In marketing analytics, there is a concept of adstock link that captures through a variable transformation, the decaying impact of an advertising stimulus on demand. Finding the optimal adstock rate is typically done through non-linear least squares and ends up distributing a stimulus in week n across weeks n, n+1, n+2, .., n+N.
In my case, I am looking not at marketing but promotions (e.g. discounts, sales, premiums) and the expectation is that a discount will produce an effect where there is a spike in demand and then a negative effect (because the demand was pulled forward in time) and then to zero. So, month 1 pay be positive, month 2 negative (to a lesser degree) and then no effect.
Is there a variable transformation like this or do I need to likely fall back on a distributed lag model (with explicit variables for promotion level now, promotion level 1 period back, promotion level 2 periods back)?