I want to simulate an AR(1) process, knowing the last observation, but not the first. This should be an easy problem, but it does not seem to be so. Specifically, I can generate sensible average behavior, but the time series of individual units running through the process have a pronounced "kink" at the time I start back-casting.
To illustrate my problem, I have the following example: There is an economy with only two firms, both of which have a log productivity that follows the known AR(1) process $$y_t=0.5 y_{t-1}+ u_{t-1}$$
(After discussion in the comments, an alternative common way to define this proccess is $y_t=0.5 y_{t-1}+ u_{t}$)
with $u_t$ being drawn from N(0,1). One firm has a log productivity of 1000 at t=0, the other of -1000. So, compared to the shocks, the initial values are huge and the firms behave almost deterministic.
Simulating this data forward in time is no issue: the log productivity of both firms decays to (almost) 0 with small random permutations.
Simulating the data backwards, I first tried $$y_{t-1} = 2 (y_t- u_{t-1})$$ which yielded a correlation between the error $u_{t-1}$ and the value $y_{t-1}$ and cannot be correct(?).
(With the alternative indexing, this would be $y_{t-1} = 2 (y_t- u_{t})$, creating a correlation between $u_{t}$ and $y_{t-1}$)
Of course, with the values chosen above, this does not matter much empirically, but it seems this cannot be the right solution. After reading up a bit, and finding some other answers here (if I interpret them correctly) I assumed that AR(1) backcasting works by applying the above process also to past values, but that would mean that I predict $y_{t-1} = (500,-500)$, which cannot be correct as well: If I forecast again with those values, I will never get the initial state. However, if I apply this logic with less extreme values, at least the distribution of productivity is stable in the past. It seems that I misunderstand/oversee an important assumption that goes into the "reversability" of AR(1). How does AR(1) backcasting actually work?
The following answers are not exactly my question, but the answers to them are very general and should imo include my special case. All seem to imply (to me) that I should just use the forward law of motion also for moving into the past:
Simulating AR(1) Process with Initial and Final Condition
Deriving the conditional distributions of a multivariate normal distribution
