This sample intentionally has wider residual spread at larger X values, which helps demonstrate heteroscedasticity tests.
| # | X | Y | Comment |
|---|---|---|---|
| 1 | 1 | 2.2 | Small early variance |
| 2 | 2 | 4.1 | Close to trend |
| 3 | 3 | 5.8 | Low spread |
| 4 | 4 | 8.5 | Moderate deviation |
| 5 | 5 | 9.4 | Lower than local trend |
| 6 | 6 | 13.7 | Spread rising |
| 7 | 7 | 12.9 | Spread rising |
| 8 | 8 | 18.6 | Larger positive error |
| 9 | 9 | 17.1 | Larger negative error |
| 10 | 10 | 23.8 | Higher variance zone |
| 11 | 11 | 20.9 | Higher variance zone |
| 12 | 12 | 29.0 | Higher variance zone |
Interpretation: a small p-value suggests the residual variance changes across observations, indicating heteroscedasticity.
- Paste paired X and Y data into the input box. One row per observation.
- Set the significance level (alpha), usually 0.05.
- Choose the Breusch-Pagan auxiliary predictor (X or fitted values).
- Set the Goldfeld-Quandt omitted middle percentage and test mode.
- Click Run Heteroscedasticity Tests to generate results above the form.
- Review test statistics, p-values, and observation diagnostics.
- Use the CSV or PDF buttons to save a report copy.
Tip: If one test is borderline, compare all three tests together and inspect residual magnitudes before deciding whether to use robust standard errors or model transformations.
Data Patterns and Variance Behavior
Heteroscedasticity occurs when residual variance changes across observations rather than staying constant. In applied modeling, this often appears when larger predictor values produce wider prediction errors. This calculator converts that visual suspicion into measurable evidence. Users paste paired X and Y data, run the tests, and review p-values, fitted values, residuals, and residual squares together. The combined output helps confirm whether ordinary least squares assumptions remain reliable before formal reporting decisions are finalized.
Breusch-Pagan Test Interpretation
The Breusch-Pagan test checks whether squared residuals are explained by a selected variance driver. This calculator lets users choose X values or fitted values for the auxiliary regression. It computes the LM statistic as sample size multiplied by auxiliary R-squared. A low p-value suggests changing variance, while a higher p-value supports homoscedasticity. The result is practical for routine diagnostics, especially when analysts expect a smooth variance relationship across ordered observations in production datasets.
White Test for Flexible Variance Structure
The White test extends variance detection by using X and X squared in the auxiliary regression. That extra flexibility helps identify curved or irregular variance patterns that simpler checks may miss. The calculator reports the White LM statistic and p-value beside Breusch-Pagan and Goldfeld-Quandt outputs for direct comparison. Because the auxiliary model uses more terms, analysts should ensure enough observations are available and interpret findings alongside the residual table and fitted-value behavior.
Goldfeld-Quandt Split Sample Analysis
Goldfeld-Quandt testing is useful when observations can be ordered by predictor scale. The calculator sorts rows by X, omits a configurable middle percentage, and fits separate regressions to lower and upper groups. It compares mean squared errors using an F statistic. One-sided mode targets increasing variance, while two-sided mode checks any major difference. This split-sample approach is effective for growth, size, pricing, or exposure data with clear ordering and widening dispersion risks.
Practical Actions After Detection
When heteroscedasticity is detected, the model is not automatically invalid. Analysts should confirm data quality, inspect outliers, and compare all three tests before changing methods. Common remedies include robust standard errors, weighted least squares, logarithmic transformations, or segmented models. This calculator supports that workflow with formulas, example data, observation diagnostics, and downloadable reports. It improves documentation quality and helps teams make consistent, variance-aware decisions during regression review and communication clearly.
1) What does a low p-value mean here?
A low p-value suggests residual variance is not constant across observations. In other words, the model likely shows heteroscedasticity, and ordinary standard errors may be unreliable for inference.
2) Which test should I trust most?
Use all three together. Breusch-Pagan is efficient for smooth variance patterns, White is broader, and Goldfeld-Quandt is useful when variance changes with ordered predictor scale.
3) How many observations are recommended?
At least eight observations are required for basic use, but more data improves stability. White and split-sample testing especially benefit from larger samples and clearer variance structure.
4) Can I use fitted values in Breusch-Pagan?
Yes. This calculator allows X or fitted values as the auxiliary predictor. Fitted values can be useful when variance appears linked to expected output magnitude.
5) What should I do after detection?
Consider robust standard errors, weighted least squares, transformations, or a revised model form. Always review residual diagnostics and data quality before selecting a correction method.
6) Does heteroscedasticity change coefficients?
OLS coefficients can remain unbiased under common conditions, but inference is affected. The main issue is distorted standard errors, confidence intervals, and significance testing.