ch5defor13

Chapter 5: Demand Forecasting

Department of Business Administration

FALL 2012-2013I see that you willget an A this semester.

Ch 5 : Demand Forecasting

© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved. Managerial Economics in a Global Economy

2

Outline: What You Will Learn . . . List the elements of a good forecast. Outline the steps in the forecasting process. Describe at least three qualitative forecasting

techniques and the advantages and disadvantages of each.

Compare and contrast qualitative and quantitative approaches to forecasting.

Briefly describe averaging techniques, trend and seasonal techniques, and regression analysis, and solve typical problems.

Describe two measures of forecast accuracy. Describe two ways of evaluating and controlling

forecasts. Identify the major factors to consider when choosing

a forecasting technique


© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved. Managerial Economics in a Global Economy3

What is meant by Forecasting and Why?What is meant by Forecasting and Why?

Forecasting is the process of estimating a variable, such as the sale of the firm at some future date.

Forecasting is important to business firm, government, and non-profit organization as a method of reducing the risk and uncertainty inherent in most managerial decisions.

A firm must decide how much of each product to produce, what price to charge, and how much to spend on advertising, and planning for the growth of the firm.



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The aim of forecastingThe aim of forecasting

The aim of forecasting is to reduce the risk or uncertainty that the firm faces in its short-term operational decision making and in planning for its long term growth.

Forecasting the demand and sales of the firm’s product usually begins with macroeconomic forecast of general level of economic activity for the economy as a whole or GNP.



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The aim of forecastingThe aim of forecasting

The firm uses the macro-forecasts of general economic activity as inputs for their micro-forecasts of the industry’s and firm’s demand and sales.

The firm’s demand and sales are usually forecasted on the basis of its historical market share and its planned marketing strategy (i.e., forecasting by product line and region).

The firm uses long-term forecasts for the economy and the industry to forecast expenditure on plant and equipment to meet its long-term growth plan and strategy.



Forecasting Process MapForecasting Process Map

MarketingMarketingSalesSalesProductProduct

Management Management & Finance& Finance

Executive Executive ManagementManagement

Production &Production & Inventory Inventory

ControlControl

Causal Causal FactorsFactors

Statistical Statistical ModelModel

Statistical Statistical ModelModel

Demand Demand HistoryHistory

Consensus Consensus ProcessProcess

Consensus Consensus ForecastForecast



Accounting Cost/profit estimates

Finance Cash flow and funding

Human Resources Hiring/recruiting/training

Marketing Pricing, promotion, strategy

MIS IT/IS systems, services

Operations Schedules, MRP, workloads

Product/service design New products and services

Uses of ForecastsUses of Forecasts



Assumes causal systempast ==> future

Forecasts rarely perfect because of randomness

Forecasts more accurate forgroups vs. individuals

Forecast accuracy decreases as time horizon increases or may otherway around

I see that you willget an A this semester.

Features of ForecastsFeatures of Forecasts



9

Elements of a Good Forecast

Timely

AccurateReliable

Mea

ningfu

l

Written

Easy

to u

se



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Steps in the Forecasting Process

Step 1 Determine purpose of forecast

Step 2 Establish a time horizon

Step 3 Select a forecasting technique

Step 4 Obtain, clean and analyze data

Step 5 Make the forecast

Step 6 Monitor the forecast

“The forecast”



Forecasting Techniques

A wide variety of forecasting methods are available to management. These range from the most naïve methodsnaïve methods that require little effort to highly complex approacheshighly complex approaches that are very costly in terms of time and effort such as econometric systems of simultaneous equations.

Mainly these techniques can break down into three parts: QQualitative approachesualitative approaches (Judgmental (Judgmental forecasts)forecasts) and QQuantitative approachesuantitative approaches (Time- (Time-series forecasts) and Associative model series forecasts) and Associative model forecasts)forecasts)..



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Judgmental - uses subjective inputs such as opinion from consumer surveys, sales staff etc..

Time series - uses historical data assuming the future will be like the past

Associative models - uses explanatory variables to predict the future

Forecasting TechniquesForecasting Techniques



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Survey Techniques Some of the best-know surveys

Planned Plant and Equipment SpendingExpected Sales and Inventory ChangesConsumers’ Expenditure Plans

Opinion PollsBusiness ExecutivesSales ForceConsumer Intentions

Qualitative ForecastsQualitative Forecasts or or Judgmental ForecastsJudgmental Forecasts



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What are qualitative forecast ?

Qualitative forecast estimate variables at some future date using the results of surveys and opinion polls of business and consumer spending intentions.

The rational is that many economic decisions are made well in advance of actual expenditures.

For example, businesses usually plan to add to plant and equipment long before expenditures are actually incurred.



Surveys and opinion pools are often used to make short-term forecasts when quantitative data are not available.

Usually based on judgments about causal factors that underlie the demand of particular products or services.

Do not require a demand history for the product or service, therefore are useful for new products/services.

Approaches vary in sophistication from scientifically conducted surveys to intuitive hunches about future events.

The approach/method that is appropriate depends on a product’s life cycle stage.




Polls can also be very useful in supplementing quantitative forecasts, anticipating changes in consumer tastes or business expectations about future economic conditions, and forecasting the demand for a new product.

Firms conduct opinion polls for economic activities based on the results of published surveys of expenditure plans of businesses, consumers and governments.




Survey Techniques– The rationale for forecasting based on surveys of economic intentions is that many economic decisions are made in advance of actual expenditures (Ex: Consumer’s decisions to purchase houses, automobiles, TV sets, furniture, vocation, education etc. are made months or years in advance of actual purchases)

Opinion Polls– The firm’s sales are strongly dependent on the level of economic activity and sales for the industry as a whole, but also on the policies adopted by the firm. The firm can forecast its sales by pooling experts within and outside the firm.




Executive Polling- Firm can poll its top management from its sales, production, finance for the firm during the next quarter or year.

Bandwagon effect (opinions of some experts might be overshadowed by some dominant personality in their midst).

Delphi Method – experts are polled separately, and then feedback is provided without identifying the expert responsible for a particular opinion.




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Consumers intentions polling- Firms selling automobiles, furniture, etc.

can pool a sample of potential buyers on their purchasing intentions. By using results of the poll a firm can forecast its sales for different levels of consumer’s future income.

Sales force polling – – Forecast of the firm’s sales in each region

and for each product line, it is based on the opinion of the firm’s sales force in the field (people working closer to the market and their opinion about future sales can provide essential information to top management).




Based on the assumption, the “forces” that generated the past demand will generate the future demand, i.e., history will tend to repeat itself.

Analysis of the past demand pattern provides a good basis for forecasting future demand.

Majority of quantitative approaches fall in the category of time series analysis.

Quantitative Forecasting Approaches



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Time Series Analysis

A time series (naive forecasting) is a set of numbers where the order or sequence of the numbers is important, i.e., historical demand

Attempts to forecasts future values of the time series by examining past observations of the data only. The assumption is that the time series will continue to move as in the past

Analysis of the time series identifies patterns Once the patterns are identified, they can be

used to develop a forecast



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Forecast Horizon

Short term Up to a year

Medium term One to five years

Long term More than five years



Reasons for Fluctuations in Time Series Data

Secular Trend are noted by an upward or downward sloping line- long-term movement in data (e.g. Population shift, changing income and cultural changes).

Cycle fluctuations is a data pattern that may cover several years before it repeats itself- wavelike variations of more than one year’s duration (e.g. Economic, political and agricultural conditions).

Seasonality is a data pattern that repeats itself over the period of one year or less- short-term regular variations in data (e.g. Weekly or daily restaurant and supermarket experiences).

Irregular variations caused by unusual circumstances (e.g. Severe weather conditions, strikes or major changes in a product or service).

Random influences (noise) or variations results from random variation or unexplained causes. (e.g. residuals)



Forecast Variations

Trend

Irregularvariation

Seasonal variations

908988

Cycles



Stable time series dataF(t) = A(t-1)

Seasonal variationsF(t) = A(t-n)

Data with trendsF(t) = A(t-1) + (A(t-1) – A(t-2))

Uses for Naïve ForecastsUses for Naïve Forecasts



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Techniques for Averaging

Moving average Weighted moving average Exponential smoothing



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Moving Averages

Moving average – A technique that averages a number of recent actual values, updated as new values become available.

Ft = MAn= n

At-n + … At-2 + At-1

Ft = WMAn=

wnAt-n + … wn-1At-2 + w1At-1

Weighted moving average – More recent values in a series are given more weight in computing the forecast.

nn=total amount of number of weights



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Simple Moving Average

35

37

39

41

43

45

47

1 2 3 4 5 6 7 8 9 10 11 12

Actual

MA3

MA5

Ft = MAn= n

At-n + … At-2 + At-1



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Simple Moving Average

An averaging period (AP) is given or selected The forecast for the next period is the arithmetic

average of the AP most recent actual demands It is called a “simple” average because each

period used to compute the average is equally weighted

It is called “moving” because as new demand data becomes available, the oldest data is not used

By increasing the AP, the forecast is less responsive to fluctuations in demand (low impulse response and high noise dampening)

By decreasing the AP, the forecast is more responsive to fluctuations in demand (high impulse response and low noise dampening)



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Exponential SmoothingExponential Smoothing

Premise--The most recent observations might have the highest predictive value. Therefore, we should give more weight to the more recent time periods when forecasting.

Weighted averaging method based on previous forecast plus a percentage of the forecast error

A-F is the error term, is the % feedback

Ft = Ft-1 + (At-1 - Ft-1)Ft = forecast for period tFt-1 = forecast for the previous period smoothing constant At-1 = actual data for the previous period



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Exponential SmoothingExponential Smoothing ForecastsForecasts

The weights used to compute the forecast (moving average) are exponentially distributed.

The forecast is the sum of the old forecast and a portion (a) of the forecast error (A t-1 - Ft-1).

The smoothing constant, , must be between 0.0 and 1.0.

A large provides a high impulse response forecast.

A small provides a low impulse response forecast.



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Example-Moving AverageExample-Moving Average

Central Call Center (CCC) wishes to forecast the number of incoming calls it receives in a day from the customers of one of its clients, BMI. CCC schedules the appropriate number of telephone operators based on projected call volumes.CCC believes that the most recent 12 days of call volumes (shown on the next slide) are representative of the near future call volumes.



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Moving Average Use the moving average method with an

AP = 3 days to develop a forecast of the call

volume in Day 13 (The 3 most recent demands)

compute a three-period average forecast compute a three-period average forecast given scenario above:given scenario above:

F13 = (168 + 198 + 159)/3 = 175.0 calls

Example-Moving AverageExample-Moving Average



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Example-Weighted Moving AverageExample-Weighted Moving Average

Weighted Moving Average (Central Call Center ) Use the weighted moving average method with an AP = 3 days and weights

of .1 (for oldest datum), .3, and .6 to develop a forecast of the call volume in Day 13.

compute a weighted average forecast given scenario above:compute a weighted average forecast given scenario above:

F13 = .1(168) + .3(198) + .6(159) = 171.6 calls

Note: The WMA forecast is lower than the MA forecast because Day 13’s relatively low call volume carries almost twice as much weight in the WMA (.60) as it does in the MA (.33).

1



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Example-Example-Exponential SmoothingExponential Smoothing Exponential Smoothing Exponential Smoothing ((Central Call Center) SupposeSuppose a smoothing constant value of .25 is used and the a smoothing constant value of .25 is used and the

exponential smoothing forecast for Day 11 was 180.76 callsexponential smoothing forecast for Day 11 was 180.76 calls .. what is the exponential smoothing forecast for Day 13?what is the exponential smoothing forecast for Day 13?

F12 = 180.76 + .25(198 – 180.76) = 185.07F12 = 180.76 + .25(198 – 180.76) = 185.07 F13 = 185.07 + .25(159 – 185.07) = 178.55F13 = 185.07 + .25(159 – 185.07) = 178.55

Ft = Ft-1 + (At-1 - Ft-1)



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Example 2-Example 2-Exponential SmoothingExponential SmoothingPeriod Actual Alpha = 0.1 Error Alpha = 0.4 Error

1 422 40 42 -2.00 42 -23 43 41.8 1.20 41.2 1.84 40 41.92 -1.92 41.92 -1.925 41 41.73 -0.73 41.15 -0.156 39 41.66 -2.66 41.09 -2.097 46 41.39 4.61 40.25 5.758 44 41.85 2.15 42.55 1.459 45 42.07 2.93 43.13 1.87

10 38 42.36 -4.36 43.88 -5.8811 40 41.92 -1.92 41.53 -1.5312 41.73 40.92

Exponential Smoothing Exponential Smoothing (Actual (Actual Demand forecasting ) SupposeSuppose a smoothing constant value of . a smoothing constant value of .1010 is used and the is used and the

exponential smoothing forecast for exponential smoothing forecast for the previous periodthe previous period was was 42 units (actual demand was 40 units).42 units (actual demand was 40 units).

what is the exponential smoothing forecast for what is the exponential smoothing forecast for the nextthe next periodsperiods??

FF33 = = 4242 + . + .1010((4040 – – 4242) = ) = 4141..88 FF44 = = 41.841.8 + . + .1010((4343 – – 41.841.8) = ) = 41.9241.92



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35

40

45

50

1 2 3 4 5 6 7 8 9 10 11 12

Period

Dem

and .1

.4

Actual

Example 2-Example 2-Exponential SmoothingExponential SmoothingGraphical presentationGraphical presentation



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Trend Projection

The simplest form of time series is projecting the past trend by fitting a straight line to the data either visually or more precisely by regression analysis.

Linear regression analysis establishes a relationship between a dependent variable and one or more independent variables.

In simple linear regression analysis there is only one independent variable.

If the data is a time series, the independent variable is the time period.

The dependent variable is whatever we wish to forecast.



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Linear Trend Equation

Ft = Forecast for period t t = Specified number of time

periods a = Value of Ft at t = 0 b = Slope of the line

Ft = a + bt

0 1 2 3 4 5 t

Ft



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Trend Projection

Linear Trend:St = S0 + b t

b = Growth per time period Constant Growth Rate(Non-linear)

St = S0 (1 + g)t

g = Growth rate Estimation of Growth Rate

ln St = ln S0 + t ln (1 + g)



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Trend Projection- Simple Linear Regression

Regression Equation This model is of the form:

Y = a + bX

Y = dependent variable (the value of time series to be forecasted for period t)

X = independent variable ( time period in which the time series is to be forecasted)

a = y-axis intercept (estimated value of the time series, the constant of the regression)

b = slope of regression line (absolute amount of growth per period)


© Dominick Salvatore; ed. 2007;2011, 2012/13, Sami Fethi, EMU, All Right Reserved. Managerial Economics in a Global Economy 42

The correlation coefficient, determination of coefficient and standard deviation

Standard deviation

Correlation Coefficient

Determination of coefficient



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Trend Projection- Calculating a and b

Constants a a and bb The constants aa and bb

are computed using the equations given:

Once the a a and b b values are computed, a future value of X can be entered into the regression equation and a corresponding value of Y (the forecast) can be calculated.

2

2 2

x y- x xya =

n x -( x)

2 2

xy- x yb =

n x -( x)

n



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Trend Projection- Calculating a and b

Or If formula b is used first, it may be used formula a in the following format:

2 2

xy- x yb =

n x -( x)

n n

XbYa



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Example 1 for Trend Projection-Electricity sales

Suppose we have the data show electricity sales in a city between 1997.1 and 2000.4. The data are shown in the following table. Use time series regression to forecast the electricity consumption (mn kilowatt) for the next four quarters.next four quarters.

Do not forget to use the formulae a and b



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Example1 for Trend Projection

2 2

xy- x yb =

n x -( x)

n

2

2 2

x y- x xya =

n x -( x)

Year Trent (t)ELECSALE

(Y) (t )SQ Y*t (y) SQ (Σt) SQ (ΣY) SQ1997Q1 1 11 1 11 121 1997Q2 2 15 4 30 225 1997Q3 3 12 9 36 144 1997Q4 4 14 16 56 196 1998Q1 5 12 25 60 144 1998Q2 6 17 36 102 289 1998Q3 7 13 49 91 169 1998Q4 8 16 64 128 256 1999Q1 9 14 81 126 196 1999Q2 10 18 100 180 324 1999Q3 11 15 121 165 225 1999Q4 12 17 144 204 289 2000Q1 13 15 169 195 225 2000Q2 14 20 196 280 400 2000Q3 15 16 225 240 256 2000Q4 16 19 256 304 361 (SUM) Σ 136 244 1496 2208 3820 18496 59536

a 11.9 b 0.394

Av 15.25



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Y = 11.90 + 0.394XY = 11.90 + 0.394X

Y17 = 11.90 + 0.394(17) = 18.60 in the first quarter of 2001Y17 = 11.90 + 0.394(17) = 18.60 in the first quarter of 2001Y18 = 11.90 + 0.394(18) = 18.99 in the second quarter of 2001Y18 = 11.90 + 0.394(18) = 18.99 in the second quarter of 2001Y19 = 11.90 + 0.394(19) = 19.39 in the third quarter of 2001Y19 = 11.90 + 0.394(19) = 19.39 in the third quarter of 2001Y20 = 11.90 + 0.394(20) = 19.78 in the fourth quarter of 2001Y20 = 11.90 + 0.394(20) = 19.78 in the fourth quarter of 2001

Note:Note: Electricity sales are expected to increase Electricity sales are expected to increase by 0.394 mn kilowatt-hours per quarter.by 0.394 mn kilowatt-hours per quarter.




Std.dev

Sxy = SQRT( [3820-(11.9) (244)- (0.394) (2208)]/(16-2))=1.82

Sxy is a measure of how historical data points have been dispersed about the trend line. If it is large (reference point in mean of the data) , the historical data points have been spread widely about the trend line and if otherway around, the data points have been grouped tightly about the trend.




Corr. Coeff.

r= ((16) (2208)- (136) (244))/SQRT( [(16) (1496)-(18496)*((16)(3820-59536)]=0.73

r lies between -1 and 1, -1 is strong negative whereas 1 is strong positive. 0 means that there is no relationship between the two variables (x and y). In this case, there is a strong positive relationship between the two variables and if an increase in independent variable, it will be a rise in dependent variable.



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R2=0.533. It varies between 0 and 1. 0 means that there is no relationship between the two variables whereas 1 indicates that there is a perfect relationship. 53.3% variation in dependent variable can be explained by the variation happened in the independent variable. It is worth to emphasize that 46.7% shows unexplained part of the relationship.



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Example 2 for Trend Projection

Estimate a trend line using regression analysis

Year

Time Period

(t)Sales

(y)

200320042005200620072008

123456

204030507065

tbby 10

Use time (t) as the independent variable:



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The linear trend model is:

Sales trend

01020304050607080

0 1 2 3 4 5 6 7

Year

sale

s

YearTime

Period (t)

Sales (y)

200320042005200620072008

123456

204030507065

t 5714.9333.12y

(continued)



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Forecast for time period 7:

Sales

01020304050607080

0 1 2 3 4 5 6 7

Year

sale

s

YearTime

Period (t)

Sales (y)

2003200420052006200720082009

1234567

204030507065??

(continued)

33.79

(7) 5714.9333.12y



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Example for Trend Projection using-Non linear form

St = S0 (1 + g)t

Running the regression above in the form of logarithms: ln St = ln S0 + t ln (1 + g) to construct the equation which has coefficients a and b.

Antilog of 2.49 is 12.06 and Antilog of 0.026 is 1.026.

CoefficientsStandard Error t StatIntercept 2.486914 0.062793 39.60489T 0.026371 0.006494 4.060874

St = 12.06(1.026)t



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Example for Trend Projection using St = S0 (1 + g)t

S17= 12.06(1.026)17 = 18.66 in the first quarter of 2001

S18= 12.06(1.026)18 = 19.14 in the second quarter of 2001

S19= 12.06(1.026)19 = 19.64 in the third quarter of 2001

S20= 12.06(1.026)20= 20.15 in the fourth quarter of 2001

These forecasts are similar to those obtained by fitting a linear trend



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Evaluating Forecast-Model Performance Accuracy

Accuracy is the typical criterion for judging the performance of a forecasting approach

Accuracy is how well the forecasted values match the actual values

Accuracy of a forecasting approach needs to be monitored to assess the confidence you can have in its forecasts and changes in the market may require reevaluation of the approach

Accuracy can be measured in several ways Standard error of the forecast (SEF) Mean absolute deviation (MAD) Mean squared error (MSE) Mean absolute percent error (MAPE) Root mean squared error (RMSE)



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Forecast Accuracy

Error - difference between actual value and predicted value

Mean Absolute Deviation (MAD)Average absolute error

Mean Squared Error (MSE)Average of squared error

Mean Absolute Percent Error (MAPE)Average absolute percent error

Root Mean Squared Error (RMSE)Root Average of squared error



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MAD, MSE, and MAPE

MAD = Actual forecast

n

MSE = Actual forecast)

-1

2

n

(

MAPE = Actual forecast

n

/ Actual)*100)

2( )t tA FRMSE

n



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MAD, MSE and MAPE

MAD Easy to compute Weights errors linearly

MSE Squares error More weight to large errors

MAPE Puts errors in perspective

RMSE Root of Squares error More weight to large errors



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Example-MAD, MSE, and MAPECompute MAD, MSE and MAP for the following data showing actual and the predicted numbers of account serviced.

Period Actual Forecast (A-F) |A-F| (A-F)^2 (|A-F|/Actual)*1001 217 215 2 2 4 0.922 213 216 -3 3 9 1.413 216 215 1 1 1 0.464 210 214 -4 4 16 1.905 213 211 2 2 4 0.946 219 214 5 5 25 2.287 216 217 -1 1 1 0.468 212 216 -4 4 16 1.89

-2 22 76 10.26

MAD= 2.75MSE= 10.86

MAPE= 1.28

22/8=2.7576/8-1=10.86

10.26/8=1.28 %



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Example-For MA TechniquesElectricity sales data from 2000.1 to 2002.4 (t=12)-Forecast

Accuracy - RMSE1 2 3 4 5 6 7 8

Quarter Firm's ams (A) Tqmaf (F) A-F sq(A-F) Fqmaf (F) A-F sq(A-F)1 202 223 234 24 21.6666667 2.333333 5.4444445 18 23 -5 256 23 21.6666667 1.333333 1.777778 21.4 1.6 2.567 19 21.6666667 -2.66667 7.111111 22 -3 98 17 20 -3 9 21.4 -4.4 19.369 22 19.6666667 2.333333 5.444444 20.2 1.8 3.2410 23 19.3333333 3.666667 13.44444 19.8 3.2 10.2411 18 20.6666667 -2.66667 7.111111 20.8 -2.8 7.8412 23 21 2 4 19.8 3.2 10.24

total 78.33333 total 62.4813 21.3333333 20.6

AP = 3 moving average AP = 5 moving average



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Example-For MA TechniquesElectricity sales data from 2000.1 to 2002.4 (t=12)-Forecast

Accuracy - RMSE

2( )t tA FRMSE

n

RMSE for 3-qma=2.95

RMSE for 5-qma=2.99

Thus three-quarter moving average forecast is marginally better than the corresponding five- moving average forecast.

Sqroot of 78.33/9=2.95

Sqroot of 62.48/7=2.98



Example-Exponential Smoothing Example-Exponential Smoothing Forecast Accuracy - Forecast Accuracy - RMSERMSE1 2 3 4 5 6 7 8

QuarterFirm's ams (A) (F) w=0.3 A-F sq(A-F) (F) w=0.5 A-F sq(A-F)1 20 21 -1 1 21 -1 12 22 20.7 1.3 1.69 20.5 1.5 2.253 23 21.09 1.91 3.6481 21.25 1.75 3.06254 24 21.663 2.337 5.461569 22.125 1.875 3.5156255 18 22.3641 -4.3641 19.04537 23.0625 -5.0625 25.628916 23 21.05487 1.94513 3.783531 20.53125 2.46875 6.0947277 19 21.63841 -2.63841 6.961202 21.76563 -2.76563 7.6486828 17 20.84689 -3.84689 14.79853 20.38281 -3.38281 11.443429 22 19.69282 2.30718 5.323078 18.69141 3.308594 10.9467910 23 20.38497 2.615026 6.838359 20.3457 2.654297 7.04529211 18 21.16948 -3.16948 10.04562 21.67285 -3.67285 13.4898412 23 20.21864 2.781363 7.735978 19.83643 3.163574 10.0082

total 87.19 total 101.513 21 21.5

F2= 21+(0.3) (20-21)=20.7 with w=α=0.3F2= 21+(0.5) (20-21)=20.5 with w=α=0.5

(20+22+...23)/12=21=F1 Ft = Ft-1 + (At-1 - Ft-1)



Example-Exponential Smoothing Example-Exponential Smoothing Forecast Accuracy - Forecast Accuracy - RMSERMSE

RMSE with α=0.3 is 2.6955 RMSE with α=0.5 is 2.908

Both exponential forecasts are better than the previous techniques in terms of average values.

2( )t tA FRMSE

n

RMSE= SQRT(87.19/12)= 2.6955RMSE= SQRT(101.5/12)=2.908



65

Example for Trend Projection- Petrol sales

Suppose we have the data show petrol sales in a city between 2004 and 2011. The data are shown in the following table. Use time series regression to forecast the petrol consumption (mn gallons) for the next fnext fourour yearyear..

Do not forget to use the formulae a and b




2 2

xy- x yb =

n x -( x)

n

2

2 2

x y- x xya =

n x -( x)

Year Trent (t) PETROLSALE (Y) (t )SQ Y*t (y) SQ (Σt) SQ (ΣY) SQ

2004 1 1 1 1 1

2005 2 3 4 6 9

2006 3 4 9 12 16

2007 4 2 16 8 4

2008 5 1 25 5 1

2009 6 3 36 18 9

2010 7 5 49 35 25

2011 8 3 64 24 9

(SUM) Σ 36 22 204 109 74 1296 484

a 1.678

b 0.238



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Example1 for Trend ProjectionY = 1.Y = 1.678678 + 0. + 0.238238XX

YY99 = = 1.6781.678 + 0. + 0.238238((99) ) = = 33..8383 in 20 in 201212Y1Y100 = 1. = 1.678678 + 0. + 0.238238((1010) = ) = 4.064.06 in 20 in 201313Y1Y111 = 1. = 1.678678 + 0. + 0.238238(1(111) = ) = 44..3030 in 20 in 201414YY1212 = 1. = 1.678678 + 0. + 0.238238((1212) = ) = 44..5454 in 20 in 201515

Note:Note: PetrolPetrol sales are expected to increase by sales are expected to increase by 0.0.238238 mn mn gallonsgallons pe per yearr year..



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Std.dev

Sxy =

Sxy is a measure of how historical data points have been dispersed about the trend line. If it is large (reference point in mean of the data) , the historical data points have been spread widely about the trend line and if otherway around, the data points have been grouped tightly about the trend.

1.36



69


Corr. coef r=0.42

r lies between -1 and 1, -1 is strong negative whereas 1 is strong positive. 0 means that there is no relationship between the two variables (x and y). In this case, there is a strong positive relationship between the two variables and if an increase in independent variable, it will be a rise in dependent variable.



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R2=0.18. It varies between 0 and 1. 0 means that there is no relationship between the two variables whereas 1 indicates that there is a perfect relationship. 18.0% variation in dependent variable can be explained by the variation happened in the independent variable. It is worth to emphasize that 82% shows unexplained part of the relationship.



71

Example for MA, WMA and ES

(a) Use a simple three-month moving average to find the next period(b) Use a weight average method conducting 0.50 (for most recent datum), 0.30 , and 0.20 to find the next period.(c) Use single exponential smoothing technique to find the next period employing smoothing constant and 5. period forecast value are 0.4 and 58.60 respectively.(d) Use RMSE error model and decide which technique is better explain the data (MA and ES).(e) Plot the monthly data, three-month moving average estimates as as well as exponential smoothing estimates. Briefly explain the patterns.

period No of law case

1 60

2 64

3 55

4 58

5 64



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Example for MA, WMA and ES a-b-c

period No of law case MA3 WMA3 ES 0.4

1 60 60.2

2 64 60.12

3 55 61.672

4 5859.66667 59.0032

5 64 59 58.60192

next period 59 60.4 60.761152



Example for MA, WMA and ES -d-

ErrorMA3 sq(ErrorMA3) ErrorES0.4 sq(ErrorES0.4)

-0.2 0.04

3.88 15.0544

-6.672 44.515584

-1.666666667 2.777777778 -1.0032 1.00641024

5 25 5.39808 29.13926769

sum 27.77777778 30.14567793

RMSE 3.726779962 2.455429817 Exponential smoothing technique is better than three-quarter moving average

forecast technique because the former one gives less error than the latter one..



Example for MA, WMA and ES -e-



75

Seasonal Variation



76

Seasonal Variation

Ratio to Trend Method

ActualTrend Forecast

Ratio =

SeasonalAdjustment =

Average of Ratios forEach Seasonal Period

AdjustedForecast =

TrendForecast

SeasonalAdjustment



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Seasonal Variation

Ratio to Trend Method:Example Calculation for Quarter 1

Trend Forecast for 2001.1 = 11.90 + (0.394)(17) = 18.60

Seasonally Adjusted Forecast for 2001.1 = (18.60)(0.887) = 16.50

YEAR Forecasted Actual Act/Forec1997Q1 12.29 11 0.8950371998Q1 13.87 12 0.8651771999Q1 15.45 14 0.9061492000Q1 17.02 15 0.881316

AV 0.886920.887

Deseasonalize data=actual sales/seasonal relative (index)



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Seasonal Variation

Select a representative historical data set. Develop a seasonal index for each season. Use the seasonal indexes to deseasonalize the

data. Perform linear regression analysis on the

deseasonalized data. Use the regression equation to compute the

forecasts. Use the seasonal indexes to reapply the

seasonal patterns to the forecasts.



79

Example: Computer Products Corp. Seasonalized Times Series Regression Analysis An analyst at CPC wants to develop next year’s quarterly

forecasts of sales revenue for CPC’s line of Epsilon Computers. The analyst believes that the most recent 8 quarters of sales (shown on the next slide) are representative of next year’s sales. Calculate the seasonal indexes



80

Example: Computer Products Corp.



81

Example: Computer Products Corp.



Unseasonalized vs. Seasonalized

QuarterSeasonalized

SalesSeasonal

IndexDeseasonalized

Sales

123456789

1011…

2340252732483337375040

0.8251.3100.9200.9450.8251.3100.9200.9450.8251.3100.920 …

27.8830.5327.1728.5738.7936.6435.8739.1544.8538.1743.48

…

0.825

2327.88

Sales: Unseasonalized vs. Seasonalized

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 11Quarter

Sale

s

Sales Deseasonalized Sales



83

Deflating a Time Series

Observed values can be adjusted to base year equivalent

Allows uniform comparison over time Deflation formula:

)100(I

yy

t

tadjt

where

= adjusted time series value at time t

yt = value of the time series at time t

It = index value at time t

tadjy



84

Deflating a Time Series: Example

Which movie made more money (in real terms)?

YearMovieTitle

Total Gross $

1939Gone With the Wind

199

1977 Star Wars 461

1997 Titanic 601

(Total Gross $ = Total domestic gross ticket receipts in $millions)



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Deflating a Time Series: Example

YearMovieTitle

Total Gross

CPI (base year =

1984)

Gross adjusted to 1984 dollars

1939Gone

With the Wind

199 13.9 1431.7

1977Star Wars

461 60.6 760.7

1997 Titanic 601 160.5 374.5

7.1431)100(9.13

199GWTW 1984adj

GWTW made about twice as much as Star Wars, and about 4 times as much as Titanic when measured in equivalent dollars.



86

Thanks

ch5defor13

Documents

central call

computer products

weighted moving

absolute percent

historical

exponential

time series

time series