ECNE610ManagerialEconomics

Week 4

MARCH 2014

1

Chapter-5

Demand Estimation and Forecasting

2

5

Lesson Objectivesknow how to specify and interpret a

regression.understand importance of forecasting

in business.describe six different forecasting

techniques.use seasonal and smoothing methods.recognize limitations of consumer

data.

3

4

Data Sources

PrimaryData Collection

SecondaryData Compilation

Observation

Experimentation

Survey

Print or Electronic

Those that are collected for your purposes.

Data collected & compiled by an outside source or by someone in your organization who provides others access to the data.

TYPES OF DATA

5

Data

Qualitative Quantitative

ContinuousDiscrete

6

Data Timing

Time series data consist of a set of ordered data values observed at successive points in time.

Cross-sectional data are a set of data values observed at a fixed point in time.

7

Data Timing (Panda’s sales reports)

Sales (in $1000’s)

2003 2004 2005 2006

Jeddah 435 460 475 490

Riyadh 320 345 375 395

Dammam

405 390 410 395

Madina 260 270 285 280

Time Series Data

Cross Section Data

8

A Population is the set of all items or individuals of interest. Examples: All likely voters in the next election.

All parts produced today.All sales receipts for November.

A Sample is a subset of the population. Examples: 1000 voters selected at random for

interview.A few parts selected for destructive testingEvery 100th receipt selected for audit.

Populations and Samples

9Introduction to Regression AnalysisRegression analysis is used to:

Predict the value of a dependent variable based on the value of at least one independent variable

Explain the impact of changes in an independent variable on the dependent variable

Dependent variable: the variable we wish to explain

Independent variable: the variable used to explain the dependent variable

15-10

The Multiple Regression Model

Idea: Examine the linear relationship between 1 dependent (y) & 2 or more independent variables (xi)

εxβxβxββy kk22110

kk22110 xbxbxbby

Population model:

Y-intercept Population slopes Random Error

Estimated (or predicted) value of y

Estimated slope coefficients

Estimated multiple regression model:

Estimatedintercept

The Least Squares Equation

The formulas for b1 and b0 are:

21 )x(x

)y)(yx(xb

xbyb 10

and

2i

ii1 )x(x

)y)(yx(xb

2

2

2ˆ

YY

YYR

i

i

ExampleA distributor of frozen desert pies

wants to evaluate factors thought to influence demand.Dependent variable: Pie sales (units per week)Independent variables: Price (in $)

Advertising ($100’s)

Data are collected for 15 weeks

April 21, 2023

Dr. Mazharul Islam

Slide 12

Formulate the ModelWeek

Pie Sales

Price($)

Advertising($100s)

1 350 5.50 3.3

2 460 7.50 3.3

3 350 8.00 3.0

4 430 8.00 4.5

5 350 6.80 3.0

6 380 7.50 4.0

7 430 4.50 3.0

8 470 6.40 3.7

9 450 7.00 3.5

10 490 5.00 4.0

11 340 7.20 3.5

12 300 7.90 3.2

13 440 5.90 4.0

14 450 5.00 3.5

15 300 7.00 2.7

  Pie Sales Price Advertising

Pie Sales 1

Price -0.44327 1

Advertising 0.55632 0.03044 1

Correlation matrix:

Estimated Regression model:

April 21, 2023

Dr. Mazharul Islam

Slide 13

Dr. Mazharul Islam15-14

Regression OutputRegression Statistics

Multiple R 0.72213

R Square 0.52148

Adjusted R Square 0.44172

Standard Error 47.46341

Observations 15

ANOVA   df SS MS FSignificance

F

Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333      

  CoefficientsStandard

Error t Stat P-value Lower 95% Upper 95%

Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404

Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392

Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888

ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales

April 21, 2023

The Regression Equation

ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales where Sales is in number of pies per week Price is in $ Advertising is in $100’s.

April 21, 2023

Dr. Mazharul Islam

Slide 15

Using The Model to Make Predictions

Predict sales for a week in which the selling price is $5.50 and advertising is $350:

Predicted sales is 428.62 pies

428.62

(3.5) 74.131 (5.50) 24.975 - 306.526

ertising)74.131(Adv ce)24.975(Pri - 306.526 Sales

Note that Advertising is in $100’s, so $350 means that x2 = 3.5

April 21, 2023

Dr. Mazharul Islam

Slide 16

15-17

Regression Statistics

Multiple R 0.72213

R Square 0.52148

Adjusted R Square 0.44172

Standard Error 47.46341

Observations 15

ANOVA   df SS MS FSignificance

F

Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333      

 Coefficient

sStandard

Error t Stat P-value Lower 95% Upper 95%

Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404

Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392

Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888

.52148056493.3

29460.0

SST

SSRR 2

52.1% of the variation in pie sales is explained by the variation in price and advertising

Coefficient of Determination(continued)

April 21, 2023 Slide 18

To test the statistical significance of the regression relation between the response variable y and the set of variables x2 and x3, i.e. to choose between the alternatives:

We use the test statistic:0 allnot :

0:0

ia

i

H

H

MSE

MSRF(cal)

Dr. Mazharul Islam

F-Test for Overall Significance of the Model

Shows if there is a linear relationship between all of the x variables considered together and y.

where numerator’s df = k and denominator’s df = (n – k – 1)

)kk,n(αFTabFCal 1;

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

15-19

6.53862252.8

14730.0

MSE

MSRF

Regression Statistics

Multiple R 0.72213

R Square 0.52148

Adjusted R Square 0.44172

Standard Error 47.46341

Observations 15

ANOVA   df SS MS FSignificance

F

Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333      

  CoefficientsStandard

Error t Stat P-value Lower 95% Upper 95%

Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404

Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392

Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888

(continued)F-Test for Overall Significance

With 2 and 12 degrees of freedom

P-value for the F-Test

5386.62252.8

14730.0

MSE

MSRF

H0: β2 = β3 = 0

HA: β2 and β3 not both zero

= 0.05df2= 2 df3 = 12

Test Statistic:

Decision:

Conclusion:

Reject H0 at = 0.05

The regression model does explain a significant portion of the variation in pie sales

(There is evidence that at least one independent variable affects y )

0

= 0.05

F0.05 = 3.885Reject H0

6.5386MSE

MSRF

Critical Value:

F = 3.885

F-Test for Overall Significance(continued)

FDo not reject H0

April 21, 2023 Slide 21

Significance tests for i

‘t’ test for a population slope• Is there a linear relationship between

x and y ? Null and alternative hypotheses

H0: βi = 0 (no linear relationship)

HA: βi 0 (linear relationship does exist)

Test statistic

iˆse

iβ

iˆ

t

Dr. Mazharul Islam

April 21, 2023 Slide 22

Reject HReject H00 if, if,

Dr. Mazharul Islam

)1;2

( kntcalculatedt

Copyright ©2011 Pearson Education, Inc. publishing as Prentice Hall

15-23

Regression Statistics

Multiple R 0.72213

R Square 0.52148

Adjusted R Square 0.44172

Standard Error 47.46341

Observations 15

ANOVA   df SS MS FSignificance

F

Regression 2 29460.027 14730.013 6.53861 0.01201

Residual 12 27033.306 2252.776

Total 14 56493.333      

 Coefficient

sStandard

Error t Stat P-value Lower 95% Upper 95%

Intercept 306.52619 114.25389 2.68285 0.01993 57.58835 555.46404

Price -24.97509 10.83213 -2.30565 0.03979 -48.57626 -1.37392

Advertising 74.13096 25.96732 2.85478 0.01449 17.55303 130.70888

(continued)

Are Individual Variables Significant?

t-value for Price is t = -2.306, with p-value 0.0398

t-value for Advertising is t = 2.855, with p-value 0.0145

d.f. = 15-2-1 = 12

= 0.05

t/2 = 2.1788

Inferences about the Slope: t Test Example

H0: βi = 0

HA: βi 0

The test statistic for each variable falls in the rejection region (p-values < 0.05)

There is evidence that both Price and Advertising affect pie sales at = 0.05

From Excel output:

Reject H0 for each variable

  Coefficients Standard Error t Stat P-value

Price -24.97509 10.83213 -2.30565 0.03979

Advertising 74.13096 25.96732 2.85478 0.01449

Decision:

Conclusion:Reject

H0

Reject H0

/2=0.025

-tα/2

Do not reject H0

0tα/2

/2=0.025

-2.1788

2.1788

Forecasting HorizonForecasting Horizon:

The number of future periods covered by a forecast.(Consists of one or more Forecasting Periods.)

It is sometimes referred to as forecast lead-time.

Forecasting Horizon, or lead time, is typically divided into four categories.

Immediate term – less than one month Short term – one to three months Medium term – three months to two years Long term – two years or more

Time-Series Components

Time-Series

Cyclical Component

Random Component

Trend Component

Seasonal Component

Upward trend

Trend Component Long-run increase or decrease over time

(overall upward or downward movement) Data taken over a long period of time

Sales

Time

Downward linear trend

Trend Component Trend can be upward or downward Trend can be linear or non-linear Can be stationary or non-stationary

Sales

Time Upward nonlinear trend

Sales

Time

(continued)

Seasonal Component Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly Recurrence period - shortest period of repetition

(must be less than one year)

Sales

Time (Quarterly)

Winter

Spring

Summer

Fall

Cyclical Component Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to

trough

Sales

1 Cycle

Year

Random Component Unpredictable, random, “residual” fluctuations Due to random variations of

Nature Accidents or unusual events

“Noise” in the time series

Multiplicative Time-Series Model

Used primarily for forecasting Allows consideration of seasonal

variation Observed value in time series is the

product of components Classical decomposition is used to

identify the various components

where Tt = Trend value at time t

St = Seasonal value at time t

Ct = Cyclical value at time t

It = Irregular (random) value at time t

ttttt ICSTy

16-33

Seasonal Adjustment

1. Compute each moving average2. Compute the centered moving averages3. Isolate the seasonal component by determining

the ratio-to-moving average values4. Determine seasonal indexes and normalize if

necessary5. Deseasonalize the time series6. Develop trend line using deseasonalized data7. Develop unadjusted forecasts using trend

projection8. Seasonally adjust the forecasts

16-34

Moving AveragesExample: Four-quarter moving average

First average:

Second average:

etc…

(continued)

4

Q4Q3Q2Q1average Moving 1

4

Q5Q4Q3Q2average Moving 2

16-35

Seasonal Data

Quarter Sales

1

2

3

4

5

6

7

8

9

10

11

etc…

23

40

25

27

32

48

33

37

37

50

40

etc…

Quarterly Sales

0

10

20

30

40

50

60

1 2 3 4 5 6 7 8 9 10 11

Quarter

Sal

es…

…

16-36

Calculating Moving Averages

Each moving average is for a consecutive block of 4 quarters

Quarter Sales

1 23

2 40

3 25

4 27

5 32

6 48

7 33

8 37

9 37

10 50

11 40

Average Period

4-Quarter Moving

Average

2.5 28.75

3.5 31.00

4.5 33.00

5.5 35.00

6.5 37.50

7.5 38.75

8.5 39.25

9.5 41.00

4

43212.5

4

2725402328.75

etc…

16-37

Centered Moving Averages

Average periods of 2.5 or 3.5 don’t match the original quarters, so we average two consecutive moving averages to get centered moving averages

Average Period

4-Quarter Moving

Average

2.5 28.75

3.5 31.00

4.5 33.00

5.5 35.00

6.5 37.50

7.5 38.75

8.5 39.25

9.5 41.00

Centered Period

Centered Moving

Average

3 29.88

4 32.00

5 34.00

6 36.25

7 38.13

8 39.00

9 40.13

etc…

16-38

Calculating the Ratio-to-Moving Average

Divide the actual sales value by the centered moving average for that quarter

16-39

Calculating Seasonal Indexes

Quarter Sales

Centered Moving Average

Ratio-to-Moving Average

1

2

3

4

5

6

7

8

9

10

11

…

23

40

25

27

32

48

33

37

37

50

40

…

29.88

32.00

34.00

36.25

38.13

39.00

40.13

etc…

…

…

0.837

0.844

0.941

1.324

0.865

0.949

0.922

etc…

…

…

88.29

25837.0

Example:

16-40

Calculating Seasonal Indexes

Quarter Sales

Centered Moving Average

Ratio-to-Moving Average

1

2

3

4

5

6

7

8

9

10

11

…

23

40

25

27

32

48

33

37

37

50

40

…

29.88

32.00

34.00

36.25

38.13

39.00

40.13

etc…

…

…

0.837

0.844

0.941

1.324

0.865

0.949

0.922

etc…

…

…

Average all of the Fall values to get Fall’s seasonal index

Fall

Fall

Fall

Do the same for the other three seasons to get the other seasonal indexes

(continued)

16-41

Interpreting Seasonal Indexes

Suppose we get these seasonal indexes:

SeasonSeasonal

Index

Spring 0.825

Summer 1.310

Fall 0.920

Winter 0.945

= 4.000 -- four seasons, so must sum to 4

Spring sales average 82.5% of the annual average sales

Summer sales are 31.0% higher than the annual average sales

etc…

Interpretation:

Since the sum 4 use a multiplier

averages of Sum

4 Multiplier

To find the Seasonal Index

Adjusted seasonal indexes are:

These seasonal indexes can now be used to remove the seasonal component from the original series

Winter Spring Summer Fall1.4014 0.6078 1.2732 0.65121.4468 0.5836 1.3592 0.64471.4754 0.6318 1.3368 0.5809

Average 1.4412 0.6077 1.3231 0.6256 3.9976 SumAdjusted SI 1.4420 0.6081 1.3239 0.6260 4.0000 Sum

16-44

Deseasonalizing

The data is deseasonalized by dividing the observed value by its seasonal index

This smooths the data by removing seasonal variation

16-45

DeseasonalizingQuarter Sales

Seasonal Index

Deseasonalized Sales

1

2

3

4

5

6

7

8

9

10

11

…

23

40

25

27

32

48

33

37

37

50

40

0.825

1.310

0.920

0.945

0.825

1.310

0.920

0.945

0.825

1.310

0.920

…

27.88

30.53

27.17

28.57

38.79

36.64

35.87

39.15

44.85

38.17

43.48

…

0.825

2327.88

etc…

(continued)

Example:

16-46

Unseasonalized vs. Seasonalized

Sales: Unseasonalized vs. Seasonalized

0

10

20

30

40

50

60

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

Sal

es

Sales Deseasonalized Sales

Fitting trend models

Once the seasonality has been removed from the data set, trend models can be applied to the deseasonalised data

Trend-Based Forecasting Estimate a trend line using regression analysis

Year

Time Period

(t)Sales

(y)

1999

2000

2001

2002

2003

2004

1

2

3

4

5

6

20

40

30

50

70

65

tbby 10

Use time (t) as the independent variable:

Trend-Based Forecasting The linear trend model is:

Sales trend

01020304050607080

0 1 2 3 4 5 6 7

Year

sale

s

Year

Time Period

(t)Sales

(y)

1999

2000

2001

2002

2003

2004

1

2

3

4

5

6

20

40

30

50

70

65

years in measured ist and 1999 1, t where

t 9.571412.333Sales

(continued)

Trend-Based Forecasting Forecast for 2005 (ie t = 7):

Sales

01020304050607080

0 1 2 3 4 5 6 7

Year

sale

s

Year

Time Period

(t)Sales

(y)

1999

2000

2001

2002

2003

2004

2005

1

2

3

4

5

6

7

20

40

30

50

70

65

??

(continued)

33.79

(7) 5714.9333.12Sales