chapter 13 analyzing and forecasting time series data
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Chapter 13Chapter 13
Analyzing and Analyzing and Forecasting Time Forecasting Time
Series DataSeries Data
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Chapter 13 - Chapter 13 - Chapter Chapter OutcomesOutcomesAfter 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.
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ForecastingForecasting
Model specificationModel specification refers to the process of selecting the forecasting technique to be used in a particular situation.
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ForecastingForecasting
Model fittingModel fitting refers to the process of determining how well a specified model fits its past data.
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ForecastingForecasting
Model diagnosisModel 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.
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ForecastingForecasting
The forecasting horizon forecasting horizon refers to the number of future periods covered by the forecast, sometimes referred to as forecast lead time.
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ForecastingForecasting
The forecasting period forecasting period refers to the unit of time for which the forecasts are to be made.
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ForecastingForecasting
The forecasting interval forecasting interval refers to the frequency with which the new forecasts are prepared.
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ForecastingForecasting
Time-Series dataTime-Series data are data which are measured over time. In most applications the period between measurements is uniform.
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Components of Time Components of Time Series DataSeries Data
• Trend Component• Seasonal
Component• Cyclical Component• Random Component
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Time Series ForecastingTime Series Forecasting
A time-series plottime-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.
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Time-Series PlotTime-Series Plot(Figure 13-1)(Figure 13-1)
0
100
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1000
Months
$ x
1,0
00
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Time Series ForecastingTime Series Forecasting
A linear trendlinear trend is any long-term increase or decrease in a time series in which the rate of change is relatively constant.
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Time Series ForecastingTime Series Forecasting
A seasonal componentseasonal component is a pattern that is repeated throughout a time series and has a recurrence period of at most one year.
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Time Series ForecastingTime Series Forecasting
A cyclical componentcyclical 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.
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Time Series ForecastingTime Series Forecasting
The random componentrandom component refers to changes in the time-series data that are unpredictable and cannot be associated with the trend, seasonal, or cyclical components.
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Trend-Based Forecasting Trend-Based Forecasting TechniquesTechniques
LINEAR TREND MODELLINEAR TREND MODEL
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, . . . )
tt ty 10
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Linear Trend ModelLinear Trend Model(Example 13-2)(Example 13-2)
Year t Sales1991 1 $300,0001992 2 $295,0001993 3 $330,0001994 4 $345,0001995 5 $320,0001996 6 $370,0001997 7 $380,0001998 8 $400,0001999 9 $385,0002000 10 $430,000
Taft Ice Cream Sales
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Linear Trend ModelLinear Trend Model(Example 13-2)(Example 13-2)
$0
$50,000
$100,000
$150,000
$200,000
$250,000
$300,000
$350,000
$400,000
$450,000
$500,000
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Year
Sa
les
Taft Sales
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Linear Trend ModelLinear Trend Model(Example 13-2)(Example 13-2)
LEAST SQUARES EQUATIONSLEAST SQUARES EQUATIONS
where:n = Number of periods in the time
seriest = Time period independent
variableyt = Dependent variable at time t
n
tyt
n
ytty
b
tt
22
1
n
tb
n
yb t 10
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Linear Trend ModelLinear Trend Model(Example 13-2)(Example 13-2)
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.955138103R Square 0.912288796Adjusted R Square 0.901324895Standard Error 14513.57776Observations 10
ANOVAdf SS MS F Significance F
Regression 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
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Linear Trend ModelLinear Trend Model(Example 13-2)(Example 13-2)Taft Linear Trend Model
y = 14576t + 277333
$0
$50,000
$100,000
$150,000
$200,000
$250,000
$300,000
$350,000
$400,000
$450,000
$500,000
1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001
Year
Sa
les
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Linear Trend ModelLinear Trend Model- Forecasting -- Forecasting -
Trend Projection:
)(76.575,1433.333,277 tFt
Forecasting Period t = 11:
69.666,437$
)11(76.575,1433.333,277 tF
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Linear Trend ModelLinear Trend Model- Forecasting -- Forecasting -
MEAN SQUARE ERRORMEAN SQUARE ERROR
where:yt = Actual value at time t
Ft = Predicted value at time t
n = Number of time periods
n
FyMSE tt
2)(
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Linear Trend ModelLinear Trend Model- Forecasting -- Forecasting -
MEAN ABSOLUTE DEVIATIONMEAN ABSOLUTE DEVIATION
where:yt = Actual value at time t
Ft = Predicted value at time t
n = Number of time periods
n
FyMAD tt
||
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Linear Trend ModelLinear Trend Model- Forecasting -- Forecasting -
MEAN ABSOLUTE DEVIATIONMEAN ABSOLUTE DEVIATION
or:n
Fy tt
)( BiasForecast
n
)(error BiasForecast
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Nonlinear Trend ModelsNonlinear Trend Models(Example)(Example)
tt ty 210
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Trend-Based ForecastingTrend-Based Forecasting
A seasonal indexseasonal index is a number used to quantify the effect of seasonality for a given time period.
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Trend-Based ForecastingTrend-Based Forecasting
MUTIPLICATIVE TIME SERIES MUTIPLICATIVE TIME SERIES MODELSMODELS
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
ttttt ICSTy
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Trend-Based ForecastingTrend-Based Forecasting
A moving averagemoving average is the average of n consecutive values in a time series.
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Trend-Based ForecastingTrend-Based Forecasting
RATIO-TO-MOVING-AVERAGERATIO-TO-MOVING-AVERAGE
tt
ttt CT
yIS
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Trend-Based ForecastingTrend-Based Forecasting
DESEASONALIZATIONDESEASONALIZATION
t
tttt S
yICT
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Trend-Based ForecastingTrend-Based Forecasting
A seasonally unadjusted seasonally unadjusted forecastforecast is a forecast made for seasonal data that does not include an adjustment for the seasonal component in the time series.
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Steps in the Seasonal Steps in the Seasonal Adjustment ProcessAdjustment 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.
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Steps in the Seasonal Steps in the Seasonal Adjustment ProcessAdjustment Process
(continued)(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.
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Forecasting Using Forecasting Using Smoothing TechniquesSmoothing Techniques
Exponential smoothingExponential 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.
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Forecasting Using Forecasting Using Smoothing TechniquesSmoothing Techniques
EXPONENTIAL SMOOTHING MODELEXPONENTIAL SMOOTHING MODEL
or::
where:Ft+1= Forecast value for period t +
1yt = Actual value for period t
Ft = Forecast value for period t
= Alpha (smoothing constant)
)(1 tttt FyFF
ttt FyF )1(1
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Forecasting Using Smoothing Forecasting Using Smoothing TechniquesTechniques
DOUBLE EXPONENTIAL SMOOTHING MODELDOUBLE EXPONENTIAL SMOOTHING MODEL
where:yt = Actual value in time t
= Constant-process smoothing constant = Trend-smoothing constantCt = Smoothed constant-process value for
period tTt = Smoothed trend value for period t
forecast value for period tFt+1= Forecast value for period t + 1
t = Current time period
))(1( 11 tttt TCyC
11 )1()( tttt TCCT
ttt TCF 1
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Key TermsKey Terms• Alpha ()• Beta ()• Cyclical Component• Deseasonalizing• Double Exponential
Smoothing• Exponential
Smoothing• Forecast Bias
• Forecast Error• Forecasting• Forecasting Horizon• Forecasting Interval• Forecasting Period• Linear Trend• Mean Absolute
Deviation (MAD)• Mean Squared Error
(MSE)
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Key TermsKey Terms(continued)(continued)
• Model Diagnosis• Model Fitting• Model Specification• Moving Average• Nonlinear Trend• Qualitative
Forecasting• Quantitative
Forecasting• Random Component
• Ratio-To-Moving-Average Method
• Residual• Seasonal
Component• Seasonal Index• Seasonally
Unadjusted Forecast• Smoothing Constant• Splitting Samples
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Key TermsKey Terms(continued)(continued)
• Time-Series Data
• Time-Series Plot• Trend