Breusch-Pagan Test Calculator

Breusch-Pagan Test

Dependent Variable (Y) Observations separated by commas, spaces, or newlines

Please enter valid numeric data for Dependent Variable Y.

Predictor X₁

Predictor X₂

Significance Level (α) Default threshold is 0.05 (5% Type I error rate)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Heteroscedasticity Diagnostic

What Is the Breusch-Pagan Test?

The Breusch-Pagan Test is a statistical diagnostic test used to determine whether heteroscedasticity (non-constant error variance) is present in a linear regression model.

Under the null hypothesis ($H_0$), error variance is homoscedastic ($\sigma_i^2 = \sigma^2$). The test regresses squared OLS residuals ($e_i^2$) on the regression predictors to test whether variance scales systematically across predictor levels.

When to Use It

Checking Regression Assumptions: Testing whether OLS residual variance remains constant across fitted values ($\hat{Y}$).
Econometric & Financial Modeling: Identifying heteroscedasticity in cross-sectional or income/price elasticity models.
Correcting Standard Errors: Directing analysis toward robust standard errors (e.g., Huber-White / HC3) when $p \le 0.05$.
Validating Inference: Ensuring hypothesis $t$-tests and $F$-tests are not compromised by unequal error dispersion.

The Mathematical Formulas

$$e_i = y_i - \hat{y}_i, \quad e_i^2 = \gamma_0 + \gamma_1 X_{1i} + \dots + \gamma_k X_{ki} + \nu_i$$ $$LM = n \cdot R^2_{\text{aux}} \sim \chi^2(k)$$ $$p = 1 - F_{\chi^2}(LM; df = k)$$

Where $R^2_{\text{aux}}$ is the coefficient of determination from the auxiliary regression of squared residuals $e_i^2$ on the original predictors, and $k$ is the number of predictors.

Worked Example

Scenario: Regressing $Y$ on $X_1$ and $X_2$ ($n=6, k=2$) yields residuals $e_i$, with auxiliary squared-residual regression $R^2_{\text{aux}} = 0.3069$ at $\alpha=0.05$.

Step 1: Compute Lagrange Multiplier Statistic ($LM$):
$LM = n \times R^2_{\text{aux}} = 6 \times 0.3069 = 1.8415$.

Step 2: Chi-Square P-Value ($df = k = 2$):
$p = 1 - F_{\chi^2}(1.8415; 2) \approx 0.3982$.

Conclusion: Since $p = 0.3982 > 0.05$, fail to reject $H_0$. The error variance is homoscedastic.

Diagnostic Audit Complete

Breusch-Pagan Test Results

Computed Lagrange Multiplier (LM) & P-Value

LM = 1.8415, p = 0.3982

Homoscedastic Error Variance (Fail to Reject H₀ at α = 0.05)

LM Statistic 1.8415

P-Value 0.3982

Auxiliary R² 0.3069

Degrees of Freedom df = 2

Sample Size (n) 6

Variance Status Constant

Breusch-Pagan Test Calculator

Breusch-Pagan Test

Breusch-Pagan Test Results

Step-by-Step Calculation

Interpretation

Chi-Square Distribution & Critical Rejection Region

Perfect for students, researchers, and professionals looking to build real statistical skills.