HELIX: Heteroskedasticity Evaluation via Localized Inference and eXtended models

HELIX is a from-scratch implementation of four heteroskedasticity tests that go beyond the standard Breusch-Pagan (BP) test, combining a neural network-based detector, White's general test, and a novel sliding-window Local BP variant that pinpoints where in the feature space variance changes -- something no classical test can do.

Overview

Classical heteroskedasticity tests (Breusch-Pagan, White) are global and rely on asymptotic chi-squared distributions. HELIX extends them along three axes:

Limitation of classical tests	How this project addresses it
Linear auxiliary regression only	HELIX Neural BP uses an MLP to detect arbitrary nonlinear variance patterns
Asymptotic p-values (breaks on small n)	HELIX bootstrap permutation test gives exact finite-sample p-values
Global scope (one number for the whole dataset)	HELIX Local BP runs sliding-window tests, revealing where heteroskedasticity lives

The Four Tests in HELIX

1. Classic Breusch-Pagan

Standard LM test. Regresses squared OLS residuals on the original predictors. Tests whether variance depends linearly on X.

• Statistic: LM = n * R-squared from auxiliary regression
• Distribution: chi-squared(k-1)
• Validated against statsmodels.het_breuschpagan (exact match)

2. White's Test

Extends BP by adding squared terms and cross-products (via PolynomialFeatures(degree=2)) to the auxiliary regression. Catches nonlinear heteroskedasticity that BP misses (e.g., U-shaped variance).

• Statistic: LM = n * R-squared from augmented auxiliary regression
• Distribution: chi-squared(q-1) where q = number of augmented features + 1

3. Neural BP

Replaces the linear auxiliary regression with a 2-layer MLP (64, 32 hidden units). If the network can predict squared residuals from X, heteroskedasticity exists and may be arbitrarily nonlinear.

• Uses bootstrap permutation (default 500 iterations) for the p-value
• No distributional assumptions -- works on any sample size
• Catches patterns that even White's test misses

4. Local BP

Runs the classic BP test in sliding windows along a chosen predictor, with 50% overlap. Outputs a per-window p-value curve showing exactly which regions of the feature space are heteroskedastic and which are not.

• Window size: configurable fraction of n (default 0.3)
• Sort column: configurable (default: first predictor)
• Unique capability: no other standard test tells you where the problem is

Visualization

plot_diagnostics() produces a 2x2 diagnostic figure:

Panel	Plot	What it shows
Top-left	Residuals vs Fitted	Classic residual plot with LOWESS trend
Top-right	Scale-Location	sqrt(abs(residuals)) vs fitted -- rising trend = heteroskedasticity
Bottom-left	Local p-value curve	Per-window BP p-values with alpha threshold and shaded rejection regions
Bottom-right	Variance heatmap	Feature-space scatter colored by abs(residual) -- shows where variance concentrates

Installation

pip install numpy scipy scikit-learn matplotlib

Optional (for cross-validation only):

pip install statsmodels

Quick Start

import numpy as np
from bp_tester import HeteroskedasticityTester

np.random.seed(42)
n = 200
x1 = np.random.uniform(0, 5, n)
x2 = np.random.uniform(0, 5, n)
X = np.column_stack([x1, x2])

eps = np.random.normal(0, 0.4 * x1)
y = 2 + 1.5 * x1 - 0.8 * x2 + eps

tester = HeteroskedasticityTester(X, y)
results = tester.run_all(alpha=0.05)

Run individual tests

tester.breusch_pagan()
tester.white_test()
tester.neural_bp(n_bootstrap=300)
tester.local_bp(window_size=0.25, sort_col=0)
tester.plot_diagnostics()

API Reference

HeteroskedasticityTester(X, y)

Parameter	Type	Description
X	array (n, p)	Feature matrix, no constant column (added internally)
y	array (n,)	Response vector

Methods

Method	Returns	Key Parameters
breusch_pagan()	{test, LM, p_value}	--
white_test()	{test, LM, p_value}	--
neural_bp(n_bootstrap, random_state)	{test, R2, bootstrap_p, n_bootstrap}	n_bootstrap=500, random_state=42
local_bp(window_size, sort_col)	{test, sort_col, windows}	window_size=0.3, sort_col=0
plot_diagnostics(alpha, sort_col)	matplotlib Figure	alpha=0.05, sort_col=0
run_all(alpha, neural_bootstrap, sort_col)	dict of all results	Runs everything, prints summary, shows plots

Project Structure

BP test/
    bp_tester.py      Core module -- HeteroskedasticityTester class (the HELIX engine)
    demo.ipynb         Demo notebook with synthetic data, full walkthrough, and comparison figures
    research_paper.md  Research paper on HELIX findings and methodology
    figures/           Auto-generated diagnostic figures from the notebook
    README.md          This file
    .gitignore

What to Do When Heteroskedasticity is Detected

Strategy	When to use it
HC3 robust standard errors	Quick fix -- keeps OLS coefficients, corrects SEs
WLS (Weighted Least Squares)	When you can estimate the variance function
Log-transform y	When variance grows proportionally with the mean
GLS (Generalized Least Squares)	Full generalized error structure
Model the variance explicitly	GARCH (time series), evidential regression

License

MIT