  Retailers use the Vector AutoRegressive (VAR) model as a standard tool to
estimate the effects of prices, promotions and sales in one product category on
the sales of another product category. Besides, these price, promotion and
sales data are available for not just one store, but a whole chain of stores.
We propose to study cross-category effects using a multi-class VAR model: we
jointly estimate cross-category effects for several distinct but related VAR
models, one for each store. Our methodology encourages effects to be similar
across stores, while still allowing for small differences between stores to
account for store heterogeneity. Moreover, our estimator is sparse: unimportant
effects are estimated as exactly zero, which facilitates the interpretation of
the results. A simulation study shows that the proposed multi-class estimator
improves estimation accuracy by borrowing strength across classes. Finally, we
provide three visual tools showing (i) the clustering of stores on identical
cross-category effects, (ii) the networks of product categories and (iii) the
similarity matrices of shared cross-category effects across stores.
