I would like to know if the $n$ realizations of a variable, say $Y$ expressed in the form of a time series constitutes $n$ random variables or just a single random variable $Y$? For example, the output of a linear regression model indexed by time, $n$ :$y(n) = ay(n-1) + by(n-2) + w(n)$
Here, $y$ is a random variable containing $n$ data points. The time ordered sequence of observations of $y$ form a time series $\{y_n\}_{n=1}^{1000}$.
Then, what is a multivariate case $Y_1,Y_2,\ldots,Y_n$? I am not clear about these concepts and shall appreciate help. Thank you.
Best Answer
It depends on your model / views. For a given time series with a timespan $T$, you can consider that you observe $T$ realizations of a given random variable $X$, or you can consider that you observe one realization of a stochastic process that is one path among many others. If you consider an independent identically distributed random process, these are the same.
It is not clear whether $Y_1,Y_2,\ldots,Y_n$ represents the $n$ variables of your random process and thus a single time series, or $n$ time series each represented by a single random variable $Y_i$, and therefore your time series data is a matrix $n \times T$, i.e. a time series for each $Y_i$ with $T$ realizations.
As long as you are consistent, it is up to you to choose your model.