# Interpretation of interaction term coefficient in fixed effects estimation

I have balanced panel data for 14 industries and 21 years. The data consists of several variables with annual observations. Based on theoretical reasoning and visual data inspection I assume that there are unobserved time-invariant individual (industry) effects that are likely to be correlated with the explanatory variables. Therefore, I conduct one-way fixed effects (within) estimation. Specifically, I used the plm package in R (model = within, effect = individual). Just a side note: I have to use FE, so please do not discuss whether or not this is justified and just focus on my question below. Thanks :)

When inspecting simple scatter plots by industry (main outcome variable against main explanatory variable), I noticed that the relationship of interest is negative for three industries and positive for all the others. So, I included an interaction term between the main explanatory variable and a dummy variable (1 if one of the three industries, 0 if else) in order to allow for differences in slopes. However, I am not sure how to interpret the coefficients. As far as I understand (from Introductory Econometrics by Wooldridge, 2018), the coefficient of the main explanatory variable (without interaction) refers to all industries except the three mentioned above, while the coefficient of the interaction term gives the difference in slope for these three industries in reference to the former. Is this correct?

For clarification: Given the model below, how can beta2 be interpreted?

$$y = \beta_1 v_1 + b_2v_1\cdot D + b_3X$$

where $$y$$ = dependent variable, $$v_1$$ = main explanatory variable, $$D$$ = dummy variable, $$X$$ = placeholder for controls.

(Note, that I omitted subscripts, intercepts/fixed effects and the error term for simplicity)

First, I'm a little leery of your interaction - making such a specific interaction after looking at the results isn't a great idea.

Second, you should include D. It's rarely a good idea to omit the main effects that are part of an interaction. One reason is that $$\beta_2$$ in the equation you have is hard to interpret.

If you change it to:

$$Y = \beta_1 v_1 + \beta_2 D + \beta_3 v_1D$$

then $$\beta_3$$ is the interaction between $$D$$ and $$v_1$$ --- that is, the change in the relationship between $$x_1$$ and $$Y when$$D$$is 1 vs. 0. (And also the change in the relationship between$$D$$and$$Y$$when$$x_1$changes). I find that graphs of the predicted values help a lot in interpreting these things. • Thanks for the quick reply @Peter Flom and thanks for editing my question @Alexis! – MaMö Commented Mar 4 at 18:53 • For clarification, I did include D in the model, but didn't include it in my oversimplified specification above, because I don't get a coefficient for D in my results. As far as I understand within estimation, the coefficient ends up in the time-constant fixed effects anyways. But, I see why I should have included it in my question. Also, I did not decide to include the interaction because of regression results, but because of scatter plots of the unfitted data. Sorry for the confusion! As you might have guessed, I am quite new to econometrics. – MaMö Commented Mar 4 at 19:03 • So, if I understand your comment on the interpretation correctly, this means that, for those three selected industries, the effect of one unit change in$v_1$on$y$equals$\beta_1 + \beta_3$, whereas, in the case of all the other industries, the effect of one unit change in$v_1$on$y$is just$\beta_1\$, right?
– MaMö
Commented Mar 4 at 19:16
• Yes, that is right. Commented Mar 4 at 19:20
• One last question, if I may. Why do you think it is not a good idea to include an interaction term after looking at the data? In my case, my tutor already approved the decision as it is plausible with respect to my theoretical framework and my hypotheses. Is there another way to allow for heterogeneous slopes in fixed effects estimation? Sorry, if this is a silly question!
– MaMö
Commented Mar 4 at 19:31