# Panel data, fixed effects, and interaction terms

I've a question concerning the specification of a panel regression with:

• unit and time fixed effects ($\alpha_i$ and $\delta_t$);
• interaction term between continuous variables ($x_{it}$ and $z_{it}$).

Let's start with a model with no interaction effect. Model (1) looks like:

$y_{it} = \alpha_i^{y1} + \delta_t^{y1} + \beta^{y1} x_{it} + \gamma^{y1} z_{it} + u_{it}^{y1}$ (1)

If the focus of the analysis is on $x_{it}$ and $z_{it}$, one could as well estimate model (2):

$x_{it} = \alpha_i^x + \delta_t^x + u_{it}^x$

$z_{it} = \alpha_i^z + \delta_t^z + u_{it}^z$

$y_{it} = \alpha_i^{y2} + \delta_t^{y2} + \beta^{y2} u_{it}^x + \gamma^{y2} u_{it}^z + u_{it}^{y2}$ (2)

In model (1) and (2) the parameters of interest are in fact identical: $\beta^{y1}=\beta^{y2}$ and $\gamma^{y1}=\gamma^{y2}$. The inclusion of unit and time dummies in model (1) allows to control for "common characteristics" (such as common shocks at time t or structural features over time) and to focus on the "residual variation" in $x_{it}$ and $z_{it}$. In model (2), $u_{it}^x$ and $u_{it}^z$ directly measure such variation.

Let's now focus on a model with interaction effect. A possible model is the following:

$y_{it} = \alpha_i^{y3} + \delta_t^{y3} + \beta^{y3} x_{it} + \gamma^{y3} z_{it} + \theta^{y3} x_{it}z_{it} + u_{it}^{y3}$ (3)

In model (3), "common characteristics" can still affect the individual factors of the product $x_{it}z_{it}$ (i.e. the interaction term).

Thus my question: assuming that the goal of the analysis requires to exclude these "common characteristics", would it not be more natural to demean all continuous variables with respect to the fixed effects groups (unit and time) before estimation?

$x_{it} = \alpha_i^x + \delta_t^x + u_{it}^x$

$z_{it} = \alpha_i^z + \delta_t^z + u_{it}^z$

$y_{it} = \alpha_i^{y4} + \delta_t^{y4} + \beta^{y4} u_{it}^x + \gamma^{y4} u_{it}^z + \theta^{y4} u_{it}^x u_{it}^z + u_{it}^{y4}$ (4)

• What do you model 4 does include "such effects"? You put them right there, at the beginning of the model. – Repmat Nov 26 '17 at 22:43