Let's start with the given that, in addition to presenting unstandardized coefficients, I am also presenting "standardized" coefficients calculated by beta*SD(x) (my Y variable is log transformed, and I wish to discuss "a one-standard-deviation change in Xs results in a ___% change in Y")

Now, one of my Xs of interest is an interaction term that consists of a continuous variable*dummy variable. Specifically, I am measuring the changes in a home price index (continuous variable) specifically on homeowners (dummy variable) in my sample (which includes both renters and owners, hence the interaction term). My Y variable is a measure of household debt. The study in part examines if homeowners accrue more debt as the home prices rise (since home equity is an asset).

For my analysis of this interaction term, I only care about how rising home prices affect the debt of homeowners.

My question is, which SD(x) do I multiply with my beta for the interaction term? Of course, the interaction term has its own SD. However, if I want to interpret the effect of a one-standard-deviation change in home prices on the debt of homeowners, shouldn't I multiply the beta for the interaction by the SD of the home price variable, that is beta(interaction)*SD(home prices)?