Squeeze Kernel Covariance Estimator

A streaming covariance estimator for panels of daily financial returns. One O(n²) update per day, positive semi-definite by construction at every step, missing values handled natively, and defaults that require no tuning. Only dependency: NumPy.

Reference: "The Squeeze Kernel Covariance Estimator: Dual-Timescale Tracking with Adaptive Shrinkage" (Kende, 2026) — SSRN abstract 6455918.

Why

Rolling-window estimators (Ledoit–Wolf, nonlinear shrinkage, RMT denoising) refit over a fixed window each day and cannot adapt within it; multivariate GARCH (DCC) adapts but needs multi-stage estimation and a fragile news coefficient. The Squeeze Kernel is a single streaming recursion that:

• is PSD at every step, structurally — never needs eigenvalue clipping, nearest-PSD projection, or a solver;
• adapts fastest exactly when it matters — a Fisher-information kernel up-weights high-dispersion (stress) days, when correlation regimes actually move;
• regularises itself — an adaptive equicorrelation shrinkage activates automatically as the asset count approaches the effective sample size, with a provable condition-number bound;
• ingests missing values natively — listings, delistings, and halts enter as NaN; no imputation or complete-case subsetting;
• is fast — a full 30-year daily pass takes ~0.75 s at n=100 and ~3.4 s at n=300 (single-threaded), 30–40× faster than rolling-window baselines at scale.

On a 30-year S&P 500 panel (n=100, ~7,600 out-of-sample days) it statistically ties DCC on one-step density forecasts and beats EWMA, Ledoit–Wolf, OAS, nonlinear shrinkage, RMT denoising, and the Gerber statistic — and it is the only method in the 90% model confidence set together with DCC. At n=300 it leads every competitor that remains statistically viable.

Installation

pip install squeeze-kernel          # NumPy only
pip install "squeeze-kernel[full]"  # + SciPy (kappa calibration, chi² kernel)

Quickstart

import numpy as np
from squeeze_kernel import SqueezeKernelEstimator

# daily_returns: array of shape (T, n) — may contain NaN for missing assets
est = SqueezeKernelEstimator(n_assets=daily_returns.shape[1])

for r_t in daily_returns:          # stream one day at a time
    est.update(r_t)

cov  = est.get_cov()               # (n, n) covariance, PSD by construction
corr = est.get_corr()              # (n, n) correlation

That is the whole API for most uses. The defaults (lambda_vol=0.98, lambda_corr=0.996, kappa=0.25) are the paper-recommended settings for daily returns, selected by time-series cross-validation and robust across a 50× parameter sweep — deploy them as-is.

Batch mode, if you prefer the full path in one call:

from squeeze_kernel import estimate_squeeze_cov

cov_path, corr_path, weights = estimate_squeeze_cov(daily_returns, with_weights=True)
# cov_path: (T, n, n) — the estimate after each day

A complete runnable walkthrough (streaming, missing data, batch) is in examples/quickstart.py.

Missing values

Pass NaN for any asset not observed on a given day — nothing else to do:

r_t = np.array([0.004, np.nan, -0.011])   # asset 2 not trading today
est.update(r_t)                            # PSD preserved, no imputation

Parameters

Useful read-only state after each update(): est.weight (last kernel weight), est.effective_sample_size (kernel-weighted T_eff), est.shrinkage_intensity (current α).

To recalibrate kappa for a different asset class (requires the full extra):

kappa = SqueezeKernelEstimator.calibrate_kappa(burn_in_returns, target_weight=0.6)

Advanced options

Adaptive scale-free correlation memory (corr_half_lives=(43, 173, 693), corr_theta=0.25): replaces the single correlation timescale with a positive combination of EWMAs on a geometric half-life ladder — each scale normalized and adaptively shrunk against its own effective sample size, then the covariances blended with weights resting at the prior ∝ half-life^corr_theta. By Bernstein's theorem the ladder approximates the power-law memory of financial correlations (the streaming analogue of HAR). For ladders of two or more rungs the blend weights are gated by a sequential surprise detector: a two-sided Page CUSUM on the studentized fast-vs-slow per-rung predictive-score drift (threshold set by Siegmund's average-run-length approximation at ~2 years, no tuned parameters) tilts the weights toward the fast or slow end of the ladder when one side accumulates statistically forced evidence, decaying back at the fastest rung's half-life. Weights equal the prior on all non-alarmed days, the blend stays convex, so PSD holds by construction; None (default) reproduces the published single-scale estimator exactly. This is the paper's headline configuration: on the S&P 500 benchmark it leads every tested method at every universe size (held-out one-step NLL −4.6 vs single-scale at n=100, −11.1 at n=300 before the cluster target), the 90% model confidence set collapses to it alone, and the detector's margin is confirmed out-of-time on an external industry panel (+0.53 NLL/day, p=1×10⁻⁴). Cost is O(K·n²) per update plus one Cholesky per rung per day for the detector scores. Composes with shrinkage_target="cluster"; mutually exclusive with lambda_corr_fast.

est = SqueezeKernelEstimator(n_assets=100, corr_half_lives=(43, 173, 693), corr_theta=0.25)
# maximal variant at high dimension:
est = SqueezeKernelEstimator(n_assets=300, corr_half_lives=(43, 173, 693), shrinkage_target="cluster")

Score-exact weighting (weight_statistic="mahalanobis", use with kappa=1.0): drives the kernel with the Mahalanobis surprise z'C⁻¹z/N against the estimator's own correlation instead of the marginal dispersion. Improves accuracy in the moderate-concentration regime — use only when n / T_eff ≲ 0.5 (e.g. n ≤ 100 at the default lambda_corr); at higher concentration the estimated inverse degrades it and the default is strictly better.

est = SqueezeKernelEstimator(n_assets=100, kappa=1.0, weight_statistic="mahalanobis")

Score-driven memory (lambda_corr_fast=0.99): lets stress days also shorten the correlation memory (decay slides from lambda_corr toward lambda_corr_fast as the kernel weight rises). Do not combine with the Mahalanobis option — they act on the same channel and the combination degrades accuracy.

OU volatility anchor (vol_anchor_phi=0.995): mean-reverts each asset's variance prediction toward a slow per-asset anchor (a ~1000-day EWMA of squared returns) before the daily update — a two-timescale, component-style volatility structure. One global parameter with a clean interpretation (deviation half-life ≈ ln 2/(1−φ) days; φ=0.995 ≈ 139 d). On the S&P 500 n=100 benchmark this improved held-out one-step NLL by 3.3 points (4.3 at φ=0.99) and five-step NLL by 3.9 (5.0), with no degradation at n=300. None (default) or φ=1 reproduces the published estimator exactly.

est = SqueezeKernelEstimator(n_assets=100, vol_anchor_phi=0.995)

Cluster shrinkage target (shrinkage_target="cluster"): generalizes the equicorrelation shrinkage target to respect the correlation matrix's own block/cluster structure — with no clustering algorithm. The target morphs with the shrinkage intensity, T = (1−α)·T_equi + α·[(1−γ)I + γ·(C∘C)], where C∘C is the Hadamard square of the current correlation (positive semi-definite by the Schur product theorem; entries are pairwise shared-variance fractions) and γ is level-matched automatically. Zero added parameters, still O(n²), and as α→0 it reduces exactly to the default estimator. Held-out one-step NLL on the S&P 500 benchmark: ±0.1 at n=100, −4.2 at n=200, −25.0 at n=300 — recommended whenever the universe size approaches the effective sample size.

est = SqueezeKernelEstimator(n_assets=300, shrinkage_target="cluster")

New-listing usability gate (min_obs=60): on an expanding universe, an asset's forecast rows are dominated by its single-observation variance initialization for its first weeks of life and are unusable for scoring or portfolio construction (measured ≈ +2,800 NLL/day on days whose scored set included such assets, on a 42-instrument multi-asset panel). min_obs gates nothing inside the estimator — states warm normally, all outputs are unchanged — it exposes a usable_mask property marking assets with at least min_obs finite observations, so deployments subset with it:

est = SqueezeKernelEstimator(n_assets=42, min_obs=60)
# ... update loop ...
m = est.usable_mask
cov_usable = est.get_cov()[np.ix_(m, m)]

Alternative kernels: pass kernel_fn=kernel_exponential (with kernel_kwargs={"gamma": ...}) or kernel_chi2_cdf, or any callable (d2, *, n_observed, **kw) -> float mapping to [0, 1). The PSD guarantee holds for any such kernel.

How it works

Three mechanisms in one recursion:

1. Dual-timescale EWMA — fast per-asset volatility (lambda_vol) is separated from slow correlation dynamics (lambda_corr), so variance shocks don't contaminate the correlation estimate.
2. Fisher kernel weighting — each day's standardized outer product enters with weight w = d²/(d² + kappa), where d² is the mean squared standardized return: calm days contribute little, dispersion shocks contribute fully.
3. Adaptive equicorrelation shrinkage — alpha = min(1, max(0, n/(2·S) − delta)) blends toward an equicorrelation target using the estimator's own kernel-weighted sample size S; it is a no-op at low dimension and provides provably bounded conditioning at high dimension.

The complete update is a natural-gradient step on the Gaussian log-likelihood, with the kernel weight acting as an adaptive Riemannian learning rate (paper, Appendix B).

Development

uv sync --extra full --extra dev
uv run python -m pytest        # test suite
uv run python -m ruff check .  # lint
uv run mypy                      # strict type check (src/squeeze_kernel)
uv build                       # build sdist + wheel

Releases: publishing a GitHub release from a v* tag triggers the publish workflow, which builds and uploads to PyPI via trusted publishing.

Citation

@article{kende2026squeeze,
  title  = {The Squeeze Kernel Covariance Estimator: Dual-Timescale Tracking with Adaptive Shrinkage},
  author = {Kende, Robert},
  year   = {2026},
  note   = {Available at SSRN: \url{https://ssrn.com/abstract=6455918}}
}

See also CITATION.cff.

License

MIT