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?


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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