CFA Level II Exam · Extensions of Multiple Regression

Serial Correlation in Regression: Tests and Corrections

Q: How do I interpret the Durbin-Watson statistic?

A value near 2 suggests no first-order serial correlation. Values below 2 suggest positive correlation and values above 2 suggest negative. Compare the statistic with dL and dU: below dL rejects no positive correlation, above dU fails to reject, and between them is inconclusive.

Q: What is the difference between Durbin-Watson and Breusch-Godfrey?

Durbin-Watson tests only first-order serial correlation and has an inconclusive region. Breusch-Godfrey tests for correlation up to any lag p using an auxiliary regression. Its statistic is n × R², compared with a chi-square with p degrees of freedom.

Q: How do you correct serial correlation?

Use Newey-West serial-correlation-consistent standard errors, which the curriculum also links to Hansen's method. They replace the original standard errors and fix the t-tests. The coefficient estimates stay the same.

Q: Why does positive serial correlation inflate t-statistics?

Correlated errors mean the observations carry less independent information than the ordinary formula assumes. The usual standard error is therefore too small. A small standard error makes the t-statistic larger, which raises the chance of a false rejection.

Updated 7 October 2026 · Fact-checked

Serial correlation means regression errors are correlated across time. Coefficients stay unbiased, but standard errors are wrong, so t-tests mislead. Detect it with the Durbin-Watson test (first order, DW ≈ 2(1 − r)) or the Breusch-Godfrey test (higher orders, n × R² against a chi-square). Fix it with Newey-West (Hansen) robust standard errors.

Understand Serial Correlation

A regression assumes the error terms are uncorrelated with each other. Serial correlation (autocorrelation) breaks this. The error in one period is linked to the error in earlier periods. It shows up mostly in time-series data, such as monthly returns or quarterly sales.

With positive serial correlation, an error of one sign tends to be followed by an error of the same sign. With negative serial correlation, an error tends to be followed by one of the opposite sign. Positive is far more common in finance data.

The damage is to inference, not to the fitted line. In a model with no lagged dependent variable, the coefficient estimates remain consistent and unbiased. But the standard errors are not reliable. Under positive serial correlation (the usual case, with positively correlated regressors), standard errors are underestimated. The t-statistics are too large and the F-statistic is inflated. You reject true null hypotheses too often (Type I errors) and variables look more significant than they are. Under negative serial correlation, standard errors tend to be overestimated (when regressors are positively correlated), so t-statistics tend to be too small and you tend to make more Type II errors. This is a tendency, not a certainty.

A warning for lagged variables: if the model includes a lagged dependent variable as a regressor and the errors are serially correlated, the coefficient estimates themselves become inconsistent. Also, the Durbin-Watson test is not valid in that setting.

You detect serial correlation with the Durbin-Watson (DW) test for first-order correlation, or the Breusch-Godfrey (BG) test for correlation at higher orders. You correct it by replacing the standard errors with Newey-West serial-correlation-consistent standard errors, which the curriculum also attributes to Hansen's method. These also handle heteroskedasticity. Only the standard errors change. The coefficients stay the same.

Key formulas to remember

Durbin-Watson approximation: DW ≈ 2(1 − r); r is the sample correlation between residuals and their first lag. DW ranges from 0 to 4. DW = 2 means no first-order correlation; below 2 points to positive; above 2 points to negative.
Durbin-Watson decision rule (positive correlation): H0: no positive serial correlation. DW < dL: reject. dL ≤ DW ≤ dU: inconclusive. DW > dU: fail to reject; dL and dU come from a table and depend on n and the number of independent variables k. The exam normally gives them in the vignette.
Durbin-Watson decision rule (negative correlation): DW > 4 − dL: reject. 4 − dU ≤ DW ≤ 4 − dL: inconclusive. DW < 4 − dU: fail to reject; Mirror image of the positive test. Only use it if the question asks about negative correlation.
Breusch-Godfrey test statistic: BG = n × R² (from the auxiliary regression), compared with χ² with p degrees of freedom; Regress the original residuals on the original independent variables plus p lagged residuals. H0: no serial correlation up to lag p. Reject if the statistic exceeds the critical value. This is a one-tailed test.
Direction of effect on standard errors: Positive serial correlation → SE too small, t too large. Negative → SE tends to be too large, t tends to be too small; The negative case is a tendency (when regressors are positively correlated). Coefficient estimates are unaffected when there is no lagged dependent variable.

How to solve Serial Correlation questions

Use this sequence for any serial correlation question in an item set.

1Identify the data type. Time-series or ordered data raises serial correlation risk. Check whether a lagged dependent variable is in the model.
2Find the test result in the exhibit: a DW statistic, a residual correlation, or a BG auxiliary R² with n and the number of lags.
3For DW, compare to dL and dU from the exhibit. If you only have r, compute DW ≈ 2(1 − r) first.
4For BG, compute n × R² and compare to the chi-square critical value with p degrees of freedom. Reject H0 only if the statistic is larger.
5State the conclusion in words: positive, negative, or no evidence of serial correlation, and at what order.
6State the consequence: coefficients are unbiased, standard errors are biased (too small if positive), t-statistics are inflated or deflated, and tests are unreliable.
7Choose the remedy: Newey-West (Hansen) robust standard errors. Remember the coefficients do not change.
8Re-check the original conclusions. A coefficient that was significant with the old standard errors may no longer be significant.

Quickest way: Three-line serial correlation check

When to use it: Use when the vignette gives a DW value, a correlation of residuals, or a BG R² and you need an answer fast.

DW near 2 means fine. Well below 2 means positive correlation. Well above 2 means negative correlation. Then confirm with dL and dU.
BG: multiply n by R² and compare with the chi-square critical value. Bigger means serial correlation is present.
Positive correlation means standard errors too small, t too high, too many false rejections. The fix is Newey-West standard errors, with the same coefficients.

Common mistakes in Serial Correlation

Saying serial correlation biases the regression coefficients.
Students mix it up with omitted variable bias or confuse 'unreliable' with 'biased'.
Fix: Remember: coefficients stay unbiased and consistent (without a lagged dependent variable). Only the standard errors, t-tests and F-test are affected.
Getting the direction of the standard error effect backwards.
The word 'positive' feels like it should inflate things.
Fix: Positive correlation makes the data look more informative than it is, so standard errors are too small and t-statistics too large. Type I errors rise. Negative correlation tends to do the opposite (when regressors are positively correlated).
Treating DW values between dL and dU as a rejection or a pass.
Students forget the inconclusive zone.
Fix: Below dL reject, above dU fail to reject, in between the test is inconclusive. Say so when the answer options include 'inconclusive'.
Using DW to test for higher-order serial correlation.
DW is the most familiar test.
Fix: DW only tests first-order correlation. For lags beyond one, use the Breusch-Godfrey test.
Comparing the BG statistic with the wrong degrees of freedom or the wrong direction.
Students use n − k − 1 or reject when the statistic is below the critical value.
Fix: Degrees of freedom equal the number of lagged residuals p. Reject when n × R² is greater than the chi-square critical value.
Saying Newey-West changes the coefficient estimates.
It is described as a 'correction' to the regression.
Fix: It only replaces the standard errors. Coefficients are identical. The t-statistics are recomputed using the new standard errors.

Worked examples

Example 1

An analyst regresses a fund's monthly excess return on two factors using 40 observations (k = 2). The correlation between residuals and their first lag is 0.35. For n = 40 and k = 2 at the 5% level, the table gives dL = 1.39 and dU = 1.60. (1) Estimate the DW statistic. (2) What is the conclusion on positive serial correlation? (3) What is the effect on the t-statistics if no action is taken?

Show the solution

(1) DW ≈ 2(1 − r) = 2(1 − 0.35) = 2 × 0.65 = 1.30.
(2) Compare 1.30 with dL = 1.39. Since 1.30 < 1.39, reject H0 of no positive serial correlation.
(3) Positive serial correlation makes the standard errors too small. The t-statistics are too large, so the analyst risks Type I errors and may call factors significant when they are not.

Answer: DW ≈ 1.30. Reject the null: there is positive serial correlation. The t-statistics are overstated, and Newey-West standard errors should be used.

Example 2

A researcher fits a regression with 60 observations and suspects serial correlation up to two lags. She regresses the residuals on the original independent variables and two lagged residuals. The auxiliary R² is 0.09. The 5% chi-square critical value with 2 degrees of freedom is 5.991. (1) Compute the test statistic. (2) State the conclusion. (3) Would the original coefficient estimates be changed by a Newey-West correction?

Show the solution

(1) BG = n × R² = 60 × 0.09 = 5.4.
(2) Compare 5.4 with 5.991. Since 5.4 < 5.991, fail to reject H0. There is no significant evidence of serial correlation up to two lags at the 5% level.
(3) Newey-West adjusts only the standard errors. The coefficient estimates would not change.

Answer: BG = 5.4, below 5.991, so fail to reject the null of no serial correlation up to lag 2. Newey-West would not change the coefficients.

Exam tips

Expect the vignette to hand you dL and dU or a chi-square critical value. Your job is to apply the decision rule and state the consequence, not to recall tables.
Questions often ask for the effect on standard errors, t-statistics and Type I or Type II error. Learn the positive case first, then reverse it for negative, remembering the negative case is a tendency.
If an option says the coefficients are biased, check whether a lagged dependent variable is in the model. If not, that option is wrong.
When you see a lag order above one, pick Breusch-Godfrey over Durbin-Watson.
If both heteroskedasticity and serial correlation appear in the same item set, Newey-West robust standard errors address both.

Serial Correlation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Serial Correlation: frequently asked questions

How do I interpret the Durbin-Watson statistic?

A value near 2 suggests no first-order serial correlation. Values below 2 suggest positive correlation and values above 2 suggest negative. Compare the statistic with dL and dU: below dL rejects no positive correlation, above dU fails to reject, and between them is inconclusive.

What is the difference between Durbin-Watson and Breusch-Godfrey?

Durbin-Watson tests only first-order serial correlation and has an inconclusive region. Breusch-Godfrey tests for correlation up to any lag p using an auxiliary regression. Its statistic is n × R², compared with a chi-square with p degrees of freedom.

How do you correct serial correlation?

Use Newey-West serial-correlation-consistent standard errors, which the curriculum also links to Hansen's method. They replace the original standard errors and fix the t-tests. The coefficient estimates stay the same.

Why does positive serial correlation inflate t-statistics?

Correlated errors mean the observations carry less independent information than the ordinary formula assumes. The usual standard error is therefore too small. A small standard error makes the t-statistic larger, which raises the chance of a false rejection.