Questions: White Test and Detection of Heteroskedasticity

Question 1

You run White's test on your regression and reject the null hypothesis at the 5% level. What is the most important diagnostic question to ask before switching to robust standard errors?

• A. Whether your sample size is large enough for the chi-squared approximation to be valid
• B. Whether your model is correctly specified, since misspecification can also produce systematic residual patterns that White's test detects
• C. Whether you should use Breusch-Pagan instead to confirm the result
• D. Whether the auxiliary regression's R² is above 0.5

Answer: (B)

White's test detects whether squared residuals are systematically explained by the regressors. Both genuine heteroskedasticity and model misspecification (e.g., omitting a quadratic term) produce systematic patterns in residuals. White's test cannot distinguish between the two causes. If the real problem is a misspecified functional form, switching to robust standard errors leaves the bias from misspecification untreated. The correct first response is to interrogate the model specification, not automatically apply a correction for heteroskedasticity.

Question 2

How does White's test fundamentally differ from Breusch-Pagan in its approach to detecting heteroskedasticity?

• A. White's test uses F-statistics, while Breusch-Pagan uses chi-squared statistics
• B. White's test includes squared regressors and cross-products in the auxiliary regression, making it general rather than assuming a specific linear form for the heteroskedasticity
• C. White's test requires a larger sample size because it tests a different null hypothesis
• D. Breusch-Pagan is the general test and White's is a restricted special case

Answer: (B)

Breusch-Pagan tests whether error variance is a linear function of the regressors — it assumes heteroskedasticity, if present, takes a specific linear form. White's test makes no such assumption: by adding squared terms and cross-products of all regressors, it searches for nonlinear and interactive patterns in error variance. This generality is its main advantage when you don't know how variance relates to the regressors. The downside is that more auxiliary variables consume more degrees of freedom.

Question 3

In White's test, the dependent variable in the auxiliary regression is the squared OLS residual ê².

• T. True
• F. False

Answer: True

Since the true error variance Var(u|X) is unobservable, the squared residual ê² serves as a proxy. The auxiliary regression asks: do the regressors, their squares, and their cross-products explain variation in ê²? If yes (significant R²), the null of homoskedasticity is rejected. The test statistic nR² follows a chi-squared distribution under the null.

Question 4

Rejecting the null in White's test conclusively establishes that the model has heteroskedastic errors rather than a misspecified functional form.

• T. True
• F. False

Answer: False

White's test cannot distinguish between these two causes. A misspecified model — for example, fitting a straight line to a relationship that is actually quadratic — will produce residuals with systematic patterns that depend on the regressors. White's test will detect this and reject the null, even though the issue is misspecification, not heteroskedasticity. This is why the test should trigger a model-specification check first, not an automatic switch to heteroskedasticity corrections.

Question 5

Why might White's test reject the null of homoskedasticity even when the true error variance is constant? What alternative explanation should you investigate first?

Model answer: White's test uses squared residuals as a proxy for error variance and tests whether the regressors, their squares, and their cross-products explain those squared residuals. But systematic patterns in squared residuals are not unique to heteroskedasticity — they also arise when the model is misspecified (wrong functional form, omitted variables, incorrect link function). A misspecified model produces residuals that systematically vary with the regressors, which White's test detects. The first step after rejection should be to check model specification: add polynomial terms, inspect residual plots, and consider whether the functional form is appropriate before concluding that heteroskedasticity is the problem.

This caution is practically important. Applying robust standard errors to a misspecified model still leaves the specification bias in the coefficient estimates — it only addresses inference. Fixing the model specification is the more fundamental correction. White's test rejection is a diagnostic signal, not a definitive diagnosis.