Determining first element of an AR1 model

Question

I have an AR1 process of the well-known form:

$y(t) = a*y(t-1) + e(t)$

And, in any computational software the elements of the time-series $y$ will be stored in a vector.

Now, how can I calculate $y(1)$ when $y(1-1=0)$ does not exist?

Answer 1

You can't. You must specify, as part of the process, an initial condition, i.e. a value for $Y_0.$ It's possible in theory (and in practice to some extent) for this to be random, i.e. drawn from some distribution.

If $|a|<1,$ the process is stationary and there is a stationary distribution for $Y.$ A natural choice might be to draw $Y_0$ from this stationary distribution (this might be familiar if you've studied Markov chains). This is similar to imagining the process 'has always been running'.

For practical simulation, when it might be impossible or too much trouble to produce variates from the stationary distribution, a typical technique that works is 'burn-off' (which some consider as wasteful as it sounds). You select some initial $Y_0$... it doesn't matter much what you pick as long is it's not too atypical relative to the stationary distribution. Then you run the simulation for some decent number of time steps till you're confident the system has evolved to its stationary state, then you start the process again "for real" with the final value from the burn-off run as your new initial $Y_0.$

It's also not necessarily the case that you want the initial value to be from the stationary distribution, it depends what your purposes are.