06217316 for arima we fit the model run

We note that even though AICc gave lowest values for the model with drift, RMSE was the least at 0.06217316 for
ARIMA . We fit the model, run checkresiduals() and use the model for the forecast.
fit <- Arima (h02, order= c ( 3 , 0 , 1 ), seasonal= c ( 0 , 1 , 2 ), lambda= 0 )

3/31/2020 Lec 10: Seasonal ARIMA (Chapter 8) and Combination Forecast (Section 12.4) (66)
checked-new2nd ed/8_ARIMA_models_Pt2_2020.html#(66) 44/66
checkresiduals (fit)
##
## Ljung-Box test
##
## data: Residuals from ARIMA(3,0,1)(0,1,2)[12]
## Q* = 23.663, df = 18, p-value = 0.1664
##
## Model df: 6. Total lags used: 24

3/31/2020 Lec 10: Seasonal ARIMA (Chapter 8) and Combination Forecast (Section 12.4) (66)
checked-new2nd ed/8_ARIMA_models_Pt2_2020.html#(66) 45/66
autoplot ( forecast (fit), ylab= "h02 sales (million scripts)" , xlab= "Year" )

3/31/2020 Lec 10: Seasonal ARIMA (Chapter 8) and Combination Forecast (Section 12.4) (66)
checked-new2nd ed/8_ARIMA_models_Pt2_2020.html#(66) 46/66
ARIMA vs ETS
It is a common 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. There are also many ARIMA models that have no exponential smoothing counterparts.
In particular, every ETS model is non-stationary, while ARIMA models must be stationary.
The ETS models with seasonality and non-damped trend 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) – please refer to the table in the next slide.

3/31/2020 Lec 10: Seasonal ARIMA (Chapter 8) and Combination Forecast (Section 12.4) (66)
checked-new2nd ed/8_ARIMA_models_Pt2_2020.html#(66) 47/66
Equivalence relationships of some of the ETS and ARIMA models

3/31/2020 Lec 10: Seasonal ARIMA (Chapter 8) and Combination Forecast (Section 12.4) (66)
checked-new2nd ed/8_ARIMA_models_Pt2_2020.html#(66) 48/66
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 models 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, and
the likelihood is computed in different ways. The examples below demonstrate selecting between these classes of models.

3/31/2020 Lec 10: Seasonal ARIMA (Chapter 8) and Combination Forecast (Section 12.4) (66)
checked-new2nd ed/8_ARIMA_models_Pt2_2020.html#(66) 49/66
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. An example would be as follows:
fets <- function (x, h) {
forecast ( ets (x), h =
}
farima <- function (x, h) {
forecast ( auto.arima (x), h= h)
}
The returned objects can then be passed into tsCV
h)

3/31/2020 Lec 10: Seasonal ARIMA (Chapter 8) and Combination Forecast (Section 12.4) (66)
checked-new2nd ed/8_ARIMA_models_Pt2_2020.html#(66) 50/66
Using ausair data
air <- window (ausair, start= 1990 )
fets <- function (air, h) {
forecast ( ets (air), h = h)
}
farima <- function (air, h) {
forecast ( auto.arima (air), h= h)
}

3/31/2020 Lec 10: Seasonal ARIMA (Chapter 8) and Combination Forecast (Section 12.4) (66)