# How to interpret B when x is squared?

My OLS regression looks like this:

$$y=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3(x_3)^2+\epsilon.$$

I do not include $$x_3$$ linear because it vifs with squared value and without squared value I have wrong specification according to RESET.

How do I interpret the $$\beta_3$$ in my equation (in general)? I know it should be something like: $$\Delta y=(2\beta_3x_3)\Delta x$$ But what should I use as an $$x_3$$? Should it be an average $$x_3$$ value?

• I would suggest you include both linear and quadratic terms otherwise we risk biasing our model. Please see: stats.stackexchange.com/questions/28730 for more details. Jun 29, 2019 at 9:05
• You may use $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_3 + \beta_{32} (x_3 - \overline{x_3})^2$ where $\overline{x_3}$ is the mean of $x_3$ or an estimate of the extreme of the parabolic relation between $x_3$ and $y$. In this latter case, it can be chosen such that $\beta_3 \approx 0$. Jun 29, 2019 at 11:54

The interpretation of the $$\beta$$'s is the same for each $$\beta_i$$; when $$y=\beta_0+\beta_1x_1+\beta_2x_2+\beta_3(x_3^2)+ε$$, we have that $$y$$ increases by $$\beta_1$$ when $$x_1$$ increases by one and all the other covariates stay the same. For $$\beta_3$$ this is similar: $$y$$ increases by $$\beta_3$$ as $$x_3^2$$ increases by one.
If you want to compute the value for $$y$$ given the values you have for $$x_1,x_2,x_3$$, you can simply use the OLS regression formula you have created but square $$x_3$$ when multiplying by $$\beta_3$$. Hope this answers your question.
$$\beta_3$$has no meaningful interpretation itself, its more so what changes with respect to the interpretation of the marginal effect. In your example, if we obtain a positive value for $$\beta_3$$then the marginal effect of $$x_3$$ on $$y$$ is increasing with $$x_3$$. That is, the effect is larger for larger values of $$x_3$$. But as mentioned in the comments, i would include a linear term whenever using a quadratic term except for in very special circumstances.