©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

©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)

©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

©2003 Thomson/South-Western 4 Power Example Figure – – – Power consumption (million kwh) | t (time) |

©2003 Thomson/South-Western 5 Employees Example Figure – – – – – – – – – – – Number of employees (thousands) | t Trend

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

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

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

©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

©2003 Thomson/South-Western 10 Seasonal Variation – – – – – – – Power consumption (millions kwh) ||||||||| Jan Jul Dec Figure 16.5

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

©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

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

©2003 Thomson/South-Western 14 Textile Example – – – – – – – Corporate taxes (millions of dollars) 123 | Figure 16.8

©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

©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

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

©2003 Thomson/South-Western 18 Trend Line using Coded Data – – 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

©2003 Thomson/South-Western 19 Trend Line using Coded Data – – 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

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

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

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

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

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

©2003 Thomson/South-Western 25 Trend and Cyclical Activity ty t y t C t  ytytytytytytytyt ^ ^ Table 16.1

©2003 Thomson/South-Western 26 Cyclical Activity – – – – – – – – – – – Number of employees (thousands) | t y t = t (trend line) Actual y t YtYtYtYt Figure 16.14

©2003 Thomson/South-Western 27 Cyclical Components – – – – – – StartEnd CtCtCtCt t |11|111 |22|222 |33|333 |44|444 |55|555 |66|666 |77|777 |88| Figure 16.15

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

©2003 Thomson/South-Western 29 Jetski Sales – – – – – – – Sales (tens of thousands of dollars) YtYtYtYt TR t = t 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 | Figure 16.17

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

©2003 Thomson/South-Western 31 Jetski Sales - Multiplicative Season Variation Figure – – – – – – – Sales (tens of thousands of dollars) YtYtYtYt TR t = t 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 |

©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

©2003 Thomson/South-Western 33 Centered Moving Averages TimeQuarterty t Moving Totals (1) (2) (3) and so on Table 16.2

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

©2003 Thomson/South-Western 35 Moving Averages for Video-Comp CenteredRatio to MovingMovingMoving YearQuarterty t TotalAverageAverage —— 2212——— —— ——— Table 16.3

©2003 Thomson/South-Western 36 Smoothing a Time Series – – – – – – Sales (number of units) Moving averages (no seasonality) |11|111 |22|222 |33|333 |44| |11|111 |22|222 |33|333 |44| |11|111 |22|222 |33|333 |44| |11|111 |22|222 |33|333 |44| t YtYtYtYt Quarters by year Figure 16.20

©2003 Thomson/South-Western 37 Ratios for Each Quarter Quarter 1Quarter 2Quarter 3Quarter Total Average —— —— Table 16.5

©2003 Thomson/South-Western 38 DeseasonalizedValuesSeasonal YeartY t Index (S t )d t = — YtYtStStYtYtStSt Deseasonalizing Data Table 16.6

©2003 Thomson/South-Western 39 Total U.S. Retail Trade Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Table 16.7

©2003 Thomson/South-Western 40 Summary of Ratios Month (Period) JanFebMarAprMayJunJulAugSepOctNovDec Average Table 16.9

©2003 Thomson/South-Western 41 Deseasonalized Data – – – – – – – |00|000 | Time (t) Deseasonalized Values (d t ) Figure 16.21

©2003 Thomson/South-Western 42 Cyclical Components (1) = — (2) = (3) = (4) = (5) = (6) = td t d t —(= C t I t ) dtdtdtdtdtdtdtdt ^ ^ 3-MonthMoving Average (C t ) Table 16.12

©2003 Thomson/South-Western 43 Plot of Cyclical Activity Figure – – – – – – – |00|000 | Cyclical Components MonthFebJanJanJan Year

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

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

©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

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