Can a variable be both a moderator and predictor? I have dependent variable Y and independent variables X and Z. I am not sure at all, if it is allowed or if it makes sense to state the following hypotheses:
H1: Variables X and Z correlate positively with each other.
H2: X predicts variable Y positively.
H3: Z predicts variable Y negatively.
H4: The association between X and Y is moderated by Z.
Is it possible to hypothesize both a moderation and a predictor role for variable Z? 
I would plan to test H1 with a correlation analysis. For H2 and H3, I would perform a multiple regression (with predictors X and Z). Could I state that I would only test for H4, if there H2 and H3 turn out to be significant, and then perform another multiple regression with including the moderation (i.e. predictors, X, Z and interaction X*Z)?
Sorry for all these questions, I hope somebody can help me to gain some clarity.  
 A: I don't really understand the question. In some regression model $Y= f(x_1, \dotsc,x_p)+\epsilon$, all the variables on the RHS are needed to make predictions for $Y$, all are predictors. A moderator is a predictor that plays a specific role, that of modifying (interacting with) the effect of some other predictor. That does not make it any less a predictor itself. 
But maybe I have misunderstood something? 
A: Regarding the OPs comment: I know that explanation might be too simple. But, the mediator has an arrow, as you can see in the picture (appendix). Indicating a direction (positive or negative) of a direct partial effect of Z (if we leave the indirect effect of X over Z, and the direct effect of X out of the equation)! The question is how strong this effect may be:

*

*Full mediation: Without the mediator there would have been no real connection between two variables, eliminating the mediator would result in no relationships between X an Y as it is 100 % dependent of Z


*partial mediation means there is some effect of Z but also some of X, taking z away not eliminates the relationship.
So in other words, A mediator is also a predictor just like the moderator, here in this case the only question is, what sort of mediator you have. Does it explain the full effect or is 100 % responsible for a prediction, or only partial.
So the answer is yes in terms of mediator and moderator (as was already answered)

https://www.statisticshowto.com/mediator-variable/
