Regression where Regressor is a Product To find how $X$ impacts $Y$, I estimate $$Y_i=\alpha+\beta X_i+\epsilon_i.$$ Suppose that I can decompose $X$ into a product of three components: $$X_i=P_iQ_iR_i.$$ Q: How can I estimate which of the individual components drives the relationship between $X$ and $Y$?
If $X_i=P_i+Q_i+R_i$, I could just include $P$, $Q$ and $R$ as independent variables, but I am unsure because of the product. Note that $Y$ and $X$ can be negative, so taking logs may not be possible.

With this model, there is no sense in which the response Y is driven more by one of the components P, Q, or R than the others, save that if one of P, Q, or R tends to be numerically larger than the others, and their values are not anti-correlated in some way, that variable would explain more of the variation in the magnitude of $PQR$. In principle you could take any regression model $Y_i = \alpha + \beta X_i + \epsilon_i$, even one where $X_i$ is a useless predictor and $\hat \beta$ is arbitrarily close to 0, and define $X_i = P_i Q_i$ where $P_i \equiv Y_i$ and $Q_i \equiv X_i / Y_i$ when $Y_i \neq 0$ (and whose value is immaterial otherwise).  No information could be extracted from the regression that would tell you that $P$ was secretly a perfect predictor.
So, if you think one or more P, Q, and R could be correlated with the response Y, but not the others, only a different regression model could tell you.
