# Are ARMA models used to model the noise in a time series?

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.

• ARMA models are sometimes used this way. Alternatively, it may be that $Y_t$ is an ARMA. Dec 21, 2022 at 18:03
• @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.) Dec 21, 2022 at 18:07
• 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. Dec 21, 2022 at 18:15

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.