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©2003 Thomson/South-Western 1 Chapter 16 – Time Series Analysis and Index Numbers Slides prepared by Jeff Heyl, Lincoln University ©2003 South-Western/Thomson.

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Presentation on theme: "©2003 Thomson/South-Western 1 Chapter 16 – Time Series Analysis and Index Numbers Slides prepared by Jeff Heyl, Lincoln University ©2003 South-Western/Thomson."— Presentation transcript:

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

2 ©2003 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 ©2003 Thomson/South-Western 3 Trend (TR) Linear Trend TR = b 0 + b 1 t Quadratic Trend TR = b 0 + b 1 t + b 2 t 2 Decaying Trend TR = b 0 + b 1 or TR = b 0 + b 1 e -1 1t

4 ©2003 Thomson/South-Western 4 Power Example Figure 16.1 300 300 – 200 200 – 100 100 – Power consumption (million kwh) |1994 1995 1996 1997 1998 1999 2000 2001 t (time) |1992 1993

5 ©2003 Thomson/South-Western 5 Employees Example Figure 16.2 11.0 11.0 – 10.0 10.0 – 9.0 9.0 – 8.0 8.0 – 7.0 7.0 – 6.0 6.0 – 5.0 5.0 – 4.0 4.0 – 3.0 3.0 – 2.0 2.0 – 1.0 1.0 – Number of employees (thousands) |1994 1995 1996 1997 1998 1999 2000 2001 t Trend

6 ©2003 Thomson/South-Western 6 Linear Trends YtYtYtYt t (a) Increasing trend YtYtYtYt t (b) decreasing trend Figure 16.3

7 ©2003 Thomson/South-Western 7 Curvilinear Models Figure 16.4 YtYtYtYt t b 2 < 0 (a) YtYtYtYt t (b)

8 ©2003 Thomson/South-Western 8 Curvilinear Models Figure 16.4 YtYtYtYt t b 2 > 0 (c) YtYtYtYt t (d)

9 ©2003 Thomson/South-Western 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 ©2003 Thomson/South-Western 10 Seasonal Variation 40 40 – 35 35 – 30 30 – 25 25 – 20 20 – 15 15 – 10 10 – Power consumption (millions kwh) ||||||||| Jan Jul Dec 1999 2000 2001 Figure 16.5

11 ©2003 Thomson/South-Western 11 Seasonal Variation Figure 16.6 4 4 – 3 3 – 2 2 – 1 1 – Sales of Wildcat sailboats (millions of dollars) |July1998 July1999 July2000 July2001 Linear trend t

12 ©2003 Thomson/South-Western 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 ©2003 Thomson/South-Western 13 Cyclical Variation Cyclical activity Z1Z1Z1Z1 P1P1P1P1 V1V1V1V1 Z2Z2Z2Z2 P2P2P2P2 V2V2V2V2 t Figure 16.7

14 ©2003 Thomson/South-Western 14 Textile Example 4.0 4.0 – 3.5 3.5 – 3.0 3.0 – 2.5 2.5 – 2.0 2.0 – 1.5 1.5 – 1.0 1.0 – Corporate taxes (millions of dollars) 123 |1975 1985 1995 2000 Figure 16.8

15 ©2003 Thomson/South-Western 15 Irregular Activity Irregular activity consists of what is “left over” after accounting for the effect of any trend, seasonality, or cyclical activity

16 ©2003 Thomson/South-Western 16 Combining Components Additive Structure y t = TR t + S t + C t + I t Multiplicative Structure y t = TR t S t C t I t

17 ©2003 Thomson/South-Western 17 Measuring Trend Linear Trend ∑t= 1 + 2 +... + T = T(T + 1) 2 ∑t 2 = 1 + 4 +... + T 2 = T(T + 1)(2T + 1) 6 t = = ∑t∑tTT∑t∑tTTT T + 1 2 b1 =b1 =b1 =b1 = 12B - 6(T + 1)A T(T 2 - 1) b 0 = - b 1 AT T + 1 2

18 ©2003 Thomson/South-Western 18 Trend Line using Coded Data 12 12 – – 9 9 – – 6 6 – – 3 3 – – |||||||| 1994 (t = 1) 1995 (t = 2) Year 2001 (t = 8) Number of employees (thousands) YtYtYtYt t Figure 16.9

19 ©2003 Thomson/South-Western 19 Trend Line using Coded Data 12 12 – – 9 9 – – 6 6 – – 3 3 – – |||||||| 1994 (t = 1) 1995 (t = 2) Year 2001 (t = 8) Number of employees (thousands) YtYtYtYt t Figure 16.9 y t = TR t = b 0 + b 1 t

20 ©2003 Thomson/South-Western 20 Excel Solution Figure 16.10

21 ©2003 Thomson/South-Western 21 Quadratic Trend Figure 16.11

22 ©2003 Thomson/South-Western 22 Illustration of Quadratic Trend Lines YtYtYtYt Time (t) t = - b1b12b22b2b1b12b22b2 YtYtYtYt Time (t) t = - b1b12b22b2b1b12b22b2 AB Figure 16.12

23 ©2003 Thomson/South-Western 23 Measuring Cyclical Activity y t = TR t C t I t Ct Ct Ct Ct  ytytytytytytytyt^

24 ©2003 Thomson/South-Western 24 Complete Cycle YtYtYtYt 1 complete cycle Trend C t > 1 C t < 1 Time Figure 16.13

25 ©2003 Thomson/South-Western 25 Trend and Cyclical Activity ty t y t C t  11.11.125.978 22.42.529.949 34.63.9331.169 45.45.3371.012 55.96.741.875 68.08.145.982 79.79.5491.016 811.210.9531.022 ytytytytytytytyt ^ ^ Table 16.1

26 ©2003 Thomson/South-Western 26 Cyclical Activity 11.0 11.0 – 10.0 10.0 – 9.0 9.0 – 8.0 8.0 – 7.0 7.0 – 6.0 6.0 – 5.0 5.0 – 4.0 4.0 – 3.0 3.0 – 2.0 2.0 – 1.0 1.0 – Number of employees (thousands) |1994 1995 1996 1997 1998 1999 2000 2001 t y t =-.279 + 1.404t (trend line) Actual y t YtYtYtYt Figure 16.14

27 ©2003 Thomson/South-Western 27 Cyclical Components 1.15 1.15 – 1.10 1.10 – 1.05 1.05 – 1.00 1.00 –.95.95 –.90.90 – StartEnd CtCtCtCt t |11|111 |22|222 |33|333 |44|444 |55|555 |66|666 |77|777 |88|888 1994199619982000 Figure 16.15

28 ©2003 Thomson/South-Western 28 Additive Seasonal Variation 100 units Trend Actual time series |Winter1999 Winter2000 Winter2001 t YtYtYtYt 2000 2000 – 1500 1500 – 1000 1000 – 500 500 – Units sold Figure 16.16

29 ©2003 Thomson/South-Western 29 Jetski Sales 700 700 – 600 600 – 500 500 – 400 400 – 300 300 – 200 200 – 100 100 – Sales (tens of thousands of dollars) YtYtYtYt TR t = 100 + 20t Estimated sales using trend and seasonality t |11|111 |22|222 |33|333 |44|444 |55|555 |66|666 |77|777 |88|888 |99|999 |10 11 12 13 14 15 16 17 18 19 20 Figure 16.17

30 ©2003 Thomson/South-Western 30 Heat Pump Sales Figure 16.18 100 units 250 units 180 units Trend Actual time series |Winter1999 Winter2000 Winter2001 t YtYtYtYt 2000 2000 – 1500 1500 – 1000 1000 – 500 500 – Units sold

31 ©2003 Thomson/South-Western 31 Jetski Sales - Multiplicative Season Variation Figure 16.19 700 700 – 600 600 – 500 500 – 400 400 – 300 300 – 200 200 – 100 100 – Sales (tens of thousands of dollars) YtYtYtYt TR t = 100 + 20t Estimated sales using trend and seasonality t |11|111 |22|222 |33|333 |44|444 |55|555 |66|666 |77|777 |88|888 |99|999 |10 11 12 13 14 15 16 17 18 19 20

32 ©2003 Thomson/South-Western 32 Four Step Procedure for Decomposition 1.Determine a seasonal index, S t, for each time period 2.Deseasonalize the data, d t = TR t C t I t 3.Determine the trend component, TR t 4.Determine the cyclical component, C t

33 ©2003 Thomson/South-Western 33 Centered Moving Averages TimeQuarterty t Moving Totals 19901185(1) 263 2241(2) 268 3392(3) 270 4445and so on 19911590 2643 3795 4847 19921992... Table 16.2

34 ©2003 Thomson/South-Western 34 Sales Data for Video-Comp YearQuarter 1Quarter 2Quarter 3Quarter 4 199820124760 199940326576 2000565085100 20017570101123 Table 16.3

35 ©2003 Thomson/South-Western 35 Moving Averages for Video-Comp CenteredRatio to MovingMovingMoving YearQuarterty t TotalAverageAverage 19981120—— 2212——— 334713937.251.26 446015942.251.42 1999154017947.00.85 263219751.25.62 376521355.251.18 487622959.201.28 2000195624764.25.87 2105026769.75.72 3118529175.131.13 41210031080.001.25 20011137533084.50.89 2147034689.38.78 315101369—— 416123——— Table 16.3

36 ©2003 Thomson/South-Western 36 Smoothing a Time Series 120 120 – 100 100 – 80 80 – 60 60 – 40 40 – 20 20 – Sales (number of units) Moving averages (no seasonality) |11|111 |22|222 |33|333 |44|444 1998 |11|111 |22|222 |33|333 |44|444 1999 |11|111 |22|222 |33|333 |44|444 2000 |11|111 |22|222 |33|333 |44|444 2001 t YtYtYtYt Quarters by year Figure 16.20

37 ©2003 Thomson/South-Western 37 Ratios for Each Quarter Quarter 1Quarter 2Quarter 3Quarter 4 1.261.42.85.621.181.28.87.721.131.25.89.78 Total2.612.123.573.95 Average0.8700.7071.1901.317 —— —— Table 16.5

38 ©2003 Thomson/South-Western 38 DeseasonalizedValuesSeasonal YeartY t Index (S t )d t = — YtYtStStYtYtStSt 1998120.85223.47 212.69217.34 3471.16640.31 4601.29046.51 1999140.85246.95 232.69246.24 3651.16655.75 4761.29058.91 2000156.85265.73 250.69272.25 3851.16672.90 41001.29077.52 2001175.85288.03 270.692101.16 31011.16686.62 41231.29095.35 Deseasonalizing Data Table 16.6

39 ©2003 Thomson/South-Western 39 Total U.S. Retail Trade 1997199819992000 Jan134.738139.935146.613158.691 Feb130.255135.538145.121164.725 Mar148.497151.118165.736183.875 Apr145.703155.820166.011178.776 May156.603162.797173.496190.753 Jun150.915159.701171.286187.868 Jul153.200161.541172.364182.891 Aug156.782162.369174.788191.647 Sep149.407155.747169.809183.229 Oct157.523164.528174.740186.550 Nov161.925169.914185.347198.706 Dec203.117215.590238.452243.255 Table 16.7

40 ©2003 Thomson/South-Western 40 Summary of Ratios Month (Period) JanFebMarAprMayJunJulAugSepOctNovDec 19970.9931.0130.9641.0131.0361.295 19980.8880.8570.9520.9791.0180.9941.0001.0010.9541.0021.0291.299 19990.8780.8640.9810.9761.0140.9920.9900.9960.9590.9801.0321.317 20000.8710.8990.9960.9631.0221.003 Average0.8790.8370.9770.9731.0180.9960.9941.0040.9590.9981.0331.304 Table 16.9

41 ©2003 Thomson/South-Western 41 Deseasonalized Data 200 200 – 190 190 – 180 180 – 170 170 – 160 160 – 150 150 – 140 140 – |00|000 |10 20 30 40 50 Time (t) Deseasonalized Values (d t ) Figure 16.21

42 ©2003 Thomson/South-Western 42 Cyclical Components 1153.349146.470 -.934(1) = 146.9791.0433— 2149.238146.470 -.934(2) = 147.9131.00901.025 3152.165146.470 +.934(3) = 148.8471.02231.011 4149.871146.470 +.934(4) = 149.7821.00061.015 5153.899146.470 +.934(5) = 150.7161.02111.007 6151.602146.470 +.934(6) = 151.6500.99971.010..... td t d t —(= C t I t ) dtdtdtdtdtdtdtdt ^ ^ 3-MonthMoving Average (C t ) Table 16.12

43 ©2003 Thomson/South-Western 43 Plot of Cyclical Activity Figure 16.22 1.03 1.03 – 1.02 1.02 – 1.01 1.01 – 1.00 1.00 – 0.99 0.99 – 0.98 0.98 – 0.97 0.97 – |00|000 |10 20 30 40 50 Cyclical Components MonthFebJanJanJan Year1997199819992000

44 ©2003 Thomson/South-Western 44 Excel Plots Figure 16.23

45 ©2003 Thomson/South-Western 45 Index Numbers 1985199019952000 Wage$7.05$8.50$10.90$12.50 Index (base = 1980)100120.6154.6177.3 Table 16.15

46 ©2003 Thomson/South-Western 46 Price Indexes Simple Aggregate Price Index = 100 ∑P1∑P1∑P0∑P0∑P1∑P1∑P0∑P0 Weighted Aggregate Price Index = 100 ∑P1Q∑P1Q∑P0Q∑P0Q∑P1Q∑P1Q∑P0Q∑P0Q Laspeyres Index = 100 ∑P1Q0∑P1Q0∑P0Q0∑P0Q0∑P1Q0∑P1Q0∑P0Q0∑P0Q0 ∑P1Q1∑P1Q1∑P0Q1∑P0Q1∑P1Q1∑P1Q1∑P0Q1∑P0Q1 Paasche Index = 100

47 ©2003 Thomson/South-Western 47 Prices of Four Items Item19902000 Eggs.75 (doz)1.35 Chicken.95 (lb)1.79 Cheese.89 (lb)1.85 Auto battery$31.00 (each)$55.00 (each) Table 16.16 LongLife

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