If variables in my data have different starting time because of availability, how shall I do? To shorten my data so that each variable has the same starting time or extend some variables data with artificial number, like letting those historical values before equal to the first available one?
You should not replicate the first observation to complete your sample. There is no rational for that, except if you want to implicitly impose some kind of unit root prior. You have two possibilities :
Shorten your sample so that you have data available for all the variables in all periods.
If for some variables data are not available at the beginning of the sample, treat them as missing observations. You just have to put some NaNs in the datafile for these missing observations.
Note that the missing observations, i.e. the NaNs, need not to be at the beginning of the sample, they can arbitrarily distributed in the sample. I would opt for the second approach.
Thank your, Professor.
I got it. Your reply helps me a lot.