Questions: Breusch-Godfrey Test for Serial Correlation

Question 1

The Breusch-Godfrey auxiliary regression regresses residuals on lagged residuals AND the original model's regressors. Why are the original regressors included?

• A. To increase the R² of the auxiliary regression and make the test more powerful
• B. To remove mechanical correlation induced when the main model contains a lagged dependent variable, making the test valid in that setting
• C. Because the original regressors serve as instrumental variables for the lagged residuals
• D. To ensure the auxiliary regression has the same degrees of freedom as the main model

Answer: (B)

This is the key design feature that makes Breusch-Godfrey superior to Durbin-Watson. When the main model includes a lagged dependent variable (e.g., y_{t-1} as a regressor), the Durbin-Watson test is invalid because the residuals are mechanically correlated with this regressor. Including the original regressors in the auxiliary regression purges this mechanical dependence, allowing the test statistic to correctly reflect genuine serial correlation in the disturbances. Without this inclusion, the test would be biased in exactly the cases where serial correlation testing is most needed.

Question 2

A researcher runs the Durbin-Watson test on a quarterly time series model and fails to reject no serial correlation. They then run Breusch-Godfrey with p=4 and reject. What does this tell us?

• A. The Breusch-Godfrey test is producing a false positive because DW already confirmed no serial correlation
• B. The model likely has higher-order serial correlation (beyond lag 1) that DW cannot detect
• C. The model must contain a lagged dependent variable, which invalidates both tests
• D. Four lags is too many — reducing p would likely also fail to reject

Answer: (B)

Durbin-Watson only tests for AR(1) serial correlation — correlation between adjacent residuals. Quarterly economic data often carries memory across multiple periods (a shock in Q1 may still affect residuals in Q4). BG with p=4 tests whether any of the first four lags carry predictive power. Failing DW but rejecting BG at p=4 is exactly the scenario BG was designed to catch: higher-order autocorrelation invisible to DW. This doesn't indicate a false positive; it reveals DW's fundamental limitation.

Question 3

The Breusch-Godfrey test can detect serial correlation at multiple lags simultaneously, unlike the Durbin-Watson test which is restricted to first-order autocorrelation.

• T. True
• F. False

Answer: True

Correct. By including p lagged residuals in the auxiliary regression and testing their joint significance, BG detects any autocorrelation structure up to order p. The choice of p is a judgment call based on data frequency and prior expectations. DW, by contrast, produces a single statistic targeting AR(1) and has well-known failure modes (inconclusive zones, invalidity with lagged regressors). This flexibility is the primary practical advantage of BG over DW.

Question 4

The Breusch-Godfrey test is invalid when the original model includes a lagged dependent variable as a regressor.

• T. True
• F. False

Answer: False

This is precisely backwards — it describes the Durbin-Watson test's limitation, not BG's. The Durbin-Watson test IS invalid with lagged dependent variables because the test statistic is biased toward 2 (no serial correlation) in that setting. The Breusch-Godfrey test was specifically designed to handle this case by including the original regressors in the auxiliary regression, removing the mechanical correlation that invalidates DW. BG is the recommended diagnostic precisely when lagged dependent variables are present.

Question 5

What does rejecting the null hypothesis in a Breusch-Godfrey test tell you, and what are the two main remedies depending on the likely source of the serial correlation?

Model answer: Rejection means at least one of the p lagged residual coefficients is significantly different from zero — the residuals carry predictable autocorrelation structure. The source diagnosis determines the remedy: if serial correlation reflects dynamic misspecification (the model's conditional mean is wrong), add lags of the dependent variable to the main equation. If it reflects pure disturbance autocorrelation (the DGP genuinely has correlated errors), use Newey-West HAC standard errors to produce valid inference without changing the point estimates.

The distinction between a misspecified model and a correctly specified model with autocorrelated errors is crucial. Adding lags fixes the former by capturing dynamics erroneously left in the residuals. HAC standard errors fix inference for the latter without altering point estimates. Applying the wrong remedy — HAC when the true problem is omitted dynamics — produces consistent but potentially inefficient estimates and leaves the misspecification in place.