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1、CHAPTER 18Models for Time Series and Forecasting,to accompanyIntroduction to Business Statisticsfourth edition,by Ronald M.WeiersPresentation by Priscilla Chaffe-Stengel Donald N.Stengel 2002 The Wadsworth Group,Chapter 18-Learning Objectives,Describe the trend,cyclical,seasonal,and irregular compon
2、ents 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 extrapolati
3、on 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,Chapter 18-Key Terms,Time seriesClassical time series modelTrend val
4、ueCyclical componentSeasonal componentIrregular componentTrend equationMoving averageExponential smoothing,Seasonal indexRatio to moving average methodDeseasonalizingMAD criterionMSE criterionConstructing an index using the CPIShifting the base of an index,2002 The Wadsworth Group,Classical Time Ser
5、ies Model,y=T C S Iwhere 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,w
6、hich 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,Trend Equations,Linear:=b0+b1xQuadratic:=b0+b1x+b2x2=the trend line estimate of y x=
7、time periodb0,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.,2002 The Wadsworth Group,Smoothing Techniques,Smooth
8、ing 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
9、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
10、 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 n
11、umber 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,Moving Average-An Example,Time Period Data Value1997,Quarter I8181997,Quarter II8611997,Quarter III8441997,Quarter IV9061998,Quart
12、er I8671998,Quarter II8993-Quarter Centered Moving Average for 1997,Quarter IV:4-Quarter Centered Moving Average for 1997,Quarter IV:,2002 The Wadsworth Group,Exponential Smoothing,Et=ayt+(1 a)Et1whereEt=exponentially smoothed value for time period tEt1=exponentially smoothed value for time period t
13、 1yt=actual time series value for time period ta=the smoothing constant,0 a 1The 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,Exponential Smoothing-An Example,Data Smoothed ValueSmoothed V
14、aluePeriod Value(a=0.2)(a=0.8)1818818818 2861826.6852.4 3844830.1845.7 4906845.3893.9Calculation 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,Seasonal Indexes,A seasonal index is a factor that adjusts a trend value to compensate for typical
15、 seasonal fluctuation in that period of a seasonal cycle.A seasonal index is expressed as a percentage with a value of 100%corresponding to an average position in a seasonal cycle.,2002 The Wadsworth Group,Seasonal Indexes-An Example,SeasonIndex(Annual Quarter)ValueI 84.5II102.3III 95.5IV117.7If the
16、 trend value for Quarter I in the given year was 902,the value with seasonal fluctuation would bey=T S=902 84.5%=762.2,2002 The Wadsworth Group,Ratio to Moving Average Method,A technique for developing a set of seasonal index values from the original time series.Steps:1.Construct a centered moving a
17、verage of the time series.Set N=number of periods in the seasonal cycle.2.Express each original time series value as a percentage of the corresponding centered moving average.The result is the ratio to moving average.Example:If the original data value is 906 and the corresponding centered moving ave
18、rage is 872.3,Ratio to moving average=(906/872.3)100=103.86,2002 The Wadsworth Group,Ratio to Moving Average Method,Steps,cont.:3.For each period in the seasonal cycle,average all the ratio to moving average values(from Step 2)corresponding to that period in the seasonal cycle.The result is the unad
19、justed seasonal index for that period in the seasonal cycle.Example:If ratios corresponding to Quarter I are 80.4,87.3,82.1,89.5,and 78.7,the unadjusted seasonal index value is,2002 The Wadsworth Group,Ratio to Moving Average Method,Steps,cont.:4.The average of the seasonal index values should be 10
20、0.0 or their sum should be N100.If not,multiply all seasonal index values by the appropriate adjustment factor,N100 divided by the sum of unadjusted seasonal index values.Example:Unadjusted AdjustedSeason Seasonal Index Seasonal IndexI 83.60 83.83II102.07102.35III 95.42 95.68IV117.81118.34,2002 The
21、Wadsworth Group,Deseasonalizing a Time Series,This procedure involves use of a seasonal index to remove the effect of typical seasonal fluctuation from a time series data value.The result is also called a seasonally-adjusted value.Example:If the original data value for the first quarter of a given y
22、ear is 1124 and the seasonal index for Quarter I is 83.4,the seasonally-adjusted value is:,2002 The Wadsworth Group,Forecasting with Classical Time Series Models,To forecast a value in a future time period:1.Use the trend equation to forecast the trend value for that time period.2.Adjust the data va
23、lue using the cyclical and seasonal index values.If there is no cyclical index,do not do a cyclical adjustment.,2002 The Wadsworth Group,Forecasting with Classical Time Series Models-An Example,Example-Trend Equation:where=trend value x=number of quarters to 1997,Quarter IVTo forecast the value for
24、1999,Quarter IIForecast of trend=970.2+12.3(6)=1044.0If the seasonal index for Quarter II is 102.35,the forecast with seasonal fluctuation is:,2002 The Wadsworth Group,Forecasting with Exponential Smoothing,A technique for generating a forecast for the next time period using the forecast and actual
25、data value for the current time period.This technique is not valid if there is a significant upward or downward trend.Ft+1=a yt+(1 a)Ft Ft+1=forecast for period t+1 yt=actual value for period t Ft=forecast for period t a=smoothing constant,(0 a 1),2002 The Wadsworth Group,Forecasting with Exponentia
26、l Smoothing-An Example,If the forecast for the current time period was 842 and the actual value was 872,using a smoothing constant of a=0.6,the forecast for the next period is:(0.6)(872)+(0.4)(842)=860,2002 The Wadsworth Group,Evaluating Time Series Models,Models can be evaluated using past data by
27、examining the differences(or errors)between the values predicted from the models and the actual data values.The errors can be summarized and accuracy measured using either of the following criteria:Mean Absolute Deviation(MAD)Criterion:1.Express each difference as a positive number.2.Find the averag
28、e of the differences from Step 1.Mean Squared Error(MSE)Criterion:1.Square each error difference.2.Find the average of the squared error differences from Step 1.,2002 The Wadsworth Group,Evaluating Time Series Models-An Example,ValueActualComputed Data Absolute SquaredBy Model Value Deviation Error1
29、4401436 4 1614561461 5 2514721480 8 6414881472 1615615041495 9 81Sums:42342MAD=42/5=8.4MSE=342/5=68.4,2002 The Wadsworth Group,What are index numbers?,Index numbers:are time series that focus on the relative change in a count or measurement over time.express the count or measurement as a percentage
30、of the comparable count or measurement in a base period.,2002 The Wadsworth Group,Base Periods for Index Numbers,The base period is arbitrary but should be a convenient point of reference.The value of an index number corresponding to the base period is always 100.The base period may be a single peri
31、od or an average of multiple adjacent periods.,2002 The Wadsworth Group,Applications of Index Numbers in Business and Economics,A price index shows the change in the price of a commodity or group of commodities over time.A quantity index shows the change in quantity of a commodity or group of commod
32、ities used or purchased over time.A value index shows a change in total dollar value(price quantity)of a commodity or group of commodities over time.,2002 The Wadsworth Group,Simple Relative Index,A simple relative index shows the change in the price,quantity,or value of a single commodity over time
33、.Calculation of a simple relative index:Index in period t=,2002 The Wadsworth Group,Example:Simple Relative Price Index,Price Index Price IndexYear Price 1985 as base year1995 as base year1985$140 100.0 58.31990 195 139.3 81.31995 240 171.4 100.02000 275 196.4 114.6Computation of index for 1990(1985
34、 as base year):,2002 The Wadsworth Group,Consumer Price Index,A weighted aggregate price index used to reflect the overall change in the cost of goods and services purchased by a typical consumer.Applications:Indicator of rate of inflationUsed to adjust wages to compensate for lost purchasing power
35、due to inflationUsed to convert a price or wage to a real price or real wage to show the equivalent amount in a base period after adjusting for inflation.,2002 The Wadsworth Group,Example:The CPI as Deflator,Suppose a person was earning$50,000 per year in June 2001,when the CPI was 178.0(base year:1
36、982-84).What was the persons real income in its 1982-84 equivalent?Real income in period t=Income in period t Real earnings in 2001=$50,000 100/178.0=$28,090,2002 The Wadsworth Group,Example:The CPI as Deflator,Suppose the same person was earning$46,500 per year in 1997,when the CPI was 160.5(base y
37、ear:1982-84).What was the persons real income in its 1982-84 equivalent?Real earnings in 1997=$46,500 100/160.5=$28,972The purchasing power of the persons earnings was higher in 1997 than in 2001.,2002 The Wadsworth Group,Shifting the Base of an Index,For useful interpretation,it is often desirable
38、for the base year to be fairly recent.To shift the base year to another year without recalculating the index from the original data:,2002 The Wadsworth Group,Example:Shifting a Base Year,To shift a base year from 1985 to 1995:Price Index Price IndexYr1985 as base yr 1995 as base yr1985100.0 58.31990139.3 81.31995171.4100.02000196.4114.6An Illustration:,2002 The Wadsworth Group,
