Regression: IV - DV non.sig. But moderator highly sig I am researching the effect of an innovation type on firm performance. In my quantile regression model, the relationship between my IV "innovation" and and DV "firm performance" is non-significant (p=,7). However, the moderator "firm size" is highly significant (p<,001). 
Is this possible? And how should this be interpreted? I don't get how firm size should influence the relationship of innovation and firm performance, if there initially is no relationship... Thanks!
 A: When you consider firms of different sizes together and investigate the effect of "innovation" on "firm performance" among these firms, one of the reasons you might fail to see an effect is the heterogeneity of firms with respect to firm sizes. This heterogeneity may introduce "noise" which obscures the effect of interest because you are not controlling for it. Of course, it is also possible that there is no such effect for these firms when considered together, which is why you are not able to detect it. 
However, when you look only at firms of the same size, you might see that, for those firms, there is an effect of "innovation" on "firm performance". This effect may be independent of firm size or may depend on firm size. 
The model where you ignore "firm size" (model 1)  is therefore different from the model where you account for it (model 2). 
In model 1, you are concerned with the effect of "innovation" on "firm performance" among all firms, irrespective of their size.
In model 2, you are concerned with the effect of "innovation" on "firm performance" among all firms of the same size. 
Model 2 can allow for the effect of "innovation" on "firm performance" to be the same among small firms as it is among moderate firms or among moderate firms (in which case, the model would include only "innovation" and "firm size", but not their interaction). Alternatively, it can allow for this effect to depend on firm size (in which case, the model would include "innovation", "firm size" and their interaction as predictor variables). 
