demand forecasting information

8/11/2019 Demand Forecasting Information

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Demand Est imat ion and Forecast ing



Types of data



Research Approaches to Demand

Estimation & Forecasting

• Survey Methods – Consumer surveys

• Complete Enumeration Method

• Sample Method

– Expert Surveys (Delphi Method)

• Statistical Methods – Trend Projection

• Graphical Method

• Trend Fitting using Least Squares Method

• Box-Jenkins Method

– Econometric Methods• Regression Method

– Simple Regression

– Multiple Regression

• Simultaneous Equations



Complete Enumeration Method

• Potential users are contacted and asked about their

future plan of purchasing the product.

If n out f m households in a city report the quantity

(d) they are willing to purchase of a commodity,

then total probable demand (Dp) will be

Dp = d1+d2+d3+….+dn



Sample Method

Dp

= (HR

/HS

)( H.AD

)

Dp = Probable Demand

HR = No. of households reporting demandfor the product.

HS = No. of households surveyed.

AD = Average expected consumption by thereporting households.

H = Census no. of the households from the

relevant market.

HHS

HR



Delphi Method

Delphi method uses expert opinion about futuredevelopments. It was developed for long-range economicalpredictions by scientists of the Rand Corporation (1950s). It isan iterative process to collect and refine the anonymous

judgments of experts using a series of data collection andanalysis techniques interspersed with feedback.

The Delphi method is well suited as a research instrumentwhen there is incomplete knowledge about a problem orphenomenon. Feedback of results follows each step of

questioning. The process continues until no furtherconvergence of the experts' opinion is to be expected.



Trend Projection: Graphical Method



Trend Fitting using Least Squares Method

Yt = a + b t

Sales =f (time)

St = a + b t



Box-Jenkins Method

• Moving Average Forecasts

• Exponential Smoothing Forecasts



Moving Average Forecasts

Forecast is the average of data from w

periods prior to the forecast data point.

1

w

t i

t

i

A F

w



Accuracy of a Forecasting Method

2( )t t A F

RMSE n

Root Mean Square Error (RMSE):

(Measures the accuracy of a forecasting Method)



Three-quarter and Five-quarter Moving Average Forecast

?



Exponential Smoothing

Forecasts

1 (1 )t t t F wA w F

Forecast is the weighted average of the forecast

and the actual value from the prior period.

0 1w



Root Mean Square Error

2( )t t A F RMSE

n

To measure the accuracy of the forecast



Regression Analysis



Regression Analysis

• Regression Line: Line of Best Fit

• Ordinary Least Squares (OLS) Method

• Regression Line: Minimizes the sum of

the squared vertical deviations (et) of

each point from the regression line.



Scatter Diagram

Regression Analysis

Year X Y

1 10 44

2 9 40

3 11 42

4 12 46

5 11 48

6 12 52

7 13 54

8 13 58

9 14 56

10 15 60



Ordinary Least Squares (OLS)

Model: t t t Y a bX e

ˆˆ ˆt t Y a bX

ˆ

t t t

e Y Y

Properties:

(i) = 0

(ii) is minimum.




Objective: Determine the slope and intercept

that minimize the sum of the squared errors.

2 2 2

1 1 1

ˆˆ ˆ( ) ( )n n n

t t t t t

t t t

e Y Y Y a bX

Method used for this: Maxima Minima



Ordinary Least Squares (OLS)Estimation Procedure

1

2

1

( )( )ˆ

( )

n

t t

t

n

t

t

X X Y Y

b

X X

ˆa Y bX

f



Data on sales and Advertising expenditure for 10

years for a firm.

Year Ad. Expenses Sales 1 10 44

2 9 40

3 11 42

4 12 46 5 11 48

6 12 52

7 13 54

8 13 58 9 14 56

10 15 60




Estimation Example

1 10 44 -2 -6 12

2 9 40 -3 -10 30

3 11 42 -1 -8 8

4 12 46 0 -4 0

5 11 48 -1 -2 2

6 12 52 0 2 0

7 13 54 1 4 4

8 13 58 1 8 8

9 14 56 2 6 12

10 15 60 3 10 30

120 500 106

4

9

1

0

1

0

1

1

4

9

30

Time t X

t Y

t X X

t Y Y ( )( )

t t X X Y Y

2( )

t X X

10n

1

12012

10

nt

t

X X

n

1

50050

10

nt

t

Y Y

n

1

120n

t

t

X

1

500n

t

t

Y

2

1

( ) 30n

t

t

X X

1

( )( ) 106n

t t

t

X X Y Y




Estimation Example

10n 1

12012

10

nt

t

X X

n

1

50050

10

nt

t

Y Y

n

1

120n

t

t

X

1

500n

t

t

Y

2

1

( ) 30n

t

t

X X

1

( )( ) 106n

t t

t

X X Y Y

106ˆ 3.53330

b

ˆ 50 (3.533)(12) 7.60a

Y = 7.60+3.533 X



Y = 7.60+3.533 X

Tests of significance?



Test for Significance

Under the validity of H 0 , t statistic will be used,

where

SE b denotes the standard deviation of b and

is called the standard error .

H0: 1 = 0H1: 1 0

t = b

SE b

α = 0.05

With d.f. = n-2



Standard Error of the Slope Estimate (b)

2 2

ˆ 2 2

ˆ

( )( ) ( ) ( ) ( )

t t

b

t t

Y Y e s

n k X X n k X X

(k -1) (k -1)



Tests of Significance

2 2

1 1

ˆ( ) 65.4830n n

t t t

t t

e Y Y

2

1

( ) 30n

t

t

X X

2

ˆ 2

ˆ( ) 65.48300.52

( ) ( ) (10 2)(30)

t

bt

Y Y s

n k X X

1 10 44 42.90

2 9 40 39.37

3 11 42 46.43

4 12 46 49.96

5 11 48 46.43

6 12 52 49.96

7 13 54 53.49

8 13 58 53.49

9 14 56 57.02

10 15 60 60.55

1.10 1.2100 4

0.63 0.3969 9

-4.43 19.6249 1

-3.96 15.6816 0

1.57 2.4649 1

2.04 4.1616 0

0.51 0.2601 1

4.51 20.3401 1

-1.02 1.0404 4

-0.55 0.3025 9

65.4830 30

Time t

X t

Y ˆt

Y ˆt t t

e Y Y 2 2ˆ

( )t t t e Y Y

2

( )t X X




Example Calculation

2

ˆ 2

ˆ( ) 65.48300.52

( ) ( ) (10 2)(30)

t

bt

Y Y s

n k X X

2

1

( ) 30n

t

t

X X

2 2

1 1

ˆ( ) 65.4830n n

t t t

t t

e Y Y

(k -1)

T t f Si ifi




Calculation of the t Statistic

ˆ

ˆ 3.536.79

0.52b

bt

s

Degrees of Freedom = (n-k) = (10-2) = 8

Critical Value at 5% level =2.306

Since calculated t is higher than the critical(tabulated) t, therefore, the Reg. coefficient is

significant.



Y = 7.60+3.533 X

Hence we can say that b is a significant

regression coefficient which infers thatX is a significant explanatory variable

for Y.



Two tail Hypothesis test with

rejection region in both tails

• The rejection region is split equally between the two tails.

T t il t il t t



One-Tail Test

(left tail)

Two-Tail Test One-Tail Test

(right tail)

Two tail vs. one tail test

α α α/2 α/2



Test of Significance of R2



Decomposition of Variation in Dependent

Variable

2 2 2ˆ ˆ( ) ( ) ( )t t t Y Y Y Y Y Y

Total Variation = Explained Variation + Unexplained

Variation

n-1 = k-1 + n-k



Test of Significance

Coefficient of Determination

2

22

ˆ( )

( )t

Y Y Explained Variation R Total Variation Y Y

2 373.84

0.85440.00 R



Significance of Coefficient of Determination



Significance of Coefficient of Determination

H0: R 2 = 0

H1: R 2 > 0

Under the validity of H0, the appropriate test statistic is the F statistic:

which has an F distribution with 1 and n - 2 degrees of freedom.

F = S SR /(k-1)

S SE / ( n -k )

05.0



Source Sum of Squares D.F. Mean Square F

Regression SSR k-1

Error SSE n-k

Total SST n-1

If is accepted,

otherwise significant regression.

1

k

SSR MSR

k n

SSE MSE

MSE

MSR F

k nk

F F

,1

ANOVA Table



Multiple Regression Analysis

Model:



Multiple Regression Analysis

Analysis of Variance and F Statistic

/( 1)

/( )

Explained Variation k F

Unexplained Variation n k

2

2

/( 1)

(1 ) /( )

R k F

R n k

Significance Testing of Overall



Significance Testing of Overall

Regression

H0 : R 2 = 0

This is equivalent to the following null hypothesis:

H0: 1 = 2 = 3 = . . . = k = 0

The overall test can be conducted by using an F statistic:

R 2 / K-1 ( 1 - R 2 ) / ( n - k)

which has an F distribution with k-1 and (n - k ) degrees of freedom.

F =



Problems in Regression Analysis

• Multicollinearity: Two or more

explanatory variables are highly

correlated.• Heteroscedasticity: Variance of error

term is not independent of the Y

variable.• Autocorrelation: Consecutive error

terms are correlated.



Multicollinearity (MC)



Multicollinearity (MC)

Multicollinearity inflates the variances of the

parameter estimates leading to insignificant t-ratios even when R 2 is significant.

Measures to detect:

• Bivariate Correlation Coefficients b/w the independentvariables.

• VIF (Variance Inflation factor)

VIF more than 10 indicates high multicollinearity

Remedial Measures for MC



Remedial Measures for MC

• Increase the sample size and check.

• Check with the specification of the model (linear vs. Non-linear).

• If single variable causing MC, can be dropped, if theoretically

permitted.

• The specification of the individual variables can be changedsuch as per capita Income rather than total income.

• Centering of the variables Replacing the values by ( )

• Principal Component Analysis

X X

Durbin Watson Statistic



Durbin-Watson StatisticTest for Autocorrelation

21

2

2

1

( )n

t t

t

n

t

t

e e

d

e

If d = 2, autocorrelation (AC) is absent.

If d= 0, perfect +ve AC.

If d= 4, perfect -ve AC.

0-2: High +ve AC.

2-4: High -ve AC.



H0: R= 0



H0: R= 0

H1: R> 0

If d > dU conclude H0 (R= 0)

if dL <= d <= dU the test is inconclusive

if d < dL conclude H1 (R > 0)

21

2

2

1

( )

n

t t

t

n

t

t

e e

d

e



The Durbin-Watson Test:

Interpreting the Results

D-W Statistic Table



No. of independent variables



Steps in Demand Estimation

• Model Specification: Identify Variables

• Collect Data

• Specify Functional Form

• Estimate Function

• Test the Results

F ti l F S ifi ti



Functional Form Specifications

Linear Function:

Power Function:1 2( )( )b b

X X Y Q a P P

Estimation Format:

1 2ln ln ln ln X X Y Q a b P b P

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