## 10.9 Exercises

1. 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.

2. 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.

3. 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?

4. Model the aggregate series for Australian domestic tourism data vn2 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.

5. 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.

6. Define as a test-set the last two years of the vn2 Australian domestic tourism data. Generate, bottom-up, top-down and optimally reconciled forecasts for this period and compare their forecasts accuracy.