# Regression: Why does using quadratic expressions work with linear estimators? [duplicate]

My questions is, that I see people using R´s lm() (linear regression model) with Y ~ X^2 e.g. here: Simple non-linear regression problem

But I dont see how and why it works, hence lm() is clearly stated as a linear estimator (using OLS), why does it work for quadratic, clearly non-linear estimation, as well?

Thank you.

• If you let $Z=X^2$ then you have a simple linear regression of $Y$ on $Z$. Hence the name. The method does not care that you may have used a non-linear function of independent variables: it does not see that – Henry Aug 11 '19 at 21:07
• Nitpick: Y ~ X^2 is exactly the same as Y ~ X. You should study help("formula"). Apparently you eman Y ~ I(X^2). – Roland Aug 12 '19 at 6:31
• @Henry, but how can I interpret U-shaped relationships with the output of my lm regression, which tells me that, for instance, increasing X by one leads to an increase of Y bei 5? – Tw3Ak3r Aug 12 '19 at 12:34
• @Tw3Ak3r If the output of your regression is that $Y \approx 5X^2+k$ then an increase of $X^2$ by $1$ (not an increase in $X$) is associated with an increase in $Y$ of $5$ – Henry Aug 12 '19 at 12:55