# 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!