1. Time series Model assessment

2. Tourist arrivals to NZ Period is quarterly

4. Draw graph of raw data

5. Write down the components of the time series in context.

6. Identify and discuss any trend(s) in your series. The number of tourists arriving in NZ increases with time. NOT there is an increasing trend

7. Identify and describe at least 2 further features of your series. When discussing features you need to give reasons that could account for them. There is a strong seasonal pattern shown where the no. of tourist arrival is highest during the 1 st quarter for each year and lowest in the 2 nd quarter. NOT there are seasonal effects Check out where the highs and lows are by going back to the original data set.

8. Identify and describe at least 2 further features of your series. When discussing features you need to give reasons that could account for them. Amplitude of seasonal variation is changing.(not constant) i.e. There are random fluctuations. NOT there are random effects

9. Identify and describe at least 2 further features of your series. When discussing features you need to give reasons that could account for them. It seems that the data break into three sections follows: 1980 – 1987 1987 – 1997 1997 - NOT there are ramps or irregular components Find out where changes in the pattern are happening.

10. Find MM and CMM

11. Add CMM to chart

12. Investigate models

13. Linear model OK - not perfect, but no one model is likely to be perfect in these circumstances.

14. Quadratic model Models overall quite well but doesn’t reflect the most recent data very well.

15. Polynomial model-cubic Not really suitable as the end is starting to curve downwards and this does not reflect what is happening.

16. Polynomial model-quartic Not really suitable as the end is starting to curve downwards and this does not reflect what is happening.

17. Polynomial model-5th power Worse than the last two.

18. Polynomial model-6th power Not good here but that doesn’t matter too much as we want to use the model for forecasting and so we are more interested in how the model reflects the most recent data OK-may be rising too rapidly

19. Exponential Good overall but not a very good model at the end here.

20. Decide your model and state the trend in terms of the equation of the model All four of these could be used. It is not about one being better than the other. It is about you explaining why you made the choice and then evaluating your choice once you have finished the analysis. Choosing on the basis of the R 2 being higher is not recommended- all of these have high values and all are acceptable on this basis. R 2 supports your choice.

21. Do you think it is additive or multiplicative?

22. Probably multiplicative as seasonal effects are increasing.

23. Choose your analysis Linear additive Linear multiplicative Quadratic additive Quadratic multiplicative

24. Linear - additive

25. Calculate trend, individual seasonal effects and averaged seasonal effects

26. Calculate forecast and seasonally adjusted values. Then round values to the same level as the original data for clarity.

27. Add forecast and seasonally adjusted data to a copy of the graph. Remember to take off the trend line.

28. Two things to notice.. How well does the forecast reflect the situation? How reliable will it be to use this model? Identify SAV values that are higher or lower than the CMM.

29. Name your graphs. Print two graphs and hand them in.

30. Name your worksheet. Fit your data sheet to 1 page, print in landscape and hand it in.

31. Drop your workbook into your teacher’s drop box. Conclusions

32. Linear - multiplicative

33. Calculate trend, individual seasonal effects and averaged seasonal effects

34. Calculate forecast and seasonally adjusted values. Then round values to the same level as the original data for clarity.

35. Add forecast and seasonally adjusted data to a copy of the graph. Remember to take off the trend line.

36. Two things to notice.. How well does the forecast reflect the situation? How reliable will it be to use this model? Identify SAV values that are higher or lower than the CMM.

37. Name your graphs. Print two graphs and hand them in.

38. Name your worksheet. Fit your data sheet to 1 page, print in landscape and hand it in.

39. Drop your workbook into your teacher’s drop box.

40. A linear model y = 4.1861x + 74.022 (y is the number of tourists and x represents the period) is used for making the prediction since the R 2 value is 0.9787 which is quite close to 1 indicating the CMM are closely clustered around the trend line. Explain R 2 in terms of the model (98% of the variability in the number of tourists is explained by the regression model y = 4.1861x + 74.022 ) Also comment on the goodness of fit from your forecast data. This is judged visually. Conclusions Goodness of fit

41. You will write down your forecast predictions making sure you have taken notice if figures are in thousands. Interpret the gradient (4.1861) of the linear model as ‘in each quarter numbers of tourists are increasing by approximately 4200. Forecast(s)

42. Calculation of seasonally adjusted data together with an interpretation of that data.

43. If the seasonally adjusted value is lower than the CMM, then the no. of tourist arriving to NZ for that period is lower than expected. Give a specific example. On the other hand if the seasonally adjusted value is higher than the CMM, then the no. of tourist arriving to NZ during that period is more than expected. Give a specific example.

44. A discussion of the relevance and usefulness of the forecast - An indication that conditions need to remain substantially the same for the forecast to be valuable Include a comment like - In order for the forecast to be accurate, we have to assume that the conditions for the tourist arrival to NZ need to remain substantially the same. That is no volcanic eruptions/ earthquakes occur in NZ that will decrease the number of tourist arriving. Talk about how well you think your model performs.

45. A discussion of the relevance and usefulness of the forecast The forecast estimates may be reasonably reliable for short times into the future when predicting the ‘low’ quarters, but for longer times, the estimate is liable to become less accurate. Give a comment on how the model shows this referring to your graph. The forecasts are not reliable for the ‘high’ quarters as the model does not reflect the situation. This is evident in the over-plot of forecast values.

46. Potential sources of bias The data collected are the actual numbers entering the country as tourists and hence there is not likely to be any bias.

47. Possible improvements to your model - additive Obviously this model is not a good one for predictions as it does not reflect the most recent situation in the peak seasons. An improvement would be to analyse the data using a multiplicative model as the seasonal effects are increasing in amplitude. or An improvement would be to just model (using an additive model) the most recent data but mention should be made that there appears to be a shift every 10 years and hence this model should be used with limitations.

48. Possible improvements to your model - multiplicative By using a multiplicative model, you already have displayed an improvement but add. This model is a reasonably good one for predictions in the short term however there may be a change in trend as indicated by the last 3 quarters of data compared to those forecast. An improvement would be to just model (using an additive model) the most recent data but mention should be made that there appears to be a shift every 10 years and hence this model should be used with limitations.

49. Limitations of your analysis- linear additive The model was quite accurate for the period 1986 to 1994 but is not a good one for forecasting beyond this time. This is obvious when comparing the forecast plot to the original data. The seasonally adjusted data also suggest that the model is more likely to be multiplicative as every value above 1994 is either above or below the centered moving mean and hence would be deemed unusual. This is unlikely to be the case.

50. Read pages 21 to 27 in your booklet. ‘Revision Notes and Procedure’ It will help you to understand exactly what is required and also helps with how you should word your answers. If you read these pages, you may not make careless errors. Now compare all four models

51. Quadratic - additive R 2 = 0.9831

52. Calculate trend, individual seasonal effects and averaged seasonal effects

53. Calculate forecast and seasonally adjusted values. Then round values to the same level as the original data for clarity.

54. Add forecast and seasonally adjusted data to a copy of the graph. Remember to take off the trend line.

55. Two things to notice.. How well does the forecast reflect the situation? How reliable will it be to use this model? Identify SAV values that are higher or lower than the CMM.

56. Name your graphs. Print two graphs and hand them in.

57. Name your worksheet. Fit your data sheet to 1 page, print in landscape and hand it in.

58. Drop your workbook into your teacher’s drop box. Read the conclusions for linear Models and adapt answers for The quadratic model.

59. Quadratic - multiplicative R 2 = 0.9831

60. Calculate trend, individual seasonal effects and averaged seasonal effects

61. Calculate forecast and seasonally adjusted values. Then round values to the same level as the original data for clarity.

62. Add forecast and seasonally adjusted data to a copy of the graph. Remember to take off the trend line.

63. Two things to notice.. How well does the forecast reflect the situation? How reliable will it be to use this model? Identify SAV values that are higher or lower than the CMM.

64. Name your graphs. Print two graphs and hand them in.

65. Name your worksheet. Fit your data sheet to 1 page, print in landscape and hand it in.

66. Drop your workbook into your teacher’s drop box. Read these conclusion and adapt for the quadratic multiplicative model.

67. Compare all four models

68. Using only the last few years’ data from 1997 - additive model

69. Use quadratic model

70. Forecasts and SAV

71. This is probably the best model to forecast from.