《商务统计学概论》（英文版） CHAPTER 18 Models for Time Series and Forecasting

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