Suppose I have a simple linear model with two variables and their interactions:
$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2 + \beta_4 x_1^2 + \beta_5 x_2^2 + \epsilon$
where the $\beta_i$ can, of course, be estimated by OLS. Assume that the standard Gauss-Markov assumptions of OLS hold, and that data points are sampled i.i.d. My question is the following: Is there a way to approximate this model such that the approximation is separable in $x_1, x_2$?
Clearly, I could just omit the term $\beta_3 x_1 x_2$ and obtain a separable model at the cost of accuracy, but I'm wondering if there is an approach which would better approximate the original model? I'm only interested in predicting $y$, so the metric by which an approximation would be assessed would be the additional error in introduces in predicting $y$. In other words, I'm interested in an approximation which minimizes the difference in sum-squared-residuals relative to OLS.
The approximation need not be linear in its parameters (or even parametric). This seems related to the idea behind partitioned regression. I also know that orthogonal polynomials are commonly applied to function approximation, but again, am not sure if it would be useful here.
If anyone has any suggestions about where to look for some ideas/theory I'd be happy to hear them.