estimating demand function (1)

8/18/2019 Estimating Demand Function (1)

http://slidepdf.com/reader/full/estimating-demand-function-1 1/45

Estimating Demand

Function• Where do demand functions come from?• Sources of information for demand

estimation• Cross-sectional versus time series data• Estimating a demand speci cation using

the ordinary least squares ( !S" method#• $oodness of t statistics#•

Forecasting demand %y using regressionequation



&he 'urpose of Estimating Demand Function

# &o determine factors that in)uence the demand

of the product (empirically signi cant factors"*# &he analy+e the impact of the factors on

demand of the product (elasticity"

,# &o forecast demand of the product that arecrucial for planning and also in managing our%usiness



Demand function .ecall

Q = 20 + 2Y + P” –3P

!ets say that the demand function /as as follo/s

Where 0 is the quantity demanded1units of goods sold2

3 is a level of income4

'5 is the price of our rival product4and4 ' is the price of our o/n product#

The issue is how we estimatedthis demand equation?



0uestions managers should

as6 in estimating demand equations

# What is the 7%est5 equation that

can %e o%tained (estimated" fromthe availa%le data?

*# What does the equation note8plain?

,# What can %e said a%out theli6elihood and magnitude ofrelationship %et/een varia%les?

9# What are the consequences offorecast errors? (if /ant to use for



:o/ do get the data to estimate

demand equations?

• Customer surveys and intervie/s#• Controlled mar6et studies#• ;ncontrolled mar6et data#• sales record1&ime Series Data



Survey pitfallsSample %ias.esponse %ias.esponse accuracyCost



Time -series data : historical data--i.e., the data sample consists of a series ofdaily, monthly, quarterly, or annual data for variables such as prices, income ,

employment , output , car sales, stock market indices, exchange rates, and soon.

Cross-sectional data : All observations in the sample are taken from the same point in time and represent different individual entities (such as households,houses, etc.)

hich one is better!

"anel #ata$longitudinal #ata%

&ypes of data



Example of Time series data: #ailyobservations,



Student ID Sex Age Height Weight

<<<=<*9, > * = 5 <@ l%s#

*, AB@<= > *@ 5 *A l%s#AAA F B @5 * l%s#

@B@A=B@9 F ** 95 B@ l%s#

AAA,9 *,9 > *A = *5 @, l%s

Example of cross sectional data



Estimating demand equations

using regression analysis.egression analysis is a statistical technique thatallo/s us to quantify the relationship %et/een adependent varia%le and one or moreindependent or e8planatory varia%les#

f only one independent varia%le simple

regressionf more than one independence varia%les multiple regression

Which one is %etter?



Our model i !e"i#ed $ %ollo& 'Q = % (P)

&here Q i ti"*et $le $nd P i the%$re

Specifying a simple regressionmodel

0 is the dependent varia%le that is4 /e

thin6 that variations in 0 can %e e8plained%y variations in '4 the 7e8planatory5varia%le



ii P Q &' β β +=

β 0 and β 1 are called parameters orpopulation parameters#

We estimate these parameters

using the data /e have availa%le(data from sam le"

iii P Q ε β β ++= &'

Estimating the single varia%le model of demandfunction (simple regression"

ince the data points are unlikely to fall

exactly on a line, (&)must be modifiedto include a disturbance

term ( εi)



Simple !inear .egression Equation(sample"

&he estimated simple linear regressionequation

AG y b b x = +

• is the estimated value of y for a given x value#G y • b is the slope of the line#

• b A is the y intercept of the line#

•

&he graph is called the estimated regression line#



Estimation 'rocess

.egression >odel y H β A I β x I ε

.egression EquationE( y " H β A I β x

;n6no/n 'arametersβ A4β

Sample Data x y x y

. . . . x n y n

b A and bprovide estimates of

β A and β

Estimated

.egression Equation Sample Statistics

b A4b

AG y b b x = +

E



Y

X '

$nd Y $re not!er%e"tl, "orrel$ted-Ho&e.er/ there ion average $ !o iti.erel$tion hi!

et&een Y $nd

& *

Est mat on >et o r nary!east Square ( !S"



ε&

+ &

(+ &)

Y

X '&

E(Y |X i ) = β 0 + β 1 X i

ε& = Y 1 - E(Y|X 1 )

We assume thatexpected conditional values of Y associated with

alternative values of Xfall on a line.



&he line of %est t is the one thatminimi+es the squared sum of the

vertical distances of the sample pointsfrom the line

!S !ine of %est t



!east Squares >ethod

!east Squares Criterion

min ( y y i i−∑ "*

/here y i H o%served value of the dependent varia%le for the ith o%servation

J y i H estimated value of the dependent varia%le for the ith o%servation



Slope for the Estimated .egressionEquation

*( "( "

( "i i

i

x x y y b x x − −

=−

∑∑




y - ntercept for the Estimated .egressionEquation


Ab y b x = −

/here x i H value of independent varia%le for ith o%servation

n H total num%er of o%servations

K y H mean value for dependent varia%le

K

x H mean value for independent varia%le

y i H value of dependent varia%le for ith o%servation



1. Specification

. Estimation

!. Evaluation

". #nal$%in&'(orecastin&

&he 9 steps of demandestimation usingregression



:o/ to determine that /e are correctlyspecify the demand function? (linear ornon-linear"

Simple linear regression %egins %y plotting0-' values on a scatter diagram todetermine if there e8ists an appro8imatelinear relationship

Speci cation



Ticket Prices and TicketSales along an Air Route



Scatter plot diagram

Passengers

16014012010080604020

F a r e

290

280

270

260

250

240

230

220

210



Scatter plot diagram /ith possi%le line of %est t

Average One-way Fare

7

65

4

3

2

$ 2 0

2 02 0

2 0

2 0

2 0

Demand curve: Q = 330-

500 100 150

!um"er # %ea&' %#(d )er F(*g+&



Computing the !S

estimators We can estimated the equation using thestatistical soft/are pac6age SPSS (Excel alsocan). t generated the follo/ing output S'SSLLLLL

Coefficients a

478,690 88,036 5,437 ,000-1,633 ,367 -,766 -4,453 ,001

.#n'&an&/FA

#de(1

%&d, rr#r

n'&andard* ed.#e *c*en&'

e&a

%&andard*ed

.#e *c*en&'

& %*g,

De)enden& ar*a"(e: A%%a,



.eading the S'SS utput

&hus our estimated demandequation is given %y

ii P Q /.&0.1023 −=

From the ta%le /e see that our estimate of βA is1 - and our estimate of β is –4-53-



Evaluation Mo/ /e /ill evaluate the estimated

equation using standard goodness of tstatistics4 including

# &he standard errors of the estimates#

*# &he t-statistics of the estimates of the

coeNcients#,# &he standard error of the regression ( s "

9# &he coeNcient of determination ( R2"



• We assume that the regression coeNcients are normallydistri%uted varia%les#

• &he standard error (or standard deviation" of the estimates isa measure of the dispersion of the estimates around theirmean value#

• Os a general principle4 the smaller the standard error4 the%etter the estimates (in terms of yielding accurate forecasts ofthe dependent varia%le"#

Standard errors of

the estimates

Rule-of-thumb is useful: The standard error of theregression coefficient should be less than half of thesize of the corresponding regression coefficient



Coefficients a

478,690 88,036 5,437 ,000-1,633 ,367 -,766 -4,453 ,001

.#n'&an&/FA

#de(1

%&d, rr#r

n'&andard* ed.#e *c*en&'

e&a

%&andard*ed

.#e *c*en&'

& %*g,

De)enden& ar*a"(e: A%%a,

4y reference to the " output, 5e seethat the standard error of our estimateof β& is './ 0, 5hereas the (absolute value)ourestimate of β

&is &. / 6ence our estimate is about 1 7

times the si8e of its standard error.



9o test for the significance of our estimate of β&, 5e set thefollo5ing null hypothesis, 6 ' , and the alternative hypothesis, 6 &

6': β

&≥ '

6 &: β& '

9he t distribution is used totest for statistical significance ofthe estimate:

1;.1'1<.'

'/.&3

&3

&&−≅

−−=

−=

β

β β s

t

&he t test



&he coeNcient ofdetermination4 . *4 is de ned asthe proportion of the totalvariation in the dependentvaria%le (3" Pe8plainedP %y theregression of 3 on theindependent varia%le (Q"#

CoeNcient of determination (. *"

ANOVA b



We see from the S'SS modelsummary ta%le that R2 for this model

is # @=

ANOVA b

6863,624 1 6863,624 19,826 ,001 a

4846,816 14 346,20111710,440 15

egre''*#ne'*dua(#&a(

#de(1

%um #$ % uare' d$

ean% uare F %*g,

red*c&#r': .#n'&an&/ FAa,

De)enden& ar*a"(e: A%%",

Model Summary

,766 a ,586 ,557 18,60652#de(1

%9uare Ad;u'&ed

%9uare

%&d, rr#r #$ &+e'&*ma&e

red*c&#r': -.#n'&an&/ FAa,



⇒Mote that A ≤ . * ≤

⇒ f .*

H A4 all the sample points lie on ahori+ontal line or in a circle⇒ f . * H 4 the sample points all lie on the

regression line

⇒ n our case4 . * ≅ A# @=458.6 percent of the variation in the epen ent

variable is explaine by the re!ression # s itgood?

Motes on .*



⇒ &he model summary tells us that s H @#=⇒ .egression is %ased on the assumption that the

error term is normally distri%uted4 so that =@#<R ofthe actual values of the dependent varia%le (seatssold" should %e /ithin one standard error ( ± @#= inour e8ample" of their tted value#⇒Olso4 B #9 R of the o%served values of seats sold

should %e /ithin * standard errors of their ttedvalues ( ±,<#*"#

Model Summary

,766 a ,586 ,557 18,60652#de(1

%9uare Ad;u'&ed

%9uare

%&d, rr#r #$ &+e'&*ma&e

red*c&#r': -.#n'&an&/ FAa,

Standard error of the regression



Forecasting

ii P Q /.&0.1023 −=

.ecall the equation o%tained from theregression results is

Ot the most %asic level4 forecasting consists of insertingforecasted values of the e8planatory varia%le ' (fare" into the

estimated equation to o%tain forecasted values of thedependent varia%le 0 (passenger seats sold"#



Year and Predicted ActualQuarter Sales (Q*) Sales (Q) Q* - Q (Q* - Q)s

97-1 64,8 70,44 5,64 31,8197-2 33,6 45,94 12,34 152,2897-3 37,8 45,94 8,14 66,2697-4 83,3 86,77 3,47 12,0498-1 111,7 103,1 -8,6 73,9698-2 137,5 111,26 -26,24 688,54

98-3 109,6 111,26 1,66 2,7698-4 96,8 119,43 22,63 512,1299-1 59,5 103,1 43,6 1900,9699-2 83,2 94,94 11,74 137,8399-3 90,5 78,61 -11,89 141,3799-4 105,5 86,77 -18,73 350,81

00-1 75,7 70,44 -5,26 27,6700-2 91,6 86,77 -4,83 23,3300-3 112,7 86,77 -25,93 672,3600-4 102,2 94,94 -7,26 52,71

%um #$ % uared rr#r' 4846,80

n-Sample Forecast of Oirline Sales

I S l F t f Ai li S l



In-Sample Forecast of Airline Sales

Year/Quarter

00,300,199,399,198,398,197,397,1

P a s s e

n g e r s

160

140

120

100

80

60

40

20

Ac&ua(

F*&&ed



ur a%ility to generate accurate forecasts of the dependentvaria%le depends on t/o factors

• Do /e have good forecasts of the e8planatory varia%le?

• Does our model e8hi%it structural sta%ility4 i#e#4 /ill the causalrelationship %et/een 0 and ' e8pressed in our forecastingequation hold up over time?

• While the past may %e a servicea%le guide to the future in thecase of purely physical phenomena4 the same principle doesnot necessarily hold in the realm of social phenomena (to/hich economy %elongs"#

Can /e ma6e a

good demand forecast?



Single Taria%le .egression ;sing

E8cel

Class Discussion %y using

e8cel estimate an equationand use it to predict homeprices in t/o cities# &he

data set is on the ne8t slide#



Cit$ )ncome *ome +rice

Alor etar 01.& &&1.<

g "etani 2*.1 &* .<

"enang 0&.* &/'.<=poh *.2 <*.2

>uala >angsar 0<.* &/;.2

?a5ang .2 && .0

hah Alam 2*. & &.<

eremban 2;./ &1;

@elaka 0;.2 &1;./

ohor 4ahru 2<.& & *.&

@uar 0;.* &*;.<

>ota 4ahru 02.2 &1;.*

kuala Bumpur &'' &0/.

>uala 9erengganu 00./ &*;.<

>angar 20 &;&.;

"ekan 0.2 &'2.&

>ucing 0&.* &'&.&

>ota >inabalu <0.1 &<&.<

• ncome (3" isaveragefamilyincome in*A ,

• :ome 'rice(:'" is theaverageprice of ane/ ore8istinghome in*A ,#



>odel Speci cation

Y bb HP &' +=

∧



Regression tatisti!s

@ultiple ? '.<' <2/110

? quare '.2** &2<0/

AdCusted ?quare '.2&&;/* ;<

tandard rror &&.**2021&

Dbservations &2

"oe##i!ientstandard Error t tat

=ntercept -12.&&'/00*1 *&.;21;</* -*.**2<**&&1

=ncome *.//*;'10 < '.*0'02'&& 2. &1'&02<;

AEDFA

d#

?egression &</;;.0&;;'

*

?esidual &*'&0./ <1<

2

9otal &0 &&/0/.'2;

E8cel utput



Y HP //.*&&.12 +−=∧

Gity =ncome "redicted 6"

egamat ;<, '' ?@

&/2,2&<.2<

"utraCaya &*&,''' ?@

*2&,22&.2<

Equation and prediction

estimating demand function (1)

Documents