Chapter 13 Forecasting

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Chapter 13Forecasting

MGS3100Julie Liggett De Jong

It is difficult toforecast,especially in

regards to thefuture.

It isn’t difficultto forecast,just to forecastcorrectly.

Numbers, iftorturedenough, willconfess tojust aboutanything.

Economic Forecasts Influence:

Government policies & businessdecisions



Insurance companies’ investmentdecisions in mortgages and bonds

Service industries’ forecasts ofdemand as input for revenuemanagement

FEATURES

• Regression

• Solver

• Sorting

FUNCTIONS

• SUMPRODUCT( )

• SUMXMY2( )

• YEAR( )

• MONTH( )• RIGHT( )

Excel Features & Functions

Quantitative vs Qualitative

Forecasting Models

Quantitative Forecasting Models Expressed in mathematical notation



Based on an amazing quantity of data

1. Causal (Curve Fitting)a. Linear

b. Quadratic

2. Moving Averages (Naive)a. Simple n-Period Moving Averageb. Weighted n-Period Moving Average

3. Exponential Smoothinga. Basic modelb. Holt’s Model (exponential smoothing with trend)

4. Seasonality

Quantitative Forecasting Models

Important Variables

Average value of dependent variable (Y bar)Y

Predicted or forecasted dependent variable(Y hat)Y

True value of dependent variableY

Independent variable(s)X

Causal vs Time Series Models

Causal Forecasting Models Requirements



Independent and dependentvariables must share a relationship

We must know the values of theindependent variables when wemake the forecast

Curve FittingSelf Service Gas Stations

Oil company wants to expand itsnetwork of self-service gas stations

We’ll use historical data for fivestations to calculate average trafficflow and sales



Traffic flow: average # of cars / hour

Sales: average dollar sales / hour

Plot the averages in a scatterplot.

Figure 1, p274

Sales & Traffic Data

-

50

100

150

200

250

300

0 50 100 150 200 250

Cars/hour

S

a l e s / h o u r ( $ )

Figure 2, p274

Scatter Plot of Sales & Traffic Data

-

50

100

150

200

250

300

0 50 100 150 200 250

Cars/hour

S

a l e s / h o u r ( $ ) y = a + bx

Method of Least Squares

Figure 3, p275

Use Regression to fit a Linear Function

Figure 5, p277

TSS = ESS + RSS

TSS = Σi=1

n

(Y i – Y )2 –

ESS = Σi=1

n

(Y i – Y i )2^

Σi=1

n

(Y i – Y )2^ –RSS =

R2 =RSSTSS

Regression computes three types of errors

Total

Residual

Regression



Should we build a gas station atBuffalo Grove where traffic is 183cars/hour?

y = a + b * x ^

Sales/hour = 57.104 + 0.92997 * 183 cars/hr

= $227.29

How confident are we in thisforecast?

Confidence intervals use the following statistics:

1.00 =68% 1.96 = 95.0% 3.00 = 99.7%

+- 2 * Standard Error (Se)Y Se =

Σi=1

n

(Y i – Y i )2^

n – k -1=

n – k -1

ESS



Excel calculates the Standard Errorfor us. [227.29 – 2(44.18); 227.29 + 2(44.18)]

[$138.93; $315.65]

The 95% confidence interval is:

Other important information:

T-statistic and its p-valueUpper & Lower 95%F significanceR2 and Adjusted R2

Figure 5, p277

Fitting a QuadraticFunction

-

50

100

150

200

250

300

0 50 100 150 200 250

Cars/hour

S a l e s / h o u r ( $ )

Fitting a Quadratic Function

Figure 10, p283

y = a0 + a1x + a2x 2

y = a0 + a1x + a2x 2 Figure 7, p281

Use Solver to fit a Quadratic Function



We could create a formula that exactlypasses through every data point…..

But, why wouldn’t we want to do that?

Which curve to fit?Goodness of fit statistics:Sum of Squared Errors (SSE)

n

SSE = Σi=1

(Y i – Y i )2

^ Goodness of fit statistics:Mean Squared Error (MSE)



MSE =Sum of Squared Errors

(# of points – # of parameters)

Regression: SSE = 5854Quadratic: SSE = 4954

Regression: MSE = 5854 / (5 - 2) = 1951.3Quadratic: MSE = 4954 / (5 - 3) = 2477.0

y = 5x + 5

0

20

40

60

80

100

120

140

1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5

Time

P r o f i t

y = -5x + 135

0

20

40

60

80

100

120

140

1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5

Time

C u s t o m e r s

Causal Forecasting Models

Positive Slopeindicates upwardtrend

Negative Slopeindicatesdownward trend

Time-Series Forecasting Models

Time is the independent variable

1. Curve Fitting:a) Linearb) Quadratic



2. Moving Averages (Naive)a) Simple n-Period Moving Avgb) Weighted n-Period Moving Avg

3. Exponential Smoothinga) Basic modelb) Holt’s Model (trend)

Seasonality

Curve Fitting

Plot historical values as function oftime and draw a linear “trend line”.

Use trend line to predict futurevalue.

The Bank of Laramie

Moving Averages (Naïve):

Use previous period’s actual valueto forecast the current period (i.e.,use 12th value to predict 13th value).



Moving Averages:

Use average of past 12 values asbest forecast for 13th value.

Simple n-Period Moving Averages:

Use average of the most recent 6values to predict 13th value.

Steco: asimple n-periodmovingaveragesforecastingmodel

MONTH

ACTUAL

SALES

($000s)

THREE-MONTH

SIMPLE MOVING

AVERAGE FORECAST

FOUR-MONTH SIMPLE

MOVING AVERAGE

FORECAST

Jan. 20

Feb. 24

Mar. 27

Apr. 31 (20 + 24 + 27)/3 = 23.67

May 37 (24 + 27 + 31)/ 3 = 27. 33 (20 + 24 + 27 + 31)/ 4 = 25. 50

June 47 (27 + 31 + 37)/ 3 = 31. 67 (24 + 27 + 31 + 37)/ 4 = 29. 75

July 53 (31 + 37 + 47)/ 3 = 38. 33 (27 + 31 + 37 + 47)/ 4 = 35. 50

A ug. 62 (37 + 47 + 53)/ 3 = 45. 67 (31 + 37 + 47 + 53)/ 4 = 42. 00

S ep. 54 (47 + 53 + 62)/ 3 = 54. 00 (37 + 47 + 53 + 62)/ 4 = 49. 75

Oc t. 36 (53 + 62 + 54)/ 3 = 56. 33 (47 + 53 + 62 + 54)/ 4 = 54. 00

Nov. 32 (62 + 54 + 36)/ 3 = 50. 67 (53 + 62 + 54 + 36)/ 4 = 51. 25

Dec . 29 (54 + 36 + 32)/ 3 = 40. 67 (62 + 54 + 36 + 32)/ 4 = 46. 00

Three- and Four- Month Simple Moving Averages

Table 1, p2904

12131415

16ˆ

y y y y y

+++=

Goodness of fit statistics

forecastsofnumber

salesforecastsalesactual

MADforecastsall

−

=

∑

forecastsofnumber

%100salesactual

salesforecastsalesactual

MAPEforecastsall

∑ ∗−

=

STECO: Simple n-Period Moving Average

MONTH

ACTUAL

SALES

($000s)

THREE-MONTH

SIMPLE MOVING

AVERAGE

FORECAST

ABSOLUTE

ERROR

FOUR-MONTH

SIMPLE MOVING

AVERAGE

FORECAST

ABSOLUTE

ERROR

Jan. 20

Feb. 24

Mar. 27Apr. 31 23.67$ 7.33

May 37 27.33$ 9.67 25.50$ 11.50

June 47 31.67$ 15.33 29.75$ 17.25

July 53 38.33$ 14.67 35.50$ 17.50

Aug. 62 45.67$ 16.33 42.00$ 20.00

Sep. 54 54.00$ 0.00 49.75$ 4.25

Oct. 36 56.33$ 20.33 54.00$ 18.00

Nov. 32 50.67$ 18.67 51.25$ 19.25

Dec. 29 40.67$ 11.67 46.00$ 17.00

SUM = 114.00 SUM = 124.75

MAD = 12.67 MAD = 15.59

Figure 14, p291



Simple n-Period Moving Averageforecasting models have twoshortcomings

Philosophical ShortcomingMost recent observations receive nomore weight or importance thanolder observations.

Operational ShortcomingAll historical data used to makeforecast must be stored in someway to calculate the forecast.

Weighted n-Period MovingAverages:

Resolves philosophical

shortcoming of simple periodmoving average forecasting

Weighted n-Period MovingAverages:

Use weighted average of previousvalues & assign higher weights tomore recent observations

Recent data is more importantthan old data

425160ˆ y y y y α α α ++=



Constraints:

The (weights) are positivenumbers

s'α

Constraints:

Smaller weights are assigned toolder data

Constraints:

All the weights sum to 1

alpha2 = 0.167 Month Actual Sales (000)3month WMA Fcst Absolute Error

alpha1 = 0.333 January 20

alpha0 = 0.500 February 24

SUM OF WTS= 1.00 March 27

April 31 24.83 6.17

May 37 28.50 8.50

June 47 33.33 13.67

July 53 41.00 12.00

August 62 48.33 13.67

September 54 56.50 2.50

October 36 56.50 20.50

November 32 46.34 14.34

December 29 37.01 8.01

Sum = 99.35

MAD = 11.04

Use Solver to find the optimal

weightsFigure 16, p293

Weighted n-Period MovingAverages resolves philosophicalshortcoming of simple periodmoving average forecasting

Exponential Smoothing resolvesoperational shortcoming ofsimple period moving averageforecasting



Exponential Smoothing Exponential Smoothing

tt1t y)1(yy α−+α=+

Forecast for t + 1 Observed in t Forecast for t

Where is a user-specified constantα

Resolves operational shortcoming of theMoving Averages Model:

10,001:40,000:

5,000 * 8

5,000

ExponentialSmoothing

Model

8-period MovingAverage Model

Number ofInventory Items

to Forecast

ty000,5

ty000,5

α1

Saving alpha and the last forecasts stores all the previous forecasts.

When t = 1, the expression becomes:

tt2 y)1(yy α−+α=tt1t y)1(yy α−+α=+

alpha = 0.500 Month Actual Sales (000)Fcst Sales Absolute Error

January 20 20.00

February 24 20.00 4.00

March 27 22.00 5.00

April 31 24.50 6.50

May 37 27.75 9.25

June 47 32.38 14.63

July 53 39.69 13.31

August 62 46.34 15.66

September 54 54.17 0.17

October 36 54.09 18.09

November 32 45.04 13.04

December 29 38.52 9.52

Sum = 109.17

MAD = 9.92

Does exponential smoothing

produce a better forecast?

The value of alpha affects theperformance of the model

VARIABLE COEFFICIENT α = 0.1 α = 0.3 α = 0.5

y t α 0.1 0.3 0.5

y t-1 α(1-α) 0.09 0.21 0.25

y t-2 α(1-α)20.081 0.147 0.125

y t-3 α(1-α)30.07290 0.10290 0.06250

y t-4 α(1-α)40.06561 0.07203 0.03125

y t-5 α(1-α)50.05905 0.05042 0.01563

y t-6 α(1-α)6

0.05314 0.03529 0.00781y t-7 α(1-α)7

0.04783 0.02471 0.00391

y t-8 α(1-α)80.04305 0.01729 0.00195

y t-9 α(1-α)90.03874 0.01211 0.00098

y t-10 α(1-α)100.03487 0.00847 0.00049

Sum of the Weights 0.68619 0.98023 0.99951

Case 1: Response to Sudden Change

System Change when t = 100

-1

0

1

1

2

94 95 96 97 98 99 100 101 102 103

t

yt

A forecasting system with alpha = 0.5responds quickly to changes in the data.

Response to a Unit Change in yt

00.10.20.30.40.5

0.6

0.70.80.9

11.11.2

100 105 110 115 120 125

t



Exponential smoothing is not a goodforecasting tool in a rapidly growing or adeclining market.

Steadily Increasing Values of yt

(Linear Ramp)

0

1

2

3

4

5

6

0 2 4 6 8 10

t

yt

Case 2: Response to Steady Change

Steadily Increasing Values of yt

(Linear Ramp)

0

1

2

3

4

5

6

0 2 4 6 8 10

t

yt

Case 2: Response to Steady Change

But the model can be adjusted (Holt’smodel / exponential smoothing w/trend)

Exponential smoothing is not a good

model to use here because it ignores theseasonal pattern.

Seasonal Pattern in y t

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

1.3

1.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

t

y t

Case 3: Response to Seasonal Change

Seasonality

Takes into consideration and adjustsfor the seasonal patterns in data

1. Look at original data to see seasonalpattern. Examine the data &hypothesize an m-period seasonalpattern.



2. Deseasonalize the Data 3. Forecast using deseasonalized data

4. Seasonalize the forecast to accountfor the seasonal pattern

Gillett Coal Mine

Coal Receipts Over a Nine-Year Period

0

500

1,000

1,500

2,000

2,500

3,000

1 - 1

1 - 3

2 - 1

2 - 3

3 - 1

3 - 3

4 - 1

4 - 3

5 - 1

5 - 3

6 - 1

6 - 3

7 - 1

7 - 3

8 - 1

8 - 3

9 - 1

9 - 3

Time (Year and Quarter)

C o a l ( 0 0 0 T o n s )

1. Look at original data to see seasonalpattern. Examine the data &hypothesize an m-period seasonalpattern. Figure 27, p303

Deseasonalized Data

-

500.0

1,000.0

1,500.0

2,000.0

2,500.0

3,000.0

1-

1

1-

2

1-

3

1-

4

2-

1

2-

2

2-

3

2-

4

3-

1

3-

2

3-

3

3-

4

4-

1

4-

2

4-

3

4-

4

5-

1

5-

2

5-

3

5-

4

6-

1

6-

2

6-

3

6-

4

7-

1

7-

2

7-

3

7-

4

8-

1

8-

2

8-

3

8-

4

9-

1

9-

2

9-

3

9-

4

Time (Year & Qtr)

C o a l ( 0 0 0 T o n s )

2. Deseasonalize the Data

Figure 32, p306



2.Deseasonalize the Data

a) Calculate a series of m -period movingaverages, where m is the length of the

seasonal pattern.b) Center the moving average in the middle of

the data from which it was calculated.c) Divide the actual data at a given point in the

series by the centered moving averagecorresponding to the same point.

d) Develop seasonal index

e) Divide actual data by the seasonal index

a)Calculate a series of m -period movingaverages, where m is the length of theseasonal pattern.

Time Coal 4 Period

Year-Qtr Receipts Moving Average

1-1 2,159 -----

1-2 1,203 -----

1-3 1,094 1,613

1-4 1,996 1,594

2-1 2,081 1,626

2-2 1,332 1,721

2-3 1,476 1,856

2-4 2,533 1,898

3-1 2,249 1,948

3-2 1,533 2,063

3-3 1,935 2,060

3-4 2,523 2,050

4-1 2,208 2,066

(2,159+1,203+1,094+1,996)/4 = 1,613

Figure 28, p304

Time Coal 4 Period Centered

Year-Qtr Receipts Moving Average Moving Average

1-1 2,159

1-2 1,203

1-3 1,094 1,613 1,603

1-4 1,996 1,594 1,610

2-1 2,081 1,626 1,674

2-2 1,332 1,721 1,788

2-3 1,476 1,856 1,877

2-4 2,533 1,898 1,923

3-1 2,249 1,948 2,005

3-2 1,533 2,063 2,061

3-3 1,935 2,060 2,055

3-4 2,523 2,050 2,058

4-1 2,208 2,066 2,064

4-2 1,597 2,061 2,087

4-3 1,917 2,112 2,163

4-4 2,726 2,213 2,255

(1613 + 1594)/2 =1603

b)Center the moving average in themiddle of the data from which it was

calculated.Figure 28, p304

Data & Centered Moving Average

0

500

1,000

1,500

2,000

2,500

3,000

1-

1

1-

2

1-

3

1-

4

2-

1

2-

2

2-

3

2-

4

3-

1

3-

2

3-

3

3-

4

4-

1

4-

2

4-

3

4-

4

5-

1

5-

2

5-

3

5-

4

6-

1

6-

2

6-

3

6-

4

7-

1

7-

2

7-

3

7-

4

8-

1

8-

2

8-

3

8-

4

9-

1

9-

2

9-

3

9-

4

Time (Year & Qtr)

C o a l ( 0 0 0 T o n s )

Receipts

Centered

Moving

Average

b)Center the moving average in themiddle of the data from which it was

calculated.Figure 29, p305

c) Divide the actual data at a given pointin the series by the centered movingaverage corresponding to the samepoint.

Time Coal 4 Period Centered Ratio of Coal Receipts to

Year-Qtr Receipts Moving Average Moving Average Centered Moving Average

1-1 2,159

1-2 1,203

1-3 1,094 1,613 1,603 0.682

1-4 1,996 1,594 1,610 1.240

2-1 2,081 1,626 1,674 1.244

2-2 1,332 1,721 1,788 0.745

2-3 1,476 1,856 1,877 0.787

2-4 2,533 1,898 1,923 1.317

3-1 2,249 1,948 2,005 1.122

3-2 1,533 2,063 2,061 0.744

3-3 1,935 2,060 2,055 0.942

3-4 2,523 2,050 2,058 1.226

1,094 / 1,603 = 0.682

Figure 28, p304

d)Develop seasonal index for eachquarter• Group ratios by quarter• Average all of the ratios to moving

averages quarter by quarter• Add Seasonal Indices data to table• Normalize the seasonal index

T im e C oa l 4 P er io d C en te re d R at io o f C oa l R ec ei pt s t o S ea so na l

Year-Qtr Receipts Moving Average Moving Average Centered Moving Average Indices

1-1 2,159 1.112

1-2 1,203 0.786

1-3 1,094 1,613 1,603 0.682 0.863

1-4 1,996 1,594 1,610 1.240 1.238

2-1 2,081 1,626 1,674 1.244 1.112

2-2 1,332 1,721 1,788 0.745 0.786

2-3 1,476 1,856 1,877 0.787 0.863

2-4 2,533 1,898 1,923 1.317 1.238

3-1 2,249 1,948 2,005 1.122 1.112

3-2 1,533 2,063 2,061 0.744 0.786

3-3 1,935 2,060 2,055 0.942 0.863

3-4 2,523 2,050 2,058 1.226 1.238



e)Divide actual data by the seasonalindex

T im e C o al 4 P er io d C en te r ed R at i o o f C o al R e ce ip t s t o S e as on a l D es ea s on a li ze d

Y e ar -Q t r R e ce i pt s M o vi ng A v er ag e M o vi ng A v er ag e C en t er ed M o vi n g Av e ra ge I n di ce s D at a

1-1 2,159 1.112 1,941.0

1-2 1,203 0.786 1,529.8

1-3 1,094 1,613 1,603 0.682 0.863 1,267.7

1-4 1,996 1,594 1,610 1.240 1.238 1,611.9

2-1 2,081 1,626 1,674 1.244 1.112 1,870.9

2-2 1,332 1,721 1,788 0.745 0.786 1,693.8

2-3 1,476 1,856 1,877 0.787 0.863 1,710.3

2-4 2,533 1,898 1,923 1.317 1.238 2,045.6

3-1 2,249 1,948 2,005 1.122 1.112 2,021.9

3-2 1,533 2,063 2,061 0.744 0.786 1,949.4

3-3 1,935 2,060 2,055 0.942 0.863 2,242.2

3-4 2,523 2,050 2,058 1.226 1.238 2,037.5

Figure 31, p306

Deseasonalized Data

-

500.0

1,000.0

1,500.0

2,000.0

2,500.0

3,000.0

1 - 1

1 - 3

2 - 1

2 - 3

3 - 1

3 - 3

4 - 1

4 - 3

5 - 1

5 - 3

6 - 1

6 - 3

7 - 1

7 - 3

8 - 1

8 - 3

9 - 1

9 - 3

Time (Year & Qtr)

C o a

l ( 0 0 0 T o n s )

DeseasonalizedData

e)Divide actual data by the seasonalindex

Figure 32, p306

3. Forecast method in deseasonalizedterms• Review the graphed deseasonalized data to

reveal pattern• Use forecasting method that accounts for

the pattern in the deseasonalized data• Use Excel’s Solver to minimize the error

Time Coal 4 Period Centered Ratio of Coal Receipts to Seasonal Deseasonalized

Y ea r- Qt r R ec ei pt s M ov in g A ve ra ge M ov in g A ve ra ge C en te re d M ov in g A ve ra ge I nd ic es D at a F or ec as t

1-1 2,159 1.108 1,948.1 1,948.1

1-2 1,203 0.784 1,535.4 1,948.1

1-3 1,094 1,613 1,603 0.682 0.860 1,272.3 1,678.5

1-4 1,996 1,594 1,610 1.240 1.234 1,617.8 1,413.1

2-1 2,081 1,626 1,674 1.244 1.108 1,877.8 1,546.8

2-2 1,332 1,721 1,788 0.745 0.784 1,700.0 1,763.0

2-3 1,476 1,856 1,877 0.787 0.860 1,716.6 1,721.9

2-4 2,533 1,898 1,923 1.317 1.234 2,053.1 1,718.4

3-1 2,249 1,948 2,005 1.122 1.108 2,029.3 1,937.1

3-2 1,533 2,063 2,061 0.744 0.784 1,956.5 1,997.4

3-3 1,935 2,060 2,055 0.942 0.860 2,250.4 1,970.7

3-4 2,523 2,050 2,058 1.226 1.234 2,045.0 2,153.4

Figure 33, p307

4. Reseasonalize the forecast to account forthe seasonal pattern• Multiply the deseasonalized forecast by the

seasonal index for the appropriate period.• Graph the actual Coal Receipts and

Seasonalized Forecast

T im e C oa l 4 P er io d C en te re d R at io o f C oa l R ec ei pt s t o S ea so na l D es ea so na li ze d S ea so na li ze

Y ea r- Qt r R ec ei pt s M ov in g A ve ra ge M ov in g A ve ra ge C en te re d M ov in g A ve ra ge I nd ic es D at a F or ec as t F or ec as t

1-1 2,159 ----- ----- ----- 1.108 1,948.1 1,948.1 2,159.000

1-2 1,203 ----- ----- ----- 0.784 1,535.4 1,948.1 1,526.409

1-3 1,094 1,613 1,603 0.682 0.860 1,272.3 1,678.5 1,443.212

1-4 1,996 1,594 1,610 1.240 1.234 1,617.8 1,413.1 1,743.439

2-1 2,081 1,626 1,674 1.244 1.108 1,877.8 1,546.8 1,714.276

2-2 1,332 1,721 1,788 0.745 0.784 1,700.0 1,763.0 1,381.390

2-3 1,476 1,856 1,877 0.787 0.860 1,716.6 1,721.9 1,480.540

2-4 2,533 1,898 1,923 1.317 1.234 2,053.1 1,718.4 2,120.128

3-1 2,249 1,948 2,005 1.122 1.108 2,029.3 1,937.1 2,146.723

3-2 1,533 2,063 2,061 0.744 0.784 1,956.5 1,997.4 1,564.974

3-3 1,935 2,060 2,055 0.942 0.860 2,250.4 1,970.7 1,694.495

3-4 2,523 2,050 2,058 1.226 1.234 2,045.0 2,153.4 2,656.854

Actual & Forecast

0

500

1,000

1,500

2,000

2,500

3,000

3,500

1-

1

1-

2

1-

3

1-

4

2-

1

2-

2

2-

3

2-

4

3-

1

3-

2

3-

3

3-

4

4-

1

4-

2

4-

3

4-

4

5-

1

5-

2

5-

3

5-

4

6-

1

6-

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6-

3

6-

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7-

1

7-

2

7-

3

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4

8-

1

8-

2

8-

3

8-

4

9-

1

9-

2

9-

3

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4

1

0-

1

1

0-

2

Year-Quarter

C o a l ( 0 0 0 T o n s )

Coal Receipts

Seasonalized Forecast

4. Reseasonalize the forecast to account forthe seasonal pattern

1. Look at original data to see seasonal pattern. Examine thedata & hypothesize an m-period seasonal pattern.

2. Deseasonalize the data.

a) Calculate a series of m -period moving averages, where m

is the length of the seasonal pattern.b) Center the moving average in the middle of the data from

which it was calculated.

c) Divide the actual data at a given point in the series by thecentered moving average corresponding to the same point.

d) Develop seasonal indexe) Divide actual data by the seasonal index

3. Forecast method in deseasonalized terms.

4. Reseasonalize the forecast to account for the seasonalpattern.



Returns the sum of squares of differences ofcorresponding values in two arrays.

Syntax: SUMXMY2(array_x,array_y), whereArray_x is the first array or range of values.Array_y is the second array or range of values.

The equation for the sum of squared differences is:

( )∑ −=2

yx2SUMXMY

SUMXMY2( ) Measures of Comparison

forecastsof number

sales forecast salesactual

MADforecastsall

−

=

∑

forecastsof number

salesactual


MAPE forecastsall

∑ ∗−

=

%100

forecastsof number


MSE

n

t

∑=

−

= 1

2)(

Model Validation Create experience by simulating thepast.

Create the model with a portion of thehistorical data.

Use remaining data to see how wellthe model would have performed.



QualitativeForecasting

Models

ExpertJudgment

ConsensusPanel

DelphiMethod

Coordinator requests forecasts

Coordinator receivesIndividual forecasts

Coordinator determines(a) Median response(b) Range of middle

50% of answers

Coordinator requestsexplanations from any

expert whose estimateis not in the middle

50%

Coordinator sends to all experts(a) Median response

(b) Range of middle 50%(c) Explanations

Delphi

Method

Grassroots Forecasting



Market Research

Documents

Chapter 13 Forecasting