I have a linear regression model such as $Income_i = \alpha + \beta_1 Primary_i + \beta_2 Secondary_i + \beta_3 Tertiary_i + u_i$ where my predictors $Primary, Secondary, Tertiary$ are dummy variables to indicate the education level. Of course those who have completed tertiary education also completed secondary and primary.

If it's correct to build such a model, then how should I interpret the coefficients and the corresponding $p$-values? Or instead would it be more appropriate to recode the variable $Primary$ so to exclude those who have completed also higher levels of education?


Assuming everything is correctly coded (e.g. you don't have any one recorded as completing tertiary but not primary or some other weirdness like that), it doesn't matter which of the two codings are used.

The interpretation of the coefficients change with the two different codings, but the fit will be identical, and you can compute one fit from the other.

If the coding is such that a 1 in primary means "primary only" and a 1 in secondary means "up to secondary but not tertiary", then the coefficients on these variables represent a comparison with the baseline (being someone with not even primary education, I presume, though if everyone has at least primary you'll need to drop a dummy or constrain the coefficients).

On the other hand if a person with tertiary education has 1's in primary and secondary, the coefficients represent the additional effect over the next lower level (the coefficient of tertiary represents the difference between tertiary and secondary).

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