Interaction between time-variant and time-invariant variable in FE model

Question

I want to estimate the effect of several variables $x_{1,it}$, $x_{2,it}$, $\dots$ on $y_{it}$. All of these variables vary across countries $i$ and time $t$. I use OLS to estimate a model with country and year dummies $D_i$ and $D_t$:

$y_{it} = \beta_1 x_{1,it} + \beta_2 x_{2,it} + \gamma_i D_i + \delta_t D_t + \epsilon_{it}$

Additionally, I am interested in the moderating effect of a time-invariant variable $z_i$ on the relationship between $x_{1,it}$ and $y_{it}$.

My intuition is to include $\eta x_{1,it} z_{i}$ in the above estimation. While $z_i$ does not vary across time, $x_{1,it}$ does and $\eta$ should pick up the effect of interest.

Is this intuition correct? If so, are there any caveats? If not, what am I overlooking?

Answer 1

Your intuition is fine. When you take the partial derivative with respect to $x_{1,it}$, then you get exactly what you were looking for. $$\frac{\partial y_{it}}{\partial x_{1,it}} = \beta_1 + \eta z_i $$

This is particularly convenient if $z_i$ is a dummy variable. Wooldridge (2010) "Econometric Analysis of Cross Section and Panel Data" has a similar example where he interacts a time-invariant female dummy with time dummies. So even though one cannot estimate the female coefficient directly, its interaction with the time dummies still has a meaning as it shows the increase in the gender wage gap over time. So what you propose is perfectly valid under the usual assumptions, e.g. $z_i$ and $x_{1,it}$ are uncorrelated with the error $\epsilon_{it}$.