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Outline of the lecture
-Introduction
Statistical estimation of the demand
function Model
OLS estimation techniue
Interpretation of the results !esting
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!he preceding chapter de"eloped the theory ofdemand# including the concepts of priceelasticity# income elasticity# and cross-elasticityof demand$
% manager &ho is contemplating an increase inthe price of one of the firm's products needs to(no& the impact of this increase on:
)*+ uantity demanded ),+ total re"enue
)+ profits
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.hat uestions should themanager ans&er/
- Is the demand elastic# inelastic# or unit elastic&ith respect to price o"er the range ofcontemplated price increase/
-.hat &ill happen to demand if consumerincomes increase or decrease as a result of aneconomic e0pansion or contraction$
Managers face these types of pro1lemse"eryday &hether in a profit-see(ing enterprises#not-for-profit organi2ations or go"ernments$
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E0ample
.hat &ill 1e the impact of cigarette ta0eson my uantity demanded of my product/
.hat effect &ill a tuition increase ha"e onlocal state uni"ersity re"enues/
!hese are the types of uestions theempirical in"estigation attempt to ans&er
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Statistical Estimation of theDemand 3unction
Econometrics is a collection of statisticaltechniues a"aila1le for testing economictheories 1y empirically measuring relationships
among economic "aria1les$ !he measurement of economic relationships is a
necessary step in using economic theories andmodels to o1tain estimates of the numerical
"alues of "aria1les that are of interest to thedecision ma(er$
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!he estimation of a demand function usingeconometric techniues in"ol"es the
follo&ing steps - Identification of the "aria1les
- Collection of the data
- 3ormulation of the demand model
-Estimation of the parameters
- De"elopment of forecasts )estimates+1ased on the model
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Identification of the "aria1les
%s discussed in the pre"ious chapter# thedemand function may 1e "ie&ed as therelationship 1et&een the uantity demanded )thedependent "aria1le+ and se"eral independent
"aria1les$ !he first tas( in de"eloping a statistical demand
model is to identify the independent "aria1lesthat are li(ely to influence uantity demanded$
!hese might include factors such as the price ofthe good in uestion# price of competing orsu1stitute goods# population# per capita income#and ad"ertising and promotional e0penditures$
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Collection of the data
Once the "aria1les ha"e 1een identified#the ne0t step is to collect data on the"aria1les$ Data can 1e o1tained from a
num1er of different sources$
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3ormulation of the model
!he ne0t step is to specify the form of theeuation# that indicates the relationship1et&een the independent "aria1les and
the dependent "aria1le$
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E0ample
!he linear model# &hich is the mostcommon form of estimation euation:
++++=332211
XXXY
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Linear Model
# 5*#5,#5#6 are the parameters of the
model and 6 is the error term$
!he error term is included in the model toreflect the fact that the relationship is notan e0act one# i$e$# the o1ser"ed demand"alue may not al&ays 1e eual to thetheoretical "alue$
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Interpretation of the "alue of 5
!he "alue of each 5 coefficient pro"idesan estimate of the change in uantitydemanded associated &ith a one-unit
change in the gi"en independent "aria1le#holding constant all other independent"aria1les$
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Interpretation of the "alue of 5
Y
X
E
Y
X
X
Y
E
D
D
2
2
2
.
.
=
=
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Simple Linear 7egression Model
!he analysis in this section is limited to thecase of one independent and onedependent "aria1le )t&o-"aria1le case+#
&here the form of the relationship 1et&eenthe t&o "aria1les is linear$
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Estimating the simple linearregression coefficients
=
=
=
=
=
=
n
i
i
n
i
ii
n
i
i
ii
n
i
nXX
YXnYX
b
XX
YYXX
b
1
22
1
1
2
1
)(
))((
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E0ample *
Sher&in-.illiams company is attempting tode"elop a demand model for its line of e0teriorhouse paints$ !he company's chief economist
feels that the most important "aria1les affectingpaint sales )8+ )measured in gallons+ are:
)*+ promotional e0penditures)9*+ )measured indollars+$ !hese include e0penditures on
ad"ertising )radio# !# and ne&spaper+# in-storedisplays and literature# and customer re1ateprograms$
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E0ample*
),+ Selling price )9,+ )measured in dollars pergallon+$
)+ Disposa1le income per household )9+)measured in dollars+
!he chief economist decides to collect data onthe "aria1les in a sample of ten company salesregions that are roughly eual in population$Data on paint sales# promotional e0penditures#
and selling prices &ere o1tained from thecompany's mar(eting department$ Data ondisposa1le income )per capita+ &as o1tainedfrom the ;ureau of La1or Statistics
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%ns&er
8< a = 1$9
8< *,>$?@ = >$449
Interpretation: !he coefficient of 9 )>$44+indicates that for one-unit increase in 9)A*#>>> in additional promotional
e0penditures+# e0pected sales )8+ &illincrease 1y >$44 )9*#>>>+< 44 gallonsin a gi"en sales region$
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Bsing the regression euation toma(e predictions
% regression euation can 1e used toma(e predictions concerning the "alue of8# gi"en any particular "alue of 9$ !his is
done 1y su1stituting the particular "alue of9# namely 9p# into the sample regression
euation
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E0ample
Suppose one is interested in estimatingSher&in-.illiam's paint sales for ametropolitan area &ith promotional
e0penditures eual to A*@#>>> )i$e$#9p$44)*@+
8*$>4@ gallons or ,>*#>4@ gallons
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E0ample
9p> or A>>#>>>
7emar(: 9p falls outside of the series of
o1ser"ations for &hich the regression line&as calculated$
!hus# 1ecause of the a1o"e remar(# &ecannot 1e certain that the prediction ofpaint sales 1ased on the regression model&ould reasona1le$
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Standard Error of the estimate
!he error term eiis defined as the
difference 1et&een the o1ser"ed andpredicted "alue of the dependent "aria1le$
!he standard de"iation of the error termsis calculated as:
2
)(
2
1
2
1
=
=
==
n
bxay
n
e
S
n
i
ii
n
i
i
e
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!he standard error of the estimate )Se+
can 1e used to construct predictioninter"als for 8$ %n appro0imate @ percentprediction inter"al is eual to
eSY 2
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7eturning to our pre"ious e0ample
)7.22(2045.2012 =e
SY
*@@$44? to ,4$4 )that is# from *@@#44? gallons to ,4#4 gallons+$
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!esting
F>: 5 ) Go relationship 1et&een 9 and8+
Fa: 5H> ) linear relationship 1et&een 9and 8+
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!esting
!here are t&o &ays of doing the testing:
+ Calculate the t statistic and compare it tothe critical "alue
4+ Bse the p-"alue techniue
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Correlation Coefficient
2
1
2
1
)()(
))((
=
=
=
n
i
ii
n
ii
yyxx
yyxxr
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Correlation Coefficient
!he correlation coefficient measures the degreeto &hich t&o "aria1les tend to "ary together$
Correlation analysis is useful in e0planatory
studies of the relationship among economic"aria1les$ !he information o1tained in thecorrelation analysis can then 1e used as a guidein 1uilding descripti"e models of economic
phenomena that can ser"e as a 1asis forprediction and decision ma(ing$
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Correlation Coefficient
!he "alue of the correlation coefficient ranges from =* for the t&o "aria1les &ithperfect positi"e correlation to -* for t&o
"aria1les &ith perfect negati"e correlation$
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!he Coefficient of Determination
It measures the proportion of the "ariationin the dependent "aria1le that is e0plained1y the regression line )the independent
"aria1le+$ !he coefficient of determination ranges
from > ) &hen none of the "ariation in 8 ise0plained 1y the regression+ to *) &hen allthe "ariation in 8 is e0plained 1y theregression$
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E0ample
If r,$@* ) from the Sher&in-.illiam'scompany e0ample+$
Interpretation: the regression euation# &ith
promotional e0penditures as the independent"aria1le# e0plains a1out @, percent of the"ariation in paint sales in the sample$
7emar(: In the t&o-"aria1le linear regression
model# the coefficient of determination is eualto the suare of the correlation coefficient# i$e$#r,$@*@+,$
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3-ratio
It is used to test &hether the estimatedregression euation e0plains a significantproportion of the "ariation in the dependent
"aria1le$ !he decision is to reJect the null hypothesis of no
relationship 1et&een 9 and 8 ) that is# noe0planatory po&er+ at the ( le"el of significance
if the calculated 3-ratio is greater than the 3(#*#n-,"alue o1tained from the 3-distri1ution$
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E0ample
If 3
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%ssociation and Causation
!he presence of association )correlation+does not necessarily imply causation$
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Multiple Linear 7egression
% functional relationship containing t&o ormore independent "aria1les is (no&n as amultiple linear regression model$
+++++=mm
XXXY ......2211
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7egression !echniues
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Estimating Coefficients
Consider a small restaurant chain speciali2ing infresh lo1ster dinner$ !he 1usiness has collectedinformation on prices and the a"erage num1er of
meals ser"ed per day for a random sample ofeight restaurants in the chain$ !hese data aresho&n 1elo&$ Bse regression analysis toestimate the coefficients of the demand function
Kd< a =1$ ;ased on the estimated euation#calculate the point price elasticity of demand atthe mean "alues of the "aria1les$
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!he Least-Suares regressionestimation
XbY
X
a
X
YYXX
b
i
i
=
=
2(
))((
)
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Estimating the demand for lo1stersdinners using the OLS
*?>a
-4$?@1
-*?@4>
>4-,>*4*>>
>>>@**>@
->-*>*>
->-,>**,>
-,>4-,*>*4**>
-4@-*@*@
-,>4,-*>*>
>*-*>*@*>>
)i-)1ar++)Ki-K)1ar++)i-)1ar++,i-)1ar+Ki-K)1ar+iKi
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Estimating the demand for lo1sterdinners
Bsing the ordinary Least Suaresregression# &e find the estimates of thedemand for lo1sters$
.e can no& use our results to determinethe point elasticity of demand for lo1sters$
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.ritten assignment
ro1lem page *>
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