Ordinary Least Squares (OLS) Regression

Key highlights

Best linear unbiased estimator (BLUE) under the Gauss-Markov assumptions.
Simple, fast, and computed in one closed-form step.
Produces directly interpretable coefficients: each βⱼ is the expected change in y for a one-unit increase in xⱼ, holding the other predictors constant.

Intuition

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How it works

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When to use it

OLS is a good fit when the outcome is continuous and the relationship is approximately linear in the parameters, with a reasonable sample size (at least about 30 observations). It is most reliable when the Gauss-Markov assumptions hold: linearity in the parameters, exogenous regressors that are uncorrelated with the error term, errors with no autocorrelation, predictors that are not strongly collinear, and a constant (homoscedastic) error variance. When these conditions are met OLS gives unbiased, efficient, and directly interpretable coefficients. It is less suitable for small samples, strongly collinear predictors, or non-constant error variance.

Strengths & limitations

Strengths

Best linear unbiased estimator (BLUE) under the Gauss-Markov assumptions.
Simple, fast, and computed in one closed-form step.
Produces directly interpretable coefficients: each βⱼ is the expected change in y for a one-unit increase in xⱼ, holding the other predictors constant.

Limitations

Coefficients become unstable and standard errors inflate in small samples (n < 30).
Multicollinearity (VIF > 10) and heteroscedasticity invalidate the standard errors, making the t and F tests misleading.
Assumes a linear relationship and exogenous, well-behaved errors; violations bias inference.

Common pitfalls

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Applications

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Frequently asked

What does R² mean in OLS?

R² is the proportion of the variance in the outcome explained by the model. Because plain R² always rises when predictors are added, judge overall fit with adjusted R² and the F-test instead.

What is heteroscedasticity and why does it matter?

Heteroscedasticity means the error variance is not constant across observations, violating a Gauss-Markov assumption. It does not bias the coefficients but invalidates the standard errors, so the t and F tests become misleading. A Breusch-Pagan test detects it; robust standard errors or quantile regression are remedies.

How do I detect multicollinearity?

Compute the variance inflation factor (VIF) for each predictor; a VIF above 10 signals strong collinearity. Collinear predictors inflate standard errors and make individual coefficients hard to interpret.

When should I use ridge or lasso instead of OLS?

When predictors are strongly collinear or the sample is small, OLS coefficients become unstable. Ridge regression shrinks coefficients to stabilise them, and lasso regression can also drop redundant predictors, trading a little bias for much lower variance.

Sources

1.
Wooldridge, J. M. (2019). Introductory Econometrics: A Modern Approach (7th ed.). Cengage Learning.
ISBN 978-1337558860

Related methods

Same family

Panel Fixed Effects Quantile Regression

Related: see also

Lasso Regression Logistic Regression Ridge Regression

Referenced by

2SLS Regression ARCH-LM Test ARDL Bounds Test ARFIMA Model ARIMA Augmented Mean Group Estimator Bayesian Linear Regression

+141 more

Referenced by

Bayesian Multiple linear regression Bayesian OLS Bayesian Random Effects Model Bayesian Regression Bayesian Robust Regression Bayesian Simple linear regression Bayesian VAR Beta Regression Black-Litterman Model Block Bootstrap Breakdown Point Analysis Breusch-Godfrey Test Breusch-Pagan Test CAPM Causal Discovery Algorithms Causal Mediation Analysis CCEMG Estimator CGE Model Chow Test Cluster-Robust Standard Errors Condition Index Conditional Process Analysis Conformal Prediction (Time Series)Croston's Method Difference-in-Differences Difference-in-Discontinuities Doubly Robust Estimation Durbin-Watson Test Dynamic OLS Elastic Net Regression Event Study Factor Risk Model FAVAR Fixed Effects Model Fixed Effects Panel Model FMOLS Estimator Fourier OLS Fourier WLS Gamma Regression GARCH Model Generalized Linear Model Geographically Weighted Regression Global Spatial Error Model GMM Estimation Granger Causality HAR-RV Model Hausman Test Heckman Selection Model Heteroscedasticity-Robust Standard Errors Hierarchical Linear Model Huber Regression Hurdle Model Influence Diagnostics Interest Rate Models Interrupted Time Series Jackknife Kriging Least Median of Squares Least Trimmed Squares Liquidity Risk Models Long-Memory Models M-Estimator MAD Estimation Markov-Switching Model MGWR MM-Estimator Moderation Analysis Multinomial Logit Multivariate Regression NARDL Model Negative Binomial Regression Newey-West HAC Nonlinear NARDL Nonlinear OLS Nonlinear WLS Nonparametric Quantile Regression Ordered Logit Ordinal Logistic Regression Ordinal Regression Pairs Trading Panel Cointegration Tests Panel Fixed Effects Panel OLS Panel Simple Linear Regression Panel VAR Poisson Regression Polynomial Regression Pooled OLS Principal Component Risk Factors Probit Model Prophet Quantile Regression Ramsey RESET Test Random Effects Model Random Effects Panel Model Randomization Inference RANSAC Regression Regime-Switching Model Regression Discontinuity Regression Discontinuity Design Regression Kink Design Robust ANOVA Robust Correlation Robust GLS Robust Hausman Test Robust Logistic Regression Robust Mixed Model Robust Multiple linear regression Robust NARDL Robust OLS Robust Quantile Regression Robust Regression Robust Simple linear regression Robust Time Series Analysis Robust WLS S-Estimator Seemingly Unrelated Regression Spatial Durbin Model Spatial Error Model Spatial Lag Model Spatial Panel Model Spatial Regression STAR Model Stochastic Frontier Analysis Structural Break OLS System GMM Tail Risk Measures Theil-Sen Estimator Theta Method Three-Stage Least Squares Threshold Regression Time-varying parameter OLS Tobit Model Two-Stage Least Squares (2SLS)VaR Backtesting VAR Model Variance Inflation Factor VECM W-Estimator White Test Wild Bootstrap

Cite this page

ScholarGate. (2026, June 1). OLS Regression. ScholarGate. https://scholargate.app/econometrics/ols-regression