Time series (theoretical concept)

A time series is a collection of random variables {Y1,Y2,…YT} ordered in time. There is a stochastic process {Yt} that generates the series. Each element Y1,Y2,…YT of the series is a random draw from a probability distribution. However we can only observe one particular realization of the stochastic process {y1,y2,..yT} in reality.



The stochastic process {Yt} is then described by a T-dimensional joint probability distribution. The(unknown) parameters (mean, variance, covariance) of the joint probability distribution of {Yt} are, for each t=1,2…T:

For a weakly stationary time series it holds that:

To infer the parameters from a particular observed realization, we assume the process to be ergodic (the sample moments approach the population moments as T becomes infinite).