How to derive MA(1) from Wold decomposition I want to derive the moving average process 1 from the Wold decomposition theorem.  I don't know how to define the coefficient of MA(1), though it seems that it should be easy.
 A: It's not clear what you mean by 'derive from' in this case. I can explain how an MA(1) fits into what the theorem says.
The MA(1) process is a process of the form:
$$ Y_t - \mu = \varepsilon_t - \theta_1 \varepsilon_{t-1} $$
or it is sometimes written:
$$ Y_t - \mu = \varepsilon_t + \theta_1 \varepsilon_{t-1} $$
(e.g. see here)
Now Wold's decomposition theorem says every covariance-stationary time series $Y_{t}$ can be written as the sum of two time series, one deterministic and one stochastic:
$$Y_t=\sum_{j=0}^\infty b_j \varepsilon_{t-j}+\eta_t$$
Here $\eta_t$ is the deterministic part.
Let $\eta_t=\mu$. Let $b_0 = 1$ and $b_j=0$ for all $j>1$.
Then starting from the Wold decomposition:
$$Y_t=\sum_{j=0}^\infty b_j \varepsilon_{t-j}+\eta_t\\
= 1\cdot\varepsilon_{t-0}+b_1 \varepsilon_{t-1}+\mu\\
=\mu + \varepsilon_t+b_1 \varepsilon_{t-1}$$
which is an MA(1) with $\theta_1=-b_1$ (if the MA(1) is written in the first form above) or with $\theta_1=b_1$ (if written in the second form above). This makes it clear that the Wold decomposition holds for an MA(1) process.
