A twist on handling the interaction effect

One of the topics which often results in debate is the interaction effect. Recall that in performance attribution, specifically with the "Brinson models," we have two effects: allocation and selection. Depending on how we calculate these effects we may have a third: interaction. The formula for interaction is:

I.e., the difference in returns (portfolio minus benchmark), which reflecdts the selection decision, and the difference in weights (portfolio minus benchmark), which reflects allocation.

There are some who argue that the interaction effect should never be shown individually; that since no one is making an "interaction decision," it has no value and is unworthy of any attention. That the interaction effect properly belongs with selection: end of story!

On the other hand, there are folks like me who see value in the interaction effect and believe that the allocation effect should not be influenced by the selection decision (by using the portfolio's returns rather than the benchmark's in the formula), and likewise the selection effect shouldn't be influenced by allocation decision (by using the portfolio's weight rather than the benchmark's). As a result we're left with the interaction effect.

I, and one of my esteemed and well regarded colleagues, recently got a note from someone from a software vendor on this topic:

We have customer who wants to split a reporting period into two sub-periods. In the first sub-period they want the Interaction Effect added to the Stock Selection Effect, but in the second they want the Interaction Effect added to the Asset Allocation Effect.

Is there any reason why this may not be valid?

My colleague responded that interaction should only be included with selection, and that this approach is invalid. I strongly disagreed. If the selection decision is a good (i.e., the portfolio return is higher than the benchmark's) but the allocation decision was a poor one (where the portfolio weight is less than the benchmark weight), the interaction effect will be negative: why should the selection effect get this baggage? Why should it be lowered because of a poor allocation decision?

While I prefer to see the interaction effect analyzed to determine where it is to be placed (if it isn't going to appear separately), the approach this fellow's client wants is apparently an attempt to spread the effect across the other two, rather than arbitrarily always placing it with one or the other. Granted this method is one that employs an arbitrary assignment of the effect, but is done so in a manner that is consistent, just as always putting it with interaction is consistent. One cannot "game" this approach, so as to always optimize one effect rather than another.

To me, this is equivalent to always placing it with one or the other effects; it's just that it's alternating back-and-forth. I see nothing wrong with this approach.

Do you? Have some thoughts? Please share them below!