# Why doesn't the Moving Average model contain past time series values?

The moving average model looks like:

$$X_t=\mu + \varepsilon_t + \theta_1 \varepsilon_{t−1} + \dotsb + \theta_q \varepsilon_{t−q}$$

It only contains the mean and some white noise factors, with lags.

I am wondering, given a time series $\{X_1, X_2, \dotsb , X_t\}$, how could you estimate the white noise part? I just don't understand the link between the past time series values, and the white noise. For auto-regression, that is easy to understand.

That is due to the definition of the moving average (MA) model. The dependent variable $x$ is supposed to be a linear combination of the current and lagged innovations (shocks) $\varepsilon$. That is how the model is defined.
There is an intuition to the MA model, too. If a process $x$ reacts to a shock $\varepsilon$ with a delay, then a lagged shock is a natural variable to have in the model -- and that happens to be the case with the MA model.