1 Forecasting Models CHAPTER 7 2 9.1 Introduction to Time Series Forecasting Forecasting is the...

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Forecasting ModelsForecasting Models

CHAPTER 7

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9.1 Introduction to Time Series Forecasting

• Forecasting is the process of predicting the future.

• Forecasting is an integral part of almost all business enterprises.

• Examples– Manufacturing firms forecast demand for their product, to

schedule manpower and raw material allocation.

– Service organizations forecast customer arrival patterns to maintain adequate customer service.

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• More examples– Security analysts forecast revenues, profits, and debt

ratios, to make investment recommendations.

– Firms consider economic forecasts of indicators

(housing starts, changes in gross national profit)

before deciding on capital investments.

Introduction

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• Good forecasts can lead to– Reduced inventory costs.– Lower overall personnel costs.– Increased customer satisfaction.

• The forecasting process can be based on:– Educated guess.– Expert opinions.– Past history of data values, known as a time series.

Introduction

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Components of a Time Series

– Long-term trend • A time series may be stationary or exhibit trend over time.• Long-term trend is typically modeled as a linear, quadratic

or exponential function.– Seasonal variation

• When a repetitive pattern is observed over some time horizon, the series is said to have seasonal behavior.

• Seasonal effects are usually associated with calendar or climatic changes.

• Seasonal variation is frequently tied to yearly cycles.

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– Cyclical variation• An upturn or downturn not tied to seasonal variation.• Usually results from changes in economic conditions.

– Random effects


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A stationary time series

Linear trend time series

Linear trend and seasonality time series

Time

Time seriesvalue

Future


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• The goal of a time series forecast is to identify factors that can be predicted.

• This is a systematic approach involving the following steps.

– Step 1: Hypothesize a form for the time series model.

– Step 2: Select a forecasting technique.

– Step 3: Prepare a forecast.

Steps in the Time Series Forecasting Process

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Step 1: Identify components included in the time series

– Collect historical data.

– Graph the data vs. time.

– Hypothesize a form for the time series model.

– Verify this hypothesis statistically.


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• Step 2: Select a Forecasting Technique– Select a forecasting technique from among several techniques

available. The selection includes• Determination of input parameter values• Performance evaluation on past data of each technique

• Step 3: Prepare a Forecast using the selected technique


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• In a stationary model the mean value of the time series is assumed to be constant.

• The general form of such a model is

Where:

yt = the value of the time series at time period t.

0 = the unchanged mean value of the time series.

t = a random error term at time period t.

7.2 Stationary Forecasting Models

yt = 0 + t

•The values of t

are assumed to be independent• The values of t are assumed to have a mean of 0.

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• Checking for trend • Use Linear Regression iftis normally distributed.

• Use a nonparametric test iftis not normally distributed.

• Checking for seasonality component• Autocorrelation measures the relationship between the values of the time

series in different periods.• Lag k autocorrelation measures the correlation between time series

values which are k periods apart.– Autocorrelation between successive periods indicates a possible trend.– Lag 7 autocorrelation indicates one week seasonality (daily data).– Lag 12 autocorrelation indicates 12-month seasonality (monthly data).

• Checking for Cyclical Components

Stationary Forecasting Models

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• The last period techniqueThe forecast for the next period is the last observed value.

t t+1 t t+1t t+1 t t+1

Moving Average Methods

t1t yF t1t yF

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t+1 t+1tt-2 t-1 t-2 t-1 t t-2 t-1 t t-2 t-1 t t-2 t-1 t t-2 t-1 t

• The moving average methodThe forecast is the average of the last n observations of the time series.

ny...yy

F 1nt1tt1t

n

y...yyF 1nt1tt

1t


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• The weighted moving average method– More recent values of the time series

get larger weights than past values when performing the forecast.


1tF = w1yt + w2yt-1 +w3yt-2 + …+ wnyt-n+1

w1 w2 … wn

wi = 1

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• Forecasts for Future Time Periods

The forecast for time period t+ 1 is the forecast for all future time periods:

This forecast is revised only when new data becomes available.

...,3,2,1kforyF tkt ...,3,2,1kforyF tkt


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• Galaxy Industries is interested in forecasting weekly demand for its YoHo brand yo-yos.

• The yo-yo is a mature product. This year demand pattern is expected to repeat next year.

• To forecast next year demand, the past 52 weeks demand records were collected.

YOHO BRAND YO - YOsMoving Average Methods -

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• Three forecasting methods were suggested:– Last period technique - suggested by Amy Chang.

– Four-period moving average - suggested by Bob Gunther.

– Four-period weighted moving average - suggested by Carlos

Gonzalez.

• Management wants to determine:– If a stationary model can be used.

– What forecast will be obtained using each method?

YOHO BRAND YO - YOsMoving Average Methods -

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• Construct the time series plotWeek Demand Week Demand Week Demand Week Demand

1 415 14 365 27 351 40 2822 236 15 471 28 388 41 3993 348 16 402 29 336 42 3094 272 17 429 30 414 43 4355 280 18 376 31 346 44 2996 395 19 363 32 252 45 5227 438 20 513 33 256 46 3768 431 21 197 34 378 47 4839 446 22 438 35 391 48 41610 354 23 557 36 217 49 24511 529 24 625 37 427 50 39312 241 25 266 38 293 51 48213 262 26 551 39 288 52 484

Week Demand Week Demand Week Demand Week Demand1 415 14 365 27 351 40 2822 236 15 471 28 388 41 3993 348 16 402 29 336 42 3094 272 17 429 30 414 43 4355 280 18 376 31 346 44 2996 395 19 363 32 252 45 5227 438 20 513 33 256 46 3768 431 21 197 34 378 47 4839 446 22 438 35 391 48 41610 354 23 557 36 217 49 24511 529 24 625 37 427 50 39312 241 25 266 38 293 51 48213 262 26 551 39 288 52 484

0100200300400500600700

De

man

d

Weeks

Series1

Neither seasonality nor cyclical effects can be observed

YOHO BRAND YO YOs- Solution

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• Run linear regression to test 1 in the model yt=0+1t+t

• Excel resultsCoeff. Stand. Err t-Stat P-value Lower 95%Upper 95%

Intercept 369.27 27.79436 13.2857 5E-18 313.44 425.094Weeks 0.3339 0.912641 0.36586 0.71601 -1.49919 2.166990.71601

This large P-value indicates that there is little evidence that trend exists

• Conclusion: A stationary model is appropriate.

Is trend present?

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• Last period technique (Amy’s Forecast)

– Four-period moving average (Bob’s forecast)

– Four period weighted moving average (Carlo’s forecast)

53 = (y52 + y51 + y50 + y49) /4 =

(484+482+393+245) / 4 = 401 boxes.

= 484 boxes.53 = y52y

y

y53 =0.4y52 + 0.3y51 + 0.2y 50 + 0.1y49 = 0.4(484) + 0.3(482) + 0.2(393) + 0.1(245) = 441.3 boxes.

Forecast for Week 53

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• Since the time series is stationary, the forecasts for weeks 54 and 55 remain as the forecast for week 53.

• These forecasts will be revised pending observation of the actual demand in week 53.

Forecast for Weeks 54 and 55

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• This technique is used to forecast stationary time series.

• All the previous values of historical data affect the forecast.

The Exponential Smoothing Technique

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• For each period create a smoothed value Lt of the time series, that represents all the information known by t.

• The smoothed value Lt is the weighted average of– The current period’s actual value (with weight of ).– The forecast value for the current period (with weight of 1-).

• The smoothed value Lt becomes the forecast for period t+1.


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An initial “forecast” is needed to start the process.

ttt1t F)1(yLF ttt1t F)1(yLF

Define:Ft+1 = the forecast value for time t+1 yt = the value of the time series at time t = smoothing constant


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– Approach 1:

Continue from t=3 with the recursive formula.

– Approach 2:

112 yLF

• Average the initial “ n ” values of the time series.

• Use this average as the forecast for period n + 1

• Begin using exponential smoothing from that time period

onward and so on.

The Exponential Smoothing Technique – Generating an initial forecast

).yF( n1n

],y)1(yF)1(yF[ nn1nn2n

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• Since this technique deals with stationary time series, the forecasts for future periods does not change.

• Assume N is the number of periods for which data are available. Then

FN+1 = yN + (1 – )FN,FN+k = FN+1, for k = 2, 3, …

The Exponential Smoothing Technique – Future Forecasts

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• An exponential smoothing forecast is suggested, with = 0.1.

• An Initial Forecast is created at t=2 by F2 = y1 = 415.

• The recursive formula is used from period 3 onward:

F3 = .1y2 + .9F2 = .1(236) + .9(415) = 397.10

F4 = .1y3 + .9F3 = .1(348) + .9(397.10) = 392.19

and so on, until period 53 is reached (N+1 = 52+1 = 53).

F53 = .1y52 + .9F52 = .1(484) + .9(382.32) = 392.49

F54 = F55 = 392.49 ( = F52)

YOHO BRAND YO - YOs The Exponential Smoothing Technique

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Period Series Forecast1 415 #N/A2 236 4153 348 397.14 272 392.195 280 380.17149 245 382.5884742

50 393 368.829626851 482 371.246664152 484 382.321997753 392.489854 392.489855 392.4898


(Excel)

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0

100

200

300

400

500

600

700

0 10 20 30 40 50 60

Notice the amount of smoothing Included in the smoothed series


(Excel)

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• Relationship between exponential smoothing and simple moving average– The two techniques will generate forecasts having the same average age of

information if

– This formula is a useful guide to the appropriate value for • An exponential smoothing forecast “based on large number of periods” should

have a small .

• A small provides a lot of smoothing.

• A large provides a fast response to the recent changes in the time series and a smaller amount of smoothing.

The Exponential Smoothing Technique Average age of information

2

k

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7.3 Evaluating the performance of Forecasting Techniques

• Several forecasting methods have been presented.

• Which one of these forecasting methods gives the “best” forecast?

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• Generally, to evaluate each forecasting method:

– Select an evaluation measure.

– Calculate the value of the evaluation measure using the forecast error equation

– Select the forecast with the smallest value of the evaluation measure.

ttt Fy

Performance Measures

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Time 1 2 3 4 5 6 Time series: 100 110 90 80 105 115

3-Period Moving average: 100 93.33 91.6Error for the 3-Period MA: - 20 11.67 23.43-Period Weighted MA(.5, .3, .2) 98 89 85.5Error for the 3-Period WMA - 18 16 29.5

Performance Measures – Sample Example

• Find the forecasts and the errors for each forecasting technique applied to the following stationary time series.

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t2nMSE =

Performance Measures

MAD =t|n

MAPE =n

t

n yt

LAD = max |t|

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= 361.24(-20)2+(11.67)2+(23.4)2

3MSE = =tn Divide by 3, not by 6 periods.Period 1, 2, 3 do not have a forecast

Performance Measures – MSE for the Sample Example

MSE for the moving average technique:

MSE for the weighted moving average technique:

(-18)2 + (16)2 + (29.5)2

3MSE = =tn

= 483.4

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MAD for the moving average technique:

MAD for the weighted moving average technique:

= 21.17

= 18.35|-20| + |11.67| + |23.4|

3MAD = =t|n

|-18| + |116| + |29.5|3MAD = =t|

n

Performance Measures – MAD for the Sample Example

t Yt|

n

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MAPE for the moving average technique:

MAPE for the weighted moving average technique:

= .211

= .188|-20/80| + |11.67/105|+ |23.4/115|3MAPE= =

|-18/80| + |16/105| + |29.5/115|3MAPE= =

Performance Measures – MAPE for the Sample Example

t Yt|

n

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LAD for the moving average technique:

LAD for the weighted moving average technique:

= 23.4

= 29.5

Performance Measures – LAD for the Sample Example

|-20|, |11.67|, |23.4|

LAD= max =|t| max {|-18|, |16|, |29.5|}

LAD= max =|t|

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Performance Measures –YOHO BRAND YO - YOs

=B4Drag to Cell C56

=E5/B5

=D5^2=ABS(D5

)=B5-C5

Highlight Cells D5:G5 and Drag to Cells D55:G55

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Performance Measures –YOHO BRAND YO - YOs

=AVERAGE(B4:B7)Drag to Cell C56

Highlight Cells D8:G8 and Drag to Cells D55:G55

=E8/B8=D8^2=ABS(D8

)=B8-C8

Forecast begins at period 5.

=C56Drag to C58

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• Use the performance measures to select a good set of values for each model parameter.– For the moving average:

• the number of periods (n).– For the weighted moving average:

• The number of periods (n),• The weights (Wi).

– For the exponential smoothing:• The exponential smoothing factor ().

• Excel Solver can be used to determine the values of the model parameters.

Performance Measures –Selecting Model Parameters

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Weights for the Weighted Moving Average

Minimize MSE using Solver-Cell containing MSEMinimize

Weights

Weights Are Nonincreasin

gTotal of Weights Sum to 1

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• Key issues considered when determining the technique to be used in forecasting stationary time-series.

– The degree of autocorrelation.

– The possibility of future shifts in time series values.

– The desired responsiveness of the forecasting technique.

– The amount of data required to calculate the forecast.

Selecting Forecasting Technique

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• If we suspect trend, we should assess whether the trend is linear or nonlinear.

• Here we assume only linear trend.

yt = 0 + 1t + t

• Forecasting methods– Linear regression forecast.– Holt’s Linear Exponential Smoothing Technique.

7.4 Time Series with Linear Trend

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The Linear Regression Approach

• Construct the regression equation based on the historical data available.

• The independent variable is “time”.The dependent variable is the “time-series value”.

• Forecasts should not expand to periods far into the future.

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2 3

++

3F+

2L

3L

4

2T

3T

Y3

+++

ttt F)1(yL

1t1ttt T)1()LL(T

23 LL

++

Holt’s Technique – A qualitative demonstration

4F +2T

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• The Holt’s Linear Exponential Smoothing Technique.– Adjust the Level Lt , and the Trend Tt in each period:

=smoothing constant for the time series level.

= smoothing constant for the time series trend.= estimate of the time series for time t as of time t.

= estimate of the time series trend for time t as of time t.yt = value of the time series at time t.

= forecast of the value of the time series for time t calculated at a period prior to time t.

tT

tF

ttt F)1(yL

1t1ttt T)1()LL(T

12222 yy= T and yL

tL

The Holt’s Technique

Level:

Trend:

Initial values:

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• Forecasting k periods into the future– By linear regression

– By the Holt’s linear exponential smoothing technique

k)(tF 10kt

ttkt kTLF

Future Forecasts

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American Family Products Corp.

• Standard and Poor’s (S&P) is a bond rating firm.

• It is conducting an analysis of American Family Products Corp. (AFP).

• The forecast of year-end current assets is required for years 11 and 12, based on data over the previous 10 years.

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• The company’s assets have been increasing at a relatively constant rate over time.

• Data Year-end current assets

Year Current Assets1 1990 (Million)

2 22803 23284 26355 32496 33107 32568 35339 382610 4119

American Family Products Corp.

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Series1

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Year 1 2 3 4 5 6 7 8 9 10

Assets

Year

A linear trend seems to exists

AFP -SOLUTIONForecasting with the Linear Regression Model

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Regression StatisticsMultiple R 0.980225251R Square 0.960841543Adjusted R Square0.955946736Standard Error 149.0358255Observations 10

ANOVAdf SS MS F Significance F

Regression 1 4360110.982 4360110.982 196.2981421 6.53249E-07Residual 8 177693.4182 22211.67727Total 9 4537804.4

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept 1788.2 101.8108511 17.56394315 1.12773E-07 1553.423605 2022.976395Year 229.8909091 16.40830435 14.01064389 6.53249E-07 192.0532669 267.7285513

y t = 1788.2 + 229.89 t The Regression Equation

The p-value


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=$B$31+$B$32*A2Drag to cells C3:C13

Forecasts for Years 11 and 12


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Year Current Assets1199022280323284263553249………………………………………………

Demonstration of the calculation procedure.with = 0.1 and = 0.2Year 2: y2 = 2280

L2 = 2280.00

T2 = 2280 - 1990 = 290

F3= 2280 + 290 = 2570.00

Year 3: y3 = 2328

L3 = (.1)(2328) + (1 - 0.1)(2570.00) = 2545.80

T3 = (.2)(2545.80-2280)) + (1 - 0.2)(290.00) = 285.16

F4= 2545.80 + 285.16 = 2830.96

AFP -SOLUTIONForecasting using the Holt’s technique

ttt F)1(yL

1t1ttt T)1()LL(T

L2 =y2

T2=y2-y1

F3=L2+T2

Ft+1=Lt+Tt

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INPUTS OUTPUTS

Number of Periods of Data Collected = 10 Period Forecast =

Smoothing Constant (alpha) = 0.1

Smoothing Constant (gamma) = 0.2 MSE = 72994.37 MAPE = 7.752572Initial Forecast Value (Level) = 2280 MAD = 250.3462 LAD = 411.0682

Initial Forecast Value (Trend) = 290

OUTPUTS

Period 11 12 13 14 15 16 17 18 19Forecast 4593.378 4849.579 5105.779 5361.98 5618.18 5874.38 6130.581 6386.781 6642.982

Forecast Forecast Absolute Error AbsolutePeriod Value Level Trend Forecast Error Error Squared % Error

1 19902 2280 2280 2903 2328 2545.8 285.16 2570 -242 242 58564 0.1039524 2635 2811.364 281.2408 2830.96 -195.96 195.96 38400.32 0.0743685 3249 3108.244 284.3687 3092.605 156.3952 156.3952 24459.46 0.0481366 3310 3384.352 282.7164 3392.613 -82.613 82.61302 6824.912 0.0249597 3256 3625.961 274.4951 3667.068 -411.068 411.0682 168977 0.1262498 3533 3863.711 267.146 3900.456 -367.456 367.4564 135024.2 0.1040079 3826 4100.371 261.0488 4130.857 -304.857 304.8567 92937.63 0.0796810 4119 4337.178 256.2004 4361.42 -242.42 242.4199 58767.4 0.05885411 4134.04 164.3329 4593.378

AFP -SOLUTIONForecasting using the Holt’s technique (Excel)

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• Many time series exhibit seasonal and cyclical variation along with trend.

• Seasonality and cyclical variation arise due to calendar, climate, or economic factors.

• Two models are considered:– Additive model

yt = Tt + Ct + St + t

– Multiplicative model

7.5 Time Series with Trend, Seasonality, and Cyclical Variation

yt = Tt Ct Stt

Time series valueTrend componentCyclical component

Random errorSeasonal component

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• This technique can be used to develop an additive or multiplicative model.

• The time series is first decomposed to its components (trend, seasonality, cyclical variation).

• After these components have been determined, the series is re-composed by– adding the components - in the additive model– multiplying the components - in the multiplicative model.

The Classical Decomposition

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• Smooth the time series to remove random effects and seasonality.

• Calculate moving averages.

• Determine “period factors” to isolate the (seasonal)(error) factor.

• Calculate the ratio yt/MAt.

• Determine the “unadjusted seasonal factors” to eliminate the random component from the period factors

The Classical Decomposition- Procedure (1)

• Average all the yt/MAt that correspond to the same season.

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• Determine the “adjusted seasonal factors”.

Calculate: [Unadjusted seasonal factor] [Average seasonal factor]

• Determine “Deseasonalized data values”.

Calculate: yt

[Adjusted seasonal factors]t

• Determine a deseasonalized trend forecast.

The Classical Decomposition- Procedure (2)

Use linear regression on the deseasonalized time series.

Calculate:(yt/Mat) [Adjusted seasonal forecast].

• Determine an “adjusted seasonal forecast”.

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• The CFA is the exclusive bargaining agent for the state-supported Canadian college faculty.

• Membership in the organization has grown over the years, but in the summer months there was always a decline.

• To prepare the budget for the 2001 fiscal year, a forecast of the average quarterly membership covering the year 2001 is required.

CANADIAN FACULTY ASSOCIATION (CFA)

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• Membership records from 1997 through 2000 were collected and graphed.

CFA - Solution

63

CFA - Solution

YEAR PERIOD QUARTERAVERAGE

MEMBERSHIP

1997 1 1 71302 2 69403 3 73544 4 7556

1998 5 1 76736 2 73327 3 76628 4 7809

1999 9 1 787210 2 755111 3 798912 4 8143

2000 13 1 816714 2 790215 3 826816 4 8436

YEAR PERIOD QUARTERAVERAGE

MEMBERSHIP

1997 1 1 71302 2 69403 3 73544 4 7556

1998 5 1 76736 2 73327 3 76628 4 7809

1999 9 1 787210 2 755111 3 798912 4 8143

2000 13 1 816714 2 790215 3 826816 4 8436

Average Dues Paying Members (payroll deduction)

6500

7000

7500

8000

8500

9000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Per iod


6500

7000

7500

8000

8500

9000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Per iod1997 1998 1999 2000

The graph exhibits long term trend

The graph exhibits seasonality pattern

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First moving average is centered at quarter (1+4)/ 2 = 2.5

Centered moving average of the first two moving averages is [7245.01 + 7380.75]/2 = 7312.875

• Smooth the time series to remove random effects and seasonality.

Calculate moving averages.

Classical Decomposition – step 1:Isolating Trend and Cyclical Components

Average membership for the first 4 periods = [7130+6940+7354+7556]/4 = 7245.01

Second moving average is centered at quarter (2+5)/ 2 = 3.5

Average membership for periods [2, 5]= [6940+7354+7556+7673]/4 = 7380.75

Centered location is t = 3

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Classical Decomposition – step 2[SeasonalRandom Error] Factors

• Determine “period factors” to isolate the (Seasonal)(Random error) factor.

Calculate the ratio yt/MAt.

The Centered Moving Average only represents TtCt. The SeasonalRandom error factors are represented by

Stt = yt/TtC t

Example: In period 7 (3rd quarter of 1998):S77=7662/7643.875 = 1.00237

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Averaging the SeasonalRandom Error Factors eliminates the random factor from Stt . This leaves us with the seasonality component only for each season.

Example: Unadjusted Seasonal Factor for the third quarter.

S3 = {S3,97 + S3,98 3,98 + S3,99 3,99}/3 = {1.0056+1.0024+1.0079}/3 = 1.0053

Classical Decomposition – step 3Unadjusted Seasonal Factors

• Determine the “unadjusted seasonal factors” to eliminate the random component from the period factors

Average all the yt/MAt that correspond to the same season.

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Without seasonality the seasonal factors for each season should be equal to 1.Thus, the sum of all seasonal factors would be equal to 4.The adjustment of the unadjusted seasonal factors maintains the sum of 4. Example: The average seasonal factor is

(1.0149+.9658+1.00533+1.01624)/4=1.00057.The adjusted seasonal factor for the 3rd quarter:

S3/Average seasonal factor = 1.00053/1.00057 = 1.00472 Excel’s exact value=1.004759).

Classical Decomposition – step 4Adjusted Seasonal Factors

• Determine the “adjusted seasonal factors”.

Calculate: [Unadjusted seasonal factor] [Average seasonal factor]

Quarter 1 1.014325Quarter 2 0.965252Quareter 3 1.004759Quarter 4 1.015663

Adjusted Seasonal factors

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Classical Decomposition – step 5The Deseasonalized Time Series

Deseasonalize the Time Series by .yt/(Adjusted S)t = TtCtt

Example: Deseasonalized series value for the 2nd quarter, 1998 =

y6/[Adjusted S2] = 7332/0.9652 = 7595.94

• Determine “Deseasonalized data values”.

Calculate: yt

[Adjusted seasonal factors]t

69

Deseasonalized Time Series

7000

7200

7400

7600

7800

8000

8200

8400

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Period

Ave

rag

e M

emb

ersh

ip

(pay

roll

ded

uct

ion

)


6500

7000

7500

8000

8500

9000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Per iod


6500

7000

7500

8000

8500

9000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Per iod

Seasonality has been substantially Reduced. This graph represents TtCtt.

Classical Decomposition – step 5The Time series trend

70

• Determine a deseasonalized trend forecast. Use linear regression on the

deseasonalized time series.

Classical Decomposition – step 6The Time series trend Component

Trend factor: Tt = 7069.6677 + 78.4046t

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• Assuming no cyclic effects the forecast becomes:F(quarter i, time N+k) = TN+k

(Adjusted Si)

Classical Decomposition – step 7The forecast

Trend factor: T17 = 7069.6677 + 78.4046(17) = 8402Forecast(Q1, t=17) = (8402)(1.01433) = 8523

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Classical Decomposition – step 7The forecast

• Assuming cyclic effects, create a series of the cyclic component, as follows:

• Perform the forecast:F(quarter i, time N+k) = TN+k

CN (Adjusted Si)

• Deseasonalized time series Trend component (from the regression)

• Smooth out the error component using moving averages to isolate Ct.

TtCtt= = Ctt. Tt

73

Classical Decomposition Template

74

For a time series with trend and seasonality:

Yt = Tt +St + Rt, which translates to

Yt = 0 + 1t + 2S1 + … +kSk + t

The additive model – The Multiple Regression Approach

The trend element The seasonality element

75

Troy’s Mobil Station

• Troy owns a gas station that experience seasonal variation in sales.

• In addition, due to a steady increase in population Troy feels that average sales are increasing generally.

76

Gasoline Sales Over Five Year Period

3000

3200

3400

3600

3800

4000

4200

4400

4600

4800

5000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Period

Av

era

ge D

aily

G

aso

line S

ale

s

(gallo

ns)

• Data

Troy’s Mobil Station

F W Spg Smr

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Quarterly Input Data Sales t X1 X2 X33497 1 1 0 03484 2 0 1 03553 3 0 0 13837 4 0 0 03726 5 1 0 03589 6 0 1 0

Year 1

Year 2

Trend variable

Seasonal variables

Fall

Winter

Spring

Not Fall Not Winter Not Spring

Troy’s Mobil Station – Multiple Regression input data

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Troy’s Mobil Station Template

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Troy’s Mobil Station –Multiple regression (Excel output)

Extremely good fit

Extremely usefulAll the variableare linearly relatedto sales.

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3610.625 + 58.33125t

-155.00 -322.93

-248.27

-155.00

Troy’s Mobil Station –Multiple regression (Graphical interpretation)

Fall Winter Spring Summer

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• The forecasting additive model is:Ft = 3610.625 + 58.33t – 155F – 323W – 248.27S

• Forecasts for year 5 are produced as follows:• F(Year 5, Fall) = 3610.625+58.33(21) – 155(1) – 323(0) – 248.27(0)• F(Year 5, Winter) = 3610.625+58.33(22) – 155(0) – 323(1) – 248.27(0)• F(Year 5, Spring) = 3610.625+58.33(23) – 155(0) – 323(0) – 248.27(1)• F(Year 5, Summer) = 3610.625+58.33(24) – 155(0) – 323(0) – 248.27(0)

Troy’s Mobil Station – Performing the forecast

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