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Chapter 4 Time Series Analysis and Forecasting

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  1. KVANLI PAVUR KEELING Chapter 4Time Series Analysis and Forecasting

  2. Chapter Objectives • At the completion of this chapter, you should be able to: ∙ Understand the meaning of time series and the four components of the data ∙ Estimate the trend, seasonality, cyclical and noise components ∙ Use Excel macro to decompose time series data and know the meaning of each value. ∙ Calculate price indexes, including a Laspeyres and Paasche index

  3. What is a Time Series? • A time series consists of a variable (such as Sales) recorded across time • Example: tYearSales (millions of $) 1 1985 1.7 2 1986 2.4 3 1987 2.8 4 1988 3.4 . . 22 2006 9.6 23 2007 10.7 This is y1 This is an example of annual data This is y23

  4. Time Series Data • Time series data can be: ∙ annual (one value for each year) ∙ quarterly (4 values for each year) ∙ monthly (12 values for each year) • Each time series value is made up of 3 or 4 components. These are: ∙ Trend (TR) ∙ Seasonality (S) ∙ Cyclic (C) ∙ Irregular or noise (I) Monthly or quarterly data only

  5. Trend • Trend is the long-term growth or decline in the time series • Trend usually follows a straight line

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

  7. 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) | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 t Employees Example We’ll take a closer look at this example in the slides to follow

  8. Linear Trend TR = b0 + b1t Trend (TR) Trendis a steady increase or decrease in the time series This long-term growth or decay pattern can take a variety of shapes. If the rate of change in Y from one time period to the next is relatively constant, the trend is a linear trend.

  9. Quadratic Trend TR = b0 + b1t + b2t2 Curvilinear Trend • Trend can also be curvilinear • Curvilinear trend is also called quadratic trend • In this chapter, we will pay little attention to curvilinear trend • The macros will assume that trend is linear

  10. Yt Yt t t (b) (a) Examples of Curvilinear Trend

  11. Yt Yt t t (d) (c) Examples of Curvilinear Trend

  12. 300 – 200 – 100 – Power consumption (million kwh) | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 t (time) An Illustration of Curvilinear Trend The increase in power consumption slows down over time and the trend is not linear Figure 4.1

  13. Seasonality • Seasonality (seasonal variation) is periodic variation (increases or decreases) within a calendar year in a time series. • The key is that this variation in the time series follows the same pattern each year • For example, sales are always high in December • Seasonality only exists for non-annual data

  14. 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

  15. 4 – 3 – 2 – 1 – Sales of Wildcat sailboats (millions of dollars) Linear trend | July 2001 | July 2002 | July 2003 | July 2004 t Seasonal Variation Figure 13.6

  16. Cyclical Variation • Cyclical variation describes a gradual cyclical (non-seasonal) movement about the trend. • The length of a cycle can be measured from one peak to the next, or one valley to the next, or from the time value at which the cycle crosses the horizontal line to the value where it completes the cycle and returns to this point.

  17. Cyclical Variation • The next slide illustrates a time series containing cyclical movement (corporate taxes paid by a textile company over a 25-year period) • This time series does not exhibit a trend (long-term upward or downward growth)

  18. 4.0 – 3.5 – 3.0 – 2.5 – 2.0 – 1.5 – 1.0 – Corporate taxes (millions of dollars) 1 2 3 | 1980 | 1990 | 2000 | 2005 Textile Example

  19. P1 P2 Z1 Z2 Cyclical activity V1 V2 t Cyclical Variation Figure 13.7

  20. Irregular Activity • Irregular activity is what is left over after measuring the seasonal, trend, and cyclic activity • Engineers refer to this as “noise” • The irregular activity should contain no observable or predictable pattern.

  21. Combining the Components • If the seasonality is assumed to be additive, each yt is the sum of its four components yt = St+TRt + Ct+ It • If the seasonality is assumed to be multiplicative, each yt is the product of its four components yt = St ∙ TRt ∙ Ct ∙ It The seasonal component (St) is omitted for annual data Multiplicative seasonality is the usual situation and assumed in the Excel macros

  22. Capturing the Trend • We will illustrate this using annual data which has no seasonality This is the trend line

  23. Finding the Trend Line • The equation of the trend line is yt= b0+ b1t • b0 is called the intercept (and is fairly boring) • b1 is called the slope (and is pretty interesting) • The calculations necessary to find the slope and intercept are shown on the next slide

  24. Example in Section 2 • ytis the number of employees (in thousands) for eight years • t ytt∙yt 1 1.1 1.1 2 2.4 4.8 3 4.6 13.8 . . 8 11.289.6 48.3 276.3 Let A = the sum of the time series values So, A = 48.3 Let B = the sum of the right-hand column So, B = 276.3 Let T = the number of time periods. So, T = 8

  25. Equation of the Trend Line • First, find the slope: • Next, find the intercept:

  26. The Example in Section 2 Carry a lot of decimal places The trend line is = -.279 + 1.404t OK to round now

  27. Interpreting the Slope • The slope of the trend line is a very interesting value • Here, b1 is 1.404 • Since the number of employees each year (yt) is measured in thousands, then the number of employees in this company is increasing 1,404 (on the average) each year

  28. Forecasting – Extending the Trend • If you assume the linear growth or decline as described by the trend line continues for another year, a simple forecast can be obtained from this trend line • For example, what would be your forecast for the year 2008? • This is time period t = 9 • Use this value for t in the trend line equation

  29. Forecasting – Extending the Trend • This would be -.279 + 1.404(9) = 12.357 • The forecast for 2008 is 12,357 employees The forecast for 2008 The forecast period sample data

  30. Measuring Cyclic Activity – Annual Data • We’ll assume the multiplicative model, where each time series value is the product of its components • Since this is annual data, there is no seasonal component and yt = TRt ∙ Ct ∙ It • The trend line values ( values) contain trend only

  31. The Estimated Number of Employees • The estimated number of employees in each time period using the trend line: = -.279 + 1.404(1) = 1.125 = -.279 + 1.404(2) = 2.529 = -.279 + 1.404(3) = 3.933 = -.279 + 1.404(8) = 10.953

  32. Measuring Cyclic Activity – Annual Data • By dividing the yt values by the values, you can eliminate the trend components • We’ll call these ratios the cyclic components, even though they contain noise (It) • There is no way to separate out the noise component but it can be reduced when using monthly or quarterly data (illustrated later)

  33. tytytyt/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 Trend activity Cyclical activity

  34. Ct 1.15 – 1.10 – 1.05 – 1.00 – .95 – .90 – Start End | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 t 2000 2002 2004 2007 Plot of Cyclical Components

  35. 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) | 2000 | 2001 | 2002 | 2003 | 2004 | 2005 | 2006 | 2007 t Cyclical Activity

  36. Yt Trend 2000 – 1500 – 1000 – 500 – 100 units 100 units Actual time series Units sold 100 units | Winter 2005 | Winter 2006 | Winter 2007 t Additive Seasonal Variation Figure 4.16

  37. 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 JetskiSales – Additive Seasonality Figure 4.17

  38. Yt 2000 – 1500 – 1000 – 500 – 250 units Trend 180 units Units sold Actual time series 100 units | Winter 2005 | Winter 2006 | Winter 2007 t Heat Pump Sales – Multiplicative Seasonality Figure 4.18

  39. 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 4.19

  40. Procedure with Monthly or Quarterly Data • The Excel macros assume multiplicative seasonality, where yt = St ∙ TRt ∙ Ct ∙ It • Determine the seasonal components (St values) • Deseasonalize the data • Determine the trend components (TRtvalues) using the deseasonalized data • Determine the cyclic components (Ct values) • Determine the irregular (noise) components (It values)

  41. Section 4.5 – Measuring Seasonality • The five-step procedure: • Find the centered moving averages • Divide each yt by its centered moving average • Put the ratios into a table and find the mean for each period • Adjust the averages so that they sum to 4 (quarterly data) or 12 (monthly data) • Deseasonalize the data by dividing each yt by its seasonal index

  42. Sales Data for Video-Comp Year Quarter 1 Quarter 2 Quarter 3 Quarter 4 2004 20 12 47 60 2005 40 32 65 76 2006 56 50 85 100 2007 75 70 101 123 Table 4.3

  43. Using the KPK Macros – Excel 2007

  44. Enter the values in Column A, by year 2004 2005 2006 2007

  45. Run the Decomposition Macro

  46. The Seasonal sheet - Determining the seasonal indexes

  47. Determining the Seasonal Indexes

  48. Determining the seasonal indexes

  49. Put the ratios into a table

  50. Find the average for each time period