## 8.10 ARIMA vs ETS

It is a commonly held myth that ARIMA models are more general than exponential smoothing. While linear exponential smoothing models are all special cases of ARIMA models, the non-linear exponential smoothing models have no equivalent ARIMA counterparts. On the other hand, there are also many ARIMA models that have no exponential smoothing counterparts. In particular, all ETS models are non-stationary, while some ARIMA models are stationary.

The ETS models with seasonality or non-damped trend or both have two unit roots (i.e., they need two levels of differencing to make them stationary). All other ETS models have one unit root (they need one level of differencing to make them stationary).

The following table gives the equivalence relationships for the two classes of models.

ETS model ARIMA model Parameters
ETS(A,N,N) ARIMA(0,1,1) $$\theta_1 = \alpha-1$$
ETS(A,A,N) ARIMA(0,2,2) $$\theta_1 = \alpha+\beta-2$$
$$\theta_2 = 1-\alpha$$
ETS(A,A,N) ARIMA(1,1,2) $$\phi_1=\phi$$
$$\theta_1 = \alpha+\phi\beta-1-\phi$$
$$\theta_2 = (1-\alpha)\phi$$
ETS(A,N,A) ARIMA(0,0,$$m$$)(0,1,0)$$_m$$
ETS(A,A,A) ARIMA(0,1,$$m+1$$)(0,1,0)$$_m$$
ETS(A,A,A) ARIMA(1,0,$$m+1$$)(0,1,0)$$_m$$

For the seasonal models, the ARIMA parameters have a large number of restrictions.

The AICc is useful for selecting between models in the same class. For example, we can use it to select an ARIMA model between candidate ARIMA models16 or an ETS model between candidate ETS models. However, it cannot be used to compare between ETS and ARIMA models because they are in different model classes. The examples below demonastrate selecting between these classes of models.

### Example: Comparing auto.arima() and ets() on non-seasonal data

We can use time series cross-validation to compare an ARIMA model and an ETS model. The code below provides functions that return forecast objects from auto.arima() and ets() respectively.

fets <- function(x, h) {
forecast(ets(x), h = h)
}
farima <- function(x, h) {
forecast(auto.arima(x), h=h)
}

The returned objects can then be passed into tsCV. Let’s consider ARIMA models and ETS models for the air data as introduced in Section 7.2 where, air <- window(ausair, start=1990).

# Compute CV errors for ETS as e1
e1 <- tsCV(air, fets, h=1)
# Compute CV errors for ARIMA as e2
e2 <- tsCV(air, farima, h=1)
# Find MSE of each model class
mean(e1^2, na.rm=TRUE)
#> [1] 8.27
mean(e2^2, na.rm=TRUE)
#> [1] 10.1

In this case the ets model should be used as it has a lower tsCV statistic based on MSEs. Below we generate and plot forecasts for the next 5 years

air %>% ets() %>% forecast() %>% autoplot()

### Example: Comparing auto.arima() and ets() on seasonal data

In this case we want to compare seasonal ARIMA and ETS models applied to the quarterly cement production data qcement. Because the series is very long, we can afford to use a training and test set rather than time series cross-validation. The advantage is that this is much faster.

We create the training set from the beginning of 1988 to the end of 2007 and select an ARIMA and an ETS model using the auto.arima and ets functions.

# Use 20 years of the qcement data beginning in 1988
train <- window(qcement, start = 1988, end = c(2007,4))

# Fit an ARIMA and an ETS model to the training data
fit1 <- auto.arima(train) %>% checkresiduals()

#>
#>  Ljung-Box test
#>
#> data:  Residuals from ARIMA(2,0,0)(2,1,1)[4] with drift
#> Q* = 4, df = 3, p-value = 0.3
#>
#> Model df: 6.   Total lags used: 9
fit2 <- ets(train) %>% checkresiduals()

#>
#>  Ljung-Box test
#>
#> data:  Residuals from ETS(M,N,M)
#> Q* = 6, df = 3, p-value = 0.1
#>
#> Model df: 6.   Total lags used: 9

fc1 <- forecast(fit1, h = 1+4*(2013-2007))
#> Warning in mean.default(x, na.rm = TRUE): argument is not numeric or
#> logical: returning NA
fc2 <- forecast(fit2, h = 1+4*(2013-2007))
#> Warning in mean.default(x, na.rm = TRUE): argument is not numeric or
#> logical: returning NA

1. Note that comparing information criteria is only valid for ARIMA models of the same orders of differencing.