Polynomial Regression
Also known as: polynomial least squares, curvilinear regression, Polinom Regresyonu
Polynomial regression is a regression method that models non-linear relationships by including squared and higher-degree terms of an explanatory variable, and it is a core tool of response surface analysis. As developed in Montgomery, Peck and Vining's Introduction to Linear Regression Analysis (2012), it remains linear in its parameters even though the fitted curve bends.
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When to use it
Use polynomial regression when the outcome is continuous and the relationship with the predictor is clearly curved rather than straight, with a reasonable sample (at least about 30 observations). Apart from the linearity assumption, the standard OLS assumptions still apply. Keep the degree low: high-degree polynomials tend to overfit, so cross-validation is recommended, and the high powers introduce multicollinearity, so centring the predictor before raising it to powers is advised.
Strengths & limitations
- Captures smooth non-linear (curved) relationships while staying linear in the parameters, so it is estimated with ordinary least squares.
- Simple to fit and interpret as an extension of linear regression, with a single closed-form solution.
- Serves as the foundational model for response surface analysis.
- High-degree polynomials overfit the data, fitting noise rather than the underlying trend.
- Higher-order power terms are strongly collinear, destabilising the coefficient estimates unless the predictor is centred.
- Apart from relaxing linearity, it still relies on the standard OLS assumptions (such as normal, well-behaved errors).
Frequently asked
How do I choose the polynomial degree?
Start low and increase the degree only as long as it meaningfully improves out-of-sample fit. Use cross-validation rather than in-sample R², because higher degrees always raise in-sample fit while overfitting noise.
Why should I centre the predictor?
Raising x to successive powers makes the terms (x, x², x³, …) strongly correlated, inflating the standard errors. Centring the predictor before forming the powers reduces this multicollinearity and stabilises the coefficients.
Is polynomial regression still a linear model?
Yes. Although the fitted curve bends, the model is linear in its coefficients, so it is estimated with the same ordinary least-squares machinery as linear regression on an expanded set of predictor columns.
When should I use splines instead?
When the curve has local features or you need flexibility without high-degree instability, spline regression fits piecewise polynomials with better behaviour, especially near the edges of the data, than a single high-degree polynomial.
Sources
- Montgomery, D. C., Peck, E. A. & Vining, G. G. (2012). Introduction to Linear Regression Analysis. Wiley. ISBN: 978-0470542811
How to cite this page
ScholarGate. (2026, June 1). Polynomial Regression. ScholarGate. https://scholargate.app/en/statistics/polynomial-regression
Which method?
Set this method beside its closest kin and read them side by side — the library lays the books on the table; the choice is yours.
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