Please assist with question $4$ : Question $3$

$3.$Data set global$\_$economy contains the annual Exports from many countries. Select one country to analyse.

a$.$ Plot the Exports series and discuss the main features of the data.

b$.$ Use an ETS$($A$,$N$,$N$)$ model to forecast the series, and plot the forecasts.

c$.$ Compute the RMSE values for the training data.

d$.$ Compare the results to those from an ETS$($A$,$A$,$N$)$ model. $($Remember that the trended model is using one more parameter than the simpler model.$)$ Discuss the merits of the two forecasting methods for this data set.

e$.$ Compare the forecasts from both methods. Which do you think is best?

f$.$ Calculate a $95\%$ prediction interval for the first forecast for each model, using the RMSE values and assuming normal errors. Compare your intervals with those produced using R$.$

Question $4$

$4.$For this exercise use the quarterly number of arrivals to Australia from New Zealand, $1981$ Q$12012$ Q$3,$ from data set aus$\_$arrivals.

a$.$ Make a time plot of your data and describe the main features of the series.

b$.$ Create a training set that withholds the last two years of available data. Forecast the test set using an appropriate model for Holt$-$Winters$$ multiplicative method.

c$.$ Why is multiplicative seasonality necessary here?

d$.$ Forecast the two$-$year test set using each of the following methods:

$$ an ETS model;

$$ an additive ETS model applied to a log transformed series;

$$ a seasonal na$$ve method;

$$ an STL decomposition applied to the log transformed data followed by an ETS model applied to the seasonally adjusted $($transformed$)$ data.

e$.$ Which method gives the best forecasts? Does it pass the residual tests?

f$.$ Compare the same four methods using time series cross$-$validation instead of using a training and test set. Do you come to the same conclusions