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I am modeling a response Y against a predictor X, which is a product of two variables X1 and X2. I am interested in the coefficient of X.

Mathematically which way of modeling is correct?:

  1. Y ~ X
  2. Y ~ X1 + X2
  3. Y ~ X + X1 + X2

Thanks, maverik

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    $\begingroup$ Would you simultaneously have values of $X_1$, $X_2$, and $X$, or will you only have $X$ for predictions? $\endgroup$ – whuber Jul 1 '15 at 20:34
  • $\begingroup$ Is there some theoretical reason to consider $X$ in isolation as the only important quantity? Where did the idea to compute the product of $X_1\times X_2$ & consider that as a predictor come from? $\endgroup$ – gung - Reinstate Monica Jul 1 '15 at 20:41
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    $\begingroup$ @gung Purpose of this regression model is to assess the impact of marketing on test cities. There are 4 test cities, 6 control cities, and I know on a daily basis, Y = sales from each of 10 cities. I have defined a variable X1 = exposure = 1 if test city, 0 if control city and a variable X2 = period = 1 when campaign was going on and 0 when it is over. X = X1*X2 is the variable I am interested in since it takes into account the time period when there was marketing, and where it was carried out. There are actually more variables like account type Z which I didn't mention for simplicity. $\endgroup$ – maverik Jul 1 '15 at 20:49
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    $\begingroup$ I see. This is a FAQ. You are asking if the 'main effects' need to be included in a model w/ an interaction term. Here are a couple of threads that address the issue: Including the interaction but not the main effects in a model, & Do all interactions terms need their individual terms in regression model?. $\endgroup$ – gung - Reinstate Monica Jul 1 '15 at 20:58
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    $\begingroup$ Please read these threads. If you still have a question afterwards, come back here & edit your Q to state what you have learned & what you still need to know. Then we can provide the information you need w/o simply duplicating material elsewhere that already didn't help you. $\endgroup$ – gung - Reinstate Monica Jul 1 '15 at 20:59