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.
 A: 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.
A: 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.
