Chapter 13 Analyzing andForecasting Time Series Data
Chapter 13 - ChapterOutcomesAfter studying the material in this chapter, youshould be able to:•Apply the basic steps in developing andimplementing forecasting models.•Identify the components present in a time series.•Use smoothing-based forecasting modelsincluding, single and double exponentialsmoothing.•Apply trend-based forecasting models, includinglinear trend, nonlinear trend, and seasonallyadjusted trend.
ForecastingModel specification refers to theprocess of selecting the forecastingtechnique to be used in a particularsituation.
ForecastingModel fitting refers to the process ofdetermining how well a specifiedmodel fits its past data.
ForecastingModel diagnosis refers to the processof determining how well the modelfits the past data and how well themodel’s assumptions appear to besatisfied.
ForecastingThe forecasting horizon refers to thenumber of future periods covered bythe forecast, sometimes referred to asforecast lead time.
ForecastingThe forecasting period refers to theunit of time for which the forecastsare to be made.
ForecastingThe forecasting interval refers to thefrequency with which the newforecasts are prepared.
ForecastingTime-Series data are data which aremeasured over time. In most applicationsthe period between measurements isuniform.
Components of Time Series Data • Trend Component • Seasonal Component • Cyclical Component • Random Component
Time Series ForecastingA time-series plot is a two-dimensionalplot of the time series. The verticalaxis measures the variable of interestand the horizontal axis corresponds tothe time periods.
$ x 1,000 Ja nu 1000 100 200 300 400 500 600 700 800 900 0 ar y M ar ch M aySe Jul pt y em N b ov er em be Ja r nu ar y M ar ch M aySe Jul pt y em N b ov er em be Ja r nu ar y M ar ch M ay (Figure 13-1)Se Jul pt y em N be ov r em be Ja r nu ar y Time-Series Plot M ar ch M aySe Jul pt y em N be ov r em be r
Time Series Forecasting A linear trend is any long-term increase or decrease in a time series in which the rate of change is relatively constant.
Time Series ForecastingA seasonal component is a patternthat is repeated throughout a timeseries and has a recurrence periodof at most one year.
Time Series ForecastingA cyclical component is a patternwithin the time series that repeatsitself throughout the time series andhas a recurrence period of more thanone year.
Time Series ForecastingThe random component refers tochanges in the time-series data that areunpredictable and cannot be associatedwith the trend, seasonal, or cyclicalcomponents.
Trend-Based Forecasting Techniques LINEAR TREND MODEL yt = β 0 + β1t + ε twhere: yi = Value of trend at time t β 0 = Intercept of the trend line β 1 = Slope of the trend line t = Time (t = 1, 2, . . . )
Linear Trend Model (Example 13-2) LEAST SQUARES EQUATIONS ∑t ∑ y ∑ ty − n t t b1 = ∑t − ∑ ty2 2 n b0 = ∑y t − b1 ∑twhere: n n n = Number of periods in the time series t = Time period independent variable yt = Dependent variable at time t
Linear Trend Model (Example 13-2) SUMMARY OUTPUTRegression StatisticsMultiple R 0.955138103R Square 0.912288796Adjusted R Square 0.901324895Standard Error 14513.57776Observations 10ANOVA df SS MS F Significance FRegression 1 17527348485 17527348485 83.20841575 1.67847E-05Residual 8 1685151515 210643939.4Total 9 19212500000 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%Intercept 277333.3333 9914.661116 27.97204363 2.88084E-09 254470.069 300196.5977 254470.069 300196.5977t 14575.75758 1597.892322 9.121864708 1.67847E-05 10891.00889 18260.50626 10891.00889 18260.50626
Linear Trend Model (Example 13-2) Taft Linear Trend Model $500,000 $450,000 $400,000 $350,000 y = 14576t + 277333 $300,000Sales $250,000 $200,000 $150,000 $100,000 $50,000 $0 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 Year
Linear Trend Model - Forecasting -Trend Projection: Ft = 277,333.33 + 14,575.76(t )Forecasting Period t = 11: Ft = 277,333.33 + 14,575.76(11) $437,666.69
Linear Trend Model - Forecasting - MEAN SQUARE ERROR MSE = ∑(y t − Ft ) 2 nwhere: yt = Actual value at time t Ft = Predicted value at time t n = Number of time periods
Linear Trend Model - Forecasting - MEAN ABSOLUTE DEVIATION MAD = ∑| y t − Ft | nwhere: yt = Actual value at time t Ft = Predicted value at time t n = Number of time periods
Linear Trend Model - Forecasting - MEAN ABSOLUTE DEVIATION Forecast Bias = ∑(y t − Ft ) nor: Forecast Bias = ∑ (error) n
Trend-Based ForecastingA seasonal index is a number usedto quantify the effect of seasonalityfor a given time period.
Trend-Based ForecastingMUTIPLICATIVE TIME SERIES MODELS yt = Tt × St × Ct × I twhere: yt = Value of the time series at time t Tt = Trend value at time t St = Seasonal value at time t Ct = Cyclical value at time t It = Residual or random value at time t
Trend-Based ForecastingA moving average is the averageof n consecutive values in a timeseries.
Trend-Based Forecasting RATIO-TO-MOVING-AVERAGE yt St × I t = Tt × Ct
Trend-Based Forecasting DESEASONALIZATION yt Tt × Ct × I t = St
Trend-Based ForecastingA seasonally unadjusted forecast is aforecast made for seasonal data thatdoes not include an adjustment forthe seasonal component in the timeseries.
Steps in the Seasonal Adjustment Process• Compute each moving average from the k appropriate consecutive data values.• Compute the centered moving averages.• Isolate the seasonal component by computing the ratio-to-moving-average values.• Compute the seasonal indexes by averaging the ratio-to-moving-averages for comparable periods.
Steps in the Seasonal Adjustment Process (continued)• Normalize the seasonal indexes.• Deseasonalize the time series.• Use least-squares regression to develop the trend line using the deseasonalized data.• Develop the unadjusted forecasts using trend projection.• Seasonally adjust the forecasts by multiplying the unadjusted forecasts by the appropriate seasonal index.
Forecasting Using Smoothing TechniquesExponential smoothing is a time-seriessmoothing and forecasting technique thatproduces an exponentially weightedmoving average in which each smoothingcalculation or forecast is dependent uponall previously observed values.
Forecasting Using Smoothing Techniques EXPONENTIAL SMOOTHING MODEL Ft +1 = Ft + α ( yt − Ft )or:: Ft +1 = αyt + (1 − α ) Ftwhere: Ft+1= Forecast value for period t + 1 yt = Actual value for period t Ft = Forecast value for period t α = Alpha (smoothing constant)
Forecasting Using Smoothing Techniques DOUBLE EXPONENTIAL SMOOTHING MODEL Ct = αyt + (1 − α )(Ct −1 + Tt −1 ) Tt = β (Ct − Ct −1 ) + (1 − β )Tt −1 Ft +1 = Ct + Ttwhere: yt = Actual value in time t α = Constant-process smoothing constant β = Trend-smoothing constant Ct = Smoothed constant-process value for period t Tt = Smoothed trend value for period t forecast value for period t Ft+1= Forecast value for period t + 1 t = Current time period