1. Chapter 2 Time series Dr. A. PHILIP AROKIADOSS Assistant Professor Department of Statistics St. Joseph’s College (Autonomous) Tiruchirappalli-620 002.

2. Time series and Trend analysis A time series consists of a set of observations measured at specified, usually equal, time interval. Time series analysis attempts to identify those factors that exert influence on the values in the series.

3. . Time series analysis is a basic tool for forecasting. Industry and government must forecast future activity to make decisions and plans to meet projected changes. An analysis of the trend of the observations is needed to acquire an understanding of the progress of events leading to prevailing conditions.

4. . The trend is defined as the long term underlying growth movement in a time series. Accurate trend spotting can only be determined if the data are available for a sufficient length of time. Forecasting does not produce definitive results. Forecasters can and do get things wrong from election results and football scores to the weather.

5. Time series examples Sales data Gross national product Share prices $A Exchange rate Unemployment rates Population Foreign debt Interest rates

6. Time series components Time series data can be broken into these four components: 1.Secular trend 2.Seasonal variation 3.Cyclical variation 4.Irregular variation

7. Components of Time-Series Data 12345678910111213 Year Seasonal Cyclical Trend Irregular fluctuations Predicting long term trends without smoothing? What could go wrong? Where do you commence your prediction from the bottom of a variation going up or the peak of a variation going down………..

8. 1. Secular Trend This is the long term growth or decline of the series. In economic terms, long term may mean >10 years Describes the history of the time series Uses past trends to make prediction about the future Where the analyst can isolate the effect of a secular trend, changes due to other causes become clearer

10. . Look out While trend estimates are often reliable, in some instances the usefulness of estimates is reduced by: by a high degree of irregularity in original or seasonally adjusted series or by abrupt change in the time series characteristics of the original data

11. $A vs $US during day 1 vote count 2000 US Presidential election. This graph shows the amazing trend of the $A vs $UA during an 18 hour period on November 8, 2000

12. 2. Seasonal Variation The seasonal variation of a time series is a pattern of change that recurs regularly over time. Seasonal variations are usually due to the differences between seasons and to festive occasions such as Easter and Christmas. Examples include: Air conditioner sales in Summer Heater sales in Winter Flu cases in Winter Airline tickets for flights during school vacations

13. Monthly Retail Sales in NSW Retail Department Stores

15. 3. Cyclical variation Cyclical variations also have recurring patterns but with a longer and more erratic time scale compared to Seasonal variations. The name is quite misleading because these cycles can be far from regular and it is usually impossible to predict just how long periods of expansion or contraction will be. There is no guarantee of a regularly returning pattern.

16. Cyclical variation Example include: Floods Wars Changes in interest rates Economic depressions or recessions Changes in consumer spending

18. Cyclical variation This chart represents an economic cycle, but we know it doesn’t always go like this. The timing and length of each phase is not predictable.

19. 4. Irregular variation An irregular (or random) variation in a time series occurs over varying (usually short) periods. It follows no pattern and is by nature unpredictable. It usually occurs randomly and may be linked to events that also occur randomly. Irregular variation cannot be explained mathematically.

20. Irregular variation If the variation cannot be accounted for by secular trend, season or cyclical variation, then it is usually attributed to irregular variation. Example include: – Sudden changes in interest rates – Collapse of companies – Natural disasters – Sudden shift s in government policy – Dramatic changes to the stock market – Effect of Middle East unrest on petrol prices

21. Monthly Value of Building Approvals ACT)

23. . Now we have considered the 4 time series components: 1.Secular trend 2.Seasonal variation 3.Cyclical variation 4.Irregular variation We now take a closer look at secular trends and 5 techniques available to measure the underlying trend.

25. Why examine the trend? When a past trend can be reasonably expected to continue on, it can be used as the basis of future planning: Capacity planning for increased population Utility loads Market progress

27. Measure underlying trend freehand graph (plot)

29. Measure underlying trend Semi-averages This technique attempts to fit a straight line to describe the secular trend: i.Divide data into 2 equal time ranges ii.Calculate the average of the observations in each of the 2 time ranges iii.Draw a straight line between the 2 points iv.Extend line slightly past the end of the original observation to make predictions for future years

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32. Measure underlying trend Moving averages This method is based on the premise that if values in a time series are averaged over a sufficient period, the effect of short term variations will be reduced. That is, short term, cyclical, seasonal and irregular variations will be smoothed out resulting in an apparently smooth graph depicting the overall trend. The degree of smoothing can be controlled by selecting the number of cases to be included in an average.

33. The technique for finding a moving average for a particular observation is to find the average of the m observations before and after the observation itself. That is, a total of (2m + 1 ) observations must be averaged each time a moving average is calculated. YearSales 200113 200215 200317 200418 200519 200620 200720 200821 200922 Find: both 3 and 5 year moving averages (or mean smoothing) for this time series:

34. 13 + 15 +17 = 45 15 + 17 + 18 = 50…. Or 45 +18 - 13 = 50 (pick up the new drop off the old)

36. 13 + 15 + 17 + 18 = 63, 63 / 4 = 15.75 15.75 + 17.25 = 33 …… 33 / 2 = 16.5

37. Measure underlying trend Least squares linear regression A more sophisticated way to of fitting a straight line to a time series is to use the method of least squares regression. Sound familiar? The observations are the dependant y variables and time is the independent x variable.

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39. Least squares regression example YearYxx^2xy 200313 200415 200517 200618 200719 200820 200920 201021 201122 165 Soft Drink Sales $’000 for Carbonated Pty Ltd

41. Calculation for least squares regression example.

42. Least squares regression line. Yt = 18.3 + 1.03 x What is the predicted Sales figure for 2013? Hint: what value is x in 2013?

43. . Yt = 18.3 + 1.03 x What is the predicted Sales figure for 2013? Hint: what value is x in 2013? If 2011 = 4, then 2013 = 6, so 6 2 nd f ( = 24.53

44. A quick recap….

45. Line of Best Fit – plot points then draw a line that can be adjusted by clicking either end of the line

46. Semi Averaging Method at 2002-3 = 15.75 and 2007-8 = 20.75

47. Moving Averages (or mean smoothing)

48. The following 2 slides should be reviewed at home (with strong coffee) …

51. Measure underlying trend – exponential smoothing

52. . The trend line is calculated using this formula where: S x = the smoothed value for observation x Y = the actual value of observation x S x-1 = the smoothed value previously calculated for observation (x-1)  =the smoothing constant where 0    1

53. . The exponential smoothing approach has to have a starting point, so we choose the first smoothed value (S 1 ) to be the first observation (Y 1 ) in the time series. The smoothed value of each observation is a function of the smoothed value of the observation immediately before it. Errors made in calculating the smoothed values will carry on for each successive time period.

54. Exponential smoothing  = 0.4 YearSales (Y)S x-1 (1-  ) S x-1 YY SxSx 200412,000 200512,50012,0007,2005,000 200612,200 4,880 200713,000 5,200 200813,500 5,400 200913,400 5,360 201014,000 5,600

55. Complete the table below 2005 - 12500 x.4 = 5000, 12000 x.6 = 7200, then Sx = 7200 + 5000 = 12200 Bring Sx figure down to Sx-1 for start of 2006 (figure is 12200) 2006 - 12200 x.4 = 4880, 12200 x.6 = 7320, Sx = 4880 + 7320 = 12200 2007 - Sx-1 = 12200, 13000 x.4 = 5200, 12200 x.6=7320, Sx = 5200+7320 = 12520 2008 - Sx-1 = 12520, 13500 x.4=5400, 12520 x.6=7512, etc.

57. Self testing exercise 1  = 0.3

59. The ABS has some useful information on Seasonal Data

60. . A seasonally adjusted series involves estimating and removing the cyclical and seasonal effects from the original data.estimating Seasonally adjusting a time series is useful if you wish to understand the underlying patterns of change or movement in a population, without the impact of the seasonal or cyclical effects. For example, employment and unemployment are often seasonally adjusted so that the actual change in employment and unemployment levels can be seen, without the impact of periods of peak employment such as Christmas/New Year when a large number of casual workers are temporarily employed.

62. . Suppose there were some adverse publicity in December about ice-cream. It would be incorrect simply to compare the sales of ice-cream in June with those in December to determine the effect of the bad publicity. December would have higher sales in any case, because it is warmer. Useful comparisons of sales could only be made after seasonal variation had been allowed for. The true impact of the publicity could then be determined.

63. Show how “D = A / Si” can be re-arranged to be Si = A / D – memory trigger the word “SAID” except the I is the division symbol or the word “DAISY” with “I” as the division symbol

64. You tube clip calculating Seasonal indices http://www.youtube.com/watch?v=7Ug2zA0M-1Y&feature=related

65. SI - simple average method The simple average method takes the average for each period (period mean) over a period of at least three years, and expresses that as an index by comparing it to the average of all periods over the same period of time. Note: indices can be based on periods such as months or weeks.

67. Yearly average 1994 = 43 + 64 + 63 + 41 = 211/4 = 52.75 Yearly average 1995 = 46 + 64 + 67 + 43 = 220 /4 = 55.0 Yearly average 1996= 51 + 69 + 75 + 39 = 234/4 = 58.5 Yearly average 1997= 55 + 73 + 79 + 48 = 255 / 4 =63.75

68. . To calculate quarterly SI using same technique as video: 1994 Q1 Seasonal Index is 1994 Q1 sales/ average for 1994 = 43 / 52.75 = 0.82 Q1Q2Q3Q4 19940.821.211.190.78 19950.841.161.220.78 19960.871.181.280.67 19970.861.151.240.75 total3.394.704.932.98

69. . total3.394.704.932.98 to get SI Divide by 40.851.181.230.75

70. Alternate approach to calculate Seasonal Index number for each quarter note: same result!

71. D = A / SI x 100 60 = 51 / 84.8 x 100

73. D = A / SI x 100 393 = 362 / 92.1 x 100 So even though Actual sales in the 4 th quarter were $357 – once we take the seasonal factor out of the equation we can see it was the best performing quarter.

74. Projected figures will always be performed on deseasonalised figures D = A / SI x 100 Therefore A = D x SI / 100

76. D = A / SI x 100 D = 18530 / 94.3 x 100 = 19650

78. D = A / SI x 100 Therefore A = D x SI / 100 A = 2070 x 87.2 / 100 = 1805

80. Quarter 1 = 1.68 + 1.75 + 1.80 + 1.76 + 1.85 = 8.84 / 5 = 1.768 Quarter 2 = 1.89 + 2.84 + 2.96 +3.03 + 3.08 = 13.8 / 5 = 2.76 Quarter 3 = 2.06 + 2.1 + 2.08 + 2.14 +2.19 = 10.57 / 5 = 2.114 Quarter 4 = 3.25 + 3.3 + 3.32 + 3.39 + 3.38 = 16.64 / 5 = 3.328 Average for the quarters = 1.768 + 2.76 + 2.114 + 3.328 = 9.97 / 4 = 2.4925 Index for Quarter 1 = 1.768 / 2.4925 x 100 = 70.93

81. D = A / SI x 100 1997 Quarter 1 = D = 1.85 / 70.93 x 100 = 2.61

82. 11.24 / 4 = Quarterly deseasonalised data = 2.81 D = A / SI x 100 Therefore A = D x SI / 100 Quarter 1 = A = 2.81 x 70.93 / 100 = 1.99