The linear process is an infinite sum of white-noise terms, each multiplied by a coefficient $\phi_j$. $ X_t = \sum_{j=- \infty}^\infty \phi_j W_{t-j} $ **Properties:** - *Well-defined:* A linear process is well-defined, when the sum is finite. This is only the case, when the sum of all absolute coefficients is finite. This is true because the white noise terms have mean zero and are independent of each other. $ \sum_j \vert \phi_j \vert < \infty $ - *Causal process:* A linear process is called causal, when all $\phi_j$ of future observations are zero. This means that $X_t$ only depends on information from the past and not from the future. Both [[Autoregressive Model|AR Model]] $\text{AR}(p)$ and [[Moving Average Model|MA Model]] $\text{MA}(q)$ models can be expressed as special cases of the linear process. - $\text{MA}(q)$: A causal linear process where only the $\phi_j$ of the last $q$ last white noise terms are non-zero. - $\text{AR}(p)$: We can express the model only in terms of $\phi$ and $W_j$, by recursively replacing the $X$ terms. $ \begin{align} X_t &= W_t+ \phi X_{t-1} \\ &=W_t+\phi(W_{t-1}+\phi X_{t-2}) \\ &=W_t+ \phi W_{t-1}+ \phi^2 X_{t-2} \\ &= \dots \\ &=\sum_{j=0}^\infty \phi^j W_{t-j} \end{align} $ Again to ensure that this $\text{AR}(p)$ process is well-defined we require that the sum of coefficients (this time with exponents) is finite. $ \sum_{j=0}^\infty \phi^j <\infty $