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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

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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$

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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$

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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

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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

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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

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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

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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

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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$

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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 $@*>

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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$

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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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Ch04 Appen