interaction fixed and time effects

I've seen in panel studies, regressions with fixed effects (FE) of the unit (say, country or firm) and of time fixed effects (T-FE). But sometimes I see in papers the interaction of the FEs and the TFEs:

$y_{it} = \alpha_{i} + \lambda_t + X_{it} ... + \alpha_{i}*\lambda_t$

I understand the rationale behind FEs and T-FEs but I've never quite understood the rationale behind their interaction, and wondered whether anyone could help me out.

• Is that really the model? The coefficient of $\lambda_t$ for unit $i$ is exactly related to the fixed effect $\alpha_i$, which seems awfully restrictive. And there's no coefficient of $X_{it}$. – The Laconic Oct 22 '17 at 21:29

In the regression structure, interactions are used to show how the effect of predictor variable $X_i$ on outcome $y$, can vary according to some other (predictor) variable $X_j$. This other variable can be another characteristic, setting, time or anything for that matter.

So when looking at interactions with time specifically, basically what you are saying is that some variables' effect on the outcome is allowed to change over time. This is different from added a time effect to your model, as such a 'time' variable is averaged over the entire population, instead of the value of another variable.

An example with some graphics will help: imagine you would predict annual income ($y$) over time ($T$, (age) years) with gender ($X_1$; 0=male, 1=female) as predictor variable. The regression formula would be:

• formula 1: without interaction: $y=α + β_1*X_1 + β_2*T$
• formula 2: with interaction: $y=α + β_1*X_1 + β_2*T + β_3*X_1*T$

Now imagine the 'truth' is there is an effect of gender (sensitive subject I know, sorry about that, but I couldn't come up with another quick example to work it out), and that it varies over time, so that the true values of $α$, $β_1$, $β_2$ and $β_3$ are respectively 15, -15, 1 and 0.375.

Graphically the 2 formulas have obvious different results:

Formula 1: Formula 2:

As you can see in the first graph, the effect (difference between the two lines) of gender is constant, while in the second, the effect changes over time. That is the effect of such an interaction term and the reason some add it.