3.1 SARIMA models: estimation and forecasting

We have covered the estimation of ARIMA model in the last tutorial. Extension to SARIMA models is immediate: simply use - argument seasonal (a vector of length 3) for forecast::Arima - arguments P, D, Q and the period S in the function astsa::sarima - a list seasonal with components order and period (the latter is facultative if supplying ts objects) for stats::arima.

Estimation is done using a state-space formulation of the model; an alternative would be to fit the parameters using an artificial regression.

We have yet to address the matter of prediction. The latter proceeds by one-step ahead forecasting and emploies again the state-space representation. Note however that

The standard errors of prediction exclude the uncertainty in the estimation of the ARMA model and the regression coefficients. According to Harvey (1993, pp. 58–9) the effect is small.

The predict.Arima method gives forecast for \(h\)-lags ahead for arima objects. For objects of class Arima, the forecast function from the eponym package handles the additional features from forecast::Arima, namely Box–Cox transformation viz lambda and bootstrap prediction intervals, as a logical supplied to the argument bootstrap (more later on this). If you use astsa::sarima, the forecast function astsa::sarima.for allows you to do estimation and prediction all in one go.