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I've read here https://otexts.com/fpp2/arima-forecasting.html how we do forecasting in time series models like the ARMA model, but I'm wondering if we recalculate estimates of parameters of our model after receiving a new observation?

For instance let's say that I fitted MA(2) model: $$y_t=\mu+\varepsilon_t+\theta_1 \varepsilon_{t-1} +\theta_2 \varepsilon_{t-2}$$ based on $y_1,\ldots,y_T$ and got estimates $\hat \theta_1,\ \hat \theta_2,\ \hat \mu$. I made a prognosis $$\hat y_{T+1}= \hat \mu+\hat \theta_1 \hat \varepsilon_{T}+\hat \theta _2 \hat \varepsilon _{T-1}.$$ Am I supposed to recalculate $\hat \theta_1,\ \hat \theta_2 $ etc. after receiving the true $y_{T+1}$?

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Some people/tools do, some people/tools don't.

For models like ARIMA or Holt-Winters, I recommend refitting every time you get new values, for reasons detailed intuitively here.

The counter opinion, is to do filtering, as opposed to forecasting. But my understanding is that you would have to use the state space formulation of ARIMA or similar models, as opposed to the original formulation (take this with a grain of salt, since I haven't fully grasped the topic myself yet).

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