Technical Summary: Period-Aware Analog Retrieval for Probabilistic Financial Risk Forecasting

1. Problem Statement

The central challenge addressed is the accurate forecasting of the full distribution of financial returns, particularly the tail risk required for Value-at-Risk (VaR) and Expected Shortfall (ES) under frameworks like Basel III. Existing methods suffer from a fundamental blind spot: they typically build forecasts by fitting models to globally pooled historical data without verifying whether past observations are structurally comparable to the current market environment.

Three primary limitations in current literature are identified:

1. Parametric Methods: Models like GARCH, CAViaR, and quantile regression impose functional form assumptions that often break down in the tails or fail to capture regime shifts (Engle and Manganelli 2004; Ghysels et al. 1996).
2. Naive Analog Methods: Traditional analog retrieval uses Euclidean distance in raw feature space, leading to "regime mixing" (e.g., treating a 2009 crisis day as similar to a 2017 low-volatility day) and "calendar blindness" (ignoring seasonal heterogeneity), resulting in miscalibrated tail estimates (A¨ıt-Sahalia and Lo 1998; Brock et al. 1992).
3. Deep Learning Limitations: While architectures like LSTMs and Transformers learn powerful representations, they are optimized for pointwise prediction error rather than preserving the shape of the conditional distribution or structural similarity between market episodes (Hochreiter and Schmidhuber 1997; Vaswani et al. 2017).

The paper posits that distributional forecasting should be reframed as a retrieval problem: identifying past market days that resemble the current state in terms of macroeconomic regime, time of year, and feature structure, and using their realized outcomes to construct the forecast distribution.

2. Methodology: Period-Aware Analog Retrieval (PAAR)

The proposed framework, PAAR, treats distributional forecasting as a search problem constrained by three filters to ensure retrieved analogs come from genuinely comparable environments.

2.1 Architecture Overview

The pipeline consists of four stages: feature construction, regime identification, encoder training, and retrieval-based distribution estimation.

2.2 Feature Construction

A market feature vector $x_t \in \mathbb{R}^{25}$ is constructed from indicators spanning trend, reversal, volatility, liquidity, momentum, macroeconomic signals, and market breadth. Features are normalized using a rolling z-score over the preceding 252 days to remove secular trends. Calendar metadata (day-of-year, month, etc.) is appended as auxiliary input but excluded from the similarity metric, serving instead as retrieval constraints.

2.3 Regime Identification (Filter 1)

To prevent regime mixing, the archive is partitioned using a four-state Gaussian Hidden Markov Model (HMM). The HMM is fitted on a five-dimensional macroeconomic vector (10-day realized volatility, yield curve slope, 10-year bond yield change, FX change, and 60-day momentum) to capture the macroeconomic backdrop rather than surface-level price patterns.

• Constraint: Only archive observations assigned to the same inferred regime as the query date are eligible.
• Adaptation: If the current regime is sparsely represented, the search expands to adjacent regimes ranked by HMM transition probability.

2.4 Encoder Training

To address metric inadequacy (where Euclidean distance fails to capture structural similarity), a compact three-layer MLP encoder maps $x_t$ to a 32-dimensional embedding $e_t$. The encoder is trained with a composite loss:

• Outcome Alignment Loss: Pulls together days with similar realized return quintiles and pushes apart those with different quintiles within the same regime.
• Regime Separation Loss: Ensures embeddings from different regimes remain at least a margin of 2.0 apart.

2.5 Three-Filter Retrieval Cascade

Given a query date $t$, the retrieval process applies three successive filters:

1. Regime Gate: Restricts candidates to the same HMM regime.
2. Period-Aware Window: Within the regime-filtered pool, retains only observations where $|\ell_s - \ell_t| \le \delta$ (default $\delta=30$ days), ensuring seasonal comparability (e.g., matching late October to late October).
3. Multi-Facet Similarity: The remaining candidates are ranked by a weighted score $\phi(x_t, x_s)$ combining:
   • Price-pattern shape (trend/reversal).
   • Macroeconomic alignment.
   • Volatility profile.
   • Learned embedding distance ($\phi_{embed}$).
   • Note: The number of neighbors $k$ contracts in high-volatility regimes to preserve distributional sharpness.

2.6 Distribution Estimation

The forecast distribution $\hat{F}_{t+h}(y)$ is constructed as a weighted empirical CDF of the realized forward returns of the top-$k$ analogs.

• Smoothing: A Gaussian kernel density estimation (KDE) is applied.
• Fallback: In stressed regimes with small analog pools, the distribution falls back to a Student-t fit via maximum likelihood to accommodate heavy tails.
• Calibration: A rolling quantile recalibrator (isotonic regression) corrects systematic coverage errors over a 252-day window, specifically targeting tail quantiles.

3. Key Contributions

The paper makes three specific contributions:

1. A Regime- and Period-Aware Framework: A structured retrieval system that jointly addresses regime mixing, seasonal confounding, and metric inadequacy, shifting the locus of model error from parametric form to the quality of the retrieved neighbor set.
2. A Learned Similarity Metric: An embedding space trained specifically to align geometry with distributional outcomes (return quintiles) rather than pointwise prediction error.
3. An Evaluation Protocol: A comprehensive assessment using proper scoring rules (WQL, CRPS), tail risk calibration (VaR/ES), and economic performance (Sharpe ratio) across multiple market cycles.

4. Experimental Results

The framework was evaluated on daily S&P 500 data from January 1995 to March 2026 (6,430 test days), using a walk-forward backtest against ten baselines (classical, ML, deep learning, and foundation models).

4.1 Distributional Accuracy

• PAAR-RAF (incorporating a foundation model embedding as a similarity facet) and PAAR-KDE achieved the best Weighted Quantile Loss (WQL) and Continuous Ranked Probability Score (CRPS).
• Both PAAR variants were the only models to achieve a Mean Absolute Scaled Error (MASE) below 1.0, indicating consistent improvement over a naive benchmark.
• Statistical significance tests (Diebold-Mariano) confirmed PAAR-RAF significantly outperformed all baselines, including classical ARMA/ARIMA models.

4.2 Tail Calibration and Risk Management

• VaR Coverage: PAAR-KDE remained close to the 10% target VaR exceedance rate and was not rejected by Kupiec's test, whereas many ML and deep learning baselines exhibited severe undercoverage (e.g., Random Forest at 24%, Chronos at 25.1%).
• Economic Performance: PAAR variants achieved superior Sharpe ratios compared to deep learning baselines. PAAR-KDE accumulated the highest terminal return over the 26-year window and sustained positive trajectories through the 2008 and 2020 crises.
• Foundation Models: Zero-shot deployment of the Chronos foundation model resulted in severe tail miscalibration. However, using its embeddings as a retrieval facet (PAAR-RAF) yielded the best overall performance, suggesting foundation models are valuable as representation learners but not as direct probabilistic forecasters in this context.

4.3 Ablation Study

Removing any single component of the three-filter cascade degraded performance:

• No Multi-Facet Similarity: Largest degradation in WQL/CRPS, confirming representation quality is the primary driver of accuracy.
• No Regime Gate: Reduced Sharpe ratio, indicating that unconstrained retrieval introduces noise from structurally incompatible states.
• No Period Filter: Slightly improved WQL but significantly reduced Sharpe, showing that seasonal alignment is critical for tail behavior and risk-adjusted returns, even if aggregate metrics miss it.

5. Significance and Claims

The paper claims that reframing financial forecasting from function approximation to conditional distribution retrieval offers two practical advantages:

1. Avoidance of Misspecification: By constructing forecasts directly from realized outcomes under comparable conditions, the method avoids imposing parametric families on return distributions and avoids the need for global models to extrapolate across heterogeneous regimes.
2. Regulatory and Actionable Utility: The approach produces interpretable forecasts (supported by named historical analogs) and maintains tail coverage close to nominal targets across multiple crises. This directly addresses regulatory requirements for capital adequacy (Basel III) where accurate tail calibration is paramount.

The authors note limitations, including the restriction to a single equity index (S&P 500), potential circularity in HMM regime estimation, and the assumption that the historical archive contains sufficient analogs for every regime-calendar combination. They suggest future work could extend the retrieval archive to multiple asset classes and explore online retrieval updates.