# compare coefficients from different regression

I estimate the following models using the Hausman-Taylor estimator:

$$y_{i,t} = a_{0} + B_1 controls_{i,t} + \beta_1x_{i,t=2000} + B_2 Year_t + B_3 x_{i,t=2000}*Year_t + e_{i,t}, (1)$$

$$y_{i,t} = a_{0} + B_1 controls_{i,t} + \beta_1x_{i,t=2002} + B_2 Year_t + B_3 x_{i,t=2002}*Year_t + e_{i,t}, (2)$$

This is a panel data set with $$t$$ representing the $$2000-2010$$ period, $$controls$$ is a vector of control variables, $$x$$ is a continuous variable representing a measure (let's say unemployment) in year 2000 in (1) and 2002 in (2). I interact $$x$$ with a vector of yearly dummies ($$Year$$) to estimate the effect of $$x_{i,t=2000}$$ and $$x_{i,t=2002}$$ on $$y$$ for the years preceding $$x$$. So, the only differences between (1) and (2) is $$x$$ and because of this (2) has observations for less years than (1).

My question is whether I can compare the magnitude of the coefficients of the interaction effect between (1) and (2) and say, for example, that the effect of $$x_{i,t=2000}$$ on $$y$$ in $$2008$$ is larger than the effect of $$x_{i,t=2002}$$.