Regression Assumptions Checker

📊 The four OLS (Ordinary Least Squares) assumptions, also called the Gauss-Markov conditions, are: (1) Linearity — the relationship between X and Y is linear; (2) Independence — residuals are uncorrelated with each other; (3) Homoscedasticity — residual variance is constant across all fitted values; (4) Normality — residuals follow a normal distribution. When all four are satisfied, the OLS estimator is the best linear unbiased estimator (BLUE), meaning it has the smallest variance among all unbiased linear estimators.

📊 Ignoring assumption violations can lead to seriously misleading results. Non-linearity produces biased coefficients that systematically misrepresent the relationship. Autocorrelation inflates t-statistics and R², making the model appear more significant than it truly is.

⚠️ Heteroscedasticity makes standard errors unreliable, producing confidence intervals that are too narrow or too wide. Non-normality invalidates p-values and confidence intervals in small samples. In each case, you risk drawing incorrect conclusions — accepting effects that do not exist, missing effects that do, or making predictions with far less accuracy than the model suggests.

🔍 Test for linearity using both visual and formal methods. Visually, create a scatterplot of Y vs. X and a residual-vs-fitted plot — look for curved patterns that deviate from a random scatter.

📊 Formally, use Ramsey's RESET test, which adds squared and cubed fitted values as auxiliary regressors and tests whether their coefficients are jointly zero using an F-test. A significant F-statistic indicates that the linear specification is inadequate. If linearity is violated, add polynomial terms (x², x³) or apply a transformation such as log(Y) or √Y.

📊 The primary test for first-order autocorrelation is the Durbin-Watson (DW) test. The DW statistic ranges from 0 to 4, with d ≈ 2 indicating no autocorrelation.

📊 Values below 1.5 suggest positive autocorrelation (consecutive residuals tend to have the same sign), while values above 2.5 suggest negative autocorrelation (consecutive residuals alternate signs). For higher-order autocorrelation, use the Breusch-Godfrey LM test, which can test for autocorrelation at multiple lags simultaneously. Autocorrelation is most common in time-series data, such as monthly sales, daily stock returns, or annual economic indicators.

📊 The Breusch-Pagan test is the standard formal test for heteroscedasticity. It regresses the squared residuals on the original predictors and tests whether the regression coefficients are jointly zero. The test statistic LM = n × R² follows a chi-squared distribution.

📊 A significant result means the residual variance depends on the predictors — violating homoscedasticity. Visually, plot residuals against fitted values: a funnel shape (spread increasing or decreasing) indicates heteroscedasticity. Remedies include using robust standard errors, transforming Y (often log transformation), or switching to weighted least squares.

📊 Use both visual and formal methods. Visually, create a histogram of residuals (should look bell-shaped) and a Q-Q plot (points should fall along the diagonal).

📊 Formally, the Shapiro-Wilk test is the most powerful test for normality in small to moderate samples — it computes a statistic W near 1 for normal data, with a significant p-value rejecting normality. The Jarque-Bera test is an alternative based on skewness and kurtosis that works well for larger samples. For large datasets (n > 30), moderate non-normality is often tolerable due to the central limit theorem, but severe departures still warrant attention.

📊 Robust regression refers to estimation methods that are less sensitive to outliers and assumption violations than ordinary least squares. Common approaches include M-estimation (which downweights large residuals using a function that grows less quickly than squaring), S-estimation (which minimizes a robust measure of scale), MM-estimation (combining high breakdown point with high efficiency), and Theil-Sen estimation (based on pairwise slopes).

⚠️ Robust methods produce coefficient estimates that are not unduly influenced by a few extreme observations, making them valuable when outliers are present and cannot be removed. However, they do not eliminate the need to check assumptions — they simply make the results more resistant to certain types of violations.

📊 It depends on the type and severity of the violation. Mild violations often have limited practical impact, especially with large samples. The central limit theorem protects against moderate non-normality when n > 30, and robust standard errors can compensate for heteroscedasticity.

✅ However, severe violations — especially non-linearity and severe autocorrelation — fundamentally undermine the validity of the model and must be addressed before trusting any results. The safest approach is to diagnose violations, apply appropriate remedies, re-check assumptions, and only then interpret the regression output. If violations cannot be adequately fixed, consider alternative modeling approaches such as generalized linear models, quantile regression, or nonparametric methods.

📊 The mathematical minimum for computing a simple linear regression is n = 2, but assumption tests require far more data to be meaningful. The Durbin-Watson test needs at least n ≥ 6 for useful critical values.

❌ The Breusch-Pagan and Shapiro-Wilk tests have very low statistical power below n = 20 — violations may exist but go undetected. For reliable assumption checking across all four tests, aim for at least n = 30. With small samples, the tests may fail to detect genuine violations (Type II error), so visual diagnostics and domain knowledge become even more important.

📊 The Durbin-Watson (DW) statistic tests for first-order autocorrelation in regression residuals. It is computed as DW = Σ(eᵢ − eᵢ₋₁)² / Σeᵢ² and ranges from 0 to 4. DW ≈ 2 indicates no autocorrelation.

📊 DW < 2 suggests positive autocorrelation (consecutive residuals are similar), with values below 1.5 being strong evidence. DW > 2 suggests negative autocorrelation (consecutive residuals alternate), with values above 2.5 being concerning. Exact critical values depend on n and the number of predictors, but the rough thresholds of 1.5 and 2.5 are widely used as practical guidelines.

📊 The Breusch-Pagan test detects heteroscedasticity by testing whether the variance of residuals depends on the predictor variables. It works by regressing squared residuals on the predictors and computing LM = n × R² from this auxiliary regression.

📊 Under the null hypothesis of homoscedasticity, LM follows a chi-squared distribution with degrees of freedom equal to the number of predictors. A p-value below α (typically 0.05) rejects homoscedasticity, indicating that residual variance is not constant and that OLS standard errors are unreliable. The remedy is to use robust standard errors, transform Y, or apply weighted least squares.

📊 The Shapiro-Wilk test assesses whether a sample comes from a normally distributed population. It computes a statistic W that measures the correlation between the ordered sample values and the expected order statistics from a normal distribution. W ranges from 0 to 1, with values close to 1 indicating normality.

📊 A p-value below α (typically 0.05) rejects the null hypothesis of normality, meaning the residuals deviate significantly from a normal distribution. The Shapiro-Wilk test is the most powerful normality test for small to moderate samples (n < 50). For larger samples, even tiny deviations from normality can produce significant p-values, so visual inspection of Q-Q plots should always accompany the formal test.

📊 Homoscedasticity means the variance of the residuals is constant across all levels of the fitted values — the spread of residuals is roughly the same whether you are predicting small or large values of Y.

📐 Heteroscedasticity means the variance changes — typically, the spread of residuals increases with the magnitude of the fitted values, producing a funnel shape in the residual plot. Heteroscedasticity is common when the dependent variable spans several orders of magnitude (e.g., income, population sizes) or when measurement precision varies across the range. It does not bias the coefficient estimates, but it makes their standard errors incorrect, which invalidates t-tests, F-tests, and confidence intervals.

📊 Yes — multiple regression has the same four core assumptions as simple linear regression (linearity, independence, homoscedasticity, normality) plus a fifth assumption: no multicollinearity (predictors should not be too highly correlated with each other). Multicollinearity inflates standard errors and makes individual coefficients unstable.

📊 Detect it using variance inflation factors (VIF > 5 indicates a problem). All five assumptions should be checked before interpreting a multiple regression model. Our assumptions checker tests the first four; you should additionally calculate VIF for each predictor when using multiple regression.

📊 This checker is designed for simple linear regression with one predictor variable. For polynomial regression, the linearity assumption is automatically satisfied by including polynomial terms (the model is linear in the coefficients even though the relationship is nonlinear in X).

📊 You can still check independence, homoscedasticity, and normality of the residuals from a polynomial model. For exponential regression, the assumptions apply to the linearized model (after log transformation), not the original data. If you need to check assumptions for these more complex models, fit the model first using the appropriate calculator, then examine the residuals manually or with our tool.