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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.

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  • $\begingroup$ Do you observe the individual components, or are they latent variables? Can you make any additional assumptions? $\endgroup$
    – Tim
    Feb 16, 2022 at 22:19
  • $\begingroup$ @Tim Thanks for the questions. I do observe the three components; they are not latent. But I struggle to come up with further assumptions. $\endgroup$
    – Alex
    Feb 16, 2022 at 22:23

2 Answers 2

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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.

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  • $\begingroup$ Thanks very much for the answer!:) Sorry, I am unfamiliar with R and its notation. If I understand it correctly, you suggest to estimate the following regression? $$Y_i=\alpha+\beta_1 P_i+\beta_2 Q_i+\beta_3 R_i+\beta_4 P_iQ_i+\beta_5 P_iR_i+\beta_6Q_iR_i+\beta_7P_iQ_iR_i+\epsilon_i$$ $\endgroup$
    – Alex
    Feb 16, 2022 at 22:48
  • $\begingroup$ @Alex yes, exactly. $\endgroup$
    – Tim
    Feb 16, 2022 at 22:51
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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.

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  • $\begingroup$ What different regression model do you suggest? $\endgroup$
    – Alex
    Feb 16, 2022 at 23:05
  • $\begingroup$ There's no way I could say, as that depends entirely on your hypotheses and data. And for all I know your current model is the "right" one (that just happens not to allow something you may "incorrectly" want out of it). There must be some reason you picked PQR as the regressor originally, so you should revisit those reasons. You also could set aside a small sample of the data for exploratory analysis and try a whole bunch of things, like ones in the other answers, or log-scale regression log Y ~ log P, log Q, log R if all those quantities are positive. $\endgroup$
    – jwimberley
    Feb 17, 2022 at 14:28

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