The equation,

\(Y = \alpha X^2 + \epsilon\)

is still a linear regression since it follows the general linear form,

\(Y = \alpha X^* + \epsilon.\)

Therefore, the best predictor would be to use your standard OLS regression methods and interpret \(X^*\) appropriately (i.e., that it is the square of the \(X\) values). More appropriately, this model would be considered

*curvilinear* because the functional shape of it would be curvy instead of a straight line. A regression model is

**nonlinear** when it is nonlinear

*in the coefficients*. For instance, the following model is nonlinear:

\(Y = \alpha e^{\beta X}\)