Time series and random variable

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.

• If the thread at stats.stackexchange.com/questions/126791 does not answer this question, then please edit it to explain what your definition of "time series" actually is. At the end, where you write "multivariate case," do you mean that each of the $Y_i$ is a vector?
– whuber
Commented Jun 22, 2015 at 14:35
• Thank you for the link. However, I am not clear from the definition 2.1.4 where it says that the time series is a time indexed collection of random variables. This is what is confusing to me. What is meant be a "set of random variables" ? Say, I have only one variable $Y$ with 4 values = $[y_1 = 0.2, y_2 = 0.1,y_3 = -0.12, y_4 = 1.5]$ then are there 4 random variables or are the elements called as realization of a random variable, $Y$ ? Next, example, say I have 2 variables now $X,Y$ each containing 4 values. So, what will be the r.vs here be called multivariate?
– SKM
Commented Jun 22, 2015 at 17:17

1 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.

• Just to clarify if my understanding is clear : a time series of length $T$ (timespan) is one realization of a random variable, $X$ where $n=1$ dimension. So, if I have $n$ random variables, in other words, dimension is $n$ then also a time series will be one realization for all the variables producing a matrix of $n \times T$. For example, there is an ordered feature vector of $n$ features each feature has $T$ values.
– SKM
Commented Jun 22, 2015 at 17:22
• "a time series of length T (timespan) is one realization of a random variable, X where n=1 dimension" No. I would say, it is $T$ realizations of your variable $X$ OR it is one path-realization of a random process $(X_1,X_2,\ldots,X_T)$.
– mic
Commented Jun 22, 2015 at 17:34
• But, if you consider that your time series of length $T$ is $T$ realizations of your variable $X$. Then, writing $(X_1,X_2,\ldots, X_n)$ means that you have $n$ time series of timespan $T$, thus a matrix $n \times T$.
– mic
Commented Jun 22, 2015 at 17:35
• Thank you for the followup. I understood the concept of time series. However, so each of the $n$ time series $(\{X_1t\}_{t=1}^\infty, \{X_2t\}_{t=1}^\infty,...,\{X_nt\}_{t=1}^\infty)$ is a random variable ? Thus, there are $n$ random variables, and is this called as multivariate? Last Question : when we say $i.i.d$ from the context of distribution of a r.v, do we imply those $T$ realizations of r.v $X$ are $i.i.d$?
– SKM
Commented Jun 22, 2015 at 17:40
• Yes, the $n$ time series considered as a whole can be considered as realizations of a multivariate ($n$-variate) random variable. And, yes, those $T$ realizations of r.v. $X$ are i.i.d. For illustration of these concepts, you can look at this if you have some time to spend :)
– mic
Commented Jun 22, 2015 at 20:08