Durbin-Watson Test

Definition

Core Statement

The Durbin-Watson (DW) Test detects autocorrelation (or serial correlation) in the residuals of a regression analysis. It tests whether adjacent residuals are correlated, which violates the assumption of independence in Simple Linear Regression.

Purpose

Diagnose Assumption Violation: Ensure regression error terms are independent.
Flag Time Series Issues: Detect if a model fails to capture time-dependent patterns.

When to Use

Use Durbin-Watson When...

Data was collected over time (Time Series).
You fit a Linear Regression model.
You suspect the error at time $t$ depends on error at time $t - 1$ .

Do NOT Use When...

Data is cross-sectional (order doesn't matter).
Use Breusch-Godfrey Test for higher-order autocorrelation (lag > 1).

Theoretical Background

The Statistic ( $d$ )

d = \frac{\sum_{t = 2}^{T} (e_{t} - e_{t - 1})^{2}}{\sum_{t = 1}^{T} e_{t}^{2}}

where $e_{t}$ is the residual at time $t$ .

Interpretation Range

The $d$ statistic ranges from 0 to 4:

$d \approx 2$ : No Autocorrelation (Ideal).
$0 \leq d < 1.5$ : Positive Autocorrelation (Common in time series).
$2.5 < d \leq 4$ : Negative Autocorrelation (Rapid oscillation).

Worked Numerical Example

Stock Returns vs Market Returns

Regression: $R e t u r n_{A s s e t} = β_{0} + β_{1} R e t u r n_{M a r k e t}$
Residuals: [0.5, 0.4, 0.6, -0.2, -0.3, -0.5]

Observation: Positive residuals cluster together; Negative residuals cluster together.
Calculation:

DW Statistic $d = 0.8$ .

Conclusion: $d ≪ 2$ . Strong Positive Autocorrelation.
Implication: Standard errors are underestimated. Calculated t-stats are inflated. The relationship appears more significant than it really is.

Assumptions

First-Order Autocorrelation: Tests only for correlation between $e_{t}$ and $e_{t - 1}$ .
Interceptor: Regression must include an intercept.
No Lagged DV: Independent variables cannot include lagged response variables ( $Y_{t - 1}$ ). Use Breusch-Godfrey Test instead.

Limitations

Pitfalls

Order Matters: Calculating DW on unsorted cross-sectional data is meaningless.
Inconclusive Region: The test has "bounds" ( $d_{L}, d_{U}$ ). If $d$ falls between them, the test is inconclusive.
Only Lag 1: Does not detect seasonal patterns (e.g., Lag 4 or Lag 12 autocorrelation).

Python Implementation

import statsmodels.api as sm
from statsmodels.stats.stattools import durbin_watson

# Feature: Residuals from a fitted model
residuals = model.resid

# Calculate DW
dw_stat = durbin_watson(residuals)
print(f"Durbin-Watson: {dw_stat:.2f}")

if dw_stat < 1.5:
    print("Warning: Positive Autocorrelation")
elif dw_stat > 2.5:
    print("Warning: Negative Autocorrelation")
else:
    print("Assumption Met: Residuals are independent")

R Implementation

library(lmtest)

# Fit Model
model <- lm(Sales ~ Time, data = df)

# Durbin-Watson Test
dwtest(model)
# Output includes DW statistic and p-value

Interpretation Guide

DW Value	Interpretation	Edge Case Notes
2.0	No Autocorrelation.	Perfect independence.
1.8 - 2.2	Acceptable range.	Usually considered "close enough" to 2.
0.5	Strong Positive Autocorrelation.	Standard errors will be biased downwards. Risk of spurious regression.
1.3	Inconclusive Zone?	Check critical value tables ( $d_{L}, d_{U}$ based on n and k).

Breusch-Godfrey Test - More general test for AR(p) errors.
Ljung-Box Test - Tests white noise in ARIMA residuals.
Time Series Analysis
Stationarity (ADF & KPSS)