Sixty observations where the variance is smallest in the middle of the predictor’s range and rises at both ends. White’s auxiliary regression explains 32.887% of the squared residuals and gives 19.7319 on 2 degrees of freedom, p = 0.0000519. Breusch-Pagan, whose auxiliary regression is linear in the predictor, explains 0.571% and gives p = 0.5585 — nothing at all. The pattern is entirely quadratic and symmetric, so it has no linear component whatsoever, and adding the squared term is the whole difference between finding it and missing it.
60 rows · 2 auxiliary terms · model R² 0.77365
White statistic 19.7319 on 2 df, p = 5.1911e-5
Breusch-Pagan on the same data gives p = 0.5585 and finds nothing. The extra terms are earning their degrees of freedom here: the variance is moving in a way with little or no linear component, which a linear auxiliary regression cannot see at all.
White statistic
19.7319
2 degrees of freedom
p-value
5.191e-5
variance is not constant
Auxiliary terms
2
k(k+3)/2 with k = 1
Breusch-Pagan
0.55849
1 df, linear terms only
What each test is fitting
The auxiliary regressions behind the two tests, with their terms and results
Test
Auxiliary terms
Explains
df
p-value
White
predictors, squares, cross-products
32.887%
2
5.191e-5
Breusch-Pagan
predictors only
0.571%
1
0.55849
The “explains” column is the auxiliary regression’s R², and the gap between the two rows is exactly what the squares and cross-products add. When it is small, the extra degrees of freedom are pure cost.
Residuals against fitted values
This test will detect any pattern a quadratic surface in the predictors can approximate, which is most of what turns up in practice. It will not detect one that is orthogonal to all of those terms, and the plot is what catches those.
A significant result does not have to mean heteroscedasticity. The auxiliary regression includes squares and cross-products, so an omitted non-linear term in the original model — a missing squared predictor, a missing interaction — shows up here too. Check the specification before reaching for robust standard errors.
The auxiliary regression here uses 2 terms, which is k(k+3)/2 at k = 1. The verification suite asserts that count at every k from 1 to 5, because an off-by-one in the cross-product loop changes the degrees of freedom and so changes every p-value the test produces.
No form assumed Residual plot Breusch-Pagan alongside Costs degrees of freedom
What this tool shows
On the shipped preset White’s auxiliary regression explains 32.887% of the squared residuals and gives p = 0.0000519. Breusch-Pagan’s explains 0.571% and gives p = 0.5585. Same sixty observations. The variance rises at both ends of the predictor, which is entirely quadratic and has no linear component at all, so adding the squared term is the whole difference between finding it and missing it. The third preset shows the same generality losing a result it should have found.
White’s statistic from an auxiliary regression on the predictors, their squares and every cross-product
Breusch-Pagan on the same data, with both auxiliary R² values so the gap between them is explicit
The auxiliary term count, k(k+3)/2, and what it costs in degrees of freedom as predictors are added
A residual plot, because a quadratic surface covers most patterns and not all of them
A preset where the extra generality loses a result the linear test finds comfortably
Why a significant White statistic can mean a misspecified model rather than unequal variance
No form assumed Residual plot Breusch-Pagan alongside Costs degrees of freedom
A significant result can be a missing term, not unequal variance.
Updated 13 September 2026 · Works in any browser, no installation
White’s test regresses the squared residuals on the predictors, their squares and every cross-product between them, and asks whether that explains anything. Because the auxiliary model is a full quadratic surface, no guess about the shape of the heteroscedasticity is needed — which is the advantage over Breusch-Pagan, and which is paid for in degrees of freedom that grow as the square of the predictor count.
At a glance
Formula shown
Fit the model, take the residuals e, then regress e² on every predictor, every squared predictor and every pairwise product. The statistic is n·R² from that auxiliary regression, referred to chi-square on the number of auxiliary terms, which is k(k+3)/2 for k predictors: 2 at k = 1, 5 at k = 2, 9 at k = 3, 14 at k = 4, 20 at k = 5 and 27 at k = 6. The test is asymptotic, so the sample has to be large relative to that count.
Scenario support
Testing residuals when the shape of the variance is unknown, checking for heteroscedasticity that depends on an interaction, confirming a non-significant Breusch-Pagan before accepting constant variance, and screening a model where a missing quadratic or interaction term is plausible.
Educational estimate
Planning support from the values you enter — not professional advice.
The pattern a linear auxiliary regression cannot see
The shipped preset is chosen so the two tests give opposite answers, and the reason is arithmetic rather than power.
The variance is smallest in the middle and rises at both ends. Symmetric, which means its correlation with the predictor is zero by construction.
Breusch-Pagan’s auxiliary regression explains 0.571% of the squared residuals and returns p = 0.5585. There is nothing linear to find.
White’s explains 32.887% and returns p = 0.0000519, because the squared term captures the whole pattern.
Measured over 300 generated datasets of this shape, Breusch-Pagan misses it 88% of the time and White finds it every time. That is the case for running this test rather than only the faster one.
Generality is paid for in degrees of freedom
The third preset exists because “assumes no functional form” is usually presented as pure upside, and the cost is measurable.
Five predictors means k(k+3)/2 = 20 auxiliary terms. Six would need twenty-seven, eight would need forty-four.
On seventy observations where the variance really does depend on the first predictor, White gives 30.2319 on 20 degrees of freedom, p = 0.0662. Short of 0.05.
Breusch-Pagan concentrates the same evidence into 5 and gives p = 0.0016. A factor of forty in the p-value, purely from where the degrees of freedom went.
So the sensible order is both. The page runs them together on every input: the linear test for power when the form is simple, this one for coverage when it is not, and the disagreement between them as a diagnostic in its own right.
The cross-products earn their place
Half of White’s extra terms are products between different predictors, and the second preset shows the case they exist for.
Two predictors, and the error variance depends on their product. Not on either one alone.
White gives 14.8332 on 5 degrees of freedom, p = 0.0111. Breusch-Pagan, using the two predictors alone, gives p = 0.3736.
This shape is easy to produce by accident. An error that scales with total exposure — quantity times price, distance times load, dose times duration — is a product, and is invisible to a test that only sees the factors.
It is also the case a residual plot handles badly, since plotting against either predictor separately shows a band that looks acceptable. Plotting against the fitted value helps, and the test is more reliable than either.
A significant result may not be about variance at all
This is the most important caveat on the page, and it follows directly from what the auxiliary regression contains.
The auxiliary terms are squares and cross-products of the predictors. Exactly the terms a misspecified model is missing.
So an omitted quadratic or interaction shows up here as a significant statistic, because the residuals from a model missing a curve are large where the curve is steepest.
The test cannot separate the two explanations. Unequal variance and a missing term produce the same auxiliary regression.
Which is why specification comes first. Check whether a squared term or an interaction belongs in the model before reaching for robust standard errors — and note that Breusch-Pagan is narrower here in a useful way, since its linear auxiliary regression is far less sensitive to a missing curve.
It needs a large sample, and says so quietly
The statistic is asymptotic and the auxiliary regression is wide, which combine badly in a way that does not announce itself.
The chi-square reference distribution is a large-sample approximation. At small n the true distribution is different and the p-value is not reliable.
“Large” has to be measured against the auxiliary term count, not against n alone. Seventy observations and twenty terms is a thin auxiliary regression.
The failure is silent. The test returns a number, and it is neither obviously wrong nor trustworthy.
A reasonable rule is at least five observations per auxiliary term, which at three predictors means forty-five rows and at five means a hundred. Below that, the linear test and the residual plot are more informative than this one.
What to do when it is significant
The conclusion is narrower than “the regression is invalid”, and the remedy follows from what actually broke.
Check the specification first. A missing squared term or interaction produces this result and is fixed by adding the term rather than by adjusting the uncertainty.
If the specification is right, the coefficients are still unbiased. Only the standard errors are wrong, usually too small.
Robust standard errors are the standard fix, and the covariance estimator behind them comes from the same 1980 paper as this test — White introduced both together, which is not a coincidence.
Weighted least squares recovers the lost efficiency when the variance function is known, and this test deliberately does not tell you what it is. Knowing that something is wrong and not knowing its shape is precisely the situation robust standard errors were built for.
Reporting the result
Four items, and the second is the one that lets a reader judge whether the test had a chance.
Give the statistic, the degrees of freedom and the p-value. The degrees of freedom are the auxiliary term count and are what distinguish this from Breusch-Pagan at a glance.
Give the sample size against that term count. Twenty terms on seventy rows is a different claim from twenty terms on seven hundred.
Say whether you checked the specification. A significant White statistic with no comment on omitted terms leaves the most likely alternative explanation unaddressed.
And report both tests when they disagree. The disagreement is informative: a quiet linear test with a loud quadratic one says the variance has no linear component, which is a finding about shape rather than a contradiction.
Sources and methodology
References for the test and the estimator that came with it.
Method. The auxiliary regression is built from the predictors, every squared predictor and every pairwise product, with any column that turns out constant dropped so a dummy variable cannot make the design singular. The statistic is n times that regression’s R², referred to chi-square on the number of terms actually used — which the page prints, because dropping a degenerate column changes the degrees of freedom. Breusch-Pagan is computed alongside on every input, since the disagreement between them is the diagnostic. The verification suite asserts the term count k(k+3)/2 at every k from 1 to 5, because an off-by-one in the cross-product loop changes the degrees of freedom and therefore every p-value the test produces. It also asserts behaviour rather than transcribed numbers: on 150 generated homoscedastic datasets the statistic must be valid and must not reject more often than chance allows, and on 120 datasets with a symmetric U in the variance it must find the effect at least 114 times while Breusch-Pagan misses it at least 95. That engine is verified on every change against 103 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Breusch-PaganThe Breusch-Pagan test in both forms — Koenker's studentised statistic and the original 1979 chi-square — with White's test alongside and a residual plot.
Multiple RegressionFits several predictors with a VIF on every term, and names the configuration people misread: a model significant at p = 0.0103 where neither predictor reaches 0.05, at a VIF of only 7.11.
Linear RegressionThe least-squares line with r and r² — and the regression of x on y beside it, because those are two different lines rather than one line rearranged.
Durbin-WatsonThe Durbin-Watson statistic with dL and dU computed by Imhof inversion rather than tabled, so the bounds exist at any n and k.
LeverageHat values for every observation with the 2p/n and 3p/n cutoffs, the studentised residuals alongside, and the Σh = p identity printed as a check.
Cook's DistanceCook's distance for every row against both cutoffs in circulation, with the leverage and studentised residual it is built from and the F percentile it corresponds to.
An educational tool. White’s auxiliary regression contains exactly the terms a misspecified model is missing, so a significant result can mean an omitted squared term or interaction rather than unequal variance, and the test cannot distinguish the two. It is asymptotic with a wide auxiliary regression, so the sample must be large relative to the k(k+3)/2 terms it uses — the failure at small samples is silent.