I am working on multiple regression in order to realize a marketing mix model. However I have some concerns with the procedure. First, the idea of transforme data in order to incorporate the carryover effect : the intuition behind the carryover effect is clear however from what I have understood, the way to do it is unclear, I explain this now. Consider the model
$$ y_t = \beta_{0} + \sum_{i=1}^{n}\beta_ix_{i,t} + \epsilon_{t} $$
where $y_t$ is the sales volume of a company and the $x_{i}$'s some explanatory variables that correspond to channel media, price, distribution, etc..
The idea is to consider a variable $x_{i,t}$ that has been deployed in time $t$ with a carryover effect, that is, the variable has a kind of persistency for customers after time $t$ (one can think of a campaign TV for example) and include this persistency by considering a well-chosen transformation such that for the next period ($t+1$, $t+2$,...) the effect of the variable $x_{i}$ appears. My concern is the following : by doing that, we loose the equality in our model since we added some values for next periods that are not present in our initial data since $y_t$ has not changed ?
So I would like to know where I am wrong in my reasoning please in order to fully understand this concept.
Thank you a lot !
