𝗧𝗵𝗲 𝟲 𝗞𝗲𝘆 𝗔𝘀𝘀𝘂𝗺𝗽𝘁𝗶𝗼𝗻𝘀 𝗼𝗳 𝗟𝗶𝗻𝗲𝗮𝗿 𝗥𝗲𝗴𝗿𝗲𝘀𝘀𝗶𝗼𝗻 (𝗔𝗻𝗱 𝗛𝗼𝘄 𝘁𝗼 𝗧𝗲𝘀𝘁 𝗧𝗵𝗲𝗺)

Most people fit a linear regression, check the R² and p-values, and call it a day.

But regression results are only as reliable as the assumptions behind them.

Here are 6 key assumptions and conditions worth checking, how to diagnose them, and what happens when they are violated 👇

𝟭. 𝗖𝗼𝗿𝗿𝗲𝗰𝘁 𝗙𝘂𝗻𝗰𝘁𝗶𝗼𝗻𝗮𝗹 𝗙𝗼𝗿𝗺 / 𝗟𝗶𝗻𝗲𝗮𝗿𝗶𝘁𝘆

The model must be linear in its coefficients (parameters) — the predictors themselves can be transformed.

✅ yᵢ = β₀ + β₁xᵢ + β₂xᵢ² + uᵢ

✅ yᵢ = β₀ + β₁log(xᵢ) + uᵢ

❌ yᵢ = β₀ + xᵢ^β₁ + uᵢ

The last model is nonlinear in the parameter β₁.

If violated: OLS cannot directly estimate the model unless it can be transformed into a form that is linear in the parameters.

📊 How to check: Use residual plots and model specification diagnostics to assess whether the chosen functional form adequately captures the relationship.

𝟮. 𝗭𝗲𝗿𝗼 𝗖𝗼𝗻𝗱𝗶𝘁𝗶𝗼𝗻𝗮𝗹 𝗠𝗲𝗮𝗻 / 𝗘𝘅𝗼𝗴𝗲𝗻𝗲𝗶𝘁𝘆

This is one of the most important assumptions:

E(uᵢ | X) = 0

In simple terms, the predictors should not be systematically related to the unobserved factors captured by the error term.

This is where problems such as omitted-variable bias, reverse causality, and certain forms of measurement error can arise.

📊 Can we test it directly?

Not generally. Exogeneity is primarily justified through study design, domain knowledge, experimental design, or methods such as instrumental variables when appropriate.

If violated: OLS coefficients can be biased and inconsistent. Unlike heteroscedasticity, this is not simply a standard-error problem.

𝟯. 𝗛𝗼𝗺𝗼𝘀𝗰𝗲𝗱𝗮𝘀𝘁𝗶𝗰𝗶𝘁𝘆

The variance of the error term should be constant:

Var(uᵢ | X) = σ²

📊 Breusch-Pagan / White / Goldfeld-Quandt tests

For example, with the Breusch-Pagan test:

• H₀: Homoscedasticity
• H₁: Heteroscedasticity

A low p-value suggests rejecting H₀.

If violated: Under the usual exogeneity assumptions, OLS coefficients can still be unbiased, but the conventional standard errors become unreliable, which can invalidate t-tests, F-tests, p-values, and confidence intervals.

💡 Practical solution: Use heteroscedasticity-robust standard errors when appropriate.

𝟰. 𝗡𝗼 𝗔𝘂𝘁𝗼𝗰𝗼𝗿𝗿𝗲𝗹𝗮𝘁𝗶𝗼𝗻 𝗼𝗳 𝗥𝗲𝘀𝗶𝗱𝘂𝗮𝗹𝘀

Residuals should not be systematically correlated with one another, particularly in time-series settings.

📊 Durbin-Watson test

• H₀: No first-order autocorrelation
• H₁: First-order autocorrelation exists

The statistic is approximately:

2 → little/no first-order autocorrelation < 2 → positive autocorrelation > 2 → negative autocorrelation

Get Affan Ahmed’s stories in your inbox

Join Medium for free to get updates from this writer.

Remember me for faster sign in

If violated: Conventional standard errors can become unreliable, often leading to misleading significance tests and confidence intervals.

💡 Important: Durbin-Watson primarily addresses first-order autocorrelation; it is not a general test for every possible form of serial correlation.

𝟱. 𝗡𝗼 𝗣𝗲𝗿𝗳𝗲𝗰𝘁 𝗠𝘂𝗹𝘁𝗶𝗰𝗼𝗹𝗹𝗶𝗻𝗲𝗮𝗿𝗶𝘁𝘆

Predictors should not be exact linear combinations of one another.

📊 VIF (Variance Inflation Factor)

VIF is a diagnostic, not a hypothesis test.

A common rule of thumb:

• VIF ≈ 1 → little multicollinearity
• VIF > 5 → potentially concerning
• VIF > 10 → often considered serious

If violated: Perfect multicollinearity prevents OLS from uniquely estimating the affected coefficients.

With high but imperfect multicollinearity, coefficients can become unstable and their standard errors can become large, making individual predictors appear statistically insignificant even when the model performs well overall.

𝟲. 𝗡𝗼𝗿𝗺𝗮𝗹𝗶𝘁𝘆 𝗼𝗳 𝗘𝗿𝗿𝗼𝗿𝘀

For exact small-sample t- and F-inference, normality of the errors is typically assumed:

uᵢ | X ~ N(0, σ²)

📊 Jarque-Bera test + Q-Q plot

Jarque-Bera:

• H₀: Errors are normally distributed
• H₁: Errors are not normally distributed

A Q-Q plot provides a useful visual check: points should approximately follow the diagonal.

If violated: OLS coefficient estimates do not automatically become biased. With the appropriate exogeneity condition, OLS can still be unbiased.

The main issue is that exact small-sample t- and F-based inference may no longer be valid. With sufficiently large samples, asymptotic results often make normality much less important.

𝗧𝗵𝗲 𝗕𝗶𝗴 𝗣𝗶𝗰𝘁𝘂𝗿𝗲

One important lesson:

Not all assumption violations are equally dangerous.

🔴 Exogeneity violation → can bias the coefficient estimates themselves.

🟠 Heteroscedasticity/autocorrelation → coefficients may remain unbiased under exogeneity, but conventional standard errors and statistical inference can become unreliable.

🟡 Non-normal errors → mainly a concern for exact small-sample inference.

🟢 Multicollinearity → primarily increases coefficient uncertainty and instability; it does not automatically bias OLS coefficients.

And remember:

A p-value < 0.05 does NOT mean “the assumption is violated” in every situation.

It means you reject the specific null hypothesis of the particular statistical test you performed.

So don’t blindly run tests and look for p-values.

Combine:

📊 Statistical tests 📈 Residual plots 🧠 Domain knowledge 🔍 Model specification checks

A regression model isn’t trustworthy just because it has a high R² and significant coefficients.

You need to understand why those coefficients can be trusted.