Hotel B operates in a not-so-popular neighbourhood of a city and has hired you as an intern to help figure out an ideal room-pricing mechanism. Hotel A, a competitor operating in the more popular downtown region, has revealed monthly data on RoomNights (two rooms, each occupied for four nights equal eight RoomNights) and their total takings (in dollars) since January 2004. These are shown as RmNightsHotA and TotTakingsHotA in the graphs attached. Hotel B has also given you their RoomNights data (shown as RmNightsHotB), but is reluctant to divulge their total takings. They have, however, told you that an ETS(A,N,N) model will be the best to analyze TotTakingsHotB.

a. (3 pts) The trend-strength FT and seasonality-strength FS values (from additive decom- positions) for three out of these four time series are shown in the table below.

Table 1: Strength of trend and seasonality RmNightsHotA TotTakingsHotA RmNightsHotB TotTakingsHotB

FT 0.9566 0.9859 0.1316 ? FS 0.8692 0.7410 0.7139 ?

• i) RoomNights for Hotel A is more seasonal than Total Takings for Hotel A-True or False? If its true, please justify why. If its false, justify and suggest an apt graphical representation to compare the two seasonalities.

• ii) Since TotTakingsHotB is confidential, we do not have the real FT and FS values. Guess them.

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b. (6 pts) The Euclidean and correlation distances among the three available time series are shown next:

Table 2: Euclidean distances RmNightsHotA TotTakingsHotA

TotTakingsHotA 343087.95 RmNightsHotB 4834.24 347730.14

Table 3: Correlation distances RmNightsHotA TotTakingsHotA

TotTakingsHotA 0.3363 RmNightsHotB 1.2492 1.3115

• i) Which distance metric would you implement in clustering these three time series? Draw a free-hand dendrogram corresponding to the one you choose.

• ii) Interpret the highlighted value 0.3363, especially in the context of the story, in relation to Fig 1. How would you use this value in practice?

• iii) Hotel B claims that their Total Takings patterns are so different from those of Hotel A that the correlation distance between TotalTakingsHotA and TotalTakingsHotB is 3.1179. Discuss this value.

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c. (4 pts) i) Fig. 2 shows the lag-plots for RmNightsHotB. Use it to guess the seasonal periodicity.

Is this reasonable in the context of this story? ii) Use Fig. 2 to draw a free-hand ACF diagram for RmNightsHotB.

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d. (8 pts)

• i) Suggest the best order for a moving average designed to extract trend estimates for

  RmNightsHotB

• ii) Figures 3, 4, 5, and 6 represent four different decompositions of TotTakingsHotA. Identify (and provide details for) each decomposition and use the one you think most fit to guess the most recent trend estimate.

• iii) If we were to construct a 3 7 moving average for RmNightsHotB, how many trend estimates would go missing from the beginning?

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e. (6 pts)

• By looking at Fig. 1, would you expect the presence of outliers/level-shifters on RmNight-

  sHotA? If yes, where (check this formally using codes when you go home)?

• Even without looking at Fig. 1, we know RmNightsHotA must have a strong increasing

  trend since its FT value in Table 1 is close to 1 - True or False? Justify.

• Recently, Hotel A has been more concerned about the average revenue generated per night. An ETS(A,A,N) model was forced on this series, which led to an AIC of 1241.916. Will the best ETS(...) choice have an AIC greater than, equal to, or less than this value? Why?

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2. (43 points) The previous question dealt with the descriptive aspects of modeling. We now move on to the inferential side (fitting equations, generating forecasts, etc.) with the same story.

a. (4 pts) A naive model is fitted to both RmNightsHotB and TotTakingsHotB. The in-sample MAE for RmNightsHotB is 0.5945. Would the in-sample MAE for TotTak- ingsHotB be larger or smaller than this value? Why?

b. (10 pts) Fig. 7 shows the ETS(A,N,N) fit that Hotel B has revealed for the confidential series TotTakingsHotB.

i) Write out the fitted model.

ii) Interpret .

iii) The residual sequence from which fit is expected to have a smaller variance - the one from the naive fit or the one from the ETS(A,N,N) fit on TotTakingsHotB? Why?

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c. (8 pts) The last observed value for TotTakingsHotB and the last estimated level from the ETS(A,N,N) fit were 941.69 and 949.36 respectively. Find out the forecast for the next time point using:

i) the naive method. ii) the ETS(A,N,N) technique.

Would you expect the forecasts in (i) and (ii) to be similar or hugely different? Why? Do the values you just found confirm your expectation?

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d. (8 pts) Derive an expression for a 95% forecast interval for an ETS(A,N,N) model. Use it to check how far you can push the ETS(A,N,N) forecast in (ii) above with a margin of error 2 ( 1.96). You may use qnorm(0.025, 0, 1) = 1.96.

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e. (8 pts) Finally, using the last twelve observations as the test set, the following models were fit on each of RmNightHotA and RmNightHotB to gauge demands on each hotel:

• a) an ETS(A,A,N).

• b) the best ETS(...) choice using the minimum AIC criteria.

• c) an stl model applied to the untransformed data followed by an ETS model applied to the seasonally adjusted data.

  ThetablebelowsummarizessomeoftheaccuracymeasuresandFig8describestheETS(A,N,N) output for the training sets for both cities.

Table 4: Accuracy measures. RmNightHotA

RmNightHotB RMSE MAE

0.724 0.589 0.739 0.607 0.457 0.373 0.434 0.379 0.434 0.357 0.437 0.371

ETS(A,A,N) Training Testing

Best ETS Training Testing

stl Training Testing

RMSE 32.635 49.704 12.962 22.559 10.492 22.401

MAE MAPE 26.413 7.686 39.046 7.853

8.789 2.434 18.555 3.917 7.438 2.724 17.702 4.639

MAPE 6.009 5.823 3.804 3.742 3.629 4.643

• Comparing the AIC values in Fig 8, since 795.4 < 2120.6, the ETS(A,A,N) model does better on Hotel B than it does on Hotel A-True or False? In case its true, please justify further. In case its false, provide better quantitative comparisons.

• Which model (i.e., one for each hotel in each case) would you prefer if you had to

  forecast RoomNight values. simulate RoomNight values.

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f. (5 pts) Suggest improvements or further conditions that youd check before imple- menting you final choices (in the previous question) in practice (hint: be aware of the story context!).