Time series notation

Question

Consider a time series of time t=1...T as $$\left(y_{1},y_{2},...,y_{T}\right)$$ Here, do each of the $y_{t}$'s correspond to different realizations of a the same random variable Y? Or are each of the $y_{t}$'s thought of as random variables themselves ?

Answer 1

One way to look at the series is to consider each $y_t$ an observation of a random variable $Y_t$, then when you write $y_1,\dots,y_T$ you mean that you sampled from the joint T-dimensional distribution, i.e. each element is a distinct random variable. The problem is that you can only sample once. There's only one January 1 2023, so whatever you observed that day will be a single sample from the variable $y_{2023/1/1}$.

There's not much you could do to forecast such series or generally study its properties. Even the simplest metrics such as correlation between $y_t,y_s$ become pretty much impossible to estimate in this framework.

So, we start mental acrobatics, such as a notion of stationarity. For instance, we assume that correlation between $\rho_{t,s}=y_t,y_s$ depends only on the distance in time, i.e. $\rho_{t,s}=\rho_{|s-t|}$. All of a sudden, we got more than one sample to calculate the correlations! For instance we have T-2 observations to calculate $\rho_2$: $$y_1,y_3\\y_2,y_4\\y_3,y_5\\\dots\\y_{T-2,T}$$

There's also a concept of ergodicity in physics: if you observe the system long enough then you'll encounter it to go through every possible state at the rate that corresponds to the probabilities of states. This essentially means that if you're dealing with a system in some kind of equilibrium then $y_1,\dots,y_T$ is the same as T observations of the system $y$, i.e. T realizations of one variable.