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 $x_{i,t=2002}$.


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