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    Analyzing and Forecasting Time Series Data

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    Kanta Ganguly
    This chapter discusses time series forecasting techniques. It covers identifying components in time series data, model specification, fitting and diagnosis. Exponential smoothing models including single and double exponential smoothing are explained. Trend-based techniques like linear, nonlinear and seasonally adjusted trend models are covered. Seasonal adjustment processes and forecasting using smoothing methods are also summarized.

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    Analyzing and Forecasting Time Series Data

    Uploaded by

    Kanta Ganguly
    49 views41 pages
    This chapter discusses time series forecasting techniques. It covers identifying components in time series data, model specification, fitting and diagnosis. Exponential smoothing models including single and double exponential smoothing are explained. Trend-based techniques like linear, nonlinear and seasonally adjusted trend models are covered. Seasonal adjustment processes and forecasting using smoothing methods are also summarized.

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    ch13

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    This chapter discusses time series forecasting techniques. It covers identifying components in time series data, model specification, fitting and diagnosis. Exponential smoothing models including single and double exponential smoothing are explained. Trend-based techniques like linear, nonlinear and seasonally adjusted trend models are covered. Seasonal adjustment processes and forecasting using smoothing methods are also summarized.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PPT, PDF, TXT or read online from Scribd
    Download as ppt, pdf, or txt
    49 views41 pages

    Analyzing and Forecasting Time Series Data

    Uploaded by

    Kanta Ganguly
    This chapter discusses time series forecasting techniques. It covers identifying components in time series data, model specification, fitting and diagnosis. Exponential smoothing models including single and double exponential smoothing are explained. Trend-based techniques like linear, nonlinear and seasonally adjusted trend models are covered. Seasonal adjustment processes and forecasting using smoothing methods are also summarized.

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    Download as ppt, pdf, or txt
    of 41

    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.
    Time-Series Plot
    (Figure 13-1)
    1000

    900

    800

    700

    600
    $ x 1,000

    500

    400

    300

    200

    100

    0
    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

     ty  n
    t
    t y t

    b1 
    t   ty2
    2

    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
    y = 14576t + 277333
    $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
    - 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 )
    n
    or:

    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
    • • Forecasting Interval
    Deseasonalizing
    • Forecasting Period
    • Double Exponential
    • Linear Trend
    Smoothing
    • Mean Absolute
    • Exponential Deviation (MAD)
    Smoothing
    • Mean Squared Error
    • Forecast Bias (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 Forecasting • Seasonally
    • Quantitative Unadjusted Forecast
    Forecasting • Smoothing Constant
    • Random Component • Splitting Samples
    Key Terms
    (continued)

    • Time-Series Data
    • Time-Series Plot
    • Trend

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