Linear Regression Assumption

Question

the following website states that linear regression assumes a linear relationship between dependent and independent variables: "First, linear regression needs the relationship between the independent and dependent variables to be linear. " from: https://www.statisticssolutions.com/assumptions-of-linear-regression/

However, I have read elsewhere that linear regression can be applied to non-linear relationships, such as fitting a curved line such as f(x) = x^2 to a set of data.

Which one is correct?

Answer 1

Both are correct. You still will have a linear relationship: you just think of $x_i^2$ as a variable itself, so there is a linear relationship between $y_i$ and the variable $x_i^2$.

Answer 2

Linearity in linear regression is a linearity w.r.t. the coefficients, $\beta$. Formally, a model called linear if its gradient w.r.t. $\beta$ is independents of $\beta$, namely, let $$ y=\beta_0+\sum_{j=1}^p\beta_jg_j(x_j)+\epsilon, $$ and $$ \nabla_{\beta}\mathbb{E}[y|x_1,...,x_p]=(1, g_1(x_1),....,g_2(x_2))^T, $$ where $g_j(x_j)$ can be whatever you like, i.e., $x_j^2, \sin x_j$, etc., as long as they don't involve $\beta$s.