How o treat time when using stocks and a growth model (longitudinal data)

It's not entirely clear what you are trying to find out, but in
general having stocks traded at different periods shouldn't be a big
barrier.  A simple random-slope model with temporal autocorrelation
would look something like this:

   lme(log(value) ~ 1+year, random = ~1+company|year,
     correlation=corAR1(form=~year|company), data=stock_data)

Using lme() because lmer() still doesn't do autocorrelation easily.
You should probably center the year variable (i.e. for technical reasons
you don't want to have your intercept at Gregorian calendar year 0).

  I don't know much about financial modeling but I know that people use
much more sophisticated (e.g. GARCH) models; I don't know if anyone has
tried to do mixed GARCH models or not.  Probably.

Hello,

I have been studying multilevel models to understand how I can use it to
investigate the behavior of stocks listed on a exchange. I am still very
new with this methodology so I would like to ask if its possible to do that
using growth model and how I should treat stocks that start to be traded in
different period (for instance, stock of company A is traded since 1998 and
stock of company B is traded since 2005).

Thank you for any help.