See my earlier article on demand estimation for background.

There are two broad approaches to estimating aggregate-level demand: product-space approaches and characteristics-space approaches. Product-space approaches, such as Price Invariant Generalised Logarithmic models and Almost Ideal Demand Systems, treat individual products as the unit of analysis and endeavour to estimate demand functions using restrictions from economic theory. However, these approaches need to estimate N^2 elasticities (considering both own-price and cross-price elasticities), where N is the number of products, hence, even for a modest market with (e.g.) 20 products, over 400 parameters need to be estimated. This is computationally difficult, leading to the curse of dimensionality, and poses a key downside to product-space approaches, along with the detraction that such methods do not allow for the counterfactual estimation of new products being introduced. [...]