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I am getting confused in defining if models are nested. I know the basics that the reduced model can be derived from the full model by removing terms or setting terms to 0.

But what about a case like this:

Model_1<- lm(formula = expenditure ~ income * gender)

i.e. $$y = β_0 + β_1x_1 + β_2x_2 + β_3x_1x_2 + ε$$

Model <- lm(formula = expenditure ~ income + gender)

i.e. $$y = β_0 + β_1x_1 + β_2x_2+ ε.$$

My guess is that these are not nested. But i would like to understand better how to explain that.

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Those models are nested. The second model is the first model with $\beta_3=0$.

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