Why would you express a time series as deviations from the mean?

Question

I have been trying to learn time series analysis on my own and in a textbook I came across the idea that one could express each data point in a stationary time series as deviations from the mean in order to isolate the stochastic components. Since taking the first difference of a time series removes stochastic trend why would someone want to do this? Are the stochastic components still there after differencing?

Answer 1

Suppose, you ran into a pure unit root process: $x_t=x_{t-1}+\varepsilon_t$, where $\varepsilon_t$ is a random shock. All that differencing does is: $\Delta x_t=\varepsilon_t$, so it did isolate the stochastic component (noise) into the new differences series. Now, you can study this noise.

This was the simplest case with a unit root. Look at the random walk with a deterministic trend: $x_t=x_{t-1}+a_t+\varepsilon_t$, differencing will render: $\Delta x_t= a_t+\varepsilon_t$. In this case the differenced series had a stochastic noise and thx deterministic drift $a_t$.

The real problem in practice is not differencing and fitting a model but selecting a model. In these silly examples we knew the data generating process (dgp) then applied proper differencing to extract the stationary part from the nonstationary series. In practice we rarely know the true process. We often end up applying differencing where it is not appropriate or do not apply where it had to be applied, or we mistake the deterministic drift $a_t$ for stochastic because it often does look like it etc.