The white-noise model
Any time series can be considered to process two fundamental elements: signal and noise. We can present this mathematically as follows:
y(t) = signal(t) + noise(t)
The signal is some predictable pattern that we can model with a mathematical function. But the noise element in a time series is unpredictable and so cannot be modeled. Thinking of a time series this way leads to two consequential points:
- Before attempting to model, we should verify that the time series is not consistent with noise.
- Once we have fit a model to a time series, we should verify that the residuals are consistent with noise.
Regarding the first point, if a time series is consistent with noise, there is no predictable pattern to model, and attempting to model the time series could lead to misleading results. About the second point, if the residuals of a time-series model are not consistent with noise, then there are additional patterns we can further model, and the...