Learning Objectives
- distinguish direct features from model-based features and judge when a fitted procedure adds useful information beyond
- use fitted procedures to extract forecasts, filtered states, residuals, conditional volatility, regime probabilities,
- design a compact, interpretable set of model-based features from diagnostics, signal transforms, volatility models,
- enforce point-in-time correctness by fitting and selecting models within training windows, using filtered rather than
- transform asset-level temporal outputs into cross-sectional, benchmark-adjusted, pairwise, and universe-level features
- distinguish between exploratory time-series methods that are useful for research diagnosis and deployable features
- use uncertainty and regime outputs primarily as conditioning features, and recognize when they should not be treated
Diagnostics and stationarity features
Transforming signals to uncover hidden structure
Volatility Features
Uncertainty features
Regime features
Cross-sectional and panel features
Summary
Related Case Studies
See where these chapter concepts get applied in end-to-end trading workflows.
ETF Cross-Asset Exposures
All six model families compared across 100 ETFs spanning 9 asset classes
Crypto Perpetuals Funding
Alternative data and non-standard frequencies in 24/7 crypto markets
NASDAQ-100 Microstructure
Intraday microstructure signals across 114 stocks at 15-minute frequency
S&P 500 Equity + Option Analytics
Combining options-derived features with equity data for multi-source prediction
FX Spot Pairs
Momentum and carry factors in the world's most liquid market
CME Futures
Carry signals across 30 products — data quality as the critical variable
S&P 500 Options (Straddles)
Direct options trading and why equity-style cost models fail for options
US Equities Panel
Large-scale cross-sectional prediction across 3,200 stocks with 16 walk-forward folds
01 Visual Diagnostics
02 Structural Breaks
03 Fractional Differencing
04 Kalman Filter
05 Spectral Features
06 Path Signatures
07 Arima Features
08 Garch Volatility
09 Har Rough Volatility
10 Uncertainty Features
11 Hmm Regimes
12 Wasserstein Regimes
13 Regime As Feature
14 Panel Features
Case Study Temporal Summary
29 primer topics providing foundational concepts for this chapter.
ACF and PACF Interpretation
How total and direct lag dependence leave different signatures in time-series plots, and why those signatures are useful but never definitive.
Autoregressive, Moving-Average, and ARIMA Foundations for Feature Engineering
ARIMA is rarely the star predictor in liquid markets, but it is still one of the cleanest ways to separate level, persistence, shock, and forecast uncertainty before downstream models take over.
Bayesian Inference and MCMC for Time Series
A Bayesian time-series model produces a posterior distribution, not just a fitted line, which is why posterior uncertainty can itself become a feature.
Bayes' Theorem and Posterior Distributions
Bayesian inference updates a probability distribution over unknown quantities rather than replacing uncertainty with a single estimate.
Block Bootstrap and Permutation Testing for Dependent Data
Once observations have an order, valid resampling must preserve more than the one-period distribution.
Bootstrap Methods for Dependent Financial Time Series
Bootstrap paths are useful only if they preserve the dependence structure your downstream metric actually cares about.
Cointegration
Two price series can drift for years and still be tied together by a stationary spread. Cointegration is the formal way to express that hidden long-run anchor.
Fourier Transform Foundations
A finite time series can be rewritten exactly as a combination of cycles; the frequency-domain view is often the clearest way to see periodic structure, but it is easy to misuse in non-stationary financial data.
Fractional Differencing: Keeping Memory Without Keeping the Unit Root
How to turn differencing from a blunt preprocessing step into a tunable filter that trades stationarity against memory retention.
HAC Standard Errors and Robust Inference
When forecast errors have memory, the coefficient may stay the same while the t-statistic should not.
Hypothesis Testing and P-Values
How hypothesis tests turn noisy evidence into a structured decision, and how to read p-values without treating them as proof.
Markov Chains and the Markov Property
How a state representation can compress history into current conditions, and why that matters for regime models and decision processes.
Microstructure Noise and Bid-Ask Bounce
Tick-level predictability often reflects how trades print through the spread, not an exploitable signal about the efficient price.
Path Signatures and Log-Signatures for Financial Sequences
Two windows can end at the same price with similar volatility and still trace different paths; signatures are one way to encode that ordered shape.
Point-in-Time Data and Decision-Time Correctness
A value is usable only if it was actually knowable when the strategy had to decide.
Range-Based Volatility Estimators from OHLC Data
High and low prices reveal intrabar dispersion that the closing price alone cannot, but each OHLC estimator is only better when its assumptions match the bar structure you actually trade.
Regime Models for Feature Engineering: HMMs, Markov Switching, and Distributional Clustering
Chapter 9 already makes the operational point about filtered regime probabilities. This primer narrows to the more technical layer: the filtering recursion, state uncertainty, and the identification problems that make regime models easy to misuse.
State-Space Models and Kalman Filtering for Feature Engineering
How linear Gaussian state-space models turn noisy time series into point-in-time features such as level, trend, innovation, and uncertainty.
State Space Models from Kalman Intuition to Mamba
A latent-state view of long-context sequence models, and what "selective" memory actually changes.
Stationarity and Unit Roots
Why time-series stability matters, why random walks mislead regressions, and what differencing fixes and destroys.
Stationarity Tests: ADF, KPSS, and Rolling Stability Signals
These tests are useful in trading because they summarize how stable a series looks right now, not because they can certify once and for all that a process is stationary.
Structural Break Diagnostics and Time-Since-Break Features
How to tell whether one stable model stopped fitting the series, and how to turn that evidence into live-safe features.
Stylized Facts of Financial Time Series for Simulation and Validation
A synthetic market path is only useful if it reproduces the empirical pathologies that make financial returns hard to model in the first place.
Uncertainty as a Feature
Why two identical forecasts can imply very different decisions once you look at what the model does not know.
Volatility Forecasting Mechanics for Risk Control
Volatility is the one financial quantity that is genuinely forecastable at short horizons, and every risk overlay — from position sizing to drawdown limits — depends on getting that forecast roughly right.
Volatility Models as Feature Extractors: GARCH, EGARCH, and HAR
Chapter 9 already teaches what volatility-model outputs to extract. This primer narrows to the harder layer underneath that recipe: persistence geometry, parameter interpretation, and how to tell when the fitted risk state is statistically meaningful rather than just mechanically smooth.
Volatility: Realized, Implied, and Why It Clusters
Volatility is not one number but a family of related objects: what happened, what the options market prices, and what a model forecasts next.
Walk-Forward Validation for Time Series
Why model evaluation must preserve temporal order, and how expanding or rolling splits approximate live deployment.
Wavelets for Multi-Scale Diagnostics and Causal Feature Design
Wavelets are most useful in ML4T when they reveal which horizon matters and then disappear behind a trailing, auditable feature.
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The paper shows how a simple, data-driven two-regime (growth vs. contraction) model—detected via trend filtering on the S&P 500—materially increases estimated left-tail risk for endowments, and how contingent (drawdown-triggered) spending cuts can significantly reduce long-horizon downside risk.
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Factor Models of Asset Returns
Gregory Connor and Robert Korajczyk (2009) · 17 citations
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Christoph Bergmeir et al. (2018) — Computational Statistics & Data Analysis
The paper shows that standard random K-fold cross-validation can be valid for autoregressive time-series forecasting—if (and largely only if) the model’s forecast errors are serially uncorrelated—often making it better than single holdout out-of-sample evaluation.
Markus Pelger (2019) — Journal of Econometrics
This paper develops a statistical theory and practical estimators for large-dimensional factor models using high-frequency financial data, enabling the separation of continuous and jump risk factors and providing a robust method for determining the number of factors.
Jakob Runge et al. (2019) — Science Advances
The paper introduces PCMCI, a two-stage causal discovery method for high-dimensional, autocorrelated, nonlinear time series that improves true-link detection while controlling false positives compared with Granger/FullCI, PC, and Lasso.
Nick Baltas and Robert Kosowski (2019)
Optimizes Time-Series Momentum (TSMOM) by reducing turnover ~36% via efficient volatility/trend estimation and improving post-crisis performance through dynamic correlation-based leverage.
Omer Berat Sezer et al. (2019) — arXiv:1911.13288 [cs, q-fin, stat]
This paper provides a comprehensive systematic literature review of Deep Learning (DL) models applied to financial time series forecasting between 2005 and 2019, highlighting model choices, performance against traditional Machine Learning (ML), and future research directions.
Michael Pinelis and David Ruppert (2022) — The Journal of Finance and Data Science
The paper shows that a utility-maximizing market-timing rule that combines machine-learned forecasts of next-month equity premium and volatility ("reward-risk timing") materially improves Sharpe ratio, drawdowns, and investor utility versus buy-and-hold.
Andrew Ang (2023) — The Journal of Portfolio Management
A regime-based analysis of five major equity style factors showing that Value and Momentum have structurally deteriorated in the 21st century while Quality and Size have improved, decomposing returns into long-term trends and short-term cycles.
Michele Leonardo Bianchi et al. (2023) — The Journal of Portfolio Management · 8 citations
This tutorial-style paper explains why Gaussian (normal) return models underestimate extreme losses and surveys univariate and multivariate fat-tailed alternatives (especially Lévy-process-based) that are more suitable for pricing, risk, and portfolio construction.
Kashif Rasul et al. (2024)
Lag-Llama is a decoder-only transformer foundation model for univariate probabilistic time series forecasting, pretrained on a large, diverse corpus, demonstrating strong zero-shot generalization and state-of-the-art performance after fine-tuning on unseen datasets.
Agathe Sadeghi et al. (2024)
The paper proposes CD-NOTS, a constraint-based causal discovery algorithm tailored to nonstationary time series that explicitly models lagged dependencies, and shows in simulations and finance case studies that it can recover more reliable causal graphs than common baselines when stationarity/linearity assumptions fail.
Ronald Hua and Clare Sun (2024)
The paper proposes a practical framework to quantify how much historical factor information should be trusted out-of-sample under market nonstationarity, showing that optimal “shrinkage toward 1/N” depends strongly on market efficiency.
Miquel Noguer I Alonso (2024)
Conformal prediction is a statistical framework that generates reliable, distribution-free prediction intervals for financial data, enhancing risk management, portfolio optimization, and algorithmic trading, as demonstrated in a banking stress test case study.
Abhimanyu Das et al. (2024)
This paper introduces TimesFM, a decoder-only foundation model for time-series forecasting that achieves near state-of-the-art zero-shot accuracy across diverse datasets by pretraining on a large corpus of real-world and synthetic time-series data.
Zhen Zeng et al. (2024)
The paper turns time series into time–frequency spectrogram images (plus a row encoding the raw series) and shows a Vision Transformer can forecast better than line-plot images and standard baselines across synthetic, weather, and equity-price datasets.
Albert Gu and Tri Dao (2024) · 6280 citations
Mamba introduces a novel selective state space model (SSM) architecture that achieves Transformer-level performance on diverse modalities with linear scaling in sequence length, enabling faster inference and longer context processing.
Kevin J. DiCiurcio et al. (2024) — The Journal of Portfolio Management
A two-stage framework that first uses unsupervised learning to identify market risk regimes (Normal, Correction, Bear) and then uses supervised learning to predict winning equity factors within those regimes, outperforming static and business-cycle approaches.
Gerald Woo et al. (2024)
This paper introduces MOIRAI, a Transformer-based large time series model trained on a new 27B-observation dataset (LOTSA) to achieve competitive zero-shot forecasting across diverse time series, addressing challenges like varying frequencies, arbitrary variates, and diverse distributions.
Mononito Goswami et al. (2024)
MOMENT is an open-source family of large transformer models pre-trained on a diverse collection of time series data, demonstrating strong performance across various time series tasks with minimal fine-tuning, especially in limited supervision settings.
Xu Liu et al. (2024) · 75 citations
This paper introduces MOIRAI-MOE, a time series foundation model that uses sparse Mixture of Experts (MoE) to achieve data-driven, token-level specialization, significantly outperforming existing models in zero-shot forecasting by overcoming limitations of human-imposed frequency-level specialization.
Matt Lutey (2025) — The Journal of Portfolio Management
The paper is an instructional tutorial showing how to convert financial price time series into images (e.g., Gramian Angular Fields and Recurrence Plots) so CNNs can learn chart-like patterns for forecasting, screening, and risk monitoring.
Yong Liu et al. (2025)
This paper introduces Timer-XL, a causal, decoder-only Transformer that generalizes next token prediction to multivariate time series, achieving state-of-the-art unified and zero-shot forecasting performance by effectively handling excessively long contexts and complex inter-series dependencies.
Miquel Noguer i Alonso (2025)
This paper introduces a comprehensive framework for Financial Information Theory, applying information-theoretic concepts to financial time series and demonstrating their empirical usefulness for regime detection, market efficiency testing, and portfolio construction using S&P 500 data from 2000-2025.
Kun Feng et al. (2025)
KAIROS is a novel time series foundation model that uses adaptive tokenization and instance-specific positional embeddings to achieve superior zero-shot forecasting performance with fewer parameters on diverse and heterogeneous time series data.
Xiaoming Shi et al. (2025)
TIME-MOE introduces a scalable, sparse Mixture-of-Experts (MoE) Transformer architecture for time series forecasting, pre-trained on a 300 billion time point dataset, achieving state-of-the-art accuracy and efficiency by scaling to 2.4 billion parameters while reducing inference costs.
Dragos Gorduza et al. (2025) — EPJ Data Science
This paper introduces a novel approach to forecast stock market volatility by using the edge reconstruction accuracy of a graph auto-encoder (GAE) to measure the homogeneity of connections between stocks, showing that lower homogeneity correlates with higher volatility and improves out-of-sample volatility forecasting.
Matthieu Boileau et al. (2025)
The paper shows that basic time-series interpretation (trend, noise, extrema location) can be distilled from a large multimodal model into very small instruction-tuned LMs, enabling lightweight, natural-language explanations of temporal patterns.
Eghbal Rahimikia et al. (2025) · 1 citations
Time Series Foundation Models (TSFMs) show promise for financial forecasting but require domain-specific pre-training on large financial datasets to outperform traditional ML models and achieve economic gains.
Selin Yilmaz and Emre Sefer (2026) — The Journal of Finance and Data Science
This paper explores the application of time-series-based deep learning techniques, including LSTM, Informer, Autoformer, iTransformer, Scaleformer, and Chronos, to pairs trading strategies, demonstrating their potential to outperform traditional statistical arbitrage methods by more accurately predicting residual fluctuations between cointegrated asset pairs.
Philip Hans Franses — International Journal of Forecasting
The paper argues that MASE (Mean Absolute Scaled Error) is a forecast accuracy measure that “plays nicely” with Diebold–Mariano equal-accuracy tests because it preserves the moment conditions needed for asymptotic normality, unlike percentage-error based measures.
Xiaofan Zhou et al. — ACM Comput. Surv.
This survey provides a comprehensive, data-centric overview of Conformal Prediction (CP), a distribution-free uncertainty quantification framework, detailing its foundational concepts, recent advancements, and applications across diverse data modalities like structured, unstructured, and dynamic data, while also discussing open challenges.
Min B. Shrestha and Guna R. Bhatta — The Journal of Finance and Data Science
The paper provides a step-by-step framework for choosing the right econometric method for time-series data (unit roots, cointegration, ECM/ARDL), and shows on Nepal inflation data how naïve OLS/VAR can look “great” yet be spurious.
Zhe Li et al. — Association for Computing Machinery
This paper introduces TSFM-Bench, a comprehensive and unified benchmark for Time Series Foundation Models (TSFMs), evaluating their performance across diverse datasets and forecasting scenarios (zero-shot, few-shot, full-shot) using standardized protocols.