---
name: longbridge-quant-stats
description: |
  Quantitative statistics framework for time-series analysis using Longbridge price data — ADF unit root test (stationarity), cointegration (Engle-Granger / Johansen), GARCH volatility modelling (conditional heteroskedasticity), regression diagnostics (Durbin-Watson / Breusch-Pagan), bootstrap confidence intervals, hypothesis tests (t-test / F-test). Requires statsmodels and scipy. Triggers: "量化统计", "ADF检验", "单位根", "协整检验", "GARCH", "自相关", "异方差", "Bootstrap", "假设检验", "量化統計", "ADF檢驗", "單位根", "協整檢驗", "異方差", "假設檢驗", "quantitative statistics", "ADF test", "unit root", "cointegration", "GARCH", "autocorrelation", "heteroskedasticity", "bootstrap", "hypothesis test", "statsmodels".
license: MIT
metadata:
  author: longbridge
  version: "1.0.0"
  risk_level: read_only
  requires_login: false
  default_install: true
  requires_mcp: false
  tier: read
---

# longbridge-quant-stats

Apply rigorous statistical methods to financial time-series data retrieved from Longbridge — test assumptions before modelling, diagnose residuals, and produce statistically sound inferences.

> **Response language**: match the user's input language — Simplified Chinese / Traditional Chinese / English.

## When to use

- *"帮我做 ADF 单位根检验"*, *"run an ADF test on this price series"*, *"幫我做 ADF 單位根檢驗"*
- *"AAPL 和 MSFT 有没有协整关系"*, *"are AAPL and MSFT cointegrated"*
- *"用 GARCH 建模波动率"*, *"model volatility with GARCH"*
- *"回归残差有没有自相关"*, *"check residual autocorrelation (Durbin-Watson)"*
- *"用 Bootstrap 估计置信区间"*, *"bootstrap confidence interval for Sharpe ratio"*

For factor IC/IR testing, use `longbridge-factor-research`. For pairs-trading cointegration application, use `longbridge-pairs-trading`.

## Prerequisites

```bash
pip install statsmodels scipy numpy pandas
```

## Workflow and test catalogue

### Step 1 — Fetch price data

```bash
longbridge kline --help
longbridge kline <SYMBOL> --period day --count 252 --format json
```

Extract the `close` price series. Compute log returns: `r_t = ln(P_t / P_{t-1})`.

### Step 2 — Stationarity: ADF Unit Root Test

**When to use**: Before regression or time-series modelling — most models require stationary series.

**Python (statsmodels)**:
```python
from statsmodels.tsa.stattools import adfuller
result = adfuller(series, autolag='AIC')
# result: (adf_stat, p_value, lags, n_obs, critical_values, icbest)
```

**Interpretation**:
- p < 0.05 → reject unit root → series is stationary.
- p ≥ 0.05 → fail to reject → series has unit root → difference the series.
- Log prices: usually non-stationary. Log returns: usually stationary.

### Step 3 — Cointegration Test

**When to use**: Two non-stationary series may share a long-run equilibrium (pairs trading).

**Engle-Granger (two-series)**:
```python
from statsmodels.tsa.stattools import coint
t_stat, p_value, critical_values = coint(series_A, series_B)
# p < 0.05 → cointegrated
```

**Johansen (multivariate)**:
```python
from statsmodels.tsa.vector_ar.vecm import coint_johansen
result = coint_johansen(df, det_order=0, k_ar_diff=1)
# trace statistic vs critical values at 90%/95%/99%
```

Report: test statistic, p-value, critical values, and cointegrating vector.

### Step 4 — GARCH Volatility Modelling

**When to use**: Financial returns show volatility clustering (ARCH effects).

```python
from arch import arch_model
model = arch_model(returns * 100, vol='Garch', p=1, q=1)
res = model.fit(disp='off')
print(res.summary())
```

Note: `pip install arch` required in addition to statsmodels.

**Output**: omega, alpha (ARCH), beta (GARCH) coefficients. Persistence = alpha + beta. If > 0.95, volatility is highly persistent.

**ARCH-LM test first** (to verify ARCH effects exist):
```python
from statsmodels.stats.diagnostic import het_arch
lm_stat, p_value, f_stat, f_p = het_arch(residuals)
```

### Step 5 — Regression Diagnostics

After running OLS (`statsmodels.api.OLS`), check:

| Test | Purpose | Command |
|---|---|---|
| Durbin-Watson | Serial autocorrelation in residuals | `statsmodels.stats.stattools.durbin_watson(resid)` |
| Breusch-Pagan | Heteroskedasticity | `statsmodels.stats.diagnostic.het_breuschpagan(resid, exog)` |
| Jarque-Bera | Normality of residuals | `statsmodels.stats.stattools.jarque_bera(resid)` |
| VIF | Multicollinearity | `statsmodels.stats.outliers_influence.variance_inflation_factor` |

Interpret Durbin-Watson: ~2.0 = no autocorrelation; < 1.5 = positive autocorrelation; > 2.5 = negative autocorrelation.

### Step 6 — Bootstrap Confidence Intervals

**When to use**: Non-normal distributions; small samples; estimating CI for Sharpe ratio, IC, or any statistic.

```python
import numpy as np

def bootstrap_ci(data, stat_fn, n_boot=10000, ci=0.95):
    boots = [stat_fn(np.random.choice(data, len(data), replace=True))
             for _ in range(n_boot)]
    lo = np.percentile(boots, (1 - ci) / 2 * 100)
    hi = np.percentile(boots, (1 + ci) / 2 * 100)
    return lo, hi

# Example: Sharpe ratio CI
sharpe_lo, sharpe_hi = bootstrap_ci(returns, lambda x: x.mean() / x.std() * np.sqrt(252))
```

### Step 7 — Hypothesis Tests

| Test | Use case | Function |
|---|---|---|
| t-test (one sample) | Is mean IC > 0? | `scipy.stats.ttest_1samp(ic_series, 0)` |
| t-test (two sample) | Is long portfolio return > short portfolio? | `scipy.stats.ttest_ind(long_ret, short_ret)` |
| F-test / ANOVA | Are returns different across deciles? | `scipy.stats.f_oneway(*decile_returns)` |
| Mann-Whitney U | Non-parametric alternative to t-test | `scipy.stats.mannwhitneyu(a, b)` |

Always report: test statistic, p-value, degrees of freedom, and conclusion at 5% significance level.

## CLI

```bash
longbridge kline --help
longbridge kline <SYMBOL> --period day --count 252 --format json
```

## Output

For each test present:
1. Test name and null hypothesis.
2. Test statistic and p-value.
3. Critical values (where applicable).
4. Conclusion at 5% significance.
5. Practical implication for the user's use case.

## Error handling

| Situation | 简体回复 | 繁體回覆 | English reply |
|---|---|---|---|
| `command not found: longbridge` | 请安装 longbridge-terminal 或检查 MCP 配置。 | 請安裝 longbridge-terminal 或檢查 MCP 配置。 | Install longbridge-terminal or check MCP config. |
| `ModuleNotFoundError: statsmodels` | 请运行 `pip install statsmodels scipy numpy pandas`。 | 請執行 `pip install statsmodels scipy numpy pandas`。 | Run `pip install statsmodels scipy numpy pandas`. |
| Insufficient data (< 30 observations) | 样本量过小，统计结论可靠性有限，建议延长数据期。 | 樣本量過小，建議延長數據期。 | Sample too small; extend the data period for reliable results. |
| ARCH module missing for GARCH | 请运行 `pip install arch` 以使用 GARCH 模型。 | 請執行 `pip install arch` 以使用 GARCH 模型。 | Run `pip install arch` for GARCH modelling. |

## Related skills

- `longbridge-factor-research` — IC/IR factor testing
- `longbridge-pairs-trading` — cointegration-based pairs trading
- `longbridge-correlation` — cross-asset correlation analysis
- `longbridge-volatility-strategy` — volatility modelling and trading

## File layout

```
skills/longbridge-quant-stats/
└── SKILL.md
```