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    <div class="week-hero-badge">Week 09 &mdash; Phase 3: Research Execution</div>
    <h1>Model Assumptions<br>(IP §3b)</h1>
    <div class="week-hero-pill">Section 3b — Model Assumptions</div>
    <p class="week-hero-objective">Test and document every model assumption. This IS Section 3b of your IP. The most critical section for QT risk management.</p>
  </div>
</div>

<div class="container">

  <div class="ip-anchor">
    <span class="ip-anchor-label">IP Anchor</span>
    <span class="ip-anchor-section">Section 3b — Model assumptions</span>
    <span class="ip-anchor-desc">Mandatory every time you use a model. QT uses this section to adjust risk management. Document everything.</span>
  </div>

  <div class="section">
    <h2 class="section-title">What this week covers</h2>
    <p>This is the most technically detailed week. Section 3b is a direct reference for completing the assumption-testing portion of every IP. You will learn to test five critical assumptions (covered in Week 5), plus the special case of cointegration for pairs trading. Every test, result, and implication goes in Section 3b.</p>
  </div>

  <div class="section">
    <h2 class="section-title">Section 3b checklist</h2>
    <p>Copy this checklist into your IP. Complete every row. Report the test, the result, and what you'll do if it fails.</p>

    <table style="font-size: 0.9rem;">
      <thead>
        <tr>
          <th>Assumption</th>
          <th>Test</th>
          <th>Python</th>
          <th>Fail condition</th>
          <th>If it fails</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <td><strong>Normality of residuals</strong></td>
          <td>Jarque-Bera, Shapiro-Wilk</td>
          <td><code>scipy.stats.jarque_bera</code></td>
          <td>p &lt; 0.05</td>
          <td>Use robust methods; report fat-tail risk</td>
        </tr>
        <tr>
          <td><strong>Stationarity</strong></td>
          <td>ADF, KPSS</td>
          <td><code>statsmodels.adfuller</code></td>
          <td>ADF p &gt; 0.05</td>
          <td>Difference series; use returns not prices</td>
        </tr>
        <tr>
          <td><strong>No autocorrelation</strong></td>
          <td>Ljung-Box</td>
          <td><code>statsmodels.acorr_ljungbox</code></td>
          <td>p &lt; 0.05</td>
          <td>Use Newey-West HAC standard errors</td>
        </tr>
        <tr>
          <td><strong>Homoscedasticity</strong></td>
          <td>Breusch-Pagan</td>
          <td><code>statsmodels.het_breuschpagan</code></td>
          <td>p &lt; 0.05</td>
          <td>Use HC3/HC4 robust SE or GARCH model</td>
        </tr>
        <tr>
          <td><strong>No multicollinearity</strong></td>
          <td>VIF</td>
          <td><code>statsmodels.variance_inflation_factor</code></td>
          <td>VIF &gt; 10</td>
          <td>Drop or combine correlated features</td>
        </tr>
      </tbody>
    </table>
  </div>

  <div class="section">
    <h2 class="section-title">Deep dive: Cointegration (pairs trading)</h2>

    <p>For pairs strategies, you're not testing stationarity of individual series — you're testing stationarity of the spread (the linear combination).</p>

    <h3 style="margin-top: 1.5rem;">Engle-Granger cointegration test</h3>
    <p>Two series are cointegrated if a linear combination of them is stationary.</p>
    <p style="margin-top: 1rem;"><strong>Method:</strong></p>
    <ol style="font-size: 0.95rem;">
      <li>Regress y on x: \(y_t = \alpha + \beta x_t + \varepsilon_t\)</li>
      <li>Extract residuals ε</li>
      <li>Test residuals for stationarity (ADF test)</li>
      <li>If residuals are stationary, the pair is cointegrated</li>
    </ol>

    <p style="margin-top: 1.5rem;"><strong>Python:</strong></p>
    <pre><code>from statsmodels.tsa.stattools import coint

score, p_value, critical_values = coint(series_a, series_b)
print(f"Cointegration test p-value: {p_value:.4f}")
# p &lt; 0.05 → cointegrated at 5% significance
</code></pre>

    <h3 style="margin-top: 1.5rem;">Johansen test (multiple assets)</h3>
    <p>For strategies with 3+ assets, use Johansen's cointegration test.</p>
    <p>\[ \Delta \mathbf{y}_t = \Pi \mathbf{y}_{t-1} + \sum_{i=1}^{p-1} \Gamma_i \Delta \mathbf{y}_{t-i} + \varepsilon_t \]</p>

    <p style="margin-top: 1rem;"><strong>Interpretation:</strong> The rank of Π determines the number of cointegrating relationships (linearly independent stationary combinations).</p>
  </div>

  <div class="section">
    <h2 class="section-title">Section 3b template with real example</h2>

    <div class="card" style="background: var(--bg-elevated); font-size: 0.9rem;">
      <p style="color: var(--accent); font-weight: 600;">SECTION 3B: MODEL ASSUMPTIONS — Ridge Regression Commodity Strategy</p>

      <p style="margin-top: 1rem;"><strong>Strategy uses a ridge regression to predict corn returns from 3 factors:</strong></p>
      <ul style="margin-left: 1.5rem; font-size: 0.9rem;">
        <li>Factor 1: 30-day momentum</li>
        <li>Factor 2: 60-day seasonality (deviation from 10-year average)</li>
        <li>Factor 3: COT speculator positioning (normalized)</li>
      </ul>

      <p style="margin-top: 1rem;"><strong>3b.1 Normality of residuals</strong></p>
      <p>Test: Jarque-Bera on regression residuals (2020–2023 in-sample)</p>
      <p>Result: JB stat = 12.4, p-value = 0.002 → reject normality</p>
      <p>Implication: Residuals have fat tails. Daily returns exhibit kurtosis = 4.2 (excess = 1.2). Documented.</p>
      <p>Action: Report expected maximum drawdown assuming fat tails. Use robust standard errors (HC3).</p>

      <p style="margin-top: 1rem;"><strong>3b.2 Stationarity of predictors</strong></p>
      <p>Test: ADF on each factor</p>
      <ul style="margin-left: 1.5rem; font-size: 0.85rem;">
        <li>Momentum (30-day): ADF stat = -8.7, p &lt; 0.001 → stationary ✓</li>
        <li>Seasonality (deviation): ADF stat = -9.2, p &lt; 0.001 → stationary ✓</li>
        <li>COT positioning (normalized): ADF stat = -6.4, p &lt; 0.001 → stationary ✓</li>
      </ul>
      <p>All factors are stationary. No issues.</p>

      <p style="margin-top: 1rem;"><strong>3b.3 No autocorrelation in residuals</strong></p>
      <p>Test: Ljung-Box on residuals, lags 1–20</p>
      <p>Result: All p-values &gt; 0.05. No significant autocorrelation detected. ✓</p>

      <p style="margin-top: 1rem;"><strong>3b.4 Homoscedasticity</strong></p>
      <p>Test: Breusch-Pagan heteroscedasticity test</p>
      <p>Result: BP stat = 18.3, p = 0.0003 → heteroscedasticity present</p>
      <p>Action: Volatility clusters (expected in commodity markets). Use HC3 robust standard errors for inference.</p>

      <p style="margin-top: 1rem;"><strong>3b.5 No multicollinearity</strong></p>
      <p>Test: Variance Inflation Factor on 3 factors</p>
      <ul style="margin-left: 1.5rem; font-size: 0.85rem;">
        <li>Momentum: VIF = 1.3</li>
        <li>Seasonality: VIF = 1.1</li>
        <li>COT: VIF = 1.2</li>
      </ul>
      <p>All VIF &lt; 2. No multicollinearity concerns. ✓</p>

      <p style="margin-top: 1rem;"><strong>Summary:</strong> Ridge regression suitable. Primary concerns: fat-tail risk (documented), heteroscedasticity (robust SE applied). Model assumptions documented and acceptable for live trading.</p>
    </div>
  </div>

  <div class="section">
    <h2 class="section-title">Assumption testing code patterns</h2>

    <h3 style="margin-top: 1.5rem;">Normality (Jarque-Bera)</h3>
    <pre><code>from scipy import stats

jb_stat, jb_p = stats.jarque_bera(residuals)
print(f"Jarque-Bera: stat={jb_stat:.4f}, p={jb_p:.4f}")

if jb_p &lt; 0.05:
    print("Residuals are NOT normally distributed (fat tails likely)")
else:
    print("Residuals are consistent with normality")
</code></pre>

    <h3 style="margin-top: 1.5rem;">Stationarity (ADF)</h3>
    <pre><code>from statsmodels.tsa.stattools import adfuller

result = adfuller(series, autolag='AIC')
print(f"ADF stat: {result[0]:.4f}, p-value: {result[1]:.4f}")

if result[1] &lt; 0.05:
    print("Series is stationary (reject unit root)")
else:
    print("Series is non-stationary (unit root present)")
</code></pre>

    <h3 style="margin-top: 1.5rem;">Multicollinearity (VIF)</h3>
    <pre><code>from statsmodels.stats.outliers_influence import variance_inflation_factor
import pandas as pd

vif_data = pd.DataFrame()
vif_data["Feature"] = X.columns
vif_data["VIF"] = [
    variance_inflation_factor(X.values, i)
    for i in range(X.shape[1])
]
print(vif_data)

if (vif_data["VIF"] &gt; 10).any():
    print("Multicollinearity detected. Consider dropping features.")
</code></pre>
  </div>

  <div class="section">
    <h2 class="section-title">Common mistakes</h2>
    <div class="mistakes-list">
      <h4 style="margin-top: 0;">Five assumption-testing failures</h4>
      <ul>
        <li><strong>Skipping Section 3b entirely.</strong> It's mandatory. At minimum: test residuals for normality and autocorrelation, test series for stationarity. Report results.</li>
        <li><strong>Running ADF but not reporting it in Section 3b.</strong> If you ran the test, report it. The p-value, the interpretation, what you'll do if it fails.</li>
        <li><strong>Building a multi-factor model without checking VIF.</strong> Correlated factors will cancel each other out in live trading, even if they work in backtest. Check VIF before submitting.</li>
        <li><strong>Ignoring fat tails.</strong> "Close enough to normal" is not a valid analysis. If Jarque-Bera p &lt; 0.05, document the deviation. Report expected extreme drawdown.</li>
        <li><strong>Using price levels in regression.</strong> Prices are non-stationary. Test with ADF first. If non-stationary, use returns or cointegration.</li>
      </ul>
    </div>
  </div>

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