Durbin Watson Statistic

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Durbin–Watson Statistic

What it is

The Durbin–Watson (DW) statistic tests for autocorrelation (serial correlation) in the residuals of a regression model. Its value ranges from 0 to 4:
– DW ≈ 2: no autocorrelation detected
– DW < 2: positive autocorrelation (residuals tend to have the same sign in successive observations)
– DW > 2: negative autocorrelation (residuals tend to alternate signs)

Positive autocorrelation in time series (for example, prices) indicates momentum—past movements tend to continue—while negative autocorrelation suggests reversals.

Why it matters

Autocorrelation violates the ordinary least squares (OLS) assumption that residuals are uncorrelated. If present, standard errors, hypothesis tests, and confidence intervals may be invalid, leading to misleading inferences from regression results. Analysts often transform data (e.g., use returns instead of levels) to reduce autocorrelation.

Limitations

  • DW is not appropriate when the model includes lagged dependent variables among the regressors.
  • The test is designed for first-order autocorrelation; it may not detect higher-order patterns.
  • Many statistical packages report DW, but interpret results with sample size, model specification, and critical values in mind.

Formula

DW = [sum from t=2 to n of (e_t − e_{t−1})^2] / [sum from t=1 to n of e_t^2]

where e_t are the OLS residuals.

Step-by-step example

Given six (x, y) data points:
– (10, 1100), (20, 1200), (35, 985), (40, 750), (50, 1215), (45, 1000)

  1. Fit an OLS line of best fit:
    Y = −2.6268x + 1129.2

  2. Compute expected Y and residuals e_t = observed Y − expected Y:

  3. Expected Y: 1102.9, 1076.7, 1037.3, 1024.1, 997.9, 1011.0

  4. Residuals e_t: −2.9, 123.3, −52.3, −274.1, 217.1, −11.0

  5. Sum of squared residuals:
    Σ e_t^2 = 140,330.81

  6. Differences between successive residuals and sum of their squares:
    Differences (e_t − e_{t−1}): 126.2, −175.6, −221.9, 491.3, −228.1
    Σ (e_t − e_{t−1})^2 = 389,406.71

  7. Compute DW:
    DW = 389,406.71 / 140,330.81 ≈ 2.77

Interpretation for this example: DW ≈ 2.77 indicates negative first-order autocorrelation in the residuals.

Key takeaways

  • DW detects first-order serial correlation in regression residuals; values range from 0 to 4.
  • A value near 2 suggests no autocorrelation; values <2 signal positive autocorrelation; values >2 signal negative autocorrelation.
  • The test is unsuitable for models with lagged dependent variables and should be complemented by other diagnostics for higher-order or more complex autocorrelation patterns.
  • The statistic is named after James Durbin and Geoffrey Watson.