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CFA Level II Exam · Extensions of Multiple Regression

Heteroskedasticity in Multiple Regression for CFA Level II

Updated 7 October 2026 · Fact-checked

Heteroskedasticity means the variance of regression errors is not constant across observations. Coefficient estimates stay unbiased, but standard errors are wrong, so t-tests and F-tests mislead. Detect it with the Breusch-Pagan test, a chi-square test with k degrees of freedom, and correct it with robust (White-corrected) standard errors.

Understand Heteroskedasticity

One assumption of multiple regression is homoskedasticity: the variance of the error term is the same for every observation. Heteroskedasticity means the error variance changes. A common picture is a scatter plot of residuals that fans out as an independent variable gets larger. For example, errors in predicting company earnings may be bigger for large firms than for small firms.

There are two types. Unconditional heteroskedasticity is not related to the independent variables. It causes no major problems for inference. Conditional heteroskedasticity is related to the level of the independent variables, so the error variance rises or falls with them. This is the type that causes problems and the one the exam tests.

The consequences are specific. The coefficient estimates are still unbiased and consistent. But the standard errors are unreliable. With conditional heteroskedasticity, standard errors are usually underestimated. That makes t-statistics too large, so you reject true null hypotheses too often (more Type I errors) and see significance that is not there. The F-test is also unreliable. Sometimes the bias runs the other way, so the safe statement is that the tests are unreliable, and usually too many results look significant.

To detect it, you can look at a plot of residuals against the independent variables, or run the Breusch-Pagan test. You regress the squared residuals on the independent variables. If the independent variables explain the squared residuals, the error variance depends on them, and you have conditional heteroskedasticity.

To fix it, use robust standard errors (also called White-corrected or heteroskedasticity-consistent standard errors). They change the standard errors, not the coefficients. You then recompute t-statistics with the corrected standard errors. Generalized least squares is another method, but robust standard errors are the one to know.

Key formulas to remember

Breusch-Pagan test statistic
BP = n × R²(resid)
n = number of observations. R²(resid) is from the regression of squared residuals on the independent variables.
Breusch-Pagan decision rule
Chi-square with k degrees of freedom; reject H0 (no conditional heteroskedasticity) if BP > critical value
k = number of independent variables in the original regression. It is a one-tailed test.
Null and alternative
H0: no conditional heteroskedasticity; Ha: conditional heteroskedasticity
Rejecting H0 means you should use robust standard errors.
t-statistic
t = (b̂ − b hypothesized) ÷ standard error of b̂
If the standard error is too small, t is too large. Recompute t using the robust standard error.

How to solve Heteroskedasticity questions

Use this order for any heteroskedasticity question in an item set.

  1. 1Identify what the vignette gives: a residual plot, a Breusch-Pagan result, or a comparison of regular and robust standard errors.
  2. 2If a Breusch-Pagan test is needed, find n and the R² from the regression of squared residuals on the independent variables.
  3. 3Compute BP = n × R². Set degrees of freedom equal to k, the number of independent variables.
  4. 4Compare BP with the chi-square critical value (one-tailed). If BP is larger, reject the null and conclude conditional heteroskedasticity is present.
  5. 5If it is present, state the effect: coefficients unbiased, standard errors unreliable (usually too small), t-statistics too large.
  6. 6To correct, replace the standard errors with robust ones and recompute t = coefficient ÷ robust standard error.
  7. 7Compare the new t with the critical value and state whether the conclusion changes.

Quickest way: Three-line Breusch-Pagan check

When to use it: When the vignette gives n, the auxiliary R², and a chi-square critical value.

  1. Multiply n by the R² of the squared-residual regression.
  2. Use k, the count of independent variables, as degrees of freedom.
  3. If the product beats the critical value, conditional heteroskedasticity exists; fix with robust standard errors, and expect the original t-statistics to be inflated.

Common mistakes in Heteroskedasticity

  • Saying heteroskedasticity biases the coefficient estimates.

    Students link any assumption violation to bias.

    Fix: Remember that coefficients stay unbiased and consistent. Only the standard errors, and so the tests, are affected.

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

    Two regressions appear in the vignette and the R² values are confused.

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

  • Treating the Breusch-Pagan test as two-tailed or using wrong degrees of freedom.

    Other tests in the curriculum use t-tables and two tails.

    Fix: It is a one-tailed chi-square test with k degrees of freedom, where k is the number of independent variables.

  • Confusing conditional with unconditional heteroskedasticity.

    The names sound alike.

    Fix: Conditional means the variance is related to the independent variables and is the problem type. Unconditional is unrelated and usually harmless.

  • Correcting by changing the coefficients.

    Students assume the fix must change the estimates.

    Fix: Robust standard errors leave coefficients unchanged. Only the standard errors, t-statistics and conclusions change.

Worked examples

Example 1

An analyst regresses a firm's return on equity on three independent variables using 120 observations. To test for heteroskedasticity, she regresses the squared residuals on the same three variables and gets R² = 0.10. The chi-square critical value with 3 degrees of freedom at 5% is 7.815. (1) Compute the test statistic. (2) What is the conclusion? (3) What is the effect on the t-statistics of the original regression?

Show the solution
  1. Test statistic: BP = n × R² = 120 × 0.10 = 12.0.
  2. Degrees of freedom = k = 3. Critical value = 7.815.
  3. 12.0 > 7.815, so reject the null of no conditional heteroskedasticity.
  4. Effect: conditional heteroskedasticity usually makes standard errors too small, so the t-statistics are too large and significance may be overstated. Coefficients stay unbiased.

Answer: (1) BP = 12.0. (2) Reject the null; conditional heteroskedasticity is present. (3) Coefficients are unbiased, but standard errors are unreliable (typically understated), so t-statistics are typically inflated.

Example 2

After finding conditional heteroskedasticity, an analyst reruns a regression with robust standard errors. The slope on a leverage variable is 0.80. The conventional standard error was 0.30, and the robust standard error is 0.50. The critical t-value is 1.98. (1) Compute both t-statistics. (2) Does the conclusion change at the 5% level? (3) Does the coefficient change?

Show the solution
  1. Conventional t = 0.80 ÷ 0.30 = 2.67.
  2. Robust t = 0.80 ÷ 0.50 = 1.60.
  3. Compare with 1.98: 2.67 > 1.98, so the conventional test rejects the null; 1.60 < 1.98, so the robust test fails to reject.
  4. The coefficient 0.80 is unchanged because robust standard errors only correct the standard error.

Answer: (1) Conventional t = 2.67; robust t = 1.60. (2) Yes. The variable looked significant with conventional errors but is not significant with robust errors. (3) No, the coefficient stays at 0.80.

Exam tips

  • Expect a vignette with n and the auxiliary R². Compute n × R² and compare with the chi-square value; do not look for a formula that uses the original R².
  • Know the direction of the error: with conditional heteroskedasticity, standard errors are usually too small and t-statistics too large, which leads to too many rejections of the null.
  • When a question asks what stays valid, the answer is the coefficient estimates (unbiased and consistent).
  • If both conventional and robust standard errors are shown, recompute t with the robust one before choosing the answer.
  • Read the vignette for the word conditional. Unconditional heteroskedasticity is usually described as not causing significant problems.

Heteroskedasticity: 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 heteroskedasticity is not related to them. Only the conditional type causes serious problems for inference.

How do you perform the Breusch-Pagan test?

Run the original regression and save the residuals. Square them and regress them on the independent variables. Compute n × R² from this second regression and compare it with a one-tailed chi-square critical value with k degrees of freedom. If it is larger, reject the null of no conditional heteroskedasticity.

How does heteroskedasticity affect t-statistics?

Standard errors are unreliable, and usually too small. T-statistics are then too large, so you may call a variable significant when it is not. Coefficient estimates remain unbiased.

How do you correct heteroskedasticity?

Use robust standard errors, also called White-corrected standard errors. They give valid standard errors without changing the coefficients. You then recompute t-statistics and redo the tests.