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Chapter 13 – Time Series Analysis and Forecasting

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KVANLI PAVUR KEELING. Concise Managerial Statistics. Chapter 13 – Time Series Analysis and Forecasting. Slides prepared by Jeff Heyl Lincoln University. ©2006 Thomson/South-Western. Times Series Analysis. Time series represents a variable observed across time.

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  1. KVANLI PAVUR KEELING Concise Managerial Statistics Chapter 13 –Time Series Analysis and Forecasting Slides prepared by Jeff Heyl Lincoln University ©2006 Thomson/South-Western

  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) | 1995 | 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 t (time) Power Example Figure 13.1

  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) | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 t Employees Example Figure 13.2

  6. Yt Yt t t (a) Increasing trend (b) Decreasing trend Linear Trends Figure 13.3

  7. Yt Yt t t b2< 0 (b) b2 < 0 (a) Curvilinear Models Figure 13.4

  8. Yt Yt t t b2 > 0 (d) b2 > 0 (c) Curvilinear Models Figure 13.4

  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 2002 2003 2004 Seasonal Variation Figure 13.5

  11. 4 – 3 – 2 – 1 – Linear trend Sales of Wildcat sailboats (millions of dollars) | July 2001 | July 2002 | July 2003 | July 2004 t Seasonal Variation Figure 13.6

  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 13.7

  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 13.8

  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. Yt 12 – – – 9 – – – 6 – – – 3 – – – – Number of employees (thousands) | | | | | | | | 1997 (t = 1) 1998 (t = 2) t 2004 (t = 8) Year Trend Line using Coded Data Figure 13.9

  18. Yt 12 – – – 9 – – – 6 – – – 3 – – – – Number of employees (thousands) yt = TRt = b0 + b1t | | | | | | | | 1997 (t = 1) 1998 (t = 2) t 2004 (t = 8) Year Trend Line using Coded Data Figure 13.9

  19. T(T + 1) 2 ∑t = 1 + 2 + ... + T = ∑t2 = 1 + 4 + ... + T2 = T(T + 1)(2T + 1) 6 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

  20. Computer Solution Figure 13.10

  21. Quadratic Trend Figure 13.11

  22. Yt Yt b1 2b2 Time (t) Time (t) b1 2b2 t = − t = − (a) (b) Illustration of Quadratic Trend Lines Figure 13.12

  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 13.13

  25. ^ ^ t yt yt yt /yt 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 Trend and Cyclical Activity Table 13.1

  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) | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 t Cyclical Activity Figure 13.14

  27. Ct 1.15 – 1.10 – 1.05 – 1.00 – .95 – .90 – Start End | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 t 1997 1999 2001 2003 Cyclical Components Figure 13.15

  28. Yt Trend 2000 – 1500 – 1000 – 500 – 100 units 100 units Actual time series Units sold 100 units | Winter 2002 | Winter 2003 | Winter 2004 t Additive Seasonal Variation Figure 13.16

  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 13.17

  30. Yt 2000 – 1500 – 1000 – 500 – 250 units Trend 180 units Units sold Actual time series 100 units | Winter 2002 | Winter 2003 | Winter 2004 t Heat Pump Sales Figure 13.18

  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 13.19

  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. (1) 263 (2) 268 (3) 270 and so on Centered Moving Averages Time Quarter tyt Moving Totals 1990 1 1 85 2 2 41 3 3 92 4 4 45 1991 1 5 90 2 6 43 3 7 95 4 8 47 1992 1 9 92 . . . . . . . . . Table 13.2

  34. Sales Data for Video-Comp Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 2001 20 12 47 60 2002 40 32 65 76 2003 56 50 85 100 2004 75 70 101 123 Table 13.3

  35. Centered Ratio to Moving Moving Moving Year Quarter t yt Total Average Average 2001 1 1 20 — — 2 2 12 — — 3 3 47 37.25 1.26 4 4 60 42.25 1.42 2002 1 5 40 47.00 .85 2 6 32 51.25 .62 3 7 65 55.25 1.18 4 8 76 59.20 1.28 2003 1 9 56 64.25 .87 2 10 50 69.75 .72 3 11 85 75.13 1.13 4 12 100 80.00 1.25 2004 1 13 75 84.50 .89 2 14 70 89.38 .78 3 15 101 — — 4 16 123 — — — 139 159 179 197 213 229 247 267 291 310 330 346 369 — Moving Averages for Video-Comp Table 13.4

  36. Yt 120 – 100 – 80 – 60 – 40 – 20 – Sales (number of units) Moving averages (no seasonality) | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 t 2001 2002 2003 2004 Quarters by year Smoothing a Time Series Figure 13.20

  37. Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 — — 2001 1.26 1.42 2002 .85 .62 1.18 1.28 2003 .87 .72 1.13 1.25 2004 .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 13.5

  38. Seasonal Deseasonalized Year tYt Index (St) Values 2001 1 20 .852 23.47 2 12 .692 17.34 3 47 1.166 40.31 4 60 1.290 46.51 2002 1 40 .852 46.95 2 32 .692 46.24 3 65 1.166 55.75 4 76 1.290 58.91 2003 1 56 .852 65.73 2 50 .692 72.25 3 85 1.166 72.90 4 100 1.290 77.52 2004 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 13.6

  39. Total U.S. Retail Trade 2000 2001 2002 2003 Jan 159.708 172.213 175.583 185.874 Feb 165.178 167.020 170.519 177.619 Mar 184.442 187.256 193.154 198.509 Apr 178.753 187.385 192.719 200.147 May 190.849 200.634 206.194 211.749 Jun 188.151 193.729 195.771 203.683 Jul 182.889 188.590 196.938 208.196 Aug 191.524 199.557 204.015 214.858 Sep 183.312 181.118 187.578 202.581 Oct 186.906 192.052 200.275 213.712 Nov 198.733 202.614 209.076 218.026 Dec 243.110 243.445 251.152 268.887 Table 13.7

  40. Moving Centered Ratio to Total Moving Moving Year Month t(1) Yt(2) (3) Average (4) Average (5) 2000 Jan 1 159.708 Feb 2 165.178 Mar 3 184.442 Apr 4 178.753 May 5 190.849 Jun 6 188.151 Jul 7 182.889 188.317 0.971 Aug 8 191.524 188.915 1.014 Sep 9 183.312 189.109 0.969 Oct 10 186.906 189.586 0.986 Nov 11 198.733 190.353 1.044 Dec 12 243.110 190.994 1.273 2253.555 2266.060 2267.902 2270.716 2279.348 2289.133 2294.711 Moving Averages and Ratios Table 13.8

  41. Moving Centered Ratio to Total Moving Moving Year Month t(1) Yt(2) (3) Average (4) Average (5) 2003 Jan 37 185.874 202.762 0.917 Feb 38 177.619 203.645 0.872 Mar 39 198.509 204.685 0.970 Apr 40 200.147 205.870 0.972 May 41 211.749 206.803 1.024 Jun 42 203.683 207.914 0.980 Jul 43 208.196 Aug 44 214.858 Sep 45 202.581 Oct 46 213.712 Nov 47 218.026 Dec 48 268.887 2438.773 2448.716 2463.719 2477.156 2486.106 2503.841 Moving Averages and Ratios Table 13.8 (cont.)

  42. Summary of Ratios Month Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 2000 0.971 1.014 0.969 0.986 1.044 1.273 2001 0.899 0.870 0.974 0.974 1.041 1.004 0.977 1.032 0.935 0.989 1.041 1.248 2002 0.898 0.870 0.983 0.978 1.043 0.987 0.989 1.026 0.936 0.9997 1.038 1.244 2003 0.917 0.872 0.970 0.972 1.024 0.980 Average 0.907 0.871 0.976 0.975 1.036 0.990 0.979 1.024 0.947 0.991 1.041 1.255 Table 13.9

  43. Deseasonalized Retail Trade Year Month t Yt St dt 2000 Jan 1 159.708 0.906 176.322 Feb 2 165.178 0.872 189.530 Mar 3 184.442 0.977 188.872 Apr 4 178.753 0.976 183.224 May 5 190.849 1.037 184.060 Jun 6 188.151 0.991 189.805 Jul 7 182.889 0.980 186.617 Aug 8 191.524 1.025 186.886 Sep 9 183.312 0.948 193.415 Oct 10 186.906 0.992 188.494 Nov 11 198.733 1.042 190.723 Dec 12 243.110 1.256 193.526 Table 13.10

  44. Deseasonalized Retail Trade Year Month t Yt St dt 2003 Jan 37 185.874 0.906 205.210 Feb 38 177.619 0.872 203.806 Mar 39 198.509 0.977 203.277 Apr 40 200.147 0.976 205.153 May 41 211.749 1.037 204.217 Jun 42 203.683 0.991 205.474 Jul 43 208.196 0.980 212.440 Aug 44 214.858 1.025 209.655 Sep 45 202.581 0.948 213.745 Oct 46 213.712 0.992 215.528 Nov 47 218.026 1.042 209.238 Dec 48 268.887 1.256 214.045 Table 13.10 (cont.)

  45. t dtt •dt 1 176.322 176.322 2 189.530 379.060 3 188.872 566.616 4 183.224 732.896 . . . 45 213.745 9618.545 46 215.528 9614.288 47 209.238 9834.186 48 214.045 10274.160 9,450.532 236,838.929 Calculations for Trend Line Table 13.11

  46. Deseasonalized Data Figure 13.21

  47. dt dt 3-Month Moving Average (Ct) ^ tdt dt — (= Ct • It) ^ 1 176.322 182.778 - .575(1) = 183.363 0.9616 — 2 189.238 182.778 - .575(2) = 183.939 1.0304 1.005 3 188.872 182.778 - .575(3) = 184.514 1.0236 1.015 4 183.224 182.778 - .575(4) = 185.090 0.9899 1.002 5 184.060 182.778 - .575(5) = 185.665 0.9914 1.000 6 189.805 182.778 - .575(6) = 186.241 1.0191 1.003 . . . . . . . . . . . . . . . Cyclical Components Table 13.12

  48. Three-Month dt/dt Moving Year Month tdt dt (= Ct● It) Average (Ct) 2000 Jan 1 176.322 183.363 0.9616 — Feb 2 189.530 183.939 1.0304 1.005 Mar 3 188.872 184.514 1.0236 1.015 Apr 4 183.224 185.090 0.9899 1.002 May 5 184.060 185.665 0.9914 1.000 Jun 6 189.805 186.241 1.0191 1.003 Jul 7 186.617 186.816 0.9989 1.005 Aug 8 186.886 187.391 0.9973 1.008 Sep 9 193.415 187.967 1.0290 1.009 Oct 10 188.494 188.542 0.9997 1.012 Nov 11 190.723 189.118 1.0085 1.009 Dec 12 193.526 189.693 1.0202 1.009 ^ ^ Cyclical Components of Retail Trade Table 13.13

  49. Three-Month dt/dt Moving Year Month tdt dt (= Ct● It) Average (Ct) 2003 Jan 37 205.210 204.079 1.0055 0.995 Feb 38 203.806 204.654 0.9959 0.997 Mar 39 203.277 205.230 0.9905 0.994 Apr 40 205.153 205.805 0.9968 0.992 May 41 204.217 206.381 0.9895 0.993 Jun 42 205.474 206.956 0.9928 1.002 Jul 43 212.440 207.532 1.0237 1.008 Aug 44 209.655 208.107 1.0074 1.018 Sep 45 213.745 208.982 1.0243 1.021 Oct 46 215.528 209.258 1.0300 1.017 Nov 47 209.238 209.833 0.9972 1.015 Dec 48 214.045 210.409 1.0173 — ^ ^ Cyclical Components of Retail Trade Table 13.13 (cont.)

  50. Plot of Cyclical Activity Figure 13.22