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?
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.
