Linear and Nonlinear Components of a Time Series I am starting to develop a hybrid ARIMA-ANN model for forecasting. Most of the journals I read mention mostly a linear component for ARIMA and a nonlinear for ANN. 
How can you know which components in the data are linear and nonlinear?
What are the definitions of the linear and nonlinear components of a time series?
 A: Suppose that $y_{t}$ is a time series at time $t$ that you can express with the following equation 
\begin{align}  
y_{t} = \alpha + c_{1}y_{t-1} + \epsilon_{t},
\end{align}
where $\alpha$ is a constant, $\epsilon_{t}$ an error term and $y_{t-1}$ is the observation at time $t-1$. The above equation is known as an AR(1) model.  Notice that the term $y_{t-1}$ is linear because it just has a constant multiplying it.
Now, if we would like to consider non-linear components, we could express $y_{t}$ as 
\begin{align}
     y_{t}^2 = c_{0} + c_{1}y_{t-1}^2 + ... + c_{k}y_{t-k}^2 + \epsilon_{t}
\end{align}
where the term $y_{t-1},..., y_{t-k}$ are quadratic ($k$ is the number of lags). In fact, the prior expression is used to test if a time series is non-linear by setting the null hypothesis 
\begin{align}
    H_{0} : c_{1} = ... = c_{k} = 0
\end{align}
against the alternative 
\begin{align}
    H_{1} : c_{i} \neq 0, \: \text{for some} \: i\in{1,...,k} 
\end{align}
For a better clarification, chapter 1 of Nonlinear Time Series Analysis by Ruey Tsay and Rong Chen is recommended. In conclusion, linear or non-linear depends on the associated exponent to each term $y_{t-1}, ..., y_{t-k}$.
