# Wold Decomposition Theorem and Moving Average model - Error Terms

I'm stuck with Wold's decomposition theorem in time series analysis. The theorem says that every stationary time series can be written as a sum of two components, one being entirely deterministic (here represented with mean) and one being stochastic. Here is the formula from my book:

$$X_t - \mu = e_t + \psi_1 e_{t-1} + \psi_2 e_{t-2} + \cdots = \sum_{j=0} ^\infty \psi_j e_{t-j} \quad \quad \quad \quad \quad \psi_0 = 1.$$

A few chapters later it has the MA(q) model written as: $$X_t = e_t - \theta_1 e_{t-1} - \theta_2 e_{t-2} - \cdots - \theta_q e_{t-q}.$$

Where did these $$e_t$$s come from? I searched online for the answer and all I get is that $$e_t$$s are error terms that are also White Noise processes. So my first question when I read this is: Error terms of what? Is there some estimated model that we extract these from? What do these truly represent and how do I calculate them?

Thanks in advance

## 1 Answer

The errors are the unexplained component of the time series. The error terms for the MA are equal to the difference between the observed value and the fitted value. If you are referring to a standard time series book, it will have a detailed note on the computation of errors and estimation of θ's. This link gives a good description of how the errors are calculated

Here