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CFA Level II Exam · Evaluating Regression Model Fit and Interpreting Model Results

Heteroskedasticity vs Serial Correlation in CFA Level II

Updated 7 October 2026 · Fact-checked

Heteroskedasticity means the regression error variance is not constant. Serial correlation means errors are correlated across observations. Both leave coefficients unbiased in a correctly specified model, but standard errors are unreliable, so t-tests mislead. Detect with Breusch-Pagan or Durbin-Watson, then fix with robust (White) or Newey-West standard errors.

Understand Violations: Heteroskedasticity and Serial Correlation

A regression assumes the error terms have the same variance for every observation and are uncorrelated with each other. When either assumption fails, your coefficient estimates are still fine in the usual cases the curriculum tests, but the standard errors are wrong. Wrong standard errors mean wrong t-statistics and wrong conclusions about significance.

Heteroskedasticity means the variance of the errors changes with the level of the independent variables. Think of regressing spending on income: high-income households vary far more in spending than low-income ones. Unconditional heteroskedasticity is unrelated to the independent variables and causes no major problems. Conditional heteroskedasticity is related to the independent variables and is the serious case.

Serial correlation (autocorrelation) means errors in one period are correlated with errors in other periods. It shows up mainly in time-series data. Positive serial correlation means a positive error tends to be followed by a positive error. It is the common case in finance.

The effect on standard errors depends on the problem. With conditional heteroskedasticity, standard errors are unreliable and may be understated or overstated, and the F-test is also unreliable. In finance data the typical finding is understated standard errors, which inflate t-statistics and cause too many Type I errors. But the direction depends on how the error variance relates to the independent variables. With positive serial correlation, standard errors are typically understated, t-statistics are too high, and the F-test is unreliable.

You detect heteroskedasticity with a residual plot or the Breusch-Pagan test. You detect serial correlation with the Durbin-Watson test or the Breusch-Godfrey test. You correct with robust standard errors: White-corrected for heteroskedasticity, and Newey-West for serial correlation (it also handles heteroskedasticity).

Key formulas to remember

Breusch-Pagan test statistic
BP = n × R² (from regressing squared residuals on the independent variables)
Chi-square with k degrees of freedom, one-tailed (right tail). H0: no conditional heteroskedasticity. Reject if BP exceeds the critical value.
Durbin-Watson statistic
DW ≈ 2 × (1 − r), where r is the correlation between residuals and lagged residuals
DW = 2 means no serial correlation. DW below 2 suggests positive, above 2 suggests negative.
Durbin-Watson decision rule (positive serial correlation)
DW < dl: reject H0; DW > du: fail to reject; dl ≤ DW ≤ du: inconclusive
H0: no positive serial correlation. The table gives dl and du using n and k.
Effect on standard errors
Conditional heteroskedasticity → standard errors unreliable (often understated in finance data, but the direction depends on how the variance relates to the regressors). Positive serial correlation → standard errors typically understated → t-statistics overstated
The F-test is also unreliable. When standard errors are understated, you get too many Type I errors. When they are overstated, the reverse holds. Coefficient estimates are not biased in a correct specification.
Corrections
Heteroskedasticity: White (robust) standard errors. Serial correlation: Newey-West (also robust to heteroskedasticity)
Newey-West is the safe choice when both problems may be present.

How to solve Violations: Heteroskedasticity and Serial Correlation questions

Use this method on any item-set question about violations of regression assumptions.

  1. 1Identify the data type: cross-sectional points to heteroskedasticity, time series points to serial correlation.
  2. 2Find the test statistic or p-value in the exhibit: BP (or its n × R²), DW, or a residual plot description.
  3. 3Match the test to the problem and state H0: no conditional heteroskedasticity for BP, no positive serial correlation for DW.
  4. 4Compare the statistic with the critical value, or the p-value with the significance level. For DW use dl and du from the exhibit.
  5. 5Conclude whether the problem is present. Then state the effect: standard errors understated, t-statistics overstated, Type I errors more likely.
  6. 6Pick the fix: White standard errors for heteroskedasticity, Newey-West for serial correlation, or both.
  7. 7Re-read the question to check whether it asks about the test, the effect, or the remedy.

Quickest way: Three-line triage for violation questions

When to use it: When the vignette gives a test result and asks what it means or what to do next.

  1. Name the test: BP means heteroskedasticity, DW means serial correlation.
  2. Reject H0 means the problem exists. For DW, compare with dl and du, and check that DW is below 2 for positive correlation.
  3. Problem exists means t-stats are unreliable (usually too high), so use robust standard errors: White or Newey-West.

Common mistakes in Violations: Heteroskedasticity and Serial Correlation

  • Saying the coefficients are biased because of heteroskedasticity or serial correlation.

    Students link any violation with bias.

    Fix: In a correctly specified model the coefficients stay unbiased. The damage is to the standard errors and tests.

  • Using a two-tailed test for Breusch-Pagan.

    Most tests earlier in the reading are two-tailed.

    Fix: BP is a one-tailed chi-square test. A large statistic indicates heteroskedasticity.

  • Treating a DW value between dl and du as a rejection.

    Students forget the inconclusive zone.

    Fix: Below dl reject, above du fail to reject, in between is inconclusive.

  • Applying the positive-correlation Durbin-Watson rule directly to a DW value above 2.

    The decision rule is usually taught for positive correlation only.

    Fix: A DW well above 2 hints at negative serial correlation. For that test, compare 4 − DW with dl and du. Equivalently, the thresholds are 4 − du and 4 − dl.

  • Choosing White standard errors when the problem is serial correlation.

    Both corrections are called robust standard errors.

    Fix: White fixes heteroskedasticity only. Newey-West fixes serial correlation and heteroskedasticity.

  • Computing BP as R² alone, or using the wrong sample size.

    Students forget the multiplication by n.

    Fix: BP = n × R² of the auxiliary regression on squared residuals, with k degrees of freedom.

Worked examples

Example 1

An analyst regresses monthly returns of a fund on three factors using 60 observations. To test for conditional heteroskedasticity she regresses the squared residuals on the three factors and gets R² = 0.15. The 5% critical chi-square value with 3 degrees of freedom is 7.815. (1) Compute the Breusch-Pagan statistic. (2) State the conclusion. (3) State the remedy.

Show the solution
  1. BP = n × R² = 60 × 0.15 = 9.0.
  2. Degrees of freedom = k = 3. Critical value = 7.815.
  3. 9.0 > 7.815, so reject H0 of no conditional heteroskedasticity.
  4. Conditional heteroskedasticity makes standard errors unreliable, so t-statistics cannot be trusted.

Answer: BP = 9.0, which exceeds 7.815. Conditional heteroskedasticity is present. Use White-corrected (robust) standard errors.

Example 2

A regression of quarterly sales growth on one variable uses 40 observations and gives a Durbin-Watson statistic of 1.05. For n = 40 and k = 1 at 5%, dl = 1.44 and du = 1.54. (1) Test for positive serial correlation. (2) State the effect on t-statistics. (3) State the remedy.

Show the solution
  1. H0: no positive serial correlation. Compare DW = 1.05 with dl = 1.44.
  2. 1.05 < 1.44, so reject H0.
  3. Positive serial correlation typically understates standard errors, so t-statistics are overstated and Type I errors are more likely.
  4. The fix is Newey-West standard errors, which also correct for heteroskedasticity.

Answer: DW is below dl, so there is positive serial correlation. T-statistics are too high, so significance is overstated. Use Newey-West standard errors.

Exam tips

  • Match data type to violation: cross-section suggests heteroskedasticity, time series suggests serial correlation.
  • Memorize the DW zones. A question often gives dl, du and a DW value near the boundaries.
  • Remember the effect direction: understated standard errors, overstated t-statistics, too many Type I errors.
  • If a choice mentions Newey-West, check whether serial correlation is the issue. It is also valid for heteroskedasticity, so it is the safest all-round fix.
  • Do not spend time on calculation unless BP = n × R² is asked. Most questions are interpretation.

Violations: Heteroskedasticity and 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.

Violations: Heteroskedasticity and Serial Correlation: frequently asked questions

What is the difference between heteroskedasticity and serial correlation?

Heteroskedasticity is non-constant error variance, common in cross-sectional data. Serial correlation is correlation of errors across observations, common in time series. Both distort standard errors and hypothesis tests.

How do I interpret the Breusch-Pagan test?

Regress squared residuals on the independent variables and compute n × R². Compare it with a one-tailed chi-square critical value with k degrees of freedom. If it is larger, reject the null and conclude conditional heteroskedasticity is present.

How do I read the Durbin-Watson statistic?

A value near 2 means no serial correlation. Values well below 2 suggest positive serial correlation. Compare with dl and du: below dl reject, above du do not reject, between is inconclusive.

When should I use Newey-West standard errors?

Use them when serial correlation is present, since they adjust for it and for heteroskedasticity. Use White standard errors when only heteroskedasticity is the problem.