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I'm fitting a linear model with outcome $Y$. I have measurements for variables $X_1$ and $X_2$.

  1. I hypothesize that $X_1$ and $Y$ are linearly related. I want to know the slope and significance of this relationship.
  2. I know from world knowledge that $X_1$ and $X_2$ are related.

Specifically, regardless of whether the hypothesis in (1) is true or not, I expect that there is a relationship in the underlying generative model such that $X_1$ is a multiple of $X_2$. There's also noise during this process, but it's not clear whether (a) the multiplier itself is noisy, or (b) $X_1$ is a fixed multiple of $X_2$, and then some noise is added.

  1. From world knowledge, I don't expect $X_2$ and $Y$ to be related except via $X_1$. It's theoretically possible for $X_2$ and $Y$ to have an independent relationship, but it doesn't matter to me either way.

How should I fit this model so that I get the best estimate for $Y \sim X_1$? Should I divide $X_1$ by $X_2$ before entering it into the model? Should I include Z as a covariate ($Y \sim X_1 + X_2$ possibly with an interaction)?

I'm willing to do something more complicated to infer the estimate if necessary, but I'd like to keep it simple if possible since the actual model predicting $Y$ is mixed-effects and includes a number of other predictors that I have independent hypotheses about.

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You can look at a few things.

  • Instrumental variable (IV) regression. This is used when you suspect that there may be a feedback from Y to X1. In this case you look for X2 which is correlated with X1 but not Y.
  • Cointegration. If X1 and X2 are in some sort of linear relationship, this may cause a problem in the model when you simply throw them into the regression. You can apply cointegration techniques to deal with this issue.
  • You can use the transformation such as X1/X2. This is used a lot in econometrics, e.g. GDP deflator or inflation are often applied to GDP and real rates
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