quant-statistics

# 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.

## 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.

```python
from statsmodels.tsa.stattools import adfuller

def adf_test(series: pd.Series, significance: float = 0.05) -> dict:
    """
    ADF test: H0 = unit root exists (non-stationary), H1 = stationary

    Args:
        series: Time series
        significance: Significance level
    Returns:
        Test result
    """
    result = adfuller(series.dropna(), autolag='AIC')
    return {
        'adf_statistic': result[0],
        'p_value': result[1],
        'lags_used': result[2],
        'is_stationary': result[1] < significance,
        'critical_values': result[4],  # 1%, 5%, 10%
    }
```

**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).

```python
from statsmodels.tsa.stattools import coint

def cointegration_test(y: pd.Series, x: pd.Series) -> dict:
    """
    Engle-Granger two-step cointegration test
    H0: no cointegration relationship

    Args:
        y, x: Two price series
    Returns:
        Test result
    """
    score, p_value, critical = coint(y, x)
    return {
        'test_statistic': score,
        'p_value': p_value,
        'is_cointegrated': p_value < 0.05,
        'critical_values': {'1%': critical[0], '5%': critical[1], '10%': critical[2]},
    }
```

**Application in pair trading**:

```python
import statsmodels.api as sm

def find_hedge_ratio(y: pd.Series, x: pd.Series) -> dict:
    """
    Compute hedge ratio: y = α + β×x + ε
    Spread = y - β×x
    """
    x_const = sm.add_constant(x)
    model = sm.OLS(y, x_const).fit()

    spread = y - model.params[1] * x

    return {
        'hedge_ratio': model.params[1],
        'intercept': model.params[0],
        'spread_mean': spread.mean(),
        'spread_std': spread.std(),
        'half_life': compute_half_life(spread),  # mean-reversion speed
    }

def compute_half_life(spread: pd.Series) -> float:
    """Estimate half-life with OLS regression."""
    spread_lag = spread.shift(1)
    delta = spread - spread_lag
    model = sm.OLS(delta.dropna(), sm.add_constant(spread_lag.dropna())).fit()
    half_life = -np.log(2) / model.params[1]
    return half_life
```

**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

```python
from statsmodels.tsa.stattools import grangercausalitytests

def granger_test(data: pd.DataFrame, x_col: str, y_col: str, max_lag: int = 5):
    """
    Test whether x Granger-causes y (whether historical x helps predict y).
    Note: Granger causality is not true causality, only predictive causality.
    """
    results = grangercausalitytests(data[[y_col, x_col]].dropna(), maxlag=max_lag)
    return {lag: results[lag][0]['ssr_ftest'][1] for lag in range(1, max_lag+1)}
```

## 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))
```

```python
from arch import arch_model

def fit_garch(returns: pd.Series) -> dict:
    """
    Fit a GARCH(1,1) model.

    Args:
        returns: Daily return series (in percentage form)
    Returns:
        Model parameters and forecasts
    """
    model = arch_model(returns * 100, vol='Garch', p=1, q=1,
                       mean='Constant', dist='normal')
    result = model.fit(disp='off')

    # Forecast volatility for the next 5 days
    forecast = result.forecast(horizon=5)

    return {
        'omega': result.params['omega'],
        'alpha': result.params['alpha[1]'],
        'beta': result.params['beta[1]'],
        'persistence': result.params['alpha[1]'] + result.params['beta[1]'],
        'long_run_vol': np.sqrt(result.params['omega'] /
                        (1 - result.params['alpha[1]'] - result.params['beta[1]'])) / 100,
        'current_vol': np.sqrt(result.conditional_volatility[-1]) / 100,
        'forecast_vol_5d': np.sqrt(forecast.variance.values[-1, :]) / 100,
        'aic': result.aic,
        'bic': result.bic,
    }
```

### GARCH Variants

| Model | Characteristics | Applicable Scenario |
|------|------|---------|
| GARCH(1,1) | Baseline, symmetric shock response | Default choice |
| EGARCH | Asymmetric (leverage effect) | Down-move volatility > up-move