# Quadratic effect in OLS regression

I am currently trying to do an OLS regression using data of online product reviews and I have two questions:

1. Do I have to use both, the linear and the quadratic effect in the model or is it also okay to only keep the squared variable in the model? I read that I have to use both, but I do not really understand why. So why should this be like this?

2. I am doing a regression with a helpfulness score of online product reviews as the dependent variable and the star rating of the reviews (integers between 1 and 5) as an independent variable. I would like to incorporate a squared effect, because I hypothesize that 1 and 5 star ratings are more helpful than moderate reviews (e.g. 3 stars). When I just square the star rating I get 0, 1, 4, 9, or 25 as possible values for the squared variable. However, to me it makes much more sense to first subtract 3 from the rating and then square the variables, because this better reflects the hypothesis that the extremer a rating the higher its helpfulness score. Now, I get 4, 1, 0, 1, or 4 as possible values for the squared variable. Would it make sense to do this?

You don't have to use a linear term to use a quadratic, but it's usually a good idea. The only situation I wouldn't use it is when your theory tells you that you have a quadratic process. For instance, if you somehow are measuring kinetic energy as a function of speed, then there's no linear term in theory: $$e=m\frac{v^2}{2}$$
• +1. To make this even more explicit the principle is that (in more statistical notation) $y = bx^2$ implies that $y = 0$ when $x = 0$. Not knowing that to be true as a matter of principle usually excludes this square-only model in practice. Dec 19 '16 at 20:29