## 10.9 Exercises

Write out the \(\bm{S}\) matrices for the Australian tourism hierarchy and the Australian prison grouped structure. Use the

`smatrix`

command to verify your answers.Assume that a set of base forecasts are unbiased, i.e., \(E(\hat{\bm{y}}_h)=\bm{S}E(\bm{y}_{K,T+h})\). A set of coherent forecasts will also unbiased iff \(\bm{S}\bm{P}\bm{S}=\bm{S}\). In this case \(E(\tilde{\bm{y}}_h)=\bm{S}\bm{P}\bm{S}E(\hat{\bm{y}}_h)=\bm{S}E(\bm{y}_{K,T+h})\). Show that this is true for the bottom-up and optimal reconciliation approaches but not for any top-down or middle-out approaches.

Generate 8-step-ahead bottom-up forecasts using arima models for the

`vn2`

Australian domestic tourism data. Plot the coherent forecatsts by level and comment on their nature. Are you satisfied with these forecasts?Model the aggregate series for Australian domestic tourism data

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using an arima model. Comment on the model. Generate and plot 8-step-ahead forecasts from the arima model and compare these with the bottom-up forecasts generated in question 3 for the aggregate level.Generate 8-step-ahead optimally reconciled coherent forecasts using arima base forecasts for the

`vn2`

Australian domestic tourism data. Plot the coherent forecatsts by level and comment on their nature. How and why are these different to the bottom-up forecasts generated in question 3 above.Define as a test-set the last two years of the

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Australian domestic tourism data. Generate, bottom-up, top-down and optimally reconciled forecasts for this period and compare their forecasts accuracy.