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CFA Level II Exam · Model Misspecification

Heteroskedasticity, Serial Correlation and Multicollinearity Explained

Updated 7 October 2026 · Fact-checked

These are three regression violations. Heteroskedasticity means error variance is not constant. Serial correlation means errors are correlated across observations. Multicollinearity means independent variables are highly correlated. Detect them with the Breusch-Pagan test, the Durbin-Watson or Breusch-Godfrey test, and VIF. Fix standard errors with robust or Newey-West methods, or drop variables.

Understand Heteroskedasticity, Serial Correlation and Multicollinearity

A regression gives you coefficients and standard errors. The t-tests rely on assumptions about the error term. When those assumptions fail, the coefficients may still be fine, but your conclusions about significance can be wrong.

Heteroskedasticity means the variance of the errors is not constant. Unconditional heteroskedasticity is unrelated to the independent variables, so it causes no major problems. Conditional heteroskedasticity is related to the independent variables (for example, errors get larger as income rises). This is the serious kind. Coefficient estimates stay consistent, but standard errors are biased, usually too small in finance data. The t-statistics are then too large and you reject nulls too often (Type I errors). The F-test is also unreliable.

Serial correlation (autocorrelation) means errors are correlated over time. Positive serial correlation means an error tends to be followed by an error of the same sign. It is common in time series. If no lagged dependent variable is among the regressors, the coefficients remain consistent, but standard errors are biased (too small under positive correlation), so t-statistics are inflated. If a lagged dependent variable is a regressor, serial correlation makes the estimates inconsistent.

Multicollinearity means two or more independent variables are highly correlated with each other. The coefficients stay consistent and unbiased, but standard errors inflate. Result: low t-statistics, imprecise coefficients, yet a high R² and a significant F-test. The classic sign is that the F-test is significant while individual coefficients are not.

In an item set, you will usually be given a test result or an exhibit and asked to identify the problem, its effect on standard errors or t-statistics, and the remedy.

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 test. H0: no conditional heteroskedasticity. Reject if BP exceeds the critical value. R² is from the auxiliary regression, not the original model.
Durbin-Watson statistic
DW ≈ 2(1 − r), where r is the first-order correlation of residuals
DW ≈ 2 means no serial correlation. DW below 2 suggests positive correlation; above 2 suggests negative. Compare with lower and upper critical values dl and du: below dl reject H0; between dl and du inconclusive; above du fail to reject (for positive correlation).
Breusch-Godfrey test
BG = (n − p) × R² from regressing residuals on the independent variables and p lagged residuals; chi-square with p degrees of freedom
Tests serial correlation at higher orders. Using it, you can test more than one lag. Check the exact statistic form given in the question.
Variance inflation factor
VIF_j = 1 ÷ (1 − R_j²)
R_j² is from regressing variable j on the other independent variables. A VIF above 5 warrants investigation and above 10 is a serious concern. These are rules of thumb, not strict laws.
Corrections
Heteroskedasticity: robust (White) standard errors. Serial correlation: Newey-West (HAC) standard errors. Multicollinearity: remove or combine variables, or use more data.
Newey-West also corrects for heteroskedasticity. Robust errors change standard errors, not coefficients.

How to solve Heteroskedasticity, Serial Correlation and Multicollinearity questions

Use this sequence for any question on these three violations.

  1. 1Read the vignette and find what is reported: a test statistic, a DW value, a VIF, or a pattern of t-stats versus F-stat.
  2. 2Match the evidence to the problem. Squared residuals tied to X or a BP test points to heteroskedasticity. A DW value or residual autocorrelations point to serial correlation. High R² and significant F with insignificant t-stats, or a high VIF, points to multicollinearity.
  3. 3Run the test. For BP, compute n × R² and compare with the chi-square critical value. For DW, compare with dl and du, or use DW ≈ 2(1 − r).
  4. 4State the conclusion in terms of the null: reject or fail to reject.
  5. 5State the consequence: which estimates stay consistent, which standard errors are biased and in what direction, and what happens to t-stats and Type I or Type II errors.
  6. 6Choose the remedy that matches the problem: robust errors, Newey-West, or dropping or combining variables.
  7. 7Check the answer options for the exact wording, such as biased versus inconsistent, or too small versus too large.

Quickest way: Problem to effect to fix lookup

When to use it: Use when a question asks which violation is present, what it does, or how to correct it, and no heavy calculation is needed.

  1. Heteroskedasticity: standard errors usually too small, t-stats too high, Type I errors. Fix: robust (White) errors.
  2. Positive serial correlation: standard errors too small, t-stats too high, Type I errors. Fix: Newey-West errors.
  3. Multicollinearity: standard errors too large, t-stats low, Type II errors. Fix: drop or combine variables.
  4. For DW, think 2 is clean. Well below 2 means positive correlation.
  5. For BP, remember n × R² against the chi-square critical value, one-tailed.

Common mistakes in Heteroskedasticity, Serial Correlation and Multicollinearity

  • Saying multicollinearity biases the coefficients.

    Students link any violation with bias.

    Fix: Coefficients remain unbiased and consistent. Only standard errors inflate, so significance is understated.

  • Using the R² of the original regression in the Breusch-Pagan statistic.

    Both regressions produce an R², and the vignette may give both.

    Fix: Use the R² from the regression of squared residuals on the independent variables.

  • Treating unconditional heteroskedasticity as a serious problem.

    Students ignore the word conditional.

    Fix: Unconditional heteroskedasticity is unrelated to the independent variables and creates no major inference problems. Conditional is the one to correct.

  • Misreading the Durbin-Watson ranges.

    There are two critical values and an inconclusive zone.

    Fix: For positive correlation: below dl reject H0, between dl and du inconclusive, above du fail to reject. Near 2 means no correlation.

  • Using Newey-West for multicollinearity, or robust errors for it.

    All three are called regression violations, so students apply one fix to all.

    Fix: Match the fix: robust or Newey-West errors adjust standard errors for error problems. Multicollinearity is about the regressors and needs a change to the variables or data.

  • Thinking a high R² rules out multicollinearity.

    Students equate good fit with a sound model.

    Fix: A high R² with insignificant individual t-stats is the classic multicollinearity sign.

Worked examples

Example 1

An analyst regresses a stock's return 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% chi-square critical value with 3 degrees of freedom is 7.815. (1) Compute the Breusch-Pagan statistic and conclude. (2) State the effect on the original t-statistics if the null is rejected. (3) Name the remedy.

Show the solution
  1. BP = n × R² = 60 × 0.15 = 9.0.
  2. Compare with the critical value: 9.0 > 7.815, so reject the null of no conditional heteroskedasticity.
  3. With conditional heteroskedasticity, coefficient estimates remain consistent, but standard errors are typically underestimated, so t-statistics are overstated and Type I errors are more likely.
  4. Remedy: use heteroskedasticity-consistent (White) robust standard errors.

Answer: (1) BP = 9.0, which exceeds 7.815, so conditional heteroskedasticity is present. (2) Standard errors are typically too small and t-statistics too large. (3) Use robust (White) standard errors.

Example 2

A time-series regression of monthly sales growth on GDP growth and the unemployment rate gives a Durbin-Watson statistic of 1.10 with dl = 1.50 and du = 1.70. The residual first-order correlation is about 0.45. A second regression shows R² = 0.92 and a significant F-test, but none of its four slope coefficients is significant, and one VIF equals 14. (1) What does the first regression show and what is the effect? (2) Which problem affects the second regression, and what is a suitable fix?

Show the solution
  1. Check the DW approximation: 2(1 − 0.45) = 1.10, which matches the reported value.
  2. DW = 1.10 is below dl = 1.50, so reject the null of no positive serial correlation.
  3. Effect: with no lagged dependent variable, coefficients stay consistent but standard errors are understated, so t-statistics are inflated. A fix is Newey-West standard errors.
  4. In the second regression, a high R² and significant F with no significant slopes plus a VIF of 14 (well above 10) points to multicollinearity.
  5. Effect: standard errors are inflated, so t-stats are low. Fix: remove or combine the highly correlated variables, or collect more data.

Answer: (1) Positive serial correlation; standard errors are too small and t-stats overstated; use Newey-West standard errors. (2) Multicollinearity; drop or combine the correlated variables.

Exam tips

  • Know the direction of each effect: heteroskedasticity and positive serial correlation shrink standard errors, while multicollinearity inflates them.
  • Expect a distractor that says coefficients are biased. Only serial correlation with a lagged dependent variable makes estimates inconsistent.
  • For Breusch-Pagan, find the R² from the auxiliary regression and note the degrees of freedom equal the number of independent variables.
  • When a vignette gives a DW value and two critical values, place the statistic on the number line before reading the options.
  • Pair every problem with its fix. Newey-West handles both serial correlation and heteroskedasticity, but not multicollinearity.

Heteroskedasticity, Serial Correlation and Multicollinearity in other exams

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

Heteroskedasticity, Serial Correlation and Multicollinearity: frequently asked questions

What is the difference between conditional and unconditional heteroskedasticity?

Conditional heteroskedasticity means the error variance is related to the independent variables. Unconditional means it is not. Only the conditional type causes real problems with standard errors and hypothesis tests.

How do I detect multicollinearity in a regression?

Look for a high R² and a significant F-test while individual t-statistics are insignificant. You can also compute the variance inflation factor, 1 ÷ (1 − R_j²). Values above 5 deserve attention and above 10 are a serious concern.

How do Breusch-Pagan and Durbin-Watson tests differ?

Breusch-Pagan tests for conditional heteroskedasticity using n × R² from regressing squared residuals on the independent variables. Durbin-Watson tests for first-order serial correlation using residuals, with a value near 2 meaning none.

How do I correct serial correlation using Newey-West standard errors?

You keep the same coefficient estimates and replace the standard errors with Newey-West (HAC) standard errors. These are robust to serial correlation and also to heteroskedasticity. You then redo the t-tests with the corrected standard errors.