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.

What you are saying is that your model assumes interaction between three components. Using R formula notation, the model you are describing is
y ~ P:Q:R

On another hand, you are interested in how "individual components drive the relationship" between the dependent and independent variables. If that is the question, why not include the individual effects? What I mean is the model
y ~ P + Q + R + P:Q + P:R + Q:R + P:Q:R

If indeed the only relationship there is $Y = \alpha + \beta PQR + \varepsilon$, the remaining parameters would all be zeros, but in such a case the question about individual components doesn't make much sense since the only thing that matters is their interaction. On another hand, if the components have distinct individual impact, it would be reflected in the nonzero parameters.
