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Choosing between ARIMA and ETS

Today let's settle a question every forecaster runs into sooner or later: given the same series, do you reach for ARIMA or for ETS?

The running example is AirPassengers, a dataset built into R: the monthly count of international airline passengers, in thousands, from January 1949 through December 1960, 144 months in total. It climbs over the whole period, and the summer bump riding on top of that climb gets a little wider every year.

Here is the whole series, plotted month by month.

Look at that climb, and at the summer bump riding on top of it getting wider every year. That widening bump is going to matter a lot once we get to choosing between the two families.

Two different ways to model the same series

ETS and ARIMA both forecast a series, but they get there by completely different routes.

ETS smooths a level, a trend and a season directly on the series you actually observe. Each new month nudges the level a little, nudges the trend a little, and nudges the seasonal pattern a little, and the forecast just carries those smoothed pieces forward.

ARIMA works the other way around. It first differences the series, subtracting each value from an earlier one until what's left has no trend and no season, and then it models whatever autocorrelation is still sitting in that differenced series with AR (autoregressive) and MA (moving average) terms.

Fit both on AirPassengers and let each one search for its own best order automatically.

RInteractive R
# Build the AirPassengers tsibble and fit ETS and ARIMA with automatic model selection library(fable) library(tsibble) library(dplyr) ap <- as_tsibble(AirPassengers) names(ap) <- c("month", "passengers") fit <- ap |> model( ets = ETS(passengers), arima = ARIMA(passengers) ) fit #> # A mable: 1 x 2 #> ets arima #> <model> <model> #> 1 <ETS(M,Ad,M)> <ARIMA(2,1,1)(0,1,0)[12]>

  

  





Read ETS(M,Ad,M) as: multiplicative error, damped additive trend, multiplicative season. Read ARIMA(2,1,1)(0,1,0)[12] as: 2 non-seasonal AR terms, one ordinary difference, one non-seasonal MA term, and one seasonal difference at lag 12 with no seasonal AR or MA terms.



Two different searches, over two different kinds of model, landed on two different answers for the same 144 months. Is that always going to happen, or do the two families sometimes agree?