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Dummy (or binary) variables ($X_2$) can be used in linear regression models to help explaining a possible group effect that a continuous predictor variable ($X_1$) might present in explaining the response variable ($Y$).

Now, I am wondering if it makes sense to have only dummy variables as predictors in a linear regression model with a continuous response variable. Does it?

Example:

% of population with instruction = influenced by politics A + influenced by politics B.

Both politics A and B can assume values 1 or 0.

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    $\begingroup$ A regression with one response and a one binary predictor is in essence equivalent to Student's t test comparing two means. This is often regarded as too puzzling to be mentioned in introductory texts and too obvious to be underlined in more advanced texts. (In practice, not everything in your favourite software to do either procedure may match exactly.) $\endgroup$
    – Nick Cox
    Commented Oct 25, 2013 at 13:54

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Because linear regression does not assume any distribution of predictors, as long as

  1. They are not perfectly collinear, and
  2. None of them is a constant, it should be fine.

Your example is just like using regression as an ANOVA sans interaction (aka, not a full-factorial design.) If additional effect due to co-influence by A & B is of your interest, compute an interaction term (by multiplying your two dummy variables) and include it as a predictor as well.

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