CHAPTER 18 Models for time series and Forecastin to accompany Introduction to business statistics fourth edition, by Ronald m. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. stengel o The Wadsworth Group
CHAPTER 18 Models for Time Series and Forecasting to accompany Introduction to Business Statistics fourth edition, by Ronald M. Weiers Presentation by Priscilla Chaffe-Stengel Donald N. Stengel © 2002 The Wadsworth Group
l Chapter 18-Learning objectives Describe the trend, cyclical, seasonal, and irregular components of the time series model Fit a linear or quadratic trend equation to a time series Smooth a time series with the centered moving average and exponential smoothing techniques Determine seasonal indexes and use them to compensate for the seasonal effects in a time series Use the trend extrapolation and exponential smoothing forecast methods to estimate a future value Use mad and mse criteria to compare how well equations fit e Use index numbers to compare business or economic measures over time o 2002 The Wadsworth Group
Chapter 18 - Learning Objectives • Describe the trend, cyclical, seasonal, and irregular components of the time series model. • Fit a linear or quadratic trend equation to a time series. • Smooth a time series with the centered moving average and exponential smoothing techniques. • Determine seasonal indexes and use them to compensate for the seasonal effects in a time series. • Use the trend extrapolation and exponential smoothing forecast methods to estimate a future value. • Use MAD and MSE criteria to compare how well equations fit data. • Use index numbers to compare business or economic measures over time. © 2002 The Wadsworth Group
l Chapter 18-Key terms Time series Seasonal index Classical time series Ratio to moving model average method Trend value · Deseasonalizing Cyclical component MAD criterion Seasonal component mse criterion Irregular component Trend equation Constructing an index using the Cpi Moving average Shifting the base of an Exponential index smoothing g o 2002 The Wadsworth Group
Chapter 18 - Key Terms • Time series • Classical time series model – Trend value – Cyclical component – Seasonal component – Irregular component • Trend equation • Moving average • Exponential smoothing • Seasonal index • Ratio to moving average method • Deseasonalizing • MAD criterion • MSE criterion • Constructing an index using the CPI • Shifting the base of an index © 2002 The Wadsworth Group
l Classical Time Series model y=T°C·S° where y=observed value of the time series variable T= trend component, which reflects the general tendency of the time series without fluctuations C= cyclical component, which reflects systematic fluctuations that are not calendar-related, such as business cycles S=seasonal component, which reflects systematic fluctuations that are calendar-related, such as the day of the week or the month of the year I= irregular component, which reflects fluctuations that are not systematic o 2002 The Wadsworth Group
Classical Time Series Model y = T • C • S • I where y = observed value of the time series variable T = trend component, which reflects the general tendency of the time series without fluctuations C = cyclical component, which reflects systematic fluctuations that are not calendar-related, such as business cycles S = seasonal component, which reflects systematic fluctuations that are calendar-related, such as the day of the week or the month of the year I = irregular component, which reflects fluctuations that are not systematic © 2002 The Wadsworth Group
I Trend equations Linear: j=b0+ bix Quadratic: y=5o +61x+b2x2 j=the trend line estimate of y x= time period bo by and b2 are coefficients that are selected to minimize the deviations between the trend estimates j and the actual data values y for the past time periods. Regression methods are used to determine the best values for the coefficients o 2002 The Wadsworth Group
Trend Equations •Linear: = b0 + b1x •Quadratic: = b0 + b1x + b2x 2 = the trend line estimate of y x = time period b0 , b1 , and b2 are coefficients that are selected to minimize the deviations between the trend estimates and the actual data values y for the past time periods. Regression methods are used to determine the best values for the coefficients. y ? y ? y ? y ? © 2002 The Wadsworth Group
Smoothing techniques Smoothing techniques -dampen the impacts of fluctuation in a time series, thereby providing a better view of the trend and (possibly the cyclical components Moving average-a technique that replaces a data value with the average of that data value and neighboring data values Exponential smoothing -a technique that replaces a data value with a weighted average of the actual data value and the value resulting from exponential smoothing for the previous time period o 2002 The Wadsworth Group
Smoothing Techniques • Smoothing techniques - dampen the impacts of fluctuation in a time series, thereby providing a better view of the trend and (possibly) the cyclical components. • Moving average - a technique that replaces a data value with the average of that data value and neighboring data values. • Exponential smoothing - a technique that replaces a data value with a weighted average of the actual data value and the value resulting from exponential smoothing for the previous time period. © 2002 The Wadsworth Group
Moving average A moving average for a time period is the average of N consecutive data values, including the data value for that time period A centered moving average is a moving average such that the time period is at the center of the n time periods used to determine which values to average If n is an even number the techniques need to be adjusted to place the time period at the center of the averaged values. The number of time periods n is usually based on the number of periods in a seasonal cycle. The larger n is the more fluctuation will be smoothed out o 2002 The Wadsworth Group
Moving Average • A moving average for a time period is the average of N consecutive data values, including the data value for that time period. • A centered moving average is a moving average such that the time period is at the center of the N time periods used to determine which values to average. If N is an even number, the techniques need to be adjusted to place the time period at the center of the averaged values. The number of time periods N is usually based on the number of periods in a seasonal cycle. The larger N is, the more fluctuation will be smoothed out. © 2002 The Wadsworth Group
l Moving average- An example Time period Data value 1997, Quarter I 818 1997, Quarter II 861 1997, Quarter III 844 1997, Quarter IV 906 1998, Quarter I 867 1998, Quarter Il 899 3-Quarter Centered Moving Average for 1997, Quarter IV 844906+867=8723 4-Quarter Centered Moving Average for 1997, Quarter Iv 0.5.861+8444906+867+0.5.844906+867+89987425 o 2002 The Wadsworth Group
Moving Average - An Example Time Period Data Value 1997, Quarter I 818 1997, Quarter II 861 1997, Quarter III 844 1997, Quarter IV 906 1998, Quarter I 867 1998, Quarter II 899 • 3-Quarter Centered Moving Average for 1997, Quarter IV: • 4-Quarter Centered Moving Average for 1997, Quarter IV: 872.3 3 844 906 867 = + + = 874.25 4 844 906 867 899 0.5 4 861 844 906 867 0.5 = + + + + + + + = © 2002 The Wadsworth Group
I Exponential smoothing Et=°y+(1-a)E where Et=exponentially smoothed value for time period t t-1= exponentially smoothed value for time period t-1 Ut=actual time series value for time period t a= the smoothing constant0≤a≤1 The larger a is the closer the smoothed value will track the original data value. The smaller a is the more fluctuation is smoothed out o 2002 The Wadsworth Group
Exponential Smoothing Et = a•yt + (1 – a) Et–1 where Et = exponentially smoothed value for time period t Et–1 = exponentially smoothed value for time period t – 1 yt = actual time series value for time period t a = the smoothing constant, 0 a 1 • The larger a is, the closer the smoothed value will track the original data value. The smaller a is, the more fluctuation is smoothed out. © 2002 The Wadsworth Group
l Exponential smoothing- An example Data Smoothed value Smoothed value erio Value (a=02)(c=08) 818 818 818 1234 861 826.6 8524 844 830.1 8457 906 8453 893.9 Calculation for smoothed value for Period 2(a=0.2) E2=ay2+(1-a)E1 0.2(861)+08(818)=8266 o 2002 The Wadsworth Group
Exponential Smoothing - An Example Data Smoothed Value Smoothed Value Period Value (a = 0.2) (a = 0.8) 1 818 818 818 2 861 826.6 852.4 3 844 830.1 845.7 4 906 845.3 893.9 • Calculation for smoothed value for Period 2 (a = 0.2): E2 = a y + (1 – a ) E1 = 0.2 (861) + 0.8 (818) = 826.6 2 © 2002 The Wadsworth Group