What is the correct way to test multiple dummies? I'm discussing a topic with classmates and the textbook does not address our concerns. In the following model:
$$wage = \beta_0 + \beta_1 female + \beta_2 educ + \beta_3 (female \times educ) + u$$
it seems a proper test should be $\beta_1=0$ AND $\beta_3 = 0$. Testing $\beta_1=0$ ALONE or $\beta_3=0$ ALONE would not be correct. But I don't see why. To me:

*

*$\beta_1=0$ AND $\beta_3=0$ is about whether intercept AND slope are different for gender (and cannot tell whether one only is different)


*$\beta_1=0$ is about intercept only being different


*$\beta_3=0$ is about slope only being different
Similarly, in the following model:
$$wage = \beta_0 + \beta_1 married + \beta_2 widow + \beta_3 educ + u$$
where the base group is being single, it is not correct to test $\beta_1=0$ or $\beta_2=0$ ALONE, and the test should be JOINT. I don't see why? Anyone have an idea?
 A: In general, if you want to see if a predictor contributes to a model you need to examine all of the ways that it is involved in the model. That's a fundamental general response to your question. How that plays out in practice might differ somewhat between your two examples, however.
If there's an interaction term as in your first example, the values of the individual coefficients for each of the interacting predictors will depend on how the other is coded. Those individual coefficient values are generally reported for a situation when all the other predictors in the model are at 0 or at their reference levels. In your first example, the reported value for coefficient b1 would thus differ depending on whether you coded education level (in years) starting at 0 or instead centered them in terms of differences from the mean education level. With an interaction, a test against a null hypothesis that b1 = 0 could  be either "significant" or "non-significant" just depending on how educ is coded. Similarly, the coefficient b2 would differ if you instead coded gender as male=1 versus your (implied) female=1. See this page for how that sort of thing happens. If you properly include all the corresponding main and interaction effects in your analysis, however, the overall result for each predictor won't depend on coding.*
The second example is a bit trickier. If your interest is whether marital status (levels: single as implied reference, plus married and widowed) mattered overall, then you would need to incorporate coefficients for all levels of marital status in your analysis. If, however, you were simply interested in the difference between married and single, ignoring those who were widowed, a test on b1 alone would be OK. It's important to specify clearly just what hypothesis you want to test, and include the coefficients that correspond with that hypothesis.

*With analysis of variance (ANOVA) and unbalanced designs, you might get different results depending on the flavor of ANOVA that you use. See this page and the multiple links from it. One way around this is to do a Wald test on whether any of the coefficients associated with a predictor differs from 0, taking into account the covariances among the coefficient estimates.
