Interaction wipes out my direct effects in regression (non zero variable)

Question

I have the following regression

$children = \beta_0 + \beta_1 \log(earnings) + \beta_2 grandparents + \epsilon$

and $\beta_1>0$ with $p$=0.01 and $\beta_2>0$ with $p$=0.01, and N is large (N>10.000) and grandparents takes values 0,1,2,3,4.

Then I add the interaction term ($\log(earnings)*grandparents$) to equation 1, such that:

$children = \beta_0 + \beta_1 \log( earnings) + \beta_2 grandparents+ \beta_3 \log( earnings)*grandparents + \epsilon$

and $\beta_1>0$ with $p$=0.01, $\beta_2$ is no longer statistically significant and also $\beta_3$ is not statistically significant.

I do not understand how to interpret the results and if the interaction term wipes out the direct effect of grandparents since $\log(earnings)$ is always different from 0.

There is a way to test the stat. sign. of the effect of Grandparents in the interacted model? (Thanks Maarten for your previous answer)

Answer 1

$\beta_2$ in equation 2 is the effect of $grandparents$ when $\log(earnings) = 0$, i.e. $earnings = 1$. This is apperently outside the range of your data, so it is an extrapolation. The easiest way around that is to center $earnings$ before taking the logarithm or creating the interaction term at some meaningfull value withing the range of the data, for example, the median. That way the main effect of $grandparents$ will be the effect of grandparents when one has a median income instead of a fictional income of 1.