Regression Assumption Checker by Statssy

Regression Assumption Checker

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) Diagnostic rejection threshold (default is 0.05)

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

OLS Gauss-Markov Diagnostics

Why Test Regression Assumptions?

Ordinary Least Squares (OLS) regression relies on the Gauss-Markov assumptions to ensure that parameter estimates ($\hat{\beta}$) are the Best Linear Unbiased Estimators (BLUE).

Violating these core assumptions can invalidate $p$-values, severely distort confidence intervals, inflate standard errors, or produce misleading slope directions.

The 5 Core Assumptions

1. Linearity: The true population relationship between predictors and outcome is linear in parameters.
2. Normality of Residuals: Model error residuals follow a normal Gaussian distribution $e_i \sim \mathcal{N}(0, \sigma^2)$.
3. Homoscedasticity: Residual variance remains constant across all fitted values $\operatorname{Var}(e_i) = \sigma^2$.
4. Independence of Errors: Residuals are uncorrelated across consecutive observations ($\operatorname{Cov}(e_i, e_j) = 0$).
5. No Multicollinearity: Predictors are not linearly dependent or excessively correlated ($VIF < 5$).

Diagnostic Mathematical Formulas

$$\text{Durbin-Watson: } d = \frac{\sum_{i=2}^n (e_i - e_{i-1})^2}{\sum_{i=1}^n e_i^2}$$ $$\text{Breusch-Pagan LM: } LM = n \times R^2_{\text{aux}(e^2)} \sim \chi^2(k)$$ $$\text{Cook's Distance: } D_i = \frac{e_i^2}{(k+1) MS_{\text{res}}} \left[\frac{h_{ii}}{(1 - h_{ii})^2}\right]$$

Where $e_i = y_i - \hat{y}_i$ are OLS residuals, $h_{ii}$ represents diagonal leverage values from the hat matrix $\mathbf{H} = \mathbf{X}(\mathbf{X}^T\mathbf{X})^{-1}\mathbf{X}^T$, and $LM$ tests for heteroscedasticity.

Worked Example

Scenario: Regressing $Y = [15, 20, 24, 28, 33, 38]$ on $X_1 = [2, 4, 5, 7, 8, 10]$ and $X_2 = [1, 2, 3, 2, 4, 5]$ ($n=6, k=2$).

Step 1: Fit OLS Model & Residuals:
$\hat{Y} = 8.7908 + 2.4716 X_1 + 0.9575 X_2, \quad SS_{\text{res}} = 0.9007$.

Step 2: Durbin-Watson Autocorrelation:
$d = \frac{1.8906}{0.9007} = 2.0991 \implies \text{No Autocorrelation } (1.5 \le d \le 2.5)$.

Step 3: Shapiro-Wilk Normality of Residuals:
$W = 0.9082, p = 0.4223 > 0.05 \implies \text{Normality Satisfied}$.

Step 4: Breusch-Pagan Homoscedasticity:
$LM = 1.8415, p = 0.3982 > 0.05 \implies \text{Equal Variance Satisfied}$.

Diagnostic Audit Complete

Regression Assumptions Scorecard

Overall Model Diagnostic Health

All Core Assumptions Satisfied

OLS parameter estimates and inference p-values are reliable and BLUE.

Durbin-Watson (d) 2.0991

Normality (SW p) p = 0.4223

Homoscedasticity p = 0.3982

Max VIF 2.0815

Max Cook's D 0.5842

Model R² 0.9975

Assumption Compliance Matrix

Assumption	Diagnostic Metric	Observed Value	Acceptable Threshold	Status	Diagnostic Impact & Action

Regression Assumption Checker by Statssy