Time Series Analysis: White Noise
Definition 1
A sequence $\left\{ e_{t} \right\}_{t = 1}^{\infty}$ of independent identically distributed (iid) random variables $e_{t}$ is called White Noise.
- iid stands for independent identically distributed, meaning that they are independent from each other and share the same distribution.
Description
Following the definition of a sequence of random variables, it is naturally a stochastic process. Particularly, if $E ( e_{t} ) = 0$, then the stochastic process $\left\{ Y_{t} \right\}_{t = 1}^{\infty}$ defined as $Y_{t} : = \begin{cases} e_{1} & , t=1 \\ Y_{t-1} + e_{t} & , t \ne 1 \\ \end{cases}$ becomes a random walk.
In statistics, it’s acknowledged that a 100% perfect explanation of an observed phenomenon is impossible. If a problem could be perfectly explained, there would be no need to solve it using statistics. Irrespective of the model, errors are inevitable, which statistics interpret as a ’lack of information’. It would be beneficial to have as much information as possible, but it’s impossible to know everything about the universe, and when actually using it, cost issues arise too.
In this regard, white noise can be seen as the ‘inevitable error’ that occurs in time series analysis. As the data is obtained from reality and not created ideally, it necessarily includes some. Although it might be negligible at first, as time passes, it could accumulate and become significantly large. Therefore, the forecasts in time series analysis become less reliable the further into the future they go, eventually losing their meaning.
See Also
Cryer. (2008). Time Series Analysis: With Applications in R(2nd Edition): p17. ↩︎