How to conclude significance of moderation effects?

[Li recordyao]

I want to explore moderating effects of m1. I also have another moderator m2, and here are the two cases:

(1) reg y c.x##c.m1
(2) reg y c.x##(c.m1 c.m2)

Let's say, the coefficient of the interaction term c.x#c.m1 in Case (1) is significant, but it's insignificant in Case (2).
My question is --- does m1 moderates the effect of x on y?


I kinda guess the answer is No --- because we should follow the regression result in Case (2).
But this get me think, how far should we go? Should we control all the control variables in the interaction before we conclude moderating effect? Should we use the following:

(3) reg y c.x##(c.m1 c.m2 c.control1 c.control2 ...)

But we rarely do (3) in the published paper, the convention is do (1) and (2) in the same regression table. So my question is that why don't we do (3) ?

[Li recordyao]

I know it may sound like a silly question. But if (2) is preferred over (1), then why not (3) ?

[Carlo Lazzaro]

Li:
a temptative answer rests on the need to give a fair and true view of the data generating process you're interesting in.
I would not take it for granted that increasing the number controls and related interactions (which are controls and not independent variables) can help you out in this respect.

[Li recordyao]

Dear Carlo Lazzaro

Thanks for your reply. So we should follow the theory and how the data generating process is.
I guess this is different from control variable, in which case the more the better (as long as it doesn't cause multi-collinearity problem with the independent variable).


(1) reg y c.x##c.m1
(2) reg y c.x##(c.m1 c.m2)


In the case, (1) is significant and (2) is not significant.
We could still conclude that m1 moderates the effect of x on y right? Maybe the moderating effect is somehow disrupted while including c.x##c.m2.

[Carlo Lazzaro]

Li:
being two different models, I think your take makes sense.