I am comparing two nested fixed-effects models, which are shown as follows

y

and

y

where z

Question 1: my estimates of b

Question 2: I observed that the b

Thank you all in advance. Any comments and thoughts are appreciated.

y

_{it}= b_{0}+ b_{1}x_{it}+ b_{2}(x_{it})^{2}+..., (1)and

y

_{it}= b_{0}+ b_{1}x_{it}+ b_{2}(x_{it})^{2}+ b_{3}z_{it}x1_{it}+ b_{4}z_{it}(x1_{it})^{2 }+ b_{5}z_{it}+... (2)where z

_{it }is the moderation variable of interest and x has been mean-centered.Question 1: my estimates of b

_{3}and b_{4 }in (2) are significant. However, the within R-squared value only increased marginally in Model (2), and the increase is not statistically significant. How should I interpret the outcome, as some references seem to suggest that a significant increase in R-squared is important in a moderation analysis?Question 2: I observed that the b

_{1}and b_{2}coefficients of Model (1) are quite different from those in Model (2). I thought this is not indicative of a problem since these are two different models. Yet I got comments from a colleague, who said that, given that the R-squared of these two models are not significantly different , this result shows that the significance of b_{3}and b_{4}in Model (2) are likely to be falsely driven by multicollinearity. But later I computed vif for Model (2), and the scores didn't suggest multicollinearity problems with the augmented variables in model (2). Values of the standard errors also look quite reasonable. Should I worry about multicollinearity in this case?Thank you all in advance. Any comments and thoughts are appreciated.

## Comment