7.7 Forecasting with ETS models

Point forecasts are obtained from the models by iterating the equations for \(t=T+1,\dots,T+h\) and setting all \(\varepsilon_t=0\) for \(t>T\).

For example, for model ETS(M,A,N), \(y_{T+1} = (\ell_T + b_T )(1+ \varepsilon_{T+1}).\) Therefore \(\hat{y}_{T+1|T}=\ell_{T}+b_{T}.\) Similarly, \[\begin{align*} y_{T+2} &= (\ell_{T+1} + b_{T+1})(1 + \varepsilon_{T+1})\\ &= \left[ (\ell_T + b_T) (1+ \alpha\varepsilon_{T+1}) + b_T + \beta (\ell_T + b_T)\varepsilon_{T+1} \right] ( 1 + \varepsilon_{T+1}). \end{align*}\]

Therefore, \(\hat{y}_{T+2|T}= \ell_{T}+2b_{T},\) and so on. These forecasts are identical to the forecasts from Holt’s linear method, and also to those from model ETS(A,A,N). Thus, the point forecasts obtained from the method and from the two models that underlie the method are identical (assuming that the same parameter values are used).

ETS point forecasts are equal to the medians of the forecast distributions. For models with only additive components, the forecast distributions are normal, so the medians and means are equal. For ETS models with multiplicative errors, or with multiplicative seasonality, the point forecasts will not be equal to the means of the forecast distributions.

A big advantage of the models is that prediction intervals can also be generated — something that cannot be done using the methods. The prediction intervals will differ between models with additive and multiplicative methods.

For some models, there are exact formulae that enable prediction intervals to be calculated (see Rob J Hyndman, Koehler, et al. (2008a) for the formulas). A more general approach that works for all models is to simulate future sample paths, conditional on the last estimate of the states, and to obtain prediction intervals from the percentiles of these simulated future paths.

To obtain forecasts from an ETS model, we use the forecast function. The R code below shows the possible arguments that this function takes when applied to an ETS model. We explain each of the arguments in what follows.

forecast(object, h=ifelse(object$m>1, 2*object$m, 10),
level=c(80,95), fan=FALSE, simulate=FALSE, bootstrap=FALSE, npaths=5000,
PI=TRUE, lambda=object$lambda, biasadj=NULL, ...)
object
The object returned by the ets() function.
h
The forecast horizon — the number of periods to be forecast.
level
The confidence level for the prediction intervals.
fan
If fan=TRUE, level=seq(50,99,by=1). This is suitable for fan plots.
simulate
If simulate=TRUE, prediction intervals are produced by simulation rather than using algebraic formulae. Simulation will also be used (even if simulate=FALSE) where there are no algebraic formulae available for the particular model.
bootstrap
If bootstrap=TRUE and simulate=TRUE, then the simulated prediction intervals use re-sampled errors rather than normally distributed errors.
npaths
The number of sample paths used in computing simulated prediction intervals.
PI
If PI=TRUE, then prediction intervals are produced; otherwise only point forecasts are calculated.
lambda
The Box-Cox transformation parameter. This is ignored if lambda=NULL. Otherwise, the forecasts are back-transformed via an inverse Box-Cox transformation.
biasadj
If lambda is not NULL, the backtransformed forecasts (and prediction intervals) are bias-adjusted.
fit %>% forecast(h=8) %>%
  autoplot() +
  ylab("International visitor night in Australia (millions)")
Forecasting international visitor nights in Australia using an ETS(M,A,M) model.

Figure 7.11: Forecasting international visitor nights in Australia using an ETS(M,A,M) model.

References

Hyndman, Rob J, A B Koehler, J K Ord, and R D Snyder. 2008a. Forecasting with Exponential Smoothing. Berlin, Germany: Springer-Verlag.