# Interpretation of Model with Continuous Interaction Term

Imagine a variable $$y$$ across $$i$$ sections, i.e. $$y_i$$. For every $$y_i$$ there exist a $$x_{1,i}$$, a $$x_{2,i}$$ and a $$x_{3,i}$$. However, the following relation exists:

$$x_{1,i} *x_{2,i} = x_{3,i}$$.

For example, $$x_1$$ is some number (e.g. number of accidents in the year under observation), $$x_2$$ is the average volume associated with the number (e.g. average costs resulting from the accidents in the year under observation) and $$x_3$$ is the resulting total volume (e.g. total cost volume resulting from accidents in the year under observation).

Now someone estimates the following:

$$y_i = \alpha + \beta_1 x_{1,i} + \beta_2x_{2,i} + \beta_3x_{3,i}$$.

I am aware of how to usually interpret interaction terms of two continuous variables. But in this case, the interaction term itself is a standalone variable. I am confused - can I just interpret $$\beta_3$$ as being a normal coefficient of some variable, i.e. $$y$$ changes by $$\beta_3$$ if $$x_3$$ changes by 1? Or is that not possible due to the relation between the variable and the two variables before? Since a change of $$x_3$$ by 1 would necessarily imply a change of the two variables before aswell.

When you write:

$$y = \alpha + \beta_1 x_{1} + \beta_2x_{2} + \beta_3x_{3}$$

with your definition of $$x_{3}$$, it's the same as

$$y = \alpha + \beta_1 x_{1} + \beta_2x_{2} + \beta_3 x_{1}x_{2}$$

which is the customary form for an interaction term between $$x_{1}$$ and $$x_{2}$$. In that sense $$x_3$$ isn't really "standalone"; you've just effectively used the symbol $$x_3$$ to represent the interaction.

So you think about the $$\beta_3$$ coefficient just as you always have thought about interaction coefficients involving continuous predictors. Yes, "a change of $$𝑥_3$$ by 1 would necessarily imply a change of the two variables before as well," but so would a change of the standard interaction term $$x_{1}x_{2}$$.

Exactly as mentioned above, technically, $$\beta_3$$ is a normal coefficient just as it would, if $$x_3$$ were a completely independent variable. However, you need to be careful in interpreting the significance statistics due to the potentially high levels of multi-colinearity in such a model; expect your standard errors to skyrocket vs a model without $$x_3$$ in. You are more likely to get meaningful coefficients and t-stats if you manage to combine most of the useful information in one or two variables.