Questions: Polynomial Regression and Nonlinear Functional Forms

Question 1

A researcher estimates the model y = β₀ + β₁x + β₂x² + ε and obtains β₁ = 5 and β₂ = −0.3. What is the marginal effect of x on y when x = 10?

• A. 5, because β₁ is the coefficient on x
• B. −0.3, because the quadratic term dominates at large values of x
• C. 5 + 2(−0.3)(10) = 5 − 6 = −1
• D. 5 − 0.3 = 4.7, by summing the two coefficients

Answer: (C)

The marginal effect in a quadratic model is ∂y/∂x = β₁ + 2β₂x, which varies with x. At x = 10: 5 + 2(−0.3)(10) = 5 − 6 = −1. You cannot simply read β₁ as 'the effect of x' — that is only valid in a purely linear model. The quadratic term means the slope changes continuously. Option A is the most common error: reporting β₁ alone as if the quadratic term doesn't exist, which would be correct only at x = 0.

Question 2

A researcher adds x², x³, x⁴, and x⁵ to a model and observes that in-sample R² rises with each term. They keep all terms to maximize fit. What is the main problem with this approach?

• A. OLS cannot be applied when more than one polynomial term is present
• B. Higher-degree polynomials will overfit by chasing noise in the data, producing unreliable out-of-sample predictions and often implausible curve shapes
• C. R² decreases when additional polynomial terms are added, so this approach is mathematically impossible
• D. The model violates the linearity assumption of OLS because polynomial terms are nonlinear

Answer: (B)

R² mechanically increases with every term added — a polynomial of degree n−1 can perfectly fit any n data points. But perfect in-sample fit does not mean the model captures the true relationship; it means it has memorized the noise. A high-degree polynomial can produce wild swings between data points and collapse outside the observed range. Option D is the crucial misconception: 'linear' in OLS means linear in the parameters (β), not in x. Adding x² is fine — OLS estimates β₂ just like any other coefficient.

Question 3

Polynomial regression is still estimated with OLS because the model remains linear in the parameters, even though it captures nonlinear relationships in x.

• T. True
• F. False

Answer: True

OLS requires the model to be linear in the coefficients β — it places no restriction on the variables. Treating x², x³, etc. as new variables (call them z₁ = x², z₂ = x³) transforms the polynomial model into a standard multiple regression. OLS then estimates β₁, β₂, β₃ exactly as usual. The result is a curved fitted line in the (x, y) space, but the estimation procedure is unchanged.

Question 4

A higher-degree polynomial usually produces a better model because it increases R² and therefore captures more of the true relationship.

• T. True
• F. False

Answer: False

R² always increases (or stays the same) when you add a predictor — this is a mechanical property of OLS, not evidence of a better model. A higher-degree polynomial may fit the sample very well while fitting poorly on new data. The test of a good model is out-of-sample predictive accuracy, not in-sample R². The appropriate degree should be motivated by theory, scatter-plot inspection, and significance tests — not by maximizing R².

Question 5

Why can't you interpret the coefficient β₁ in isolation in the model y = β₀ + β₁x + β₂x² + ε, and what should you report instead?

Model answer: β₁ alone is not the effect of x on y because the quadratic term means the slope changes with x. The marginal effect is ∂y/∂x = β₁ + 2β₂x, which depends on the value of x. Report the marginal effect evaluated at a meaningful x value (e.g., the sample mean), along with how it varies across the range of x — often shown as a marginal effect plot.

Reporting β₁ as 'the effect of x' implicitly assumes the relationship is linear (β₂ = 0). When a quadratic is included, that is exactly what the model is testing against. The reason you added x² in the first place is to allow the slope to vary — so report that varying slope, not a single number that pretends otherwise. This connects directly to the broader principle that OLS coefficient interpretation depends on the model specification.