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Chapter 7

Nonlinear Relationships

Prepared by Vera Tabakova, East Carolina University

Chapter 7: Nonlinear Relationships

7.1 Polynomials

7.2 Dummy Variables

7.3 Applying Dummy Variables

7.4 Interactions Between Continuous Variables

7.5 Log-Linear Models

Slide 7-2Principles of Econometrics, 3rd Edition

7.1 Polynomials

7.1.1 Cost and Product Curves

Figure 7.1 (a) Total cost curve and (b) total product curve


7.1 Polynomials

Figure 7.2 Average and marginal (a) cost curves and (b) product curves


7.1 Polynomials


(7.1)

(7.2)

(7.3)

21 2 3AC Q Q e

2 31 2 3 4TC Q Q Q e

2 3

( )2

dE ACQ

dQ

(7.4)2

2 3 4

( )2 3

dE TCQ Q

dQ

7.1.2 A Wage Equation


(7.5)

(7.6)

1 2 3WAGE EDUC EXPER+ EXPER e 24

3 4

( )2

E WAGEEXPER

EXPER

If Deriv < 0 => Exper < -β3/2β4 , or Exper > β3/2β4 .





18

( ).3409 2( .0051)18 .1576

EXPER

E WAGE

EXPER

3 42 .3409 / 2( .0051) 33.47EXPER

Where 33.47 is the ‘turning point’

7.2 Dummy Variables


(7.7)

(7.8)

1 2PRICE SQFT e

1 if characteristic is present

0 if characteristic is not presentD

1 if property is in the desirable neighborhood

0 if property is not in the desirable neighborhoodD

7.2.1 Intercept Dummy Variables


(7.9)

(7.10)

1 2PRICE D SQFT e

1 2

1 2

( ) when 1( )

when 0

SQFT DE PRICE

SQFT D

7.2.1 Intercept Dummy Variables

Figure 7.3 An intercept dummy variable


7.2.1a Choosing The Reference Group


1 if property is not in the desirable neighborhood

0 if property is in the desirable neighborhoodLD

1 2PRICE LD SQFT e

1 2PRICE D LD SQFT e

7.2.2 Slope Dummy Variables


(7.11)1 2 ( )PRICE SQFT SQFT D e

1 21 2

1 2

( ) when 1( )

when 0

SQFT DE PRICE SQFT SQFT D

SQFT D

2

2

when 1 ( ) when 0

DE PRICEDSQFT


Figure 7.4 (a) A slope dummy variable. (b) A slope and intercept dummy variable




(7.12)1 2 ( )PRICE D SQFT SQFT D e

1 2

1 2

( ) ( ) when 1 ( )

when 0

SQFT DE PRICE

SQFT D

7.2.3 An Example: The University Effect on House Prices




(7.13)

1 1 2

3 2 3 +

PRICE UTOWN SQFT SQFT UTOWN

AGE POOL FPLACE e





(24.5 27.453) (7.6122 1.2994) .1901

4.3772 1.6492

51.953+8.9116 .1901 4.3772 1.6492

PRICE SQFT AGE

POOL FPLACE

SQFT AGE POOL FPLACE

24.5 7.6122 .1901 4.3772 1.6492PRICE SQFT AGE POOL FPLACE


Based on these regression results, we estimate

the location premium, for lots near the university, to be $27,453

the price per square foot to be $89.12 for houses near the university,

and $76.12 for houses in other areas.

that houses depreciate $190.10 per year

that a pool increases the value of a home by $4377.20

that a fireplace increases the value of a home by $1649.20


7.3 Applying dummy variables

7.3.1 Interactions Between Qualitative Factors


(7.14) 1 2 1 2WAGE EDUC BLACK FEMALE BLACK FEMALE e

1 2

1 1 2

1 2 2

1 1 2 2

( )

EDUC WHITE MALE

EDUC BLACK MALEE WAGE

EDUC WHITE FEMALE

EDUC BLACK FEMALE





( ) /

/ ( )R U

U

SSE SSE JF

SSE N K

4.9122 1.1385

(se) (.9668) (.0716)

WAGE EDUC

( ) / (31093 29308) / 320.20

/ ( ) 29308 / 995R U

U

SSE SSE JF

SSE N K

7.3.2 Qualitative Factors with Several Categories


(7.15)1 2 1 2 3WAGE EDUC + SOUTH MIDWEST WEST e

1 3 2

1 2 2

1 1 2

1 2

( )

EDUC WEST

EDUC MIDWESTE WAGE

EDUC SOUTH

EDUC NORTHEAST

7.3.2 Qualitative Factors with Several Categories


7.3.3 Testing the Equivalence of Two Regressions


(7.16)

1 2 1 2

1 2 3

4 5

WAGE EDUC BLACK FEMALE BLACK FEMALE

SOUTH EDUC SOUTH BLACK SOUTH

FEMALE SOUTH BLACK FEMALE SOUTH e

1 2 1 2

1 1 2 2 1 3

2 4 5

0

( ) ( ) ( ) 1

( ) ( )

EDUC BLACK FEMALESOUTH

BLACK FEMALE

E WAGE

EDUC BLACK SOUTH

FEMALE BLACK FEMALE





0 1 2 3 4 5: 0H

( ) / (29307.7 29012.7) / 52.0132

/ ( ) 29012.7 / 990R U

U

SSE SSE JF

SSE N K



Remark: The usual F-test of a joint hypothesis relies on the

assumptions MR1-MR6 of the linear regression model. Of particular

relevance for testing the equivalence of two regressions is assumption

MR3, that the variance of the error term is the same for all observations.

If we are considering possibly different slopes and intercepts for parts of

the data, it might also be true that the error variances are different in the

two parts of the data. In such a case the usual F-test is not valid.

7.3.4 Controlling for Time

7.3.4a Seasonal Dummies

7.3.4b Annual Dummies

7.3.4c Regime Effects


1 1962 1965,1970 1986

0ITC

otherwise

1 2 3 1t t t t tINV ITC GNP GNP e





(7.17)1 2 3PIZZA AGE INCOME e

1 2 3PIZZA AGE INCOME e

3( )E PIZZA INCOME

342.88 7.58 .0024

( ) ( 3.27) (3.95)

PIZZA AGE INCOME

t



(7.18)1 2 3 4 ( )PIZZA AGE INCOME AGE INCOME e

2 4( )E PIZZA AGE INCOME

3 4( )E PIZZA INCOME AGE

161.47 2.98 .009 .00016( )

( ) ( .89) (2.47) ( 1.85)

PIZZA AGE INCOME AGE INCOME

t



2 4

( )

2.98 .00016

6.98 for $25,000

17.40 for $90,000

E PIZZAb b INCOME

AGE

INCOME

INCOME

INCOME

7.5 Log-Linear models

7.5.1 Dummy Variables


(7.19)1 2ln( )WAGE EDUC FEMALE

1 2

1 2

ln( )EDUC MALES

WAGEEDUC FEMALES

7.5.1a A Rough Calculation


ln( ) ln( ) lnFEMALES MALESWAGE WAGE WAGE

ln( ) .9290 .1026 .2526

(se) (.0837) (.0061) (.0300)

WAGE EDUC FEMALE

7.5.1b An Exact Calculation


ln( ) ln( ) ln FEMALESFEMALES MALES

MALES

WAGEWAGE WAGE

WAGE

FEMALES

MALES

WAGEe

WAGE

1FEMALES MALES FEMALES MALES

MALES MALES MALES

WAGE WAGE WAGE WAGEe

WAGE WAGE WAGE

ˆ .2526100 1 % 100 1 % 22.32%e e

7.5.2 Interaction and Quadratic Terms


(7.20)1 2 3ln( ) ( )WAGE EDUC EXPER EDUC EXPER

3

ln( )

EDUC fixed

WAGEEDUC

EXPER

ln( ) .1528 +.1341 .0249 .000962( )

(se) (.1722) (.0127) (.0071) (.00054)

WAGE EDUC EXPER EDUC EXPER

7.5.2 Interaction and Quadratic Terms


21 2 3 4ln( )WAGE EDUC EXPER EXPER EDUC EXPER

3 4% 100 2 %WAGE EXPER EDUC

Keywords


annual dummy variables binary variable Chow test collinearity dichotomous variable dummy variable dummy variable trap exact collinearity hedonic model interaction variable intercept dummy variable log-linear models nonlinear relationship polynomial

reference group regional dummy variable seasonal dummy variables slope dummy variable

Chapter 7 Appendix


Appendix 7 Details of log-linear model

interpretation

Appendix 7 Details of log-linear model interpretation


2 21 2 1 2( ) exp( 2) exp exp 2E y x x

21 2( ) exp exp 2E y x D



1 0

0

2 21 2 1 2

21 2

1 2 1 2

1 2

% ( ) 100 %

exp exp 2 exp exp 2100 %

exp exp 2

exp exp exp100 %

exp

100 exp 1 %

E y E yE y

E y

x x

x

x x

x



21 2 3( ) exp ( ) exp 2E y x z xz

21 2 3 3

( )exp ( ) exp 2 )

E yx z xz x

z

3

( ) ( )100 100 %

E y E yx

z