Autocorrelation (Durbin-Watson)
What Autocorrelation Tests Check
Autocorrelation tests detect whether model residuals are correlated with their own past values. In time series data (which is typical in MMM), autocorrelated residuals indicate that the model hasn't fully captured temporal patterns.
Purpose: Checks if residuals are correlated with previous time periods, ensuring the independence assumption is satisfied.
Why Autocorrelation Matters
When residuals have no autocorrelation:
Standard Errors are Accurate: Coefficient standard errors correctly reflect uncertainty
Hypothesis Tests are Valid: P-values and confidence intervals have correct coverage
Model is Complete: All systematic temporal patterns have been captured
Autocorrelated residuals suggest the model is missing time-dependent effects, leading to overconfident (too narrow) standard errors and unreliable hypothesis tests.
Statistical Tests Available
MixModeler provides three autocorrelation tests:
Durbin-Watson Test
Tests for first-order autocorrelation (lag 1)
DW ≈ 2 is ideal; < 1.5 or > 2.5 indicates problems
Breusch-Godfrey Test
More general test for higher-order autocorrelation
p < 0.05 indicates autocorrelation present
Ljung-Box Test
Tests for autocorrelation at multiple lags
p < 0.05 indicates autocorrelation present
Durbin-Watson Statistic Interpretation
The Durbin-Watson (DW) statistic ranges from 0 to 4:
< 1.5
Positive autocorrelation
Add lagged variables or time trends
1.5 - 2.5
No significant autocorrelation
✓ Assumption satisfied
> 2.5
Negative autocorrelation (rare)
Check for model misspecification
Ideal Value: DW ≈ 2 indicates no autocorrelation
Positive autocorrelation (DW < 2): Current period residuals are similar to previous period residuals - most common issue in time series
Negative autocorrelation (DW > 2): Current period residuals are opposite to previous period residuals - unusual and may indicate over-differencing
Visual Diagnostics
MixModeler provides visualizations to detect autocorrelation:
Autocorrelation Function (ACF) Plot:
Shows correlation between residuals and lagged residuals
Good: All lags within confidence bounds (shown as dashed lines)
Problem: Spikes outside confidence bounds indicate significant autocorrelation
Residuals Over Time Plot:
Time series plot of residuals
Good: Random scatter around zero with no visible patterns
Problem: Cyclical patterns, trends, or clustering
Confidence Bounds: Typically set at ±1.96/√n, representing 95% confidence intervals
Interpreting Test Results
Passed Tests (✓)
What it means: No significant autocorrelation detected
Durbin-Watson near 2 (typically 1.5-2.5)
Breusch-Godfrey p-value ≥ 0.05
ACF plot shows most lags within confidence bounds
Implications:
Residuals are independent across time periods
Standard errors are reliable
Model has captured temporal patterns adequately
Action: No action needed - independence assumption is satisfied
Failed Tests (⚠)
What it means: Autocorrelation detected in residuals
Implications:
Standard errors may be too small (overconfident)
P-values may be overstated (variables appear more significant than they are)
Model predictions may be biased
Common Causes:
Missing lagged effects of marketing variables
Omitted seasonal or trend components
Persistence in dependent variable not modeled
Wrong functional form (e.g., linear when should be non-linear)
What to Do When Tests Fail
If autocorrelation tests fail, try these solutions:
1. Add Lagged Variables (Most Common Solution)
Include lagged values of the dependent variable
Add lagged marketing variables
Consider adstock transformations with higher decay rates
2. Include Time Trends
Add linear or quadratic time trend
Include monthly or quarterly dummy variables
Model seasonality explicitly
3. Add Omitted Variables
Include variables that capture temporal persistence
Add external factors that vary over time (e.g., macroeconomic indicators)
Consider competitive activity or market dynamics
4. Use Different Model Specification
Try different adstock parameterizations
Consider distributed lag models
Explore autoregressive structures
5. When Autocorrelation is Acceptable
Mild autocorrelation (DW between 1.3-1.5 or 2.5-2.7)
Focus is on prediction rather than hypothesis testing
Business insights are robust to minor violations
Using robust standard errors in analysis
Practical Guidelines
Acceptable Scenarios:
Slight autocorrelation with DW between 1.3 and 2.7
Weekly data with minor persistence
Large sample sizes where effect on inference is minimal
Critical Issues:
DW < 1.0 or > 3.0 (severe autocorrelation)
Multiple lags showing significant autocorrelation in ACF plot
Clear cyclical or trending patterns in residual plots
Using model for forecasting (autocorrelation severely biases predictions)
Example Interpretation
Scenario 1 - Passed:
Durbin-Watson: 1.95
Breusch-Godfrey p-value: 0.42
ACF plot: All lags within confidence bounds
Interpretation: No significant autocorrelation detected. The independence assumption is satisfied, and standard errors are reliable.
Scenario 2 - Mild Autocorrelation:
Durbin-Watson: 1.35
Breusch-Godfrey p-value: 0.03
ACF plot: Lag 1 slightly outside bounds
Interpretation: Mild positive autocorrelation detected. Consider adding a lagged dependent variable or checking adstock specifications. The violation is moderate and may be acceptable depending on business use case.
Scenario 3 - Severe Autocorrelation:
Durbin-Watson: 0.85
Breusch-Godfrey p-value: < 0.001
ACF plot: Multiple lags significantly outside bounds
Interpretation: Severe positive autocorrelation. The model is missing important temporal dynamics. Add lagged variables, time trends, or reconsider model specification before using for business decisions.
Marketing Mix Modeling Context
In MMM, autocorrelation often indicates:
Incomplete Adstock Modeling: Marketing effects may persist longer than captured
Missing Baseline Trends: Organic growth or decay not fully modeled
Seasonal Patterns: Regular cyclical patterns not accounted for
Competitive Dynamics: Reaction to competitor activity not included
Related Diagnostics
After reviewing autocorrelation:
Check Residual Normality as autocorrelation can affect normality tests
Review Actual vs Predicted to see if temporal patterns are visible
Examine Variable Testing to optimize adstock parameters
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