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Say we have a dependent variable $Y$ and two independent variables $X_1$ and $X_2$.

If we are doing the linear regression with interacted variables, do we need to include both $X_1$ and $X_2$ as independent variables? so that. $$Y = \alpha + \beta_1X_1 + \beta_2X_2 + \beta_3X_1X_2$$

Is it wrong to drop one variable like the following? $$Y = \alpha + \beta_1X_1 + \beta_2X_1X_2$$

What's the difference? How is interpretation different?

user3346
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  • Similar questions have been asked before, see https://stats.stackexchange.com/questions/199113/model-construction-how-to-build-a-meaningful-gam-model-generalized-additive-m, https://stats.stackexchange.com/questions/11009/including-the-interaction-but-not-the-main-effects-in-a-model, https://stats.stackexchange.com/questions/432864/interaction-term-is-significant-without-main-affects-and-main-effects-are-sig, https://stats.stackexchange.com/questions/5450/what-if-interaction-wipes-out-my-direct-effects-in-regression – kjetil b halvorsen Sep 21 '22 at 13:46

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