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If we have a time series $y_t$ and we want to model it, is the following intuition about the ARMA model correct?

$$Y_t=f(\beta,t) + e_t $$ where $e_t \sim$ ARMA. So there is a decomposition of a deterministic part $f(\beta,t)$ and a stochastic part (the noise) which can be modelled by an ARMA process.

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    $\begingroup$ ARMA models are sometimes used this way. Alternatively, it may be that $Y_t$ is an ARMA. $\endgroup$ Dec 21, 2022 at 18:03
  • $\begingroup$ @RichardHardy Thank you for your reaction! But if $Y_t$ is an ARMA model itself it has to be stationary while many time series we observe in reality are not stationary (or contain seasonality, etc.) $\endgroup$ Dec 21, 2022 at 18:07
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    $\begingroup$ It might be ARIMA instead, so trying in differences of the series might help. Alternatively, you could indeed suggest a model if differentiation does not work. For instance, you can use structural models with a polynomial trend and ARMA errors. I think the package UComp in R lets you adjust this type of models easily. For seasonality, you could use SARIMA models or structural models with a seasonal component. Both can be implemented in R. $\endgroup$ Dec 21, 2022 at 18:15

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In most application cases your observed $Y_t$ is nonstationary but you can still model it using the generalized form of ARIMA such as SARIMA. If your $f(\beta,t)$ is simply a linear function of $t$ and the noise part $e_t$ is assumed to follow an ARMA process, then you can difference once your time series to model $Y_t$ as a ARIMA(p,d=1,q) model where d= is the order of difference. If $f(\beta,t)$ is a general polynomial trend then you can difference more times to arrive at a stationary ARMA model as shown in Brockwell et al's Introduction to Time Series and Forecasting. If $f(\beta,t)$ also contains seasonal trend then you can additionally do seasonal difference to arrive at SARIMA model which is a more generalized form of ARIMA and can account for both autoregressive lagged difference and seasonal difference mentioned above.

Of course In most applications you cannot even be sure your noise is really generated by a (stationary) ARMA process (white noise included since it's a special case of ARMA), albeit it's a common assumption in applications such as un-systematic measurement error or consecutive daily temperature series.

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