# Regression with a quadratic term of a variable that has both negative and positive values

I am running a linear regression with a quadratic regressor of this form:

$$Y_t= \alpha + \beta_0 X_t + \beta_1 X^2_t + \gamma W_t + \eta_t.$$

Since my $$X$$ variable measures the difference between the rainfall of a given year and the long-term rainfall average, it can take both positive and negative values. I was wondering whether I should enter the quadratic term as it is (therefore containing only positive values) or manually replace the squared values with their negative for the negative X values.

• Why would you seek to negate the squared terms when the variable is negative? What's that for? (also possibly relevant -- why did you include a quadratic term?) Commented Jul 27, 2016 at 23:42
• I need to include a quadratic term because I expect the effect to be non-linear (the variable may play a role especially at very high positive values or very low negative values). I was wondering whether to replace the quadratic value by its negative when the base value is negative because conceptually a value of -5 and a value of +5 represent very different situations while they yield the same quadratic value of +25. From the answers below it seems that this is not an issue and that the coefficients can be interpreted as if the polynomial was fit to a variable including only positive values. Commented Jul 28, 2016 at 1:14
• I'm not sure I see the difficulty with positive and negative $x$ yielding the same $x^2$, since they have different $x$ -- the combined effect is different for +ve and -ve; I asked in case there was particular theory that gives different behavior for the -ve side. If you think about a relationship being curved in a parabolicish way, then you probably just want a quadratic. If you want to have more complicated things than a simple quadratic then unless theory provides a specific functional form you should probably consider moving to a different framework than that (e.g. natural cubic splines) Commented Jul 28, 2016 at 1:37