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When modelling a linear regression model for a data set there are three choices for assumptions about the influence of categorical data:

1: Assume there is no interaction effect. Regression slope results in $y = β_0 + ... + β_i*d $. Parallel slopes, varying y-intersections.

2: Assume there is an interaction effect. Regression slope results in $y = β_0...+ β_i(d*x)$. Same y-intercept. Different slopes depending on presence of dummy variable.

3: $y = β_0 + ... + β_i(d+d*x)$. Varying y-intercepts, and varying slopes.

Interaction effects could be discovered with an ANOVA analysis, but when should the third model be used?

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You should use the third option, whenever you think it is possible that a certain group will start at a different level than the other group(s), and the it will develop/grow with different slope(s).

For instance, being black (skincolor) might start you at a lower wage than being white in, say, 1970. However, over time the wage of both groups will grow, and a usually encountered hypohthesis is that wage development of white and blacks from 1970, til today, is not the same. In this case there is a need for different intercepts AND different slopes

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