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The Durbin-Watson statistic DW = Σ(ûₜ - ûₜ₋₁)² / Σûₜ² approximates 2(1 - ρ̂) where ρ̂ is the first-order autocorrelation. Values near 2 suggest no autocorr, < 2 suggests positive autocorr, and > 2 suggests negative autocorr, providing a quick diagnostic.
From your study of autocorrelation and lag structures, you know that when regression residuals are correlated over time, OLS standard errors are biased and t-statistics are unreliable. The Durbin-Watson statistic is the standard first-pass diagnostic for detecting this problem. Its formula — the sum of squared *differences* between consecutive residuals, divided by the sum of squared residuals — is designed to measure exactly how much each residual resembles the one that came before it.
The key insight is the relationship DW ≈ 2(1 − ρ̂). If there is no autocorrelation, ρ̂ ≈ 0, so DW ≈ 2. If residuals are strongly positively autocorrelated (ρ̂ close to +1, meaning each residual is similar to the previous one), then consecutive differences are small, making the numerator small, and DW approaches 0. If residuals are strongly negatively autocorrelated (ρ̂ close to −1, meaning residuals alternate in sign), then each difference is large, and DW approaches 4. So the full scale runs 0 to 4, with 2 as the "clean" value.
In practice, you compare the computed DW statistic to critical bounds dL and dU from the Durbin-Watson tables (which depend on sample size n and the number of regressors k). If DW < dL, reject the null of no positive autocorrelation. If DW > 4 − dL, reject the null of no negative autocorrelation. Between dL and dU is an inconclusive zone — not evidence of no autocorrelation, but not decisive evidence against it either. This inconclusive region is one of the test's known limitations.
The Durbin-Watson statistic has two important restrictions to internalize. First, it only tests for first-order autocorrelation — whether ûₜ is correlated with ûₜ₋₁. It will miss higher-order patterns (e.g., quarterly seasonality where ûₜ correlates with ûₜ₋₄). Second, it is invalid when a lagged dependent variable appears as a regressor, because in that case the residuals are mechanically correlated with the regressor, violating the test's assumptions. For those more complex situations, you will next encounter the Breusch-Godfrey test, which handles both higher-order autocorrelation and lagged-dependent-variable models.