Purpose of negative indices in time series In the book Time-series analysis by Hamilton we find the following passage:
A time series is a collection of observations indexed by the date of each observation. Usually we have collected data beginning at some particular date ( say $t = 1$) and ending at another (say, $t = T$) in the following fashion:
$(y_{1} ,y_{2} ..., y_{T})$
We often imagine that we could have obtained earlier observations $(y_{0} , y_{-1} , y_{-2}...)$ or later observations $(y_{T + 1} , y_{T + 2} , y_{T + 3}...)$ had the process been observed for more time.
Typically, a time series $\{y_{t}\}^{\infty}_{t = -\infty}$ is identified by describing the $t$-th element of the time series.
Why do we need negative indices in our time series? Is it correct to say that $y_{-1}$ is the $-1\:th$ element of the time series $y_{t}$? That sounds weird and wrong. Is the negative indices only there to denote unobserved observations in the past?

The reason for this is that we might potentially want to look arbitrarily far back in time and so we set up the mathematics of our problem in a way that allows us to do that.  This doesn't even hinge on making $t=1$ the current time --- even if we were to take some higher positive value $t_0 > 1$ as the current time, if we wanted to look back $t_0+2$ time periods before the current time we would get a negative index.
If you were to proceed in the contrary fashion, by limiting the time-series to non-negative time indices then your time index would be bounded from below, so there would always be a limit on how far backward you could look from any time period.  We generally don't wish to limit ourselves in this way a priori and so it is usual to define time series over the unbounded set of time values $t \in \mathbb{Z}$.
