Time Seriesadvanced

Testing for Serial Correlation

Two standard tests detect serial correlation. The Durbin-Watson statistic targets AR(1) errors and satisfies $DW \approx 2(1 - \hat{\rho})$ , so a value near 2 signals no autocorrelation, near 0 strong positive autocorrelation, and near 4 negative. Its drawbacks are an inconclusive region and invalidity when a lagged dependent variable is present. The more flexible Breusch-Godfrey (LM) test regresses the OLS residuals on the regressors and $p$ lagged residuals, then uses $n R^2$ as a chi-square statistic to test for AR( $p$ ). Breusch-Godfrey handles higher-order autocorrelation and remains valid with lagged dependent variables, which is why it is the workhorse test.

Why it matters

Both tests ask the same question. Do this period’s residuals predict next period’s? Durbin-Watson packages that into a single number scaled around 2, easy to read but limited to one lag. Breusch-Godfrey runs an explicit auxiliary regression of the residuals on their own lags, so you can look for several orders of autocorrelation at once and still trust the result when the model contains lagged outcomes.

Formulas

Durbin-Watson statistic

DW = \frac{\sum_{t=2}^{n} (\hat{u}_t - \hat{u}_{t-1})^2}{\sum_{t=1}^{n} \hat{u}_t^2} \approx 2(1 - \hat{\rho})

DW near 2 means no AR(1); near 0 is strong positive, near 4 strong negative serial correlation.

Breusch-Godfrey LM statistic

LM = n R^2 \sim \chi^2_{p}

R-squared from regressing residuals on the regressors and p lagged residuals; tests AR(p) jointly.

Worked examples

Scenario

You want a formal check for serial correlation after estimating a dynamic time-series model that includes a lagged dependent variable.

Solution

Run the regression, then `estat bgodfrey, lags(1 4)` for the Breusch-Godfrey test at orders 1 and 4. A small p-value rejects the null of no serial correlation. Because the model contains a lagged dependent variable, prefer Breusch-Godfrey over Durbin-Watson, whose distribution is invalid here.

NoteIn Stata, `estat dwatson` reports Durbin-Watson and `estat bgodfrey` reports Breusch-Godfrey after `regress`.

Common mistakes

✗A Durbin-Watson statistic near 2 proves the errors are independent. It only indicates no first-order autocorrelation; higher-order patterns can remain.
✗Durbin-Watson is valid in every regression. It is unreliable when a lagged dependent variable is a regressor; use Breusch-Godfrey instead.
✗Durbin-Watson gives a clean accept-or-reject decision. It has an inconclusive region between its lower and upper bounds where no conclusion is drawn.
✗Breusch-Godfrey only tests for AR(1). It tests for AR( $p$ ) and lets you check several lag orders jointly through the auxiliary regression.

Revision bullets

•Durbin-Watson targets AR(1); $DW \approx 2(1-\hat{\rho})$
•DW near 2 no autocorrelation, near 0 positive, near 4 negative
•Durbin-Watson has an inconclusive region and fails with a lagged dependent variable
•Breusch-Godfrey is an LM test using $n R^2 \sim \chi^2_p$
•Breusch-Godfrey handles AR( $p$ ) and lagged dependent variables

Quick check

A Durbin-Watson statistic close to 2 indicates:

Why is the Breusch-Godfrey test often preferred to Durbin-Watson?

Connected topics

F test Hetero Tests TS.1-TS.6 Serial Corr.Fix Serial

Sources

Wooldridge (2019), §12.3
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Presents the Durbin-Watson and Breusch-Godfrey tests for serial correlation, including the lagged-dependent-variable caveat.
Breusch (1978); Godfrey (1978)
Breusch, T.S. Testing for Autocorrelation in Dynamic Linear Models. Australian Economic Papers 17 (1978); Godfrey, L.G. Econometrica 46 (1978).
Original derivations of the LM test for serial correlation that remains valid with lagged dependent variables.

How to cite this page

Dr. Phil's Quant Lab. (2026). Testing for Serial Correlation. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-testing-serial-correlation

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