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I have a question about econometrics in general. If I have a regression with an interaction between two dummies, do I need to have those dummies separated in the regression too.

I have this regression: is this right?

$$ y_i=\beta_0 + \beta_1d\_sex_i*d\_country_i +\beta_2d\_country_i + \varepsilon_i $$

or do I need to do this:

$$ y_i=\beta_0 + \beta_1d\_sex_i*d\_country_i +\beta_2d\_country_i + \beta_3d\_sex + \varepsilon_i $$

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    $\begingroup$ This has been asked and answered many time. You should include the main effects along with the interaction. See here $\endgroup$
    – Robert Long
    Oct 22, 2020 at 18:34

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Generally, second version is the default version. If you consider the effect of two dummy variables, you should include them in the model. Then you think of the interaction.

However, if you may have additional information (from the theory), that the interaction could be important in the model, but the variable itself is not. In this case you can think of removing it in the next step, if it is in fact not significant. This could improve properties of the estimator (efficiency), however it is generally not a common approach (personally I have never seen such approach in any acknowledged research).

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  • $\begingroup$ define "not significant" please. $\endgroup$
    – Robert Long
    Oct 22, 2020 at 18:32
  • $\begingroup$ Thanks for help with improving the answer. I added a link to the concept of statistical significance. $\endgroup$
    – cure
    Oct 22, 2020 at 18:47
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    $\begingroup$ OK. So assuming we use the arbitrary value of 0.05 you would think of removing the main effect for a variable that had a p value of 0.050001, but not it it was 0.049999 ? $\endgroup$
    – Robert Long
    Oct 22, 2020 at 18:49
  • $\begingroup$ This is other issue, and a topic for other question. Please have in mind, that as a first reason to do such uncommon action, I pointed out theory (also unequivocal concept). Statistical significance as a confirmation in this case is not that bad if there would be strong theoretical justification. $\endgroup$
    – cure
    Oct 22, 2020 at 18:56
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    $\begingroup$ That's a good point. In that case I would suspect there to be an artifact present in the sample which isn't present in the population. I don't think I have ever encountered a situation where there was strong theoretical justification for removing a main effect, yet the main effect was "significant". I mean, it would be easy to construct a pathelogical example through simulation, but I would really like to see a real-world example ! (+1) to your answer btw :) $\endgroup$
    – Robert Long
    Oct 22, 2020 at 19:15

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