Non-Stationarity

Financial time series have means, variances, and correlations that change over time. A model trained on 2015-2019 low-volatility data will underperform in a 2020 regime shift if it assumes fixed parameters.

The Problem

Global normalization (subtracting the full-sample mean and dividing by the full-sample standard deviation) embeds future information into every observation. It also assumes the distribution is stable, which is false for financial data. Post-2008 interest rates, COVID volatility, and factor decay are all examples of structural shifts that invalidate fixed-parameter assumptions.

A model trained on globally normalized features will overfit to the training regime and degrade when the regime changes.

The Pattern

WRONG

# Global normalization: uses future data and assumes stationarity
X_norm = (X - X.mean(axis=0)) / X.std(axis=0)

CORRECT

import polars as pl

# Expanding normalization: only uses past data, adapts to changing distribution
features = pl.DataFrame({"feat": feat_values, "timestamp": dates})

features = features.with_columns(
    feat_norm=(
        (pl.col("feat") - pl.col("feat").shift(1).cum_mean())
        / pl.col("feat").shift(1).rolling_std(window_size=252)
    )
)

Detection: ADF + KPSS Together

Run both tests. They have opposite null hypotheses, so agreement is strong evidence:

from statsmodels.tsa.stattools import adfuller, kpss

adf_stat, adf_pval, *_ = adfuller(series)
kpss_stat, kpss_pval, *_ = kpss(series, regression="c")

stationary = (adf_pval < 0.05) and (kpss_pval > 0.05)  # both agree

ADF rejects?	KPSS rejects?	Conclusion
Yes	No	Stationary
No	Yes	Non-stationary
Yes	Yes	Trend-stationary (difference first)
No	No	Inconclusive (get more data)

Mitigation Strategies

Approach	When to use	Trade-off
Expanding window	Default safe choice	Slow to adapt, no lookahead
Rolling window (e.g., 252d)	Faster adaptation needed	More variance, loses early data
First differencing	Remove trend/unit root	Loses level information
Regime conditioning	Known structural breaks	Requires regime labels

Guardrails

• X.mean() or X.std() without .expanding() or .rolling() is a red flag in any feature pipeline.
• Shorter rolling windows adapt faster but have higher estimation variance - 126d to 504d is the typical range.
• Test stationarity on raw features before modeling; non-stationary inputs produce unstable coefficients.
• Monitor feature distributions in production - a mean shift > 2 sigma signals model retraining.

Production Implementation

ml4t-diagnostic provides stationarity testing utilities:

from ml4t.diagnostic.evaluation.stationarity import analyze_stationarity

stationarity = analyze_stationarity(feature_series, include_tests=["adf", "kpss"])
print(stationarity.consensus)
print(stationarity.summary_df)

Checklist

• Stationarity tests (ADF + KPSS) run on all features before modeling
• Normalization uses expanding or rolling window, never global statistics
• Rolling window length chosen deliberately (not default)
• Feature distributions monitored for structural breaks in production
• Non-stationary series differenced or transformed before use