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7/18/2019 Ch04 Appen
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Slide 1
Problems in Applying the
Linear Regression Model Appendix 4A
• The assumptions of the linear regression model don’t
always hold in the real world
• We now eamine statisti!al problems" whi!h is the
!entral fo!us of the e!onomi! sub#field !alled
econometrics
1$ Auto!orrelation%$ &eteros!edasti!ity
'$ Spe!ifi!ation and Measurement (rror
)$ Multi!ollinearity
*$ Simultaneous e+uation relationships and the identifi!ation problem
,$ -onlinearities
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Slide %
1. Autocorrelationalso .nown as serial !orrelation
• Problem:/ 0oeffi!ients are unbiased
/ but t#alues are unreliable
• Symptoms:
/ loo. at a s!atter of the error terms to see if there is a pattern" or
/ see if Durbin Watson statisti! is far from %$
• Cures:
1$ 2ind more ariables that eplain these patterns
%$ Ta.e first differen!es of data3 ∆4 5 a 6 b•∆P
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Slide '
S!atter of (rror TermsPositie Auto!orrelation
2igure )A$1 page 171Y
X
1
%
' ) * ,
7
8
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Slide )
2. Heteroscedasticity• Problem:
/ 0oeffi!ients are unbiased
/ t#alues are unreliable
• Symptoms:
/ different arian!es for different sub#samples
/ s!atter of error terms shows in!reasing or de!reasing dispersion
• Cures:
1$ Transform data" e.g." transform them into logs
%$ Ta.e aerages of ea!h sub#sample and use weighted least s+uares
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Slide *
S!atter of (rror Terms&eteros!edasti!ity
Height
AGE1 2 5 8
alternativelog Ht = a + b•AGE
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Slide ,
3. Specification & Measurement Error
• Salary = a + b (Strike Outs) in baseball
→ b is positie 999
Why?→ :mitted ariable whi!h is the number
of Hits
• Salary 5 ! 6 d ;Stri.e :uts< 6 e ; &its <→ here d is negatie and e is positie
7/18/2019 Ch04 Appen
http://slidepdf.com/reader/full/ch04-appen 7/20Slide 7
Specification & Measurement Error• Problem:
/ 0oeffi!ients are biased = we !an een hae the wrong sign as in the baseball eample
/ (en adding more obserations will not !ure this bias
• Symptoms:
/ The results don’t ma.e e!onomi! sense• Cure:
/ Thin. through the relationships and find the missing ariables in the
spe!ifi!ation
/ See if the new spe!ifi!ation improes the fit ;higher R %< and ma.es
e!onomi! sense$
7/18/2019 Ch04 Appen
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4. Multicollinearity
• Sometimes independent ariables aren’t
independent.
(>AMPL(3 LetQ = Es sold
4 5 a 6 b Pd 6 ! Pg
where Pd is pri!e for a
do?en eggs
and Pg is the pri!e for a
gross of eggs$
•!eression Output
Q = "" # $.% Pd #.& P
;1$%< ;1$)*<
R#s+uare 5 $87
;t#alues in parentheses<
- 5 1@@ obserations
• -oti!e that3
/ R#s+uare is 87
/ But that neither !oeffi!ient
is statisti!ally signifi!ant$
7/18/2019 Ch04 Appen
http://slidepdf.com/reader/full/ch04-appen 9/20Slide C
Multicollinearity• Problem:
/ 0oeffi!ients are unbiased
/ The t#alues are small" often insignifi!ant
• Symptoms:
/ &igh R#s+uares but low t#alues
• Cures:
1$ Drop a ariable$ Esually the remaining ariable be!omes
signifi!ant$
%$ Do nothing if fore!asting" sin!e the added R#s+uare of more
ariables is worthwhile
7/18/2019 Ch04 Appen
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5. Identification Prolem
and t!e Simultaneity Prolem • Problem3
/ 0oeffi!ients are biased
• Symptom3/ Fndependent ariables are .nown to be part of a
system of e+uations
• 0ure3/ Ese as many independent ariables as possible
7/18/2019 Ch04 Appen
http://slidepdf.com/reader/full/ch04-appen 11/20Slide 11
Graphi!al (planation of the
Fdentifi!ation Problem
• Suppose we estimate the
following demand !ure Q =
a + b P$
• Suppose Supply aries and
Demand is 'E*$
• All points lie on the demand
!ure• The demand !ure is said to be
identified$quantity
P
S1
S2
S3
Demand
7/18/2019 Ch04 Appen
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Suppose instead tat
S,PPL- is 'ied• Let D(MA-D shift
and supply is /ied
on doesn’t !hange$• All Points are on
the SEPPLH !ure$
• We say that theSEPPLH !ure isidentified$
quantity
P
D1
D2
D3
Supply
7/18/2019 Ch04 Appen
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0en bot Supply
and *emand 1ary
• :ften both supply and
demand ary$
• (+uilibrium points are in
shaded region$
• A regression of
Q = a + b P will beneither a demand nor a
supply !ure$ quantity
P
D1
D2
S1
S2
2
7/18/2019 Ch04 Appen
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Simultaneous Systems
1$ Demand is 4d 5 a 6 b P 6 ! H 6 ε1
%$ Supply is 4s 5 d 6 e P 6 f W 6 ε% Where P is pri!e" H is in!ome" W is the wage rate" and ea!h
has an error term$ -oti!e that P is in both of the demand and supply fun!tion$
P is IendogenouslyJ determined by both demand and supply$
The simultaneity problem is that pri!e is not independent" as
it is determined by the whole system The !ure for this problem is usually to hae as many
independent ariables as possible in the demand regression
to ma.e demand a!t li.e it is IfiedJ$
7/18/2019 Ch04 Appen
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". #onlinear $orms• Semi-logarithmic transformations. Sometimes taking
the logarithm of the dependent variable or an independent
variable improves the R2. Examples are:
• log Y = + ß·X.
» Here, Y grows exponentially at rate ß in X; that is, ß percent growth
per period.
• Y = + ß·log X. Here, Y doubles each time X increases by
the square of X.
>
H Ln H 5 $@1 6 $@*>
7/18/2019 Ch04 Appen
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Reciprocal Transformations
• The relationship between variables may be inverse.Sometimes taking the reciprocal of a variable improves
the fit of the regression as in the example:
• Y = α + ß·(1/X)
• shapes can be:/ de!lining slowly
• if beta positie
/ rising slowly
• if beta negatie
>
H E.g., H 5 *@@ 6 % ; 1K><
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Slide 17
• Quadratic, cubic, and higher degree polynomial
relationships are common in business and economics.
Profit and revenue are cubic functions of output.
Average cost is a quadratic function, as it is U-shaped
Total cost is a cubic function, as it is S-shaped• TC = α·Q + ß·Q2 + γ ·Q3 is a cubic total cost function.
• If higher order polynomials improve the R-square, then the
added complexity may be worth it.
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Slide 18
Multipli!atie or Double Log
•With the double log form" the !oeffi!ients areelasti!ities
• Q = 3 4 P b 4 -5 4 Ps d
/ multipli!atie fun!tional form
• So3 Ln Q = a + b4Ln P + 54Ln -+ d4Ln Ps
• Transform all ariables into natural logs
• 0alled the double log" sin!e logs are on the left and the right
hand sides$ Ln and Log are used inter!hangeably$ We useonly natural logs$
7/18/2019 Ch04 Appen
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Slide 1C
Soft Drin. 0ase" pp. 167-168
a !ross se!tion of *@ states
Cans = 676 # "8" Pri5e + 7.7& n5ome + ".&79emp
Predi!tor 0oeff StDe T P0onstant *1)$8 11'$% )$** @$@@@
Pri!e #%)1$8@ )'$,* #*$*) @$@@@
Fn!ome 1$1C* 1$,88 @$71 @$)8'
Temp %$C1', @$7@71 )$1% @$@@@
R#S+ 5 ,C$8 R#S+;ad< 5 ,7$7
The Price elasticity in Wyoming is = ((∆Q/Q/∆∆P)(P/Q) = -241.8(2.31/102)=P)(P/Q) = -241.8(2.31/102)= --
Linear Spe!ifi!ation
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Slide %@
Double Log Soft Drin. 0ase
Ln 0ans 5 %$)7 # '$17 Ln Pri!e 6 @$%@% Ln Fn!ome6 1$1% Ln Temp
Predi!tor 0oef Std De T P
0onstant %$),, 1$'8* 1$78 @$@8%
Ln Pri!e #'$1,C* @$,)8* #)$8C @$@@@
Ln Fn!ome @$%@%@ @$18') 1$1@ @$%77
Ln Temp 1$11C, @$%,11 )$%C @$@@@
R#S+ 5 ,7$) R#S+;ad< 5 ,*$1
Characterize the demand for soft drinks in the "re soft drinks ine#astic$ "re the% #&'&ries$
hich seci*cation *ts the data +etter$
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