Bài giảng Statistical Techniques in Business and Economics - Chapter 18 Time Series

Copyright â 2003 by The McGraw-Hill Companies, Inc. All rights reserved. TIMESeriesChapter 18 Define the four components of a time series1.Determine a linear trend equation2.Compute a moving average3.Compute the trend equation for a nonlinear trend 4. Use trend equations to forecast future time periods and to  develop seasonally adjusted forecasts5.Determine and interpret a set of seasonal indexes6.Deseasonalize data using a seasonal index 8.When you have completed this chapter, you will be able to:Identify cyclical fluctuations7.Compute and evaluate forecasts9.TIMESeries is a collection of data recorded over a period of time ( data may be recorded weekly, monthly, or quarterly) ComponentsSecular TrendCyclical VariationSeasonal Variationis the long run direction of the Time Seriesis the fluctuation above and below the trend lineis the pattern in a time series; these patterns tend to repeat themselves from year to year ...andEpisodic variations are unpredictable, but can usually be identified, such as a flood or hurricane Residual variations are random in nature and cannot be identified Irregular Variationis divided into two components:TIMESeriesComponentsContinuedExamples...Text Chart 18-1ExcelSecular Trend Secular Trend almost constant Text Chart 18-2ExcelText Chart 18-3ExcelSecular Trend Increasing TrendText Chart 18-4ExcelSecular Trend Declining TrendText Chart 18-5ExcelCyclical VariationFigure 18-6Text Chart 18-6ExcelSeasonal VariationLinear TrendThe long term trend equation (linear) Estimated by the least squares equation for time t is:TIMESeriesExample...abtbtyytttaynbtn()()()=+=--=-ổốỗửứữnSSSnSSSS22yContinuedTIMESeriesThe owner of Farley Homes would like a forecast for the next couple of years of new homes that will be constructed in the Edmonton area. Listed below are the sales of new homes constructed in the area for the last 5 years.Example...Year19971998199920002001Sales ($1000)4.3 5.6 7.89.29.7Year Sales t Sales*t t 21997 4.3 1 4.3 11998 5.6 2 11.2 41999 7.8 3 23.4 92000 9.2 4 36.8 162001 9.7 5 48.5 25Total 36.6 15 124.2 55TIMESeriesExample least squares equation for time tContinuedContinuedAnswerYear19971998199920002001Sales ($1000)4.3 5.6 7.89.29.7= 1.445/)15(555/)15(6.362.1242--()()()// 22=S-SSS-S=nttntytyb= 3.051544.156.36ữứửỗốổ-()=ữứửỗốổS-S=ntbnya Develop a trend equation using the least squares method by letting 1997 be the time period 1 Answer* Five year (1997 – 2001) + 2002 and 2003 The time series equation is: = 3.00 + 1.44tyThe forecast for the year 2003 is: = 3.00 + 1.44(7)* = 13.08ySeeUsingClick on ToolsClick on DATA ANALYSISSeeHighlight Regression Click OKSeeSeeUsingLine Fit Plot Data RegressionSeeUsingIf the trend is not linear but rather the increases tend to be a constant percent, the y values are converted to logarithms, and a least squares equation is determined using the lns: Non-Linear Trend ln()[ ln()][ ln()]abt=+yText Figure 18-11ExcelThe Moving-Average Method is used to smooth out a time series. This is accomplished by “moving” the arithmetic mean through the time series.the moving-average is the basic method used in measuring the seasonal fluctuationto apply the moving-average method to a time series, the data should follow a fairly linear trend and have a definite rhythmic pattern of fluctuationsThe Moving-Average MethodUsingText Chart 18-9ExcelThe method most commonly used to compute the typical seasonal pattern is called the Ratio-to-Moving-Average Methodit eliminates the trend, cyclical, and irregular components from the original data (y) the numbers that result are called the typical seasonal indexes Seasonal VariationYearWinterSpringSummerFall19966.74.66.712.719976.54.66.513.619986.95.06.914.119997.05.57.015.020007.15.77.114.520018.0.6.211.414.9Listed below are the quarterly sales (in $ millions) of Toys International for the years 1996 through 2001. Determine a quarterly seasonal index using the ratio-to-moving average method.Note that the fall quarter sales are the largest and the spring sales are the smallest each year1. determine the moving total for the time series6. apply the correction factor Steps2. determine the moving average for the time series3. the moving averages are then centered4. the specific seasonal for each period is then computed by dividing the y values with the centered moving averages5. organize the specific seasonals in a tableSeasonal VariationStepsSeasonal VariationText Chart 18-61. determine the moving total for the time seriesSeasonal VariationText Chart 18-13ExcelSeasonal VariationText Chart 18-14ExcelMoving AverageSeasonal VariationThe Moving-Average MethodThe resulting series (sales) is called deseasonalized sales or seasonally adjusted salesThe reason for deseasonalizing a series (sales) is to remove the seasonal fluctuations so that the trend and cycle can be studiedSeasoning Data DeA set of typical indexes is very useful in adjusting a series, sales, for exampleText Chart 18-17ExcelSeasonalized Sales DeTest your learning www.mcgrawhill.ca/college/lindClick onOnline Learning Centrefor quizzesextra contentdata setssearchable glossaryaccess to Statistics Canada’s E-Stat dataand much more!This completes Chapter 18