward backtest of 146 out-of-sample folds on the S&P 500 (2002--2025) is used to compute the Adjusted Information Ratio ($IR^{\ast}$) for a polynomial Support Vector Machine strategy (SVMP) and a buy-and-hold benchmark (BH) in each fold. The resulting $IR^{\ast}$ sequences are modelled jointly via a Generalised Additive Model for Location, Scale and Shape (GAMLSS) with a Zero-Adjusted Gamma (ZAGA) response, with distributional parameters conditioned on market regime covariates: realised volatility and cumulative market momentum. Strategy comparison is conducted through (i) regime-specific differences in expected $IR^{\ast}$ ($ΔE$) and its variance ($ΔVar$), derived analytically from the fitted ZAGA parameters, and (ii) parametric bootstrap tests of three null hypotheses concerning $E(IR^{\ast})$, $Var(IR^{\ast})$, and their ratio, evaluated at six representative market regimes. The results demonstrate that the dominance relationship between SVMP and BH is conditional on market regime. The proposed GAMLSS/ZAGA framework constitutes a methodologically rigorous and practically interpretable alternative to conventional strategy evaluation.

• The paper introduces a GAMLSS/ZAGA framework that conditions trading performance on market regimes.
• It compares a polynomial SVMP strategy against a buy-and-hold benchmark using detailed walk-forward backtesting.
• Empirical findings reveal regime-specific shifts in risk-adjusted returns, informing adaptive strategy selection.

Regime-Conditional Distributional Evaluation of Trading Strategies via GAMLSS/ZAGA on the S&P 500

Introduction

This study addresses a central deficiency in algorithmic trading strategy evaluation: the reduction of performance assessment to single aggregate metrics, thus completely obscuring the regime-dependent variability and distributional heterogeneity in strategy outcomes. By leveraging a Generalised Additive Model for Location, Scale, and Shape (GAMLSS) with a Zero-Adjusted Gamma (ZAGA) response, the paper introduces a framework in which the distributional parameters of strategy performance metrics are explicitly conditioned on market regime variables. This approach is demonstrated by comparing a polynomial Support Vector Machine (SVMP) strategy to a buy-and-hold (BH) benchmark over the S&P 500 index from 2002 to 2025, using walk-forward backtesting across 146 non-overlapping folds.

Methodological Framework

The empirical protocol consists of three tightly integrated components:

1. Walk-Forward Backtesting: Both SVMP and BH are evaluated across 146 out-of-sample windows, each comprising 40 trading days, providing a granular, fold-level time series of realized performance.
2. Adjusted Information Ratio (IR*) Metric: Each strategy’s risk-adjusted return is quantified via IR*, an absolute, strictly non-negative metric that penalizes both high return dispersion and maximum drawdown. This measure induces a structurally mixed distribution: a discrete mass at zero, alongside a continuous positive component.
3. GAMLSS/ZAGA Modelling: The per-fold IR* sequences for both strategies are pooled and modelled jointly. Distributional parameters—location ($\mu$), scale ($\sigma$), and zero-mass ($\nu$)—are regressed on two key regime covariates: realized volatility and 40-day momentum, allowing regime-specific inference on the full distribution of strategy performance.

Critical to the methodology is the use of dummy-variable interaction terms to capture not only level shifts but also differential sensitivities of each strategy to market regimes within each distributional parameter.

Statistical and Bootstrap Analysis

Fitted GAMLSS/ZAGA models yield regime-specific analytical expressions for the mean and variance of IR*, which are then evaluated at a grid of (momentum, volatility) regimes aligning with empirical quantiles of the observed covariates. Three parametric bootstrap tests are conducted:

• Difference in mean IR* between strategies
• Difference in variance of IR*
• Difference in risk-reward ratio, i.e., the mean/variance ratio of IR*

These bootstrap procedures fully exploit the conditional distributional model, generating synthetic samples under each regime and explicitly maintaining the observed regime-mix and dependence structure.

Empirical Findings

A formal regime-stratified analysis leads to several notable observations:

• Conditional Dominance: The SVMP strategy exhibits meaningfully higher expected IR* than BH in regimes of low or negative momentum. The bootstrapped p-values for differences in means are all 1.00 (out of 9,999 draws) from the minimum through the sample mean of momentum, decisively supporting SVMP’s dominance in these conditions.
• Reversal in High-Momentum Regimes: This dominance qualitatively reverses at the highest momentum quantile, where BH outperforms SVMP by a substantial margin (difference in expected IR* = -0.022). Here, high positive momentum, which rewards directional exposure, benefits the passive benchmark.
• Variance Comparison: SVMP yields higher IR* variance than BH in most regimes, as borne out by bootstrapped p-values. The sole reversal occurs at maximum momentum, driven by sporadic large positive BH returns.
• Signal-to-Noise (Risk-Adjusted Return): Despite the mean advantage of SVMP in specific regimes, the higher variance results in a less favorable mean/variance ratio, with the null of BH superiority in this metric being rejected from the median momentum regime upwards.

Of technical significance, the GAMLSS/ZAGA model identifies near-complete separation for the zero-mass parameter at the lowest momentum, reflecting the empirical observation that BH experiences virtually zero positive IR* in deeply negative momentum windows.

Implications and Extensions

This work fundamentally alters the interpretive lens for trading strategy comparison, emphasizing the centrality of distributional variation and regime dependence over aggregate metrics. Practically, this allows for adaptive strategy selection mechanisms: tile selection of the trading approach can be conditioned on prevailing estimates of market momentum and volatility. Theoretically, the use of GAMLSS/ZAGA frames trading performance within the framework of distributional regression, facilitating both explainable AI (XAI) analysis (via coefficient interpretability) and robust simulation-based scenario testing.

Potential extensions include using smooth, non-linear regime effects, inclusion of macro-financial or latent regime variables, and the integration of trading frictions directly in the distributional model. Application to multi-strategy and multi-asset portfolios is immediate. The regime-conditional structure naturally enables stress testing, providing a quantitative mechanism for understanding strategy failure modes under adverse market conditions.

Conclusion

The regime-conditional distributional comparison framework introduced in this study offers a statistically rigorous and practically interpretable method for trading strategy evaluation. It demonstrates that performance dominance is not absolute but depends on the intersection of strategy design and market regime—a fact that is obscured by conventional scalar metrics. This approach provides both a methodological foundation and an empirical protocol for advancing adaptive, regime-aware algorithmic trading, with significant implications for research in both quantitative finance and the intersection of financial ML and explainable AI.

Citation: "Regime-Conditional Distributional Comparison of Trading Strategies: A GAMLSS/ZAGA Framework Applied to the S&P 500" (2606.31251)