Chapter 13 Analyzing and Forecasting Time Series Data

Chapter 13Chapter 13

Analyzing and Analyzing and Forecasting Time Forecasting Time

Series DataSeries Data

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.

ForecastingForecasting

Model specificationModel specification refers to the process of selecting the forecasting technique to be used in a particular situation.


Model fittingModel fitting refers to the process of determining how well a specified model fits its past data.


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.


The forecasting horizon forecasting horizon refers to the number of future periods covered by the forecast, sometimes referred to as forecast lead time.


The forecasting period forecasting period refers to the unit of time for which the forecasts are to be made.


The forecasting interval forecasting interval refers to the frequency with which the new forecasts are prepared.


Time-Series dataTime-Series data are data which are measured over time. In most applications the period between measurements is uniform.

Components of Time Components of Time Series DataSeries Data

• Trend Component• Seasonal

Component• Cyclical Component• Random Component

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.

Time-Series PlotTime-Series Plot(Figure 13-1)(Figure 13-1)

0

100

200

300

400

500

600

700

800

900

1000

Months

$ x

1,0

00


A linear trendlinear trend is any long-term increase or decrease in a time series in which the rate of change is relatively constant.


A seasonal componentseasonal component is a pattern that is repeated throughout a time series and has a recurrence period of at most one year.


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.


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.

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

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


$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


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


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

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

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


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)(


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

||


MEAN ABSOLUTE DEVIATIONMEAN ABSOLUTE DEVIATION

or:n

Fy tt

)( BiasForecast

n

)(error BiasForecast

Nonlinear Trend ModelsNonlinear Trend Models(Example)(Example)

tt ty 210

Trend-Based ForecastingTrend-Based Forecasting

A seasonal indexseasonal index is a number used to quantify the effect of seasonality for a given time period.


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


A moving averagemoving average is the average of n consecutive values in a time series.


RATIO-TO-MOVING-AVERAGERATIO-TO-MOVING-AVERAGE

tt

ttt CT

yIS


DESEASONALIZATIONDESEASONALIZATION

t

tttt S

yICT


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.

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.

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.

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.

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

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

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)

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

Key TermsKey Terms(continued)(continued)

• Time-Series Data

• Time-Series Plot• Trend

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Chapter 13 Analyzing and Forecasting Time Series Data