Questions: Testing for Autocorrelation: Durbin-Watson and Breusch-Godfrey

Question 1

A researcher runs OLS on time-series data and finds a Durbin-Watson statistic of 0.8, indicating positive serial correlation. What is the PRIMARY consequence for the regression results?

• A. The coefficient estimates are biased and inconsistent, making them unreliable
• B. The coefficient estimates are unbiased, but the standard errors are unreliable so t-statistics and p-values are invalid
• C. The regression has omitted variable bias due to the correlated residuals
• D. The model will systematically underfit the data by ignoring temporal patterns

Answer: (B)

Autocorrelation does NOT bias OLS coefficient estimates (under standard conditions). The coefficients remain unbiased and consistent. The real damage is to inference: OLS underestimates standard errors when residuals are positively correlated, inflating t-statistics and making results appear more significant than they are. Option A is the most common misconception — conflating 'wrong standard errors' with 'biased estimates.'

Question 2

A researcher estimates a distributed lag model that includes a lagged dependent variable (Yₜ₋₁) as a regressor. They want to test for autocorrelation in the residuals. Which test should they use and why?

• A. Durbin-Watson, because it is the standard and most widely used autocorrelation test
• B. Durbin-Watson with a correction factor to account for the lagged dependent variable
• C. Breusch-Godfrey LM test, because Durbin-Watson is invalid when lagged dependent variables appear as regressors
• D. Neither test — autocorrelation cannot be detected when lagged dependent variables are present

Answer: (C)

The Durbin-Watson statistic is derived under the assumption that regressors are strictly exogenous. When a lagged dependent variable appears on the right-hand side, this assumption fails, and the DW statistic is biased toward 2, making it unreliable. The Breusch-Godfrey LM test does not rely on this assumption and remains valid with lagged dependent variables. It also tests for higher-order autocorrelation, not just first-order.

Question 3

A Durbin-Watson statistic close to 2 indicates no first-order autocorrelation in the residuals.

• T. True
• F. False

Answer: True

The DW statistic approximately equals 2(1 − r̂), where r̂ is the first-order autocorrelation of the residuals. When r̂ ≈ 0 (no autocorrelation), DW ≈ 2. Values near 0 indicate strong positive autocorrelation; values near 4 indicate strong negative autocorrelation. Note that DW only tests for first-order autocorrelation and has an inconclusive range between the lower and upper critical bounds.

Question 4

Positive autocorrelation in OLS residuals typically biases the estimated regression coefficients away from zero.

• T. True
• F. False

Answer: False

This is a key misconception. Under standard conditions, autocorrelation in residuals does not bias or invalidate the OLS coefficient estimates — they remain unbiased and consistent. The problem lies exclusively with inference: the estimated standard errors are too small (positive autocorrelation understates variance), so t-statistics are inflated and confidence intervals are too narrow. The coefficients themselves point in the right direction; the problem is that you overstate your certainty about them.

Question 5

Why does autocorrelation in OLS residuals cause problems for hypothesis testing even when the coefficient estimates remain unbiased?

Model answer: OLS standard errors assume residuals are uncorrelated. When residuals are positively autocorrelated, consecutive observations carry redundant information — the effective sample size is smaller than the nominal n. OLS ignores this and treats all n observations as independent, producing standard errors that are too small. Artificially small standard errors yield inflated t-statistics, leading to false rejections of true null hypotheses (type I errors). The coefficient point estimates are still correct on average, but their apparent precision is overstated.

Think of it this way: if your residuals follow a trend (positive autocorrelation), you are not getting n independent pieces of information — you are getting something closer to n/k independent pieces, where k is the effective autocorrelation length. OLS does not know this and divides by n, not n/k, when estimating variance. The remedy is to use HAC (heteroskedasticity and autocorrelation consistent) standard errors, such as Newey-West, which correct for both the correlation and its magnitude.