Interpretation of continuous interaction term

[Rene Jansen]

Hi all,

I have a question regarding the interpretation of a squared interaction term in the following model:

  Y=β0+β1X+ β1X^2 + β2Z+ β3 X^2Z

X and Z are continuous and X is negatively related to Y whereas Z is positively related to Y. Is it true that a negative interaction coefficient in front of the interaction term suggest that the relationship between X and Y becomes less negative as the value of Z increases? And a positive interaction term suggests that the relationship between X and Y becomes more negative as the value of Z increases?

And do I need to include the interaction effect without the squared term as well?

  Y=β0+β1X+ β1X^2 + β2Z+ β3 X^2Z + β4 XZ

Thank you very much.