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Chapter 13 Analyzing and Forecasting Time Series Data Chapter 13 - Chapter Outcomes After studying the material in this chapter, you should be able to: •Apply the basic steps in developing and implementing forecasting models. •Identify the components present in a time series. •Use smoothing-based forecasting models including, single and double exponential smoothing. •Apply trend-based forecasting models, including linear trend, nonlinear trend, and seasonally adjusted trend. Forecasting Model specification refers to the process of selecting the forecasting technique to be used in a particular situation. Forecasting Model fitting refers to the process of determining how well a specified model fits its past data. Forecasting Model diagnosis refers to the process of determining how well the model fits the past data and how well the model’s assumptions appear to be satisfied. Forecasting The forecasting horizon refers to the number of future periods covered by the forecast, sometimes referred to as forecast lead time. Forecasting The forecasting period refers to the unit of time for which the forecasts are to be made. Forecasting The forecasting interval refers to the frequency with which the new forecasts are prepared. Forecasting Time-Series data are data which are measured over time. In most applications the period between measurements is uniform. Components of Time Series Data • Trend Component • Seasonal Component • Cyclical Component • Random Component Time Series Forecasting A time-series plot is a two-dimensional plot of the time series. The vertical axis measures the variable of interest and the horizontal axis corresponds to the time periods. $ x 1,000 Ja nu 1000 0 100 200 300 400 500 600 700 800 900 ar y M ar ch M ay Se Jul pt y em No ber ve m be Ja r nu ar y M ar ch M ay Se Jul pt y em No ber ve m be Ja r nu ar y M ar ch M ay (Figure 13-1) Se Jul pt y em No ber ve m be Ja r nu ar Time-Series Plot y M ar ch M ay Se Jul pt y em No ber ve m 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 Forecasting A seasonal component is a pattern that is repeated throughout a time series and has a recurrence period of at most one year. Time Series Forecasting A cyclical component is a pattern within the time series that repeats itself throughout the time series and has a recurrence period of more than one year. Time Series Forecasting The random component refers to changes in the time-series data that are unpredictable and cannot be associated with the trend, seasonal, or cyclical components. Trend-Based Forecasting Techniques LINEAR TREND MODEL yt 0 1t t where: yi = Value of trend at time t 0 = Intercept of the trend line 1 = Slope of the trend line t = Time (t = 1, 2, . . . ) Linear Trend Model (Example 13-2) Taft Ice Cream Sales Year t Sales 1991 1 $300,000 1992 2 $295,000 1993 3 $330,000 1994 4 $345,000 1995 5 $320,000 1996 6 $370,000 1997 7 $380,000 1998 8 $400,000 1999 9 $385,000 2000 10 $430,000 Linear Trend Model (Example 13-2) $500,000 Taft Sales $450,000 $400,000 $350,000 $300,000 Sales $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 (Example 13-2) LEAST SQUARES EQUATIONS t y ty n t t b1 t ty2 2 n b0 y t b1 t where: 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 OUTPUT Regression Statistics Multiple R 0.955138103 R Square 0.912288796 Adjusted R Square 0.901324895 Standard Error 14513.57776 Observations 10 ANOVA df SS MS F Significance F Regression 1 17527348485 17527348485 83.20841575 1.67847E-05 Residual 8 1685151515 210643939.4 Total 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 300196.5977 t 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 $300,000 y = 14576t + 277333 Sales $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 n where: 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 | n where: 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 ) or: n Forecast Bias (error ) n Nonlinear Trend Models (Example) yt 0 1t t 2 Trend-Based Forecasting A seasonal index is a number used to quantify the effect of seasonality for a given time period. Trend-Based Forecasting MUTIPLICATIVE TIME SERIES MODELS yt Tt St Ct I t where: 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 Forecasting A moving average is the average of n consecutive values in a time series. 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 Forecasting A seasonally unadjusted forecast is a forecast made for seasonal data that does not include an adjustment for the seasonal component in the time series. 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 Techniques Exponential smoothing is a time-series smoothing and forecasting technique that produces an exponentially weighted moving average in which each smoothing calculation or forecast is dependent upon all previously observed values. Forecasting Using Smoothing Techniques EXPONENTIAL SMOOTHING MODEL Ft 1 Ft ( yt Ft ) or:: Ft 1 yt (1 ) Ft where: Ft+1= Forecast value for period t + 1 yt = Actual value for period t Ft = Forecast value for period t = Alpha (smoothing constant) Forecasting Using Smoothing Techniques DOUBLE EXPONENTIAL SMOOTHING MODEL Ct yt (1 )(Ct 1 Tt 1 ) Tt (Ct Ct 1 ) (1 )Tt 1 Ft 1 Ct Tt where: yt = Actual value in time t = Constant-process smoothing constant = 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 Key Terms • Alpha () • Forecast Error • Beta () • Forecasting • Cyclical Component • Forecasting Horizon • Deseasonalizing • Forecasting Interval • Double Exponential • Forecasting Period Smoothing • Linear Trend • Exponential • Mean Absolute Smoothing Deviation (MAD) • Forecast Bias • Mean Squared Error (MSE) Key Terms (continued) • Model Diagnosis • Ratio-To-Moving- • Model Fitting Average Method • Model Specification • Residual • Moving Average • Seasonal Component • Nonlinear Trend • Seasonal Index • Qualitative • Seasonally Forecasting Unadjusted Forecast • Quantitative • Smoothing Constant Forecasting • Splitting Samples • Random Component Key Terms (continued) • Time-Series Data • Time-Series Plot • Trend

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