I have only a potential statistical solution: linear ridge regression, a technique that uses a shrinkage constant (k) to make regression coefficients more stable in the case of multicollinearity. See

Ryan, T. P. (2008). Modern Regression Methods. New York, NY: John Wiley & Sons.

Can you please explain your problem better, i.e.:

• Do you want to use the physical distance between the place where the customer is located and the place where the shop is located? Or do you want to use the "semantical distance" of what the customer wants and what the shops offer?
• What are these other "shop channels"? Online, mobile, etc.? Obviously, if a customer buys in a brick&mortrar shop, s/he will not buy the same good over a different channel? Or do you mean the buying will have side effects on those other channels? Which effects would you expect?

Maybe you can reduce the complexity of your question by focus on one aspect first and not muddle inter-channel effects and intra-channel considerations (i.e. "in-store initiatives and policies").

What do you mean by "these explanatory variables are located in the same place"?

Thanks for your answers.

@Thomas

I am studying a sales organization that sells products to B2B customers using a number of different sales channels.

- a brick and mortar or physical shop (spread through the country)

- a webshop

- a sales representative (on the phone or visiting) (multiple representatives)

A customer may use one or more channels to buy products.

What is suspected is that if customers use multiple sales channels, they will be more loyal and buy more products (higher sales exploitation). So it is interesting for a business to know whether or not one sales channel can stimulate the use of other sales channels by customers. And measure the effect of that promotion.

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I want to measure two things:

Firstly: the (general) effect of the customer shopping in the physical shop on how many other shopping channels he uses.

I have cross sectional customer data for sales over the period of one year.

To measure "the (general) effect of the customer shopping in the physical shop" I will take the number of orders the orders placed in the physcial shop as a measurement of exposure to influences that might have induced the customer to start using webshop (via the website, app etc) or call a sales rep.

I know of every customer exactly how many sales channels he has used in the year (either 1 2 or 3).

"how many other shopping channels the customer uses. " is simply measured as all sales channels used, minus 1 (if the customer has purchased in the physcial shop).

Now if I do an OLS regression with "other sales channels used" as the dependent and "number of orders in the physcial shop" as the independent, I get a bad estimation, due to omitted variable bias.

There may be specific customer characteristics not accounted for, influences of marketing team that promote the use of a certain sales channels etc etc. These are unobserved variables.

To overcome this I try an IV approach.

So I choose to compute the physical distance between each customer and the physcial shop that visits. This would be the as-the-crow flies distance.

Since the closer the customer is to the shop, the more likely he is to shop there, and I think / hope that distance between shop and customer is unrelated to any other unobserved variables.

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Secondly: I want to measure the effect of specific shop policies on the customers use of other sales channels. In addition to the general effect described above.

Policies like for example: a sales associate demonstrating the use of the webshop/smart phone app, the presence of a terminal (computer or tv) that promotes using the webstore or perhaps posters and flyers that promote using the webstore or calling a sales rep.

These policies should be present in every shop, but the quality of them varies by shop.

I have yet to measure these policies but I think I can do so with survey in which various stakeholders (in and around the shops) judge the quality of these policies per shop.

These policies happen inside the physical shop.

Say I survey the quality of 3 specific policies to get a rating for these for every shop.

Now if I do an OLS regression with "other sales channels used" as the dependent and

"number of orders in the physcial shop"

"policy 1"

"policy 2"

"policy 3"

as the independents, I get an inaccurate estimation, due to omitted variable bias once again.

But if I try IV, I would now need 4 seperate instruments, because using distance as an instrument would lead to collinearity and thus would be dropped.

My question is how to deal with the collinearity. I might know of another variable I might be able to use as an instrument, but then I would still be faced with a lack of instruements.

Hi Casijin

well, that makes it a bit clearer. I just focus on the first part today (may be I find time for the second part in some days).

I wouldn't consider the three channels exchangeable, since the sales channel used by a customer depend on the good itself.

1. In a brick&mortar shop, storage capabilities are usually limited. Hence, they often have goods in their inventory with a high turn-over rate. A customer in need of a special product, will have bad luck at these shops.

2. The success of online shops (standard example Amazon) is usually based on selling products from the "long tail", i.e. specialized products which are infrequently sold.

3. From my experience when I was student working in a shop: of course, a sales representative takes orders for all kind of products, but usually s/he pops-up at a customer if new products should be offered.

Well, I think, that the type of product, its turn-over rate and its novelty will influence where the purchase will be made.

Another factor might be the price. If a product can be bought at a brick&mortar shop far away much more cheaper than at the nearby brick&mortar shop, it might be useful to drive a bit longer.

"What is suspected is that if customers use multiple sales channels, they will be more loyal and buy more products" I don't agree with this. Behind this hypothesis lurks the implicit assumption that the inventory of different suppliers is exchangeable.

Usually, customer use different suppliers, since often not a single supplier has all the products in his portfolio, which the customer needs (self-check: Or do you buy at a single shop only?). I think its more a question of convenience. If a customer needs to order some special product,

which he can get only from one supplier, then it is more likely that he also buys other products (which he could obtain from other suppliers equally well) from this supplier, in order to minimize his efforts.Of course, presupposed these products are comparable.

I think it is not so much a question of marketing or other channels, whether a customer buys at a brick&mortar shop, I think it is more a question of demand and offer.

Well, I don't know your data, but I suspect they will not contain information about the bought products or the content of the "shopping baskets". These are probably important unobserved data for you.

I don't think that a linear formula will suffice to capture the problem. I would try to identify the "network of variable dependencies" first, in order to get a understanding of them. And than I would try to figure out by "causal inference"/"counterfactural reasoning", whether the data available suffice to make the intended predictions.

Hi Thomas,

Thank you for your feedback.

It should be noted that the 3 sales channels are parts of the same sales organization. The sales reps and webstore sell from the same inventory. The shops carry a limited range of products, but can order any product for a customer from that same inventory.

I have customer data ( cross sectional - for one year ) on

• customer id
• location of customer
• industry of customer (general)
• sub-industry/trade of the customer (specific)
• date of creation customer account
• assigned sales rep (who does personal selling over the phone or in person)
• assigned sales manager (boss of the sales repps)
• number of employees of customer
• sales potential of a customer (established by a heuristic)
• sales exploitation (what % of sales potential is achieved)
• total revenue per customer
• total orders per customer
• total revenue of e-commerce per customer
• total orders in webstore by customer
• shop revenue (for every shop the customer has been in)
• number of orders in shop (for every shop the customer has been in)

Dataset is n >10k. I have access to very deep data if I want, from the exact contents of every order, for what order reason (sales details are coded with a number and recorded), and the exact day that order was recorded. But I don't want to make things to complicated for myself.

If you mean by a network of variable dependencies a correlation matrix, then yes I have done one.

The most obvious explanatory variables that estimate the number of channels used are obviously the dummies for the various channels that the customer shops in.

But then you' d be estimating coefficients for a simple addition. And the channel dummies would have coefficents of 1.

I'd be happy to find out if anyone knows of theory or models that may explain the number of sales channels chosen by a customer.