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If you fail to include for interaction terms in your model if interaction exists. Would your $R^{2}$ be less then what it would be if you included the interaction terms $R^{2}$ ?

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$R^2$ will never decrease when you add more variables, and will in practice always increase, regardless of whether the added variable belongs in the model or not. So yes, adding an interaction will increase $R^2$.

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    $\begingroup$ As standard_error is implying, R2 is not the best measure of model fit. You should be using other statistics to check the value of adding an interaction term - things like information criteria (AIC, BIC, etc), likelihood tests (Wald tests, LR tests, LM tests), F tests, etc. $\endgroup$ Dec 3, 2014 at 19:49
  • $\begingroup$ Or the adjusted $R^2$, which penalizes the addition of new variables. $\endgroup$ Dec 4, 2014 at 7:17

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