Quantitative Statistical Methods

Overview

Common statistical methodology used in quantitative investing, covering time-series testing, volatility modeling, regression diagnostics, and statistical inference. Provides the statistical foundation for strategy development and factor research.

Implementation

Every test below is already implemented and unit-tested in src.quantlib.timeseries. Import and call it — do not retype these formulas into throwaway code, which is how sign errors and double-sqrt bugs get into results.

from src.quantlib.timeseries import (
    adf_test, cointegration_test, find_hedge_ratio, compute_half_life,
    granger_test, fit_garch, heteroscedasticity_test, autocorrelation_test,
    vif_test, bootstrap_statistic, bootstrap_sharpe,
)

Optional backends: statsmodels powers everything except the two bootstrap helpers (which are pure numpy); arch powers fit_garch only. Neither is declared as a dependency of vibe-trading-ai, so both are imported lazily inside the functions. Importing the module always works; calling a function whose backend is missing raises an ImportError naming the package and the install command (pip install "statsmodels>=0.14" / pip install "arch>=6.0"). If you hit that error, report it to the user rather than silently substituting a different method.

Time-Series Tests

1. ADF Unit-Root Test (Stationarity Test)

Why it matters: regressing non-stationary series directly can produce spurious regression, making conclusions unreliable.

from src.quantlib.timeseries import adf_test

result = adf_test(prices['close'], significance=0.05)
# {'adf_statistic': -1.23, 'p_value': 0.65, 'lags_used': 4,
#  'is_stationary': False,
#  'critical_values': {'1%': -3.44, '5%': -2.87, '10%': -2.57}}

if not result['is_stationary']:
    returns = np.log(prices['close']).diff().dropna()
    adf_test(returns)  # log returns are normally stationary

Decision rules:

p-value	Conclusion	Action
< 0.01	Strongly stationary	Can be used directly for regression / modeling
0.01-0.05	Stationary	Usable
0.05-0.10	Weak evidence	Difference the series and retest
> 0.10	Non-stationary	Must difference or handle with cointegration

Stationarity of common financial series:

Series	Typical Result	Treatment
Price series	Non-stationary (unit root)	Use log returns
Log returns	Stationary	Can be used directly
PE / PB series	Usually non-stationary	Use changes or logs
Volatility series	Usually stationary	Can be used directly
Volume	May be non-stationary	Use logs or standardization

2. Cointegration Test

Purpose: determine whether two non-stationary series share a long-run equilibrium relationship (the foundation of pair trading / statistical arbitrage).

from src.quantlib.timeseries import cointegration_test

result = cointegration_test(prices_a, prices_b, significance=0.05)
# {'test_statistic': -4.52, 'p_value': 0.002, 'is_cointegrated': True,
#  'critical_values': {'1%': -3.90, '5%': -3.34, '10%': -3.05}}

Both legs must be individually non-stationary (check with adf_test first) — cointegration on two already-stationary series is meaningless.

Both legs must also share one index. Two same-length series on different indices raise ValueError rather than being zipped positionally, because a positional join of, say, an A-share calendar against a US one reports cointegration between days that never coexisted. Reindex or inner-join the two legs yourself before calling.

Application in pair trading:

from src.quantlib.timeseries import find_hedge_ratio, compute_half_life

result = find_hedge_ratio(prices_a, prices_b)
# {'hedge_ratio': 2.49, 'intercept': 0.40,
#  'spread_mean': 0.40, 'spread_std': 1.73, 'half_life': 16.7}

spread = prices_a - result['hedge_ratio'] * prices_b
z_score = (spread - result['spread_mean']) / result['spread_std']

# half_life is in observation periods (days for daily bars) and is `inf`
# when the spread does not mean-revert. Sanity-check it before trading:
# a half-life longer than your holding horizon means the spread will not
# close in time, however good the cointegration p-value looks.
compute_half_life(spread)

A perfectly flat leg (a name halted for the whole window) makes the regression degenerate, so find_hedge_ratio and compute_half_life raise ValueError rather than return a meaningless β. Treat that as "this pair has no usable data in this window", not as something to work around.

Pair-trading signal:

z_score = (spread - mean) / std

| z_score | Signal |
|---------|------|
| > 2.0 | Short spread (sell y, buy x) |
| > 1.5 | Small short spread |
| < -1.5 | Small long spread |
| < -2.0 | Long spread (buy y, sell x) |
| Back near 0 | Close position |

3. Granger Causality Test

from src.quantlib.timeseries import granger_test

p_by_lag = granger_test(df, x_col='volume', y_col='return', max_lag=5)
# {1: 0.003, 2: 0.011, 3: 0.08, 4: 0.21, 5: 0.33}
# small p at lag k -> past x at that lag helps predict y

Granger causality is predictive, not structural: it says past x improves the forecast of y, never that x causes y. A common confounder is that both respond to a third variable. Note also that testing 5 lags is 5 hypothesis tests — one small p-value among them is weak evidence.

GARCH Volatility Modeling

GARCH(1,1) Model

Returns: r_t = μ + ε_t
Volatility: σ²_t = ω + α×ε²_{t-1} + β×σ²_{t-1}

Parameter meanings:
- ω (omega): long-run variance baseline
- α (alpha): impact of yesterday's shock on today's volatility
- β (beta): persistence of yesterday's volatility into today
- α + β: volatility persistence (usually 0.95-0.99)
- Long-run volatility = sqrt(ω / (1 - α - β))

from src.quantlib.timeseries import fit_garch

# `returns` are FRACTIONS (0.01 = 1%); the function rescales to percent itself.
result = fit_garch(returns, horizon=5)
# {'omega': 0.0453, 'alpha': 0.1213, 'beta': 0.8348, 'persistence': 0.9561,
#  'long_run_vol': 0.0102, 'current_vol': 0.0149,
#  'forecast_vol': array([0.0138, 0.0137, 0.0136, 0.0134, 0.0133]),
#  'horizon': 5, 'aic': 10882.39, 'bic': 10907.57}

The forecast decays from current_vol toward long_run_vol — that mean reversion is the whole point of the model, and a forecast that does not decay signals persistence too close to 1.

All volatilities come back as daily fractions — multiply by sqrt(252) to annualise. long_run_vol is nan when persistence >= 1, which means the model has no finite unconditional variance and its long-horizon forecast is not usable.

Requires the optional arch package (pip install "arch>=6.0"); the call raises a named ImportError if it is absent.

GARCH Variants

Model	Characteristics	Applicable Scenario
GARCH(1,1)	Baseline, symmetric shock response	Default choice
EGARCH	Asymmetric (leverage effect)	Down-move volatility > up-move volatility
GJR-GARCH	Another asymmetric form	Same use case as EGARCH, easier to interpret
FIGARCH	Long memory	Volatility clustering persists for very long periods

GARCH characteristics in China A-shares / crypto:

China A-shares:
- α usually 0.05-0.15
- β usually 0.80-0.90
- Clear leverage effect (EGARCH fits better)
- Strong volatility clustering persistence

BTC:
- α usually 0.05-0.20 (shocks matter more)
- β usually 0.75-0.90
- More symmetric shocks (little difference between up/down volatility)
- Long-run volatility around 60-80% annualized

Regression Diagnostics

1. Heteroskedasticity Test

import statsmodels.api as sm
from src.quantlib.timeseries import heteroscedasticity_test

fitted = sm.OLS(y, sm.add_constant(X)).fit()
result = heteroscedasticity_test(fitted)   # pass the FITTED result, not the data
# {'white_p': 0.0001, 'bp_p': 0.0003, 'has_heteroscedasticity': True,
#  'fix': 'Use HAC standard errors (Newey-West) or WLS'}

fix tracks has_heteroscedasticity, and the verdict is white_p < α or bp_p < α — either test rejecting is enough to act on. The two disagree fairly often near the threshold (White has less power against a simple linear variance trend), so do not read "White says no" as the answer.

Heteroskedasticity fixes:

• Use model.fit(cov_type='HAC', cov_kwds={'maxlags': 5})
• Or use weighted least squares (WLS)
• Financial data is almost always heteroskedastic -> use HAC standard errors by default

2. Autocorrelation Test

from src.quantlib.timeseries import autocorrelation_test

result = autocorrelation_test(fitted.resid, lags=10)
# {'durbin_watson': 1.21, 'dw_interpretation': 'positive autocorrelation',
#  'ljung_box_p': array([0.001, 0.002, ...]),   # one p-value per lag
#  'has_autocorrelation': True,
#  'fix': 'Use Newey-West standard errors or include lag terms'}

⚠️ has_autocorrelation is any(p < significance) across all lags — that is a family of tests, not one. On pure white noise it fires about 13% of the time at lags=10 versus about 2% at lags=1 (measured over 120 seeds, n=1500). Treat a lone flag at high lags as a prompt to inspect ljung_box_p lag by lag, not as a 5%-level rejection.

3. Multicollinearity Test

from src.quantlib.timeseries import vif_test

vif_test(factors, severe_threshold=10.0, watch_threshold=5.0)
#     feature     VIF  concern
# 0     value   28.41   severe
# 1  momentum    1.12   normal
# 2      size    6.30    watch

Include the constant column if your model has one — VIF is otherwise distorted by the un-centred means.

Regression Diagnostics Checklist

□ 1. Linearity: residuals vs fitted values show no obvious pattern
□ 2. Normality: residual QQ plot is close to a straight line, Jarque-Bera p>0.05
□ 3. Heteroskedasticity: White / BP test p>0.05, or use HAC standard errors
□ 4. Autocorrelation: DW≈2, Ljung-Box p>0.05
□ 5. Multicollinearity: VIF<5
□ 6. Outliers: Cook's D < 4/n

Bootstrap Methods

Nonparametric Bootstrap

from src.quantlib.timeseries import bootstrap_statistic

result = bootstrap_statistic(returns.values, np.median,
                             n_bootstrap=10000, confidence=0.95, seed=42)
# {'point_estimate': 0.0004, 'bootstrap_mean': 0.0004, 'bootstrap_std': 0.0002,
#  'ci_lower': 0.0001, 'ci_upper': 0.0008, 'confidence': 0.95}

Pass seed whenever the number goes into a report — an unseeded bootstrap gives a slightly different interval on every run, which makes results irreproducible. Needs no optional dependency (pure numpy).

Bootstrap Applications in Quant

Scenario	Method	Purpose
Sharpe-ratio confidence interval	Bootstrap return series	Determine whether Sharpe is significantly >0
Factor return test	Bootstrap factor values	Whether factor premium is robust
Maximum drawdown distribution	Bootstrap equity paths	Probability distribution of max drawdown
Strategy comparison	Paired Bootstrap	Whether strategy A is significantly better than B

from src.quantlib.timeseries import bootstrap_sharpe

# Takes a RETURN series (fractions), not an equity curve.
result = bootstrap_sharpe(returns, n_bootstrap=10000,
                          periods_per_year=252, seed=42)
# {'point_estimate': 1.25, 'ci_lower': 0.62, 'ci_upper': 1.88,
#  'bootstrap_mean': 1.26, 'bootstrap_std': 0.32,
#  'confidence': 0.95, 'is_significant': True}

is_significant means the interval sits entirely above zero. Remember what it does not mean: the interval is centred on the realised Sharpe, so it quantifies sampling error around this sample, not whether the edge persists out of sample. With only 1000 daily bars the realised Sharpe of a zero-edge strategy already has a standard deviation of sqrt(252/1000) ≈ 0.50.

For a backtest equity curve use backtest.validation.bootstrap_sharpe_ci instead — it differences the curve itself and returns report-shaped keys. The two also use different denominators: bootstrap_sharpe divides by the sample standard deviation (ddof=1), bootstrap_sharpe_ci by the population one (ddof=0). On identical data they differ by sqrt(n / (n-1)) — about 0.2% over a year of daily bars. Report one or the other, never both as if they agreed.

Hypothesis-Testing Framework

Quick Reference for Common Tests

Testing Goal	Test Method	Null Hypothesis
Mean = 0	t-test	μ = 0
Two means are equal	Independent t-test	μ1 = μ2
Normality	Jarque-Bera	Normal distribution
Stationarity	ADF	Has unit root (non-stationary)
Autocorrelation	Ljung-Box	No autocorrelation
Heteroskedasticity	White / BP	Homoskedasticity
Cointegration	Engle-Granger	Not cointegrated

Multiple-Testing Problem

Problem: test 100 factors and filter with p<0.05 -> expect 5 false positives

Correction methods:
1. Bonferroni: p_adj = p × n_tests (most conservative)
2. Holm-Bonferroni: stepwise correction (fairly conservative)
3. Benjamini-Hochberg (FDR): control false discovery rate (recommended)

from statsmodels.stats.multitest import multipletests
reject, p_adj, _, _ = multipletests(p_values, method='fdr_bh')

Statistical Significance in Financial Backtests

Sharpe significance test:
H0: Sharpe = 0 (strategy is ineffective)
H1: Sharpe > 0

Test statistic: t = Sharpe × sqrt(n) / sqrt(1 + 0.5×Sharpe²)
where n = number of observation periods (years)

Rules of thumb:
- Sharpe > 0.5 and backtest >5 years -> may be significant
- Sharpe > 1.0 and backtest >3 years -> likely significant
- Sharpe > 2.0 -> overfitting warning (hard to sustain in reality)

Output Format

## Statistical Testing Report

### Stationarity Test
| Series | ADF Statistic | p-value | Conclusion |
|------|----------|-----|------|
| Price | -1.23 | 0.65 | Non-stationary |
| Return | -15.8 | 0.000 | Stationary *** |

### Cointegration Test
| Pair | Statistic | p-value | Cointegrated |
|------|--------|-----|------|
| 600519/000858 | -4.52 | 0.002 | Yes ** |

### GARCH Model
| Parameter | Value | Meaning |
|------|-----|------|
| α | 0.08 | Shock effect |
| β | 0.88 | Volatility persistence |
| Long-run volatility | 22.5% | Annualized |

### Bootstrap Result
| Metric | Point Estimate | 95% CI | Significant |
|------|--------|--------|------|
| Sharpe | 1.25 | [0.62, 1.88] | Yes |
| Alpha (monthly) | 0.8% | [0.1%, 1.5%] | Yes |

Notes

1. Financial data is non-normal: almost all financial return series are fat-tailed, so be careful with tests assuming normality
2. Multiple testing: when backtesting many strategies / factors, multiple-testing correction (FDR control) is mandatory
3. Out-of-sample validation: statistical significance does not guarantee profitability; out-of-sample testing is still required
4. Cointegration can break down: historical cointegration does not guarantee persistence, so pair trading needs ongoing monitoring
5. GARCH forecast horizon is limited: volatility-forecast accuracy declines rapidly beyond 5-10 days
6. Be careful with small samples: financial datasets may look large, but the number of independent observations can still be small (for example, annual data)
7. p-hacking risk: do not keep adjusting until p<0.05; predefine the testing plan