# Heteroscedasticity

### What Heteroscedasticity Tests Check

Heteroscedasticity tests examine whether the variance of model residuals is constant across all levels of the independent variables (homoscedasticity) or varies systematically (heteroscedasticity).

**Purpose:** Tests if error variance is constant across fitted values, ensuring prediction intervals and hypothesis tests are accurate.

### Why Constant Variance Matters

When residuals have constant variance (homoscedasticity):

**Standard Errors are Correct:** Coefficient standard errors accurately reflect uncertainty

**Confidence Intervals are Valid:** Intervals have correct coverage probabilities

**Hypothesis Tests are Reliable:** P-values and significance tests are trustworthy

**Predictions are Optimal:** Ordinary Least Squares (OLS) provides the most efficient estimates

**Heteroscedasticity** doesn't bias coefficient estimates, but it makes standard errors incorrect, leading to invalid confidence intervals and hypothesis tests.

### Statistical Tests Available

MixModeler provides three heteroscedasticity tests:

| Test                     | Description                                                            | Interpretation                        |
| ------------------------ | ---------------------------------------------------------------------- | ------------------------------------- |
| **Breusch-Pagan Test**   | Tests for linear relationship between squared residuals and predictors | p < 0.05 indicates heteroscedasticity |
| **White Test**           | More general test for any form of heteroscedasticity                   | p < 0.05 indicates heteroscedasticity |
| **Goldfeld-Quandt Test** | Tests whether variance differs across subsamples                       | p < 0.05 indicates heteroscedasticity |

**Interpretation:** If p-value ≥ 0.05, no significant heteroscedasticity detected. If p < 0.05, heteroscedasticity is present.

### Visual Diagnostics

MixModeler provides key visualizations for detecting heteroscedasticity:

**Residuals vs Fitted Values Plot:**

* Plots residuals against predicted values
* **Good:** Random scatter around zero with consistent spread (no pattern)
* **Problem:** Funnel shape (variance increases/decreases with fitted values)
* **Problem:** Curved pattern (indicates non-linear relationship)

**Scale-Location Plot:**

* Plots square root of absolute standardized residuals against fitted values
* **Good:** Horizontal line with random scatter
* **Problem:** Upward or downward trend indicates changing variance

### Common Patterns

**Fan Shape (Most Common):**

* Variance increases as predicted values increase
* Often seen when predicting sales (larger values have more variability)

**Funnel Shape:**

* Variance decreases as predicted values increase
* Less common but still problematic

**Grouped Heteroscedasticity:**

* Different variance for different categories or time periods
* Suggests need for categorical variables or interactions

### Interpreting Test Results

#### Passed Tests (✓)

**What it means:** No significant heteroscedasticity detected (p ≥ 0.05)

**Implications:**

* Variance of errors is reasonably constant
* Standard errors are reliable
* Confidence intervals and p-values are valid

**Action:** No action needed - homoscedasticity assumption is satisfied

#### Failed Tests (⚠)

**What it means:** Heteroscedasticity detected (p < 0.05)

**Implications:**

* Standard errors may be incorrect (typically too small)
* Confidence intervals may have wrong coverage
* Hypothesis tests may be unreliable
* **Note:** Coefficient estimates remain unbiased

**Common Causes:**

* Dependent variable has increasing/decreasing variance
* Important variables omitted
* Wrong functional form (need transformations)
* Outliers affecting variance
* Natural heteroscedasticity in the data-generating process

### What to Do When Tests Fail

If heteroscedasticity tests fail, try these solutions in order:

**1. Transform the Dependent Variable (Most Effective)**

* **Log transformation:** Reduces right skewness and stabilizes variance
  * Good for: Sales, revenue, spend data
  * Effect: Multiplicative relationships become additive
* **Square root transformation:** Moderate variance stabilization
* **Inverse transformation:** For highly skewed data

**2. Use Weighted Least Squares (WLS)**

* Give less weight to observations with higher variance
* Requires estimating variance function
* Available in advanced statistical software

**3. Use Robust Standard Errors**

* Calculate heteroscedasticity-consistent (HC) standard errors
* Corrects standard errors without changing coefficients
* Common variants: HC0, HC1, HC2, HC3

**4. Add Omitted Variables**

* Include variables that explain variance patterns
* Add interaction terms
* Consider non-linear transformations

**5. Check for Outliers**

* Review Influential Points diagnostic
* Outliers can create false heteroscedasticity patterns
* Consider robust regression methods

**6. When Heteroscedasticity is Acceptable**

* Mild heteroscedasticity with large sample sizes
* Focus on coefficient estimates rather than inference
* Using robust standard errors in practice
* Business decisions not sensitive to exact confidence intervals

### Practical Guidelines

**Acceptable Scenarios:**

* Mild heteroscedasticity (p-value between 0.01-0.05)
* Large datasets where robust standard errors are used
* Predictions are the primary goal (coefficients remain unbiased)
* Natural variation in business data

**Critical Issues:**

* Severe heteroscedasticity (p < 0.001)
* Clear fan or funnel pattern in residual plots
* Small sample sizes requiring precise inference
* Using model for confidence interval construction

### Example Interpretation

**Scenario 1 - Passed:**

* Breusch-Pagan p-value: 0.28
* White test p-value: 0.42
* Residual plot shows random scatter with consistent spread

**Interpretation:** No significant heteroscedasticity detected. The constant variance assumption is satisfied, and standard errors are reliable.

**Scenario 2 - Failed:**

* Breusch-Pagan p-value: 0.008
* White test p-value: 0.003
* Residual plot shows clear fan shape (increasing variance)

**Interpretation:** Heteroscedasticity detected. Variance increases with fitted values. Consider log-transforming the KPI or using robust standard errors. If focused on coefficient interpretation rather than p-values, this may be acceptable with appropriate caveats.

**Scenario 3 - Severe:**

* Breusch-Pagan p-value: < 0.001
* White test p-value: < 0.001
* Strong funnel pattern with extreme variance differences

**Interpretation:** Severe heteroscedasticity. Model requires transformation. Try log-transforming the dependent variable before proceeding with inference or decision-making.

### Marketing Mix Modeling Context

In MMM, heteroscedasticity often appears because:

**Sales Variability:** Larger sales periods naturally have higher variance

**Promotional Effects:** Promotions create volatility in certain periods

**Seasonal Patterns:** Different variance across seasons

**Spend Ranges:** Different variance at low vs. high spend levels

**Log transformation** of the KPI is particularly effective in MMM as it:

* Stabilizes variance
* Makes relationships multiplicative (natural for marketing effects)
* Reduces influence of outliers
* Provides coefficients interpretable as elasticities

### Relationship to Other Assumptions

Heteroscedasticity often co-occurs with:

**Non-normality:** Changing variance can cause residuals to deviate from normality

**Outliers:** Extreme values can create both heteroscedasticity and influential points

**Non-linearity:** Wrong functional form can manifest as heteroscedasticity

### Related Diagnostics

After reviewing heteroscedasticity:

* Check **Residual Normality** as heteroscedasticity can affect normality
* Review **Influential Points** to identify outliers causing variance issues
* Examine **Actual vs Predicted** for systematic patterns
