Questions: Durbin-Watson Statistic for Autocorrelation

Question 1

A time-series regression yields a Durbin-Watson statistic of 0.4. What does this indicate about the residuals?

• A. Strong negative autocorrelation — residuals alternate in sign
• B. No autocorrelation — residuals are approximately independent
• C. Strong positive autocorrelation — consecutive residuals tend to be similar in sign and magnitude
• D. The test is inconclusive and no inference can be drawn

Answer: (C)

DW ≈ 2(1 − ρ̂), so DW = 0.4 implies ρ̂ ≈ 0.8 — strong positive first-order autocorrelation. This means ûₜ and ûₜ₋₁ tend to be similar, making consecutive differences small, shrinking the numerator, and pushing DW toward 0. This violates OLS assumptions: standard errors are biased downward, inflating t-statistics and making coefficients appear more significant than they are.

Question 2

A researcher estimates Yₜ = α + βYₜ₋₁ + εₜ, computes DW = 1.95, and concludes there is no autocorrelation. Is this valid?

• A. Yes — DW near 2 always indicates no autocorrelation regardless of model specification
• B. No — DW is biased toward 2 when a lagged dependent variable is included, making it unreliable
• C. No — DW only applies to cross-sectional data, not time-series regressions
• D. Yes — the lagged dependent variable controls for autocorrelation, so DW remains valid

Answer: (B)

When a lagged dependent variable appears as a regressor, the DW statistic is biased toward 2, making it appear that there is no autocorrelation even when true autocorrelation exists. The residuals are mechanically correlated with the lagged regressor in a way that violates the test's assumptions. For models with lagged dependent variables, the Breusch-Godfrey test is the appropriate diagnostic — it explicitly handles this structure.

Question 3

A Durbin-Watson value near 4 indicates strong negative first-order autocorrelation in the residuals.

• T. True
• F. False

Answer: True

DW ≈ 2(1 − ρ̂). If ρ̂ ≈ −1, then DW ≈ 2(1 − (−1)) = 4. Negative autocorrelation means consecutive residuals alternate in sign: positive, then negative, then positive. This makes consecutive differences (ûₜ − ûₜ₋₁) large, inflating the numerator and pushing DW toward 4. Like positive autocorrelation, negative autocorrelation biases OLS standard errors and invalidates inference.

Question 4

The Durbin-Watson test can detect autocorrelation at any lag order, making it a comprehensive diagnostic for time-series residuals.

• T. True
• F. False

Answer: False

DW only tests for first-order autocorrelation — whether ûₜ is correlated with ûₜ₋₁. It has no power to detect higher-order patterns such as quarterly seasonality (correlation at lag 4) or annual cycles (lag 12). A series with AR(4) structure but no AR(1) component could produce a DW near 2, falsely suggesting clean residuals. The Breusch-Godfrey test generalizes DW to test autocorrelation at any specified lag order.

Question 5

Why does the Durbin-Watson test give invalid results when a lagged dependent variable appears as a regressor?

Model answer: The DW test assumes that, under the null of no autocorrelation, the residuals are uncorrelated with each other and with the regressors. When a lagged dependent variable (Yₜ₋₁) is a regressor, OLS residuals ûₜ are mechanically correlated with the lagged regressor by construction. This correlation causes the DW statistic to be biased toward 2, making it appear that autocorrelation is absent even when it exists. The test's asymptotic distribution is no longer valid under this specification.

Intuitively, the model already 'absorbs' some time structure through the lagged DV, making residuals look more random than they truly are. The Breusch-Godfrey test handles this by explicitly modeling the relationship between residuals and lagged regressors in an auxiliary regression, making it robust to situations where DW is not.