ARMA & MA methods: how do you know the error terms?
In the context of discrete-time statistical signal processing, an ARMA-(p,q) random process is defined as (assuming zero initial conditions)

$$ \sum_{k=0}^p a_k ~ x[n-k]  = \sum_{k=0}^q b_k ~ v[n-k] $$ 

or equivalently

$$ x[n] = - \sum_{i=1}^p a_i ~ x[n-i]  + \sum_{i=0}^q b_i ~ v[n-i] $$ 

where $v[n]$ is a white-noise (WSS) random process with variance $\sigma_v^2$ and $x[n]$ represents the resulting ARMA process.

In this context your coefficients are related as $$\phi_i = - a_i $$ and $$\theta_i = b_i$$ The process reggresses over its past values $x[n-i]$ ($X_{t-i}$ acc) and is also a moving average of the input noise $v[n-i]$ ($\epsilon_{t-i}$ acc).