Gauss-Markov Assumptions › Key Components

The Gauss-Markov theorem underpins the theoretical foundation of Ordinary Least Squares (OLS) regression. It states that under five key assumptions.linearity in parameters, random sampling, no perfect multicollinearity, zero conditional mean, and homoscedasticity.the OLS estimator is the Best Linear Unbiased Estimator (BLUE) of the regression coefficients. This means that, among all estimators that are linear in the observed outcomes and unbiased, OLS yields the smallest possible variance.

These assumptions need to be understood and checked because:

• If all assumptions are satisfied: OLS produces coefficient estimates with the lowest variance, and confidence intervals and p-values are reliable.
• If some assumptions are violated: OLS estimates may be biased, inefficient, or have standard errors that do not reflect true uncertainty, leading to invalid inference.

The practical approach involves:

• Diagnosing potential violations via plots and formal statistical tests,
• Understanding the consequences for estimation and inference,
• Applying remedies or alternative estimation techniques as needed (e.g., using robust standard errors, transforming variables, or specifying different models).

In applied regression analysis, routinely reviewing the Gauss-Markov assumptions and their diagnostics greatly increases the reliability and credibility of your statistical conclusions.

• What it means: The relationship between variables can be expressed as a linear combination of parameters
• How to check: Plot residuals vs. fitted values; look for systematic patterns
• Violation consequences: Biased and inconsistent estimates
• Remedies: Transform variables, use polynomial terms, or switch to non-linear models

• What it means: Each observation is independently drawn from the population
• How to check: Examine data collection process, check for clustering or dependencies
• Violation consequences: Standard errors may be incorrect, affecting hypothesis tests
• Remedies: Use clustered standard errors, mixed-effects models, or account for sampling design

• What it means: Independent variables are not perfectly correlated
• How to check: Calculate correlation matrix, variance inflation factors (VIF)
• Violation consequences: OLS cannot be computed; high multicollinearity leads to unstable estimates
• Remedies: Remove redundant variables, use regularization (Ridge/Lasso), or collect more data

• What it means: Error term has zero expected value given the independent variables
• How to check: Test for omitted variable bias, use instrumental variables
• Violation consequences: Biased and inconsistent estimates
• Remedies: Include omitted variables, use instrumental variables, or switch to other estimation methods

• What it means: Error variance is constant across observations
• How to check: Plot residuals vs. fitted values, use Breusch-Pagan test
• Violation consequences: Inefficient estimates, incorrect standard errors
• Remedies: Use robust standard errors, weighted least squares, or transform variables