As any statistician will tell you, however, the problem is that correlation does not imply causality. Is it not just as likely that your best customers prefer to use all your channels anyway? They were your best customers when you only had stores, they are still your best customers now you have lots of ways to shop.
This research study, which unfortunately doesn't say who the retailers in question were although there are a few big hints, contains a perfect illustration:
Yes OK, multi-channel customers do spend more. The trouble is they really are simply your highest spending customers anyway, and are probably spending more everywhere they shop. This tends to be borne out by the subtext of all those other data points suggesting multichannel customers spend more. If you read more closely, you will also find those same studies mention that they tend to be on average more affluent i.e. they have more to spend in the first place (notably on things like laptops, smart-phones, iPads, high-speed broadband etc which encourage them to use all those channels): the argument gets circular. These customers expect you to have all the channels available to them - and they'll start to go elsewhere if you haven't.
We need to go back to some basics. In another previous post I used data from the UK Office of National Statistics to illustrate a very important point: eCommerce (etc) has NOT created any new money in customers' wallets. These wonderful multichannel customers do not magically have more cash to spend.
Removing food from the picture (online grocery tends to distort UK data for comparitive purposes with most of the rest of the world) and looking at just non-food, this point becomes even more apparent. Take a look at the overall "growth" of non-food offline-only non-Food sales over the last 10 years:
Non-store sales have essentially remained static over that time. Any growth has come from Online. If you factor in some measure of inflation (I've used RPI because the data is easily accessible), then the picture is that non-store sales have seriously declined:
Basically, allowing for complicated effects like cheaper prices of manufactured goods from China, the big picture is of an approximately zero sum game (no magic extra money in customer's pockets), with Online taking an ever increasing share. And factoring in the "multichannel customers spend more" data, the share that Online takes is a disproportionate share of your best, and probably most profitable, customers.
I often get asked "won't setting up a new channel cannibalise our store sales?". The answer is yes of course it will. But if you don't do it, someone else will - and what's more they'll cannibalise your best customers.
It's not really a halo effect - more a set of horns and a long pointy tail effect.