I'm performing a case of polynomial regression. I use a power $k$ for the regressors (e.g. marketing spend), which helps me determine the nature of the response curve.
I also need to estimate the coefficient for each regressor.
Consider the simplistic case: $y = ax^k + c$ ; $c$ constant, $a$ a coefficient.
The values of $k$ and $a$ need to be determined (if polynomial, $k$, $x$ and $a$ would be vectors). I vary $k$ between $-2$ and $2$ and find the value of $k$ for which
pow(x,k) correlates best with $y$ using a SAS macro. I take the top three $k$ which help $x$ correlate with $y$.
I start regressing $y$ on
pow(x,k) and vary $k$ between the top values in priority and observe model fit and error structure to decide.
This is a slightly approximate approach (depending on the intervals of $k$ which I choose to iterate over, 0.01/0.1 etc.), but has worked well in polynomial situations because it is a SAS macro and runs pretty fast.
Is there a better approach?
Editing to add some more context as suggested by @Nick-Cox. The dependent is the sales of a product. The regressors (x) are marketing spends.
There is a strong hypothesis backing interaction effects between the x's.
Another requirement is that not all marketing spends should be forced to have a diminishing impact on sales.