Durbin-Watson Calculator

Regression Residuals Dataset (e_i) Enter sequential OLS residuals separated by commas, spaces, or newlines (n ≥ 3)

Please enter at least 3 valid numeric residual values.

Residual Autocorrelation Diagnostic

What Is the Durbin-Watson Statistic?

The Durbin-Watson Statistic ($d$) is a test statistic used in linear regression to detect autocorrelation (serial correlation) in the error residuals from a regression analysis across consecutive time periods or sequence order.

The statistic ranges from $0$ to $4$. A value of $d \approx 2.0$ indicates complete absence of autocorrelation. Values significantly less than 2 signal positive serial correlation, while values greater than 2 indicate negative serial correlation.

When to Use It

Time-Series Regression: Checking whether successive error terms are correlated in econometric and financial forecasting.
Spatial & Sequential Ordering: Validating the independence assumption across ordered observations.
Pre-Inference Auditing: Ensuring OLS standard errors are not artificially deflated by autocorrelated residuals.
Correcting Standard Errors: Directing analysis toward Cochrane-Orcutt or Newey-West robust estimators when $d$ falls outside acceptable bounds.

The Mathematical Formula

$$d = \frac{\sum_{i=2}^n (e_i - e_{i-1})^2}{\sum_{i=1}^n e_i^2}$$ $$\text{Approximate Autocorrelation } (\rho) \approx 1 - \frac{d}{2}$$

Where $e_i$ represents the OLS residual for observation $i$, and $n$ is the total sample size of residuals.

Worked Example

Scenario: 6 sequential regression residuals: $[1.2, 0.8, -0.3, -1.1, -0.5, 0.4]$.

Step 1: Compute Squared Consecutive Differences:
$(0.8 - 1.2)^2 = 0.16, \quad (-0.3 - 0.8)^2 = 1.21, \quad (-1.1 - (-0.3))^2 = 0.64$
$(-0.5 - (-1.1))^2 = 0.36, \quad (0.4 - (-0.5))^2 = 0.81$
$\sum (e_i - e_{i-1})^2 = 0.16 + 1.21 + 0.64 + 0.36 + 0.81 = 3.1800$.

Step 2: Compute Sum of Squared Residuals ($\sum e_i^2$):
$(1.2)^2 + (0.8)^2 + (-0.3)^2 + (-1.1)^2 + (-0.5)^2 + (0.4)^2 = 3.7900$.

Step 3: Calculate Durbin-Watson d:
$d = \frac{3.1800}{3.7900} = 0.8391$.

Conclusion: Since $d = 0.8391 < 1.5$, strong **positive autocorrelation** is detected.

Autocorrelation Audit Complete

Durbin-Watson Results

Computed Durbin-Watson Statistic

d = 0.8391

Strong Positive Autocorrelation Detected (d < 1.5)

DW Statistic (d) 0.8391

Numerator (ΣΔ²) 3.1800

Denominator (Σe²) 3.7900

Est. Autocorrelation (ρ) 0.5805

Sample Size (n) 6

Independence Status Violated

Durbin-Watson Calculator

Durbin-Watson Calculator

Durbin-Watson Results

Step-by-Step Calculation

Interpretation

Durbin-Watson 0 to 4 Autocorrelation Spectrum Gauge

Perfect for students, researchers, and professionals looking to build real statistical skills.