# Panel regressions with an interaction term between a time dummy variable and a time invariant variable

I have estimated the coefficients of the following equation, using the fixed-effect model:

$Y_{it}=\alpha _i+ \rho _t + \beta _1 X_{it}+\beta _2 C_i*D_t+\epsilon_{it}$

I have observations from 1980 to 2010 for $Y_{it}$ and $X_{it}$.I am interested in the interaction term. The dummy variable $D_t$ is equal to $1$ for years 2000 to 2010 (and $0$ otherwise) and is interacted with the continuous variable $C_i$ which is time-invariant (but varies over the units $i$).

My question is: can $\beta _2$ be interpreted as the differentiated effect of $C$ on $Y$ during the period 2000-2010 compared to 1980-2000?

Thanks!

• Yes, this is possible and your interpretation is correct as well. If you are interested, there are other ways of doing this: How to keep time invariant variables in a fixed effects model – Andy May 5 '14 at 16:08
• What specification should I use to capture the same effect (i.e. the differentiated effect of C on Y during the period 2000-2010 compared to 1980-2000), but using the first-difference model? – Economist1111 Jun 6 '14 at 12:26
• That's more tricky and requires more data manipulation. If you average your data into two periods, i.e. the time-average of your data for the period 1980-2000 and the time-average for the period 2000-2010, you can then interact $C$ with a time dummy that is 1 for period 2 and zero otherwise, and use it in a first differences regression. – Andy Jun 7 '14 at 12:27