Calculating the fitted values in the Principles of forecasting

[nickcox896]

Suppose that I run this hypothetical model

Model<-data %>% model(ARIMA(y))

And that I obtain the fitted values using

fitted(Model, h=1)

The fitted values, as defined here, are the one-step ahead forecasts.
In order to obtain them, we first estimate the model using all data and then we use the estimated model to obtain the 1-step ahead forecasts. Every time we obtain the next 1-step ahead forecast, say t+1, we use only the previous historic data y_t, 1,2,…,t (in which case the forecast is static) and so no previous forecasts are used to obtain the next ahead forecasts. And the accuracy measures are averaged over all these 1-step ahead forecasts.Is that correct?

On the contary,when we use multi-step ahead forecasts, say

fitted(Model, h=5)

this means that if we are at time t, we use all observations up to time t to predict y_{t+1}. Then we use the predicted y_{t+1} to predict y_t+2} and so on up to time t+5. So in multi-step ahdead forecasts we use the previous forecasts to obtain the next forecasts (in which case, the forecast is dynamic or recursive) . Is that correct?

many thanks in advance

[robjhyndman]

h-step forecasts use data up to time t to predict observations at time t+h. The data at times t+1,...,t+h-1 will not be used. So h-step fitted values at time t, use only data to time t-h. For most models, including ARIMA, multi-step forecasts are computed recursively. That is, the 2-step forecast uses the 1-step forecast, etc.

[nickcox896]

OK, if understand correctly

fitted(Model, h=5)

means that at time t, I forecast t+5 without using the data at t+1, ...t+4, although these observations have been used in the estimation of the model.

whereas in the training/spit approach the multi-step forecasts of the test data are computed recursively. Is this what you mean?

many thanks in advance

[robjhyndman]

Yes, all observations are used in the estimation of the model. And multi-step forecasts for ARIMA models are always computed recursively, whether for training/test splits, or for multi-step fitted values.

[nickcox896]

Thank you!