White Test for Heteroskedasticity: Null Hypothesis, Formula, Interpretation, SPSS, R, Python and Excel Guide

What Is the White Test?

The White Test is a statistical test for detecting heteroskedasticity in regression residuals. Heteroskedasticity means that the variance of the residuals is not constant across fitted values or predictor levels. A well-behaved ordinary least squares regression model usually assumes that residuals have approximately constant variance. When this assumption fails, coefficient estimates may still be unbiased under standard OLS conditions, but the usual standard errors, t-tests, confidence intervals and p-values can become unreliable.

The White Test is more general than many simple heteroskedasticity tests because it does not only check a straight-line relationship between residual variance and predictors. It uses an auxiliary regression where squared residuals are regressed on the original predictors, squared predictors and cross-product interaction terms. This allows the test to detect many different variance patterns, including curved variance patterns and interaction-based variance patterns.

For regression diagnostics, the White Test should be read together with residual plots and related assumption checks. If your main concern is variance equality across groups, you may also compare it with Levene Test and Brown-Forsythe Test. If your concern is changing variance across ordered fitted values or time-like order, the Goldfeld-Quandt Test can also be useful. For normality context, residual plots can be compared with Q-Q Plot, P-P Plot, Kolmogorov-Smirnov Test, Lilliefors Test and DAgostino Pearson Test.

When Should You Use the White Test?

Use the White Test when you fit a regression model and want to test whether the residual variance is constant. It is especially useful when you suspect that residual spread may change in a nonlinear way or may depend on combinations of predictors. Because the White Test includes squares and interaction terms in its auxiliary regression, it can detect more complex heteroskedasticity patterns than a very simple residual variance test.

Null and Alternative Hypothesis for White Test

The White Test is a hypothesis test. The decision is based on whether the auxiliary regression explains enough variation in squared residuals to reject the idea of constant residual variance.

White Test Formula and LM Statistic

The White Test is usually calculated from an auxiliary regression. The steps are simple in concept:

White LM statistic = n × R²(auxiliary regression)

where:
n = sample size
R²(auxiliary regression) = R-squared from the regression of squared residuals on auxiliary terms

For this analysis, n = 649 and the auxiliary model summary chart gives R² = 0.1164. Therefore, the LM statistic is approximately:

LM ≈ 649 × 0.1164
LM ≈ 75.54

The auxiliary regression included 27 auxiliary terms in the full model. The p-value from the chi-square decision is 1.753e-06. This very small p-value means the squared residuals are not random noise around a constant variance level. Instead, their size is related to the predictor structure.

Dataset and Regression Model Used

This worked example uses the student performance data structure. The dependent variable is G3 final grade. The base regression predicts G3 from G1, G2, studytime, failures, absences and age. The White Test is then applied to the residuals from that base regression.

External dataset source: UCI Machine Learning Repository: Student Performance dataset.

Verified SPSS, Python and R Results

The SPSS output, Python charts and R charts all support the same decision: the White Test is statistically significant. The base model has strong predictive fit, but the residual variance is not constant. The auxiliary regression explains enough squared residual variation to reject homoscedasticity.

Base Regression Output

Base Regression Coefficients

White Auxiliary Regression Output

Final White Test Decision

Python Charts and Interpretation

1. White Test: Residuals vs Fitted Values

This chart is the first visual diagnostic for the White Test. If residual variance were constant, the points would form a fairly even horizontal band around zero across the fitted-value range. Instead, the chart shows an uneven residual spread. Some fitted-value regions have tight residual bands, while the middle fitted range contains large negative residuals and wider vertical spread. Several residuals drop below -6 and even approach about -9, while positive residuals are smaller in comparison. This visual pattern supports the White Test result because residual variance is not evenly distributed across fitted values. The dashed zero line confirms that the residuals are centered around zero overall, but the problem is the changing spread, not the average residual.

2. White Test: Squared Residuals vs Fitted Values

This chart directly connects to the White auxiliary regression because the auxiliary regression uses squared residuals as the dependent variable. Most squared residuals are close to zero, but several observations rise sharply, with the largest squared residuals above 80. These large values are concentrated mainly in the mid fitted-value range rather than being evenly distributed across all fitted values. That is exactly the type of pattern that can make the White Test significant. The chart shows that the residual variance is not simply random and flat; instead, some fitted-value regions produce much larger error sizes than others.

3. White Test: Absolute Residual Spread

The absolute residual chart removes the positive or negative sign and focuses only on error size. This makes spread easier to read. A constant-variance model would show a roughly similar range of absolute residuals across fitted values. Here, the largest absolute residuals appear around the middle fitted-value range, reaching around 9. Other fitted-value regions have much smaller absolute residuals. This uneven spread supports the decision to reject the null hypothesis of homoscedasticity. It also explains why a model can have a high base R² but still fail the constant variance assumption: prediction accuracy and residual variance stability are different diagnostics.

4. White Test: p-value Decision

This chart gives the formal hypothesis decision. The alpha threshold is .05, while the White LM p-value is 1.753e-06. Because the p-value is much smaller than .05, the test rejects the null hypothesis. In reporting language, this means there is statistically significant evidence of heteroskedasticity. The chart is useful for readers because it makes the decision rule visible: p-values below alpha reject constant residual variance.

5. White Test: Strongest Auxiliary Terms

This chart shows which auxiliary terms contributed most strongly to explaining squared residuals. The largest terms are G2_sq, G1_x_G2, G1_x_failures, absences_sq and failures_sq. These terms suggest that the residual variance pattern is connected to grade-related nonlinear effects and interactions involving previous grades and failures. In practical terms, the error spread is not constant across the grade structure. Some combinations of prior grades, failures and absences create larger prediction errors than others. This supports the White Test decision because the squared residuals are meaningfully associated with auxiliary terms.

6. White Test: Residual Distribution

The residual distribution is centered near zero, which is expected in an OLS regression with an intercept. However, the distribution is not perfectly symmetric. The left tail is longer because several residuals are strongly negative. This chart does not replace the White Test because residual normality and residual variance are different assumptions. Still, it helps explain why the residual plots show large negative errors and why squared residuals have extreme values. The main White Test decision remains based on the variance pattern and p-value, not only on histogram shape.

7. White Test: Base and Auxiliary Model Fit

This chart separates two different ideas. The base model R² of 0.8506 means the predictors explain G3 very well. The auxiliary R² of 0.1164 means the auxiliary terms explain part of the squared residual variation. A high base model R² does not protect the model from heteroskedasticity. The auxiliary R² is smaller than the base R², but for a White Test it is large enough to produce a significant LM p-value because the sample size is 649. The chart therefore supports the final conclusion: the model predicts G3 well, but the residual variance assumption is violated.

R Validation Charts and Interpretation

The R charts validate the same diagnostic story shown by Python. They use a different plotting style but show the same fitted-value range, residual spread, squared residual behavior, p-value decision, auxiliary term importance, residual distribution and model-fit comparison.

1. R Residuals vs Fitted Values

The R residuals-vs-fitted chart confirms that the residual spread is not uniform. Several large negative residuals appear in the middle fitted-value region, while many other fitted ranges have smaller error bands. This validates the Python interpretation that the residual variance is not constant across the fitted-value scale.

2. R Squared Residuals vs Fitted Values

The R squared-residual plot again shows that most squared residuals are small, but a few are very large. These high squared residuals create the variance pattern detected by the auxiliary regression. The chart is especially important because White’s auxiliary regression directly models squared residuals.

3. R Absolute Residual Spread

The R absolute residual chart confirms the uneven error size. The largest absolute residuals appear around the middle fitted range. This means the model error is not equally spread across all predictions, which supports the rejection of homoscedasticity.

4. R p-value Decision

The R p-value chart repeats the same hypothesis decision: the White LM p-value is below the alpha threshold. Therefore, the null hypothesis of constant residual variance is rejected.

5. R Strongest Auxiliary Terms

The R auxiliary-term chart confirms that variance structure is most strongly connected with grade-related squared and interaction terms such as G2_sq, G1_x_G2 and G1_x_failures. This supports the interpretation that heteroskedasticity is related to nonlinear and interaction patterns in the prediction structure.

6. R Residual Distribution

The R residual histogram confirms that residuals are centered near zero but have tail irregularities. This supports the broader diagnostic story while keeping the main focus on variance. The White Test result is about heteroskedasticity, not merely residual normality.

7. R Base and Auxiliary Model Fit

The R model-fit chart confirms that the base regression fit is strong, while the auxiliary regression still explains enough squared residual variation to trigger a significant White Test. The final interpretation remains unchanged: the regression has strong predictive power, but the residual variance is not constant.

SPSS Output Interpretation

The SPSS output PDF contains the base regression, auxiliary regression, residual descriptives and residual normality/spread context. SPSS confirms that the base regression is strong and that the auxiliary regression for squared residuals is statistically significant.

Base Regression in SPSS

The SPSS base regression used G3 as the dependent variable and entered G1, G2, studytime, failures, absences and age as predictors. The model summary shows R = .922, R² = .851, adjusted R² = .849 and a standard error of estimate of 1.254. The ANOVA table shows F(6, 642) = 609.353, p < .001. This means the base regression is statistically significant and explains a large amount of variation in G3.

Residual Statistics in SPSS

The SPSS residual statistics show predicted values ranging from about .31 to 19.58. Raw residuals range from -9.045 to 5.817, with mean approximately 0 and standard deviation about 1.249. The residual range is wide enough to justify visual residual diagnostics. The standardized residual minimum of -7.210 also shows that some cases are far from the fitted line.

White Auxiliary Regression in SPSS

The SPSS auxiliary regression used resid_sq as the dependent variable. The final auxiliary model included predictors, squared predictors and interaction terms. The final model summary shows R = .341, R² = .116 and adjusted R² = .078. The ANOVA table for the final auxiliary model shows F(27, 621) = 3.030, p < .001. This supports the White Test conclusion that the squared residuals are related to the auxiliary terms.

Important Auxiliary Terms in SPSS

Residual Normality and Spread Context

The SPSS Explore output shows that raw residuals have skewness = -2.864 and kurtosis = 18.667. Squared residuals have skewness = 8.651 and kurtosis = 80.487. Both Kolmogorov-Smirnov and Shapiro-Wilk tests are significant for residuals and squared residuals. This does not replace the White Test, but it confirms that residual behavior is not simple and normal-looking. For residual normality interpretation, compare this with Kolmogorov-Smirnov Test and Lilliefors Test.

How to Run White Test in Python, R, SPSS and Excel

Python Method

In Python, the easiest method is to fit an OLS model with statsmodels, extract residuals and use the White Test diagnostic function. The following code structure matches the logic used for this post.

import pandas as pd
import statsmodels.api as sm
from statsmodels.stats.diagnostic import het_white

df = pd.read_csv("student-por.csv", sep=";")

y = df["G3"]
X = df[["G1", "G2", "studytime", "failures", "absences", "age"]]
X = sm.add_constant(X)

model = sm.OLS(y, X).fit()

white_lm, white_lm_p, white_f, white_f_p = het_white(model.resid, model.model.exog)

print("Base R-squared:", model.rsquared)
print("White LM statistic:", white_lm)
print("White LM p-value:", white_lm_p)
print("White F statistic:", white_f)
print("White F p-value:", white_f_p)

if white_lm_p < 0.05:
    print("Reject H0: evidence of heteroskedasticity")
else:
    print("Fail to reject H0: no strong evidence of heteroskedasticity")

R Method

In R, you can manually build a White-style auxiliary regression by modeling squared residuals with predictors, squared predictors and interaction terms.

df <- read.csv("student-por.csv", sep = ";")

base_model <- lm(G3 ~ G1 + G2 + studytime + failures + absences + age, data = df)

df$resid_G3 <- resid(base_model)
df$resid_sq <- df$resid_G3^2

aux_model <- lm(
  resid_sq ~ G1 + G2 + studytime + failures + absences + age +
    I(G1^2) + I(G2^2) + I(studytime^2) + I(failures^2) + I(absences^2) + I(age^2) +
    G1:G2 + G1:studytime + G1:failures + G1:absences + G1:age +
    G2:studytime + G2:failures + G2:absences + G2:age +
    studytime:failures + studytime:absences + studytime:age +
    failures:absences + failures:age + absences:age,
  data = df
)

n <- nobs(aux_model)
aux_r2 <- summary(aux_model)$r.squared
df_white <- length(coef(aux_model)) - 1
lm_stat <- n * aux_r2
p_value <- pchisq(lm_stat, df = df_white, lower.tail = FALSE)

print(summary(base_model))
print(summary(aux_model))
print(lm_stat)
print(p_value)

SPSS Method

In SPSS, run the base regression first, save predicted values and residuals, compute squared residuals, compute squared predictors and interactions, then run the auxiliary regression with resid_sq as the dependent variable.

REGRESSION
  /DEPENDENT G3
  /METHOD=ENTER G1 G2 studytime failures absences age
  /SAVE PRED(pred_G3) RESID(resid_G3).

COMPUTE resid_sq = resid_G3 ** 2.
COMPUTE G1_sq = G1 ** 2.
COMPUTE G2_sq = G2 ** 2.
COMPUTE studytime_sq = studytime ** 2.
COMPUTE failures_sq = failures ** 2.
COMPUTE absences_sq = absences ** 2.
COMPUTE age_sq = age ** 2.

COMPUTE G1_x_G2 = G1 * G2.
COMPUTE G1_x_studytime = G1 * studytime.
COMPUTE G1_x_failures = G1 * failures.
COMPUTE G1_x_absences = G1 * absences.
COMPUTE G1_x_age = G1 * age.
COMPUTE G2_x_studytime = G2 * studytime.
COMPUTE G2_x_failures = G2 * failures.
COMPUTE G2_x_absences = G2 * absences.
COMPUTE G2_x_age = G2 * age.
COMPUTE studytime_x_failures = studytime * failures.
COMPUTE studytime_x_absences = studytime * absences.
COMPUTE studytime_x_age = studytime * age.
COMPUTE failures_x_absences = failures * absences.
COMPUTE failures_x_age = failures * age.
COMPUTE absences_x_age = absences * age.
EXECUTE.

REGRESSION
  /DEPENDENT resid_sq
  /METHOD=ENTER G1 G2 studytime failures absences age
    G1_sq G2_sq studytime_sq failures_sq absences_sq age_sq
    G1_x_G2 G1_x_studytime G1_x_failures G1_x_absences G1_x_age
    G2_x_studytime G2_x_failures G2_x_absences G2_x_age
    studytime_x_failures studytime_x_absences studytime_x_age
    failures_x_absences failures_x_age absences_x_age.

OUTPUT EXPORT
  /CONTENTS EXPORT=ALL
  /PDF DOCUMENTFILE="D:\DATA ANALYSIS\B Normality and Assumption Tests\White Test\SPSS_Output\White-Test-SPSS-Output.pdf".

Excel Method

Excel does not have a one-click White Test button, but the method is still possible with the Data Analysis ToolPak. First run the base regression and save residuals. Then square the residuals. Next create columns for squared predictors and interaction terms. Finally regress squared residuals on those auxiliary columns and calculate the LM statistic.

How to Report the White Test

A good White Test report should include the null hypothesis, alternative hypothesis, base model context, auxiliary regression result, p-value decision and practical conclusion. It should also explain what to do after detecting heteroskedasticity.

What to Do After a Significant White Test

Common Mistakes When Interpreting the White Test

Related Guides

These related Salar Cafe guides help you complete the full regression assumption workflow and reporting process.

Downloads and Resources

The following hosted resources support this White Test guide. The SPSS PDF contains the regression output, auxiliary regression and residual diagnostic context. The chart links show the Python and R visual outputs used in the interpretation sections above.

External References

• White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroskedasticity.
• Statsmodels documentation: het_white
• UCI Machine Learning Repository: Student Performance dataset