Good day,
For simplicity, lets say I have a model that looks like this:
Y= X1+ X1*X1+X2 in which, clearly, X1 is a quadratic term. Now, I want to estimate the interaction: X1*X2.
I have a heated discussion with my coauthor. I believe that the full model should then be:
1) Y= X1+ X1*X1+X2 + X1*X2 + (X2)*(X1*X1) i.e. adding not only the interaction with he low level X1 but also the interaction with the quadratic term X1*X1. Whereas, my coauthor is happy with:
2) Y= X1+ X1*X1+X2 + X1*X2 i.e. just adding the interaction between X2 and the lower level of X1.
I believe that this is a purely econometric question rather than a methodological one. Just like when you add higher order variables, you should include ALL lower order ones. Or if you introduce a triple interaction you should always include the single variables + all of the two interaction combinations.
Can someone provide any clarity in this and help settle this nasty argument? is it 1) or 2) ?
Thanks a lot.
If someone is still interested, I have SOME support for my claim (answer number 1) in the attached paper.
Article Moderation in Management Research: What, Why, When, and How
Adrian,
a good textbook about this topics is:
Hayes, A. F. (2013). Introduction to mediation, moderation, and conditional process analysis. New York: The Guilford Press.
ana
Thank you everyone for your answers, they have all been really helpful. I thought this was a purely econometric question, but some of you seem to think it is more methodological, i.e.: are there good reasons to hypothesise the quadratic relation in the variable interaction the squared variable?
Thank you very much and anything that can be included, I am sure, will be helpful for all.
Extra information: my squared variable is Wealth and I am interacting Wealth with Gender to predict depressive symptoms. In my mind, there is NO way why I could justify wealth being quadratic, but gender*wealth just linear.
The equation Y = a + bX1 + cX1^2 makes sense when there are clear nonlinearities in the relationship between Y and X1. You should be able to see this in a scatter plot between Y and X1, where X1 is a continuous variable (income or wealth). The coefficient of the square of X1 captures the effect of higher wealth. The inclusion of an interaction term is a different but complementary issue: Y = a + bX1 + cX1^2 + dX2 + eX2X1 + fX2X1^2 + u, where X2 is a binary variable (sex). When X2=0 then Y = a + bX1 + cX1^2. When X2=1 then Y = (a+d) + (b+e)X1 + (c+f)X1^2 + u. This setup allows both sexes to be influenced by the higher level of wealth, not just one of them.
Adrián Villaseñor hello, can you guide me now which is the right one
This question is really realistic when we touch on data analysis. From your answers, it opened up my way to treat my current data. Thanks a lot.
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