How to check if the effect from independent variable is changing over time? I have panel data and want to find out whether the effect of key independent variable is changing over time.
At first, I intended to run OLS for each year separately, then compare coefficients.
But then I started to have doubts. Maybe there is a better strategy?
 A: Looking at the estimates at each year separately is a reasonable descriptive strategy for what you're doing. If you're looking for a formal test, you could use a varying coefficient regression model. 
Briefly, the varying coefficient model extends the standard regression model: 
$$ y_i = \beta_0 + \beta_1 x_i + \epsilon_i $$ 
to allow the coefficient (say, the slope) to be a function of time: 
$$ y_i = \beta_0 + \beta_1(t) x_i + \epsilon_i $$ 
In general, there are not parametric specifications on the functional form of $\beta_1(t)$, but the practicality of a nonparametric estimate depends on how much data you have and the level of variability in $t$.  When you assume a linearly varying coefficient, i.e. $\beta_1(t) = \beta t$, this is equivalent to a linear interaction by time. You can test the null hypothesis $\beta_1(t) = c$ (i.e. a non-varying coefficient) by comparing the two models above. The are nested, so the likelihood ratio test is applicable.  
These models were first introduced as a special case of generalized additive models in 
Hastie, T.; R. Tibshirani (1993). "Varying-Coefficient Models". Journal of the Royal Statistical Society, Series B. 55: 757–796.
If you use R, you can fit varying coefficient models using mgcv, e.g. gam(y~s(x,by=z)) estimates how the coefficient for $z$ varies as a function of $x$. See the help file for more info. 
