Estimation of a power function in regression $y = ax^k$

Question

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.

Answer 1

Although you are sensibly keeping a careful eye on what fits, your approach can fairly be described as rather home-grown or ad hoc. Depending on your target audience or readership, the consequence may range from practitioner puzzlement to statistician flak.

Your general model I take to be a sum of power functions. With some change of notation that could be

$y = b_0 + \sum_{j=1}^J b_j x_j^{k_j}$

with additive error. As @Glen_b comments, the usual approach to fitting such a model would be to use nonlinear least squares, which I take to be well supported in SAS, although I can't advise on details.

In many ways a simpler model is a multiplicative power function

$y = B_0 \prod_{j=1}^J x_j^{b_j}$, from which

$\ln y = b_0 + \sum_{j=1}^J b_j \ln x_j$, where $\ln B_0 =: b_0$.

That model is easy to fit by least squares as it is just multiple regression on the logged variables. Error with this is taken to be multiplicative on the original scale and additive on a logarithmic scale, which is often about right. A virtue of this model, as with power functions taken singly, is that it can be consistent with the limiting behaviour that $y$ tends to 0 as all the $x$s tend to 0, often important economically (physically, biologically). (However, the assumption is that all data are positive. Your own approach appears consistent with occasional zeros but not with negative values.)

More generally, however, we have no sight of your data and only a hint of what the regressors or predictors are, so it is difficult to say much more except to guess that your response variable is probably something zero or positive. If so, it is helpful to ensure that predictions are always positive for all data points. I can't see that is guaranteed by your present approach.

On a terminology question: I'd advise against calling any of these models a polynomial, even if you spell out very clearly that the powers are in general not integers. Either people don't know what a polynomial is or they will expect the powers to be integers, at least as a mathematical default: there is some obscurity either way, better avoided.

Edit:

The model

$\ln y = b_0 + \sum_{j=1}^J b_j \ln x_j$

can naturally be complicated according to taste and need, e.g.

$\ln y = b_0 + \sum_{j=1}^J b_j \ln x_j + \text{extra terms}$,

where the extra terms could be in the $x_j$, the $\ln x_j$ or both.