It depends on where the purchase is made and how identity is tracked. You could have a more complicated attribution model that doesn't rely solely on the last interaction, like using clock/view-through windows.
However you also have the problem of identity tracking. The simplest way is cookie tracking but this had cross device and IRL problems. You can use services like Facebook or Google top track identity across devices, even in store purchases if you track data well enough.
Problem in both cases is that the data collection and analyses becomes increasingly complex and inefficient. However this data is a lot more sophisticated than what you could achieve in old media. On the other hand it's up to debate whether the ads themselves are efficient in the digital medium.
Personally, I believe far too many advertisers obsess over ROI without having a clue how to do proper attribution modelling. This leads to everyone trying to grab that last interaction dollar, clickbaiting, creating a divisive environment and this is why we can't have nice things.
One of the major issues I'm coming up against at the minute is the view through window though. When combined with a multi-channel strategy, and retargeting, view throughs can just inflate numbers, masking true effectiveness.
I've lived my life doing digital advertising work, but if I turned off 3/4s of ours tomorrow, I'd be loathed to pin my cap and predict volume would drop by that.
Interestingly, I've seen a lot more focus on different attribution models recently, I think people are finally asking the questions now it's been ingrained in a lot of teams for a few years.
However you also have the problem of identity tracking. The simplest way is cookie tracking but this had cross device and IRL problems. You can use services like Facebook or Google top track identity across devices, even in store purchases if you track data well enough.
Problem in both cases is that the data collection and analyses becomes increasingly complex and inefficient. However this data is a lot more sophisticated than what you could achieve in old media. On the other hand it's up to debate whether the ads themselves are efficient in the digital medium.
Personally, I believe far too many advertisers obsess over ROI without having a clue how to do proper attribution modelling. This leads to everyone trying to grab that last interaction dollar, clickbaiting, creating a divisive environment and this is why we can't have nice things.