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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
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                                                                                    (Figure 13-1)




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                                                                                                Time-Series Plot




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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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