Interpretation of Model with Continuous Interaction Term

Question

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.

Answer 1

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 $x_{1}x_{2}$.

Answer 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.