Wold's theorem

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In statistics, Wold's theorem or Wold representation theorem, named after Herman Wold, says that every covariance-stationary time series can be written as a infinite moving average (MA( $\infty$ )) process of its forecast errors. This is also called Wold decomposition.

Formally

$y_{t}=\sum_{j=1}^{\infty}\theta_{j}\varepsilon_{t-j}+\varepsilon_{t}+\eta_{t},$

where $y t$ is the observed time series, $\varepsilon_{t}$ is the forecast error (actual minus predicted value), $θ$ is a coefficient of which the root is between −1 and 1 (i.e. inside the unit circle) and $η t$ is an additional error.