Whether to include $x$ and $x^2$ in regression model examining diminishing returns when only $x^2$ is significant?

Question

I have a data set with sales as $Y$ and total retail shelf space ($x$).

I want to investigate the diminishing return, i.e. whether additional shelf space would contribute to more sales. I did regression or Arima model where I found that the squared shelf space ($x^2$) was significant. Thus there is a diminishing return.

My question is how to proceed from here?

Should I transform my data to

$$\log Y = \log x\times \text{shelf_space} + 2\log x \times \text{shelf_space}^2$$

or should I simply ignore the $\text{shelf_space}$ (the one which was not significant i.e. the non squared one) and only do my analysis with shelf space squared?

How to do regression which takes diminishing return into account?

Answer 1

Consider the simple linear model $Y = \beta_0 + \beta_1X + \beta_2 X^2$. Now think of a change of scale (in your case, that could be for instance measuring in square feet instead of square meters, or viceversa) so that the new variable is $Z = aX + b$. The very least you would ask of an statistical model is that the inferences are invariant to the scale of measure. See how the model changes when writen in terms of $Z$.

\begin{eqnarray*} Y &=& \beta_0 + \beta_1Z + \beta_2Z^2 \\ &=& (\beta_0 + \beta_1b + \beta_2b²) + (\beta_1a + 2ab\beta_2)X + a^2\beta_2X^2 \\ &=& \beta_0^* + \beta_1^*X + \beta_2^*X^2 \end{eqnarray*}

Assume you want to test that the quadratic effect is zero. You can do it in either model, as $\beta_2 = 0$ implies (and is implied by) $\beta_2^* = 0$, since $\beta_2^* = a^2\beta_2$; so the model gives invariant inference about the quadratic term in either scale of measure when you include the linear term. This is no longer the case if you do not include the linear term, as can be checked easily.

So to answer your question, yes, you want to include the linear term even if interest focuses on the quadratic term, as otherwise you might get different answers in different scales. Or, put in another way, you want to fit a well structured hierarchical model.

Notice this reasoning also underlies the inclusion of an intercept, even if per se the intercept terms is of no interest.

Regarding your last question, fit the model that seems best and then obtain the first derivative, as has been advised in a comment.

Answer 2

(wanted to make this a comment, but don't have the reputation. Still think it's important.)

In case you do include both $x$ and $x^2$, you should center $x$ first. This avoids collinearity between $x$ and $x^2$, as explained in this answer.

Answer 3

I think regression which has the purpose to take diminishing return into account may involve dynamics, with ARIMA you already hint to time-series. When for example investigating advertising on sales you can include lagged sales (sales from the previous years) in your model. I don't know however if it makes sense conceptually to include lagged shelf space.

However, in technical terms, what you can do for example is a log-log (all variables log-transformed, sometimes called power model) or semi-log (only logs on the right side of the regression equation) model (similar to the model which you present in your question).

Also, you could use dummy variables to check. However, more efficient may be the Koyck specification, which imposes that the effect is decreasing geometrically. An alternative is the ADSTOCK (http://en.wikipedia.org/wiki/Advertising_adstock) model.

All approaches have their assumptions, degree of flexibility, and advantages and disadvantages. I'm not very experienced with these models myself but I'm sure you can find more information about them in Google or academic literature. I hope this helps getting some specifications and solutions on your radar.