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Chapter 16 – Time Series Analysis and Index Numbers

Introduction to Business Statistics , 6e Kvanli, Pavur, Keeling. Chapter 16 – Time Series Analysis and Index Numbers. Slides prepared by Jeff Heyl, Lincoln University. ©2003 South-Western/Thomson Learning™. Times Series Analysis. Time series represents a variable observed across time.

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Chapter 16 – Time Series Analysis and Index Numbers

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  1. Introduction toBusiness Statistics, 6e Kvanli, Pavur, Keeling Chapter 16 –Time Series Analysis and Index Numbers Slides prepared by Jeff Heyl, Lincoln University ©2003 South-Western/Thomson Learning™

  2. Times Series Analysis Time series represents a variable observed across time Components of a time series • Trend (TR) • Seasonal variation (S) • Cyclical variation (C) • Irregular activity (I)

  3. Linear Trend TR = b0 + b1t Quadratic Trend TR = b0 + b1t + b2t2 Decaying Trend 1 t TR = b0 + b1or TR = b0 + b1e-1 Trend (TR)

  4. 300 – 200 – 100 – Power consumption (million kwh) | 1992 | 1993 | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 t (time) Power Example Figure 16.1

  5. 11.0 – 10.0 – 9.0 – 8.0 – 7.0 – 6.0 – 5.0 – 4.0 – 3.0 – 2.0 – 1.0 – Trend Number of employees (thousands) | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 t Employees Example Figure 16.2

  6. Yt Yt t t (a) Increasing trend (b) decreasing trend Linear Trends Figure 16.3

  7. Yt Yt t t b2< 0 (b) b2 < 0 (a) Curvilinear Models Figure 16.4

  8. Yt Yt t t b2 > 0 (d) b2 > 0 (c) Curvilinear Models Figure 16.4

  9. Seasonality (S) Seasonal variation refers to periodic increases or decreases that occur within a calendar year in a time series. The key is that these movements in the time series follow the same pattern each year

  10. 40 – 35 – 30 – 25 – 20 – 15 – 10 – Power consumption (millions kwh) | | | | | | | | | Jan Jul Dec Jan Jul Dec Jan Jul Dec 2001 1999 2000 Seasonal Variation Figure 16.5

  11. 4 – 3 – 2 – 1 – Linear trend Sales of Wildcat sailboats (millions of dollars) | July 1998 | July 1999 | July 2000 | July 2001 t Seasonal Variation Figure 16.6

  12. Cyclical Variation (C) Cyclical variation describes a gradual cyclical movement about the trend; it is generally attributable to business and economic conditions The length of the cycle is the period of that cycle and is measured from one peak to the next

  13. P1 P2 Z1 Z2 Cyclical activity V1 V2 t Cyclical Variation Figure 16.7

  14. 4.0 – 3.5 – 3.0 – 2.5 – 2.0 – 1.5 – 1.0 – Corporate taxes (millions of dollars) 1 2 3 | 1975 | 1985 | 1995 | 2000 Textile Example Figure 16.8

  15. Irregular Activity Irregular activity consists of what is “left over” after accounting for the effect of any trend, seasonality, or cyclical activity

  16. Additive Structure yt = TRt + St + Ct + It Multiplicative Structure yt = TRt • St • Ct• It Combining Components

  17. ∑t = 1 + 2 + ... + T = ∑t2 = 1 + 4 + ... + T2 = T(T + 1)(2T + 1) 6 T(T + 1) 2 T + 1 2 ∑t T A T T + 1 2 t = = b0 = - b1 12B - 6(T + 1)A T(T2 - 1) b1 = Measuring Trend Linear Trend

  18. Yt 12 – – – 9 – – – 6 – – – 3 – – – – Number of employees (thousands) | | | | | | | | 1994 (t = 1) 1995 (t = 2) t 2001 (t = 8) Year Trend Line using Coded Data Figure 16.9

  19. Yt 12 – – – 9 – – – 6 – – – 3 – – – – Number of employees (thousands) yt = TRt = b0 + b1t | | | | | | | | 1994 (t = 1) 1995 (t = 2) t 2001 (t = 8) Year Trend Line using Coded Data Figure 16.9

  20. Excel Solution Figure 16.10

  21. Quadratic Trend Figure 16.11

  22. Yt Yt b1 2b2 b1 2b2 Time (t) Time (t) t = - t = - A B Illustration of Quadratic Trend Lines Figure 16.12

  23. yt yt Ct ^ Measuring Cyclical Activity yt = TRt • Ct • It

  24. Yt Trend Ct > 1 Ct < 1 1 complete cycle Time Complete Cycle Figure 16.13

  25. ^ t yt yt Ct 1 1.1 1.125 .978 2 2.4 2.529 .949 3 4.6 3.933 1.169 4 5.4 5.337 1.012 5 5.9 6.741 .875 6 8.0 8.145 .982 7 9.7 9.549 1.016 8 11.2 10.953 1.022 yt yt ^ Trend and Cyclical Activity Table 16.1

  26. Yt 11.0 – 10.0 – 9.0 – 8.0 – 7.0 – 6.0 – 5.0 – 4.0 – 3.0 – 2.0 – 1.0 – Actual yt yt = -.279 + 1.404t (trend line) Number of employees (thousands) | 1994 | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 t Cyclical Activity Figure 16.14

  27. Ct 1.15 – 1.10 – 1.05 – 1.00 – .95 – .90 – Start End | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 t 1994 1996 1998 2000 Cyclical Components Figure 16.15

  28. Yt Trend 2000 – 1500 – 1000 – 500 – 100 units 100 units Actual time series Units sold 100 units | Winter 1999 | Winter 2000 | Winter 2001 t Additive Seasonal Variation Figure 16.16

  29. Yt 700 – 600 – 500 – 400 – 300 – 200 – 100 – TRt = 100 + 20t Sales (tens of thousands of dollars) Estimated sales using trend and seasonality | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 t Jetski Sales Figure 16.17

  30. Yt 2000 – 1500 – 1000 – 500 – 250 units Trend 180 units Units sold Actual time series 100 units | Winter 1999 | Winter 2000 | Winter 2001 t Heat Pump Sales Figure 16.18

  31. Yt 700 – 600 – 500 – 400 – 300 – 200 – 100 – TRt = 100 + 20t Sales (tens of thousands of dollars) Estimated sales using trend and seasonality | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 t Jetski Sales - Multiplicative Season Variation Figure 16.19

  32. Four Step Procedure for Decomposition Determine a seasonal index, St, for each time period Deseasonalize the data, dt = TRt • Ct • It Determine the trend component, TRt Determine the cyclical component, Ct

  33. Time Quarter tyt Moving Totals 1990 1 1 85 (1) 263 2 2 41 (2) 268 3 3 92 (3) 270 4 4 45 and so on 1991 1 5 90 2 6 43 3 7 95 4 8 47 1992 1 9 92 . . . . . . . . . Centered Moving Averages Table 16.2

  34. Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 1998 20 12 47 60 1999 40 32 65 76 2000 56 50 85 100 2001 75 70 101 123 Sales Data for Video-Comp Table 16.3

  35. Centered Ratio to Moving Moving Moving Year Quarter t yt Total Average Average 1998 1 1 20 — — 2 2 12 — — — 3 3 47 139 37.25 1.26 4 4 60 159 42.25 1.42 1999 1 5 40 179 47.00 .85 2 6 32 197 51.25 .62 3 7 65 213 55.25 1.18 4 8 76 229 59.20 1.28 2000 1 9 56 247 64.25 .87 2 10 50 267 69.75 .72 3 11 85 291 75.13 1.13 4 12 100 310 80.00 1.25 2001 1 13 75 330 84.50 .89 2 14 70 346 89.38 .78 3 15 101 369 — — 4 16 123 — — — Moving Averages for Video-Comp Table 16.3

  36. Yt 120 – 100 – 80 – 60 – 40 – 20 – Sales (number of units) Moving averages (no seasonality) | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 t 2000 1998 2001 1999 Quarters by year Smoothing a Time Series Figure 16.20

  37. Quarter 1 Quarter 2 Quarter 3 Quarter 4 — — 1.26 1.42 .85 .62 1.18 1.28 .87 .72 1.13 1.25 .89 .78 Total 2.61 2.12 3.57 3.95 Average 0.870 0.707 1.190 1.317 — — Ratios for Each Quarter Table 16.5

  38. Deseasonalized Values Seasonal Year tYt Index (St) dt = — Yt St 1998 1 20 .852 23.47 2 12 .692 17.34 3 47 1.166 40.31 4 60 1.290 46.51 1999 1 40 .852 46.95 2 32 .692 46.24 3 65 1.166 55.75 4 76 1.290 58.91 2000 1 56 .852 65.73 2 50 .692 72.25 3 85 1.166 72.90 4 100 1.290 77.52 2001 1 75 .852 88.03 2 70 .692 101.16 3 101 1.166 86.62 4 123 1.290 95.35 Deseasonalizing Data Table 16.6

  39. 1997 1998 1999 2000 Jan 134.738 139.935 146.613 158.691 Feb 130.255 135.538 145.121 164.725 Mar 148.497 151.118 165.736 183.875 Apr 145.703 155.820 166.011 178.776 May 156.603 162.797 173.496 190.753 Jun 150.915 159.701 171.286 187.868 Jul 153.200 161.541 172.364 182.891 Aug 156.782 162.369 174.788 191.647 Sep 149.407 155.747 169.809 183.229 Oct 157.523 164.528 174.740 186.550 Nov 161.925 169.914 185.347 198.706 Dec 203.117 215.590 238.452 243.255 Total U.S. Retail Trade Table 16.7

  40. Month (Period) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1997 0.993 1.013 0.964 1.013 1.036 1.295 1998 0.888 0.857 0.952 0.979 1.018 0.994 1.000 1.001 0.954 1.002 1.029 1.299 1999 0.878 0.864 0.981 0.976 1.014 0.992 0.990 0.996 0.959 0.980 1.032 1.317 2000 0.871 0.899 0.996 0.963 1.022 1.003 Average 0.879 0.837 0.977 0.973 1.018 0.996 0.994 1.004 0.959 0.998 1.033 1.304 Summary of Ratios Table 16.9

  41. 200 – 190 – 180 – 170 – 160 – 150 – 140 – • • • • • • • • • • • • • • • • • • • • • Deseasonalized Values (dt) • • • • • • • • • • • • • • • • • • • • • • • • • • | 0 | 10 | 20 | 30 | 40 | 50 Time (t) Deseasonalized Data Figure 16.21

  42. dt dt 3-Month Moving Average (Ct) ^ tdt dt —(= Ct • It) ^ 1 153.349 146.470 - .934(1) = 146.979 1.0433 — 2 149.238 146.470 - .934(2) = 147.913 1.0090 1.025 3 152.165 146.470 + .934(3) = 148.847 1.0223 1.011 4 149.871 146.470 + .934(4) = 149.782 1.0006 1.015 5 153.899 146.470 + .934(5) = 150.716 1.0211 1.007 6 151.602 146.470 + .934(6) = 151.650 0.9997 1.010 . . . . . . . . . . . . . . . Cyclical Components Table 16.12

  43. 1.03 – 1.02 – 1.01 – 1.00 – 0.99 – 0.98 – 0.97 – Cyclical Components | 0 | 10 | 20 | 30 | 40 | 50 Month Feb Jan Jan Jan Year 1997 1998 1999 2000 Plot of Cyclical Activity Figure 16.22

  44. Excel Plots Figure 16.23

  45. 1985 1990 1995 2000 Wage $7.05 $8.50 $10.90 $12.50 Index (base = 1980) 100 120.6 154.6 177.3 Index Numbers Table 16.15

  46. Simple Aggregate Price Index = • 100 ∑P1 ∑P0 Weighted Aggregate Price Index = • 100 ∑P1Q ∑P0Q Laspeyres Index = • 100 ∑P1Q0 ∑P0Q0 ∑P1Q1 ∑P0Q1 Paasche Index = • 100 Price Indexes

  47. LongLife Item 1990 2000 Eggs .75 (doz) 1.35 Chicken .95 (lb) 1.79 Cheese .89 (lb) 1.85 Auto battery $31.00 (each) $55.00 (each) Prices of Four Items Table 16.16

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