# How to decide which main variable is modified by the interaction term?

Given the following linear model $Y=int+aX1+bX2+c(X1*X2)+e$, where $X1$ and $X2$ are the main variables, ($X1*X2$) is the interaction term, and $a$, $b$, and $c$ are the corresponding coefficients.

Question 1: Is there is a way to determine whether the interaction effect $c$ reflects whether the effect of $X1$ on $Y$ varies by $X2$, or if instead the effect of $X2$ on $Y$ varies by $X1$?

Question 2: If this were a multi-level model (with observations across time nested within country) where $X1$ is a country-level, time-invariant variable, and $X2$ is a country-level variable that varies with time (country-year companion), then the interaction will then be country-year too. In this case, does that mean $c$ can only be added to $b$ as $X2$ and $(X1*X2)$ are both time-variant while $X1$ is not? Is it possible that significant $c$ actually means that the effect of $X1$ on $Y$ varies by $X2$?

Regarding to question 2, sorry I might be a bit confusing here. Let me put it in this way. As an example, there are two countries and we look at the effect of the legal origin (X1) on its stock market performance (Y). We also try to test the moderator effect from the country's GDP (X2) to see whether GDP would alter the effect of X1 on Y. Now, the legal origin is time-invariant (i.e. it will always be common law system), while GDP is different every year. When we interact legal origin with GDP, the interaction term will become time-variant, just like GDP. My question is, it seems like that we cannot add the interaction term to X1 due to different data structure. How could we interpret the GDP as a moderator effect of X1 and Y then?

• On your first question, it's like asking "when I have 2 things and 3 things, and I put them all into a bucket, is 2 being added to 3 or is 3 being added to 2?" Mar 2, 2015 at 11:59

See Glen_B comment. To elaborate only a little, there is no way to tell whether x1 effect on y varies by x2 or the other way around, because IF x1 effect on y varies by x2, THEN x2 effect on y varies by x1, AND the strength of the effect is identical (c in your model). You cannot separate these hypotheses, because they are the same hypothesis. This is reflected by the fact that you have one parameter estimate for the interaction (c).

Consider a scenario with 2 countries (A & B), and people in these countries are either poor (P) or rich (R). Let's measure the incidence of diabetes in each of the four categories:

AP = 5 per thousand
AR = 3
BP = 15
BR = 3


Organized like this, it's easiest to say that the difference between rich and poor depends on which country you're in. But organized like this:

PA = 5
PB = 15
RA = 3
RB = 3


it's easier to say that the difference between countries depends on wealth.

I can't say I completely understand your second question, but it looks like you're again using fallacious logic about what an interaction term means. Reconsider and recast.

• Thanks a lot. I elaborated question 2 with an example and hope it would clarify the problem. Sorry that I might be a bit confusing here.
– Chay
Mar 4, 2015 at 10:05
• IF x1 effect on y varies by x2, THEN x2 effect on y varies by x1. Why is this the case? Couldn't x2 affect x1, but x1 doesn't affect x2? Jul 5, 2021 at 23:27
• good question @NoName! yes, of course, you're correct! however, you're asking about two causal structures that this linear model cannot distinguish. when you know a lot about the things being modeled, interaction effects may resolve a causal structure for you, but that level of inference is beyond the scope of the model itself. check out eg path analysis if you're interested in methods that explicitly model causal structure in more complex ways. Jul 7, 2021 at 0:29