@Holly Since the Moderation is also based in regression hence the assumptions for Moderation would be same as of regression analysis. Thus, to ensure that the data is not skewed beyond the prescribed range, it's would be appropriate to check for Skewness for Moderator as well.

better use it. coefficients based on the skewness and kurtosis to check the normality of all variables used

The normality assumption in OLS linear regression applies to the residuals, not the predictors. You want to check normality of the residuals as usual.

There is no assumptions in OLS regressions which state that the dependent or the independent variables have to be normally distributed, so no, no need to check this. And a slight correction from Christian Geiser comment, it is about the errors that need to be normally distributed, not the residuals, but since we only know the residuals, we check them.

Skewness and kurtosis of a specific type is never a requirement. Think of the interaction of a 0-1 variable with a numeric one.

Also technically any assumption on the errors is useful if we want to use plain vanilla standard errors, which in 2021, sounds very retro is you ask me @Rainer

If we were to compute a regression by hand i would understand but again it's not 1806 anymore 🤷🏼‍♂️

Stefano Nembrini yes, but isn't that a completely different topic? I mean, it does not make sense to introduce whole new concepts (and there are plenty, since there is not only one solution and one philisophy how to handle data), if appatently basic concepts and assumptions are not understood?

Thank you everyone for your comments

Some use the terms error and residual, but I'm with the residual, estimated residual set, as "error" isn't quite right.

Anyway, the estimated residuals are distributed nearly as the residuals (see hat-value info.). You only hope for approximately normal anyway, as this is not a strong requirement.

I suggest a "graphical residual analysis" for checking fit to your sample, including heteroscedasticity, and a "cross-validation" to study whether you have fit your model too closely to a particular sample to generalized well to the population or subpopulation to which you want to apply your model. This works well, and if you want to see the impact of one variable in a set, you can compare results with and without it.