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I know that traditional ARIMA models cannot handle non-linear data but I was unable to find any place where it states whether if VARMA can handle non-linear data or only linear. Please clarify this for me.

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VARMA is just a multivariate counterpart of ARMA, just like VAR is a multivariate counterpart of AR. All of these models are linear in parameters. A VARMA(p,q) model postulates that a multivariate time series $y_t$ (a vector) is a linear function of its own past values $y_{t-l}$ for $l=1,\dots,p$ and past values of the error vector $\varepsilon_{t-m}$ for $m=1,\dots,q$: $$ y_t = \sum_{l=1}^p \Phi_l y_{t-l} + \sum_{m=1}^q \Theta_m \varepsilon_{t-m} + \varepsilon_t. $$ ($\Phi_l$ and $\Theta_m$ are square matrices.) In this sense, VARMA is as linear as ARMA (or VAR, or AR).

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