Nonlinear Relationships
Prepared by Vera Tabakova, East Carolina University

7.1 Polynomials

7.2 Dummy Variables

7.3 Applying Dummy Variables

7.4 Interactions Between Continuous Variables

7.5 Log-Linear Models
Principles of Econometrics, 3rd Edition
Slide 7-2
7.1.1 Cost and Product Curves
Figure 7.1 (a) Total cost curve and (b) total product curve
Principles of Econometrics, 3rd Edition
Slide 7-3
Figure 7.2 Average and marginal (a) cost curves and (b) product curves
Principles of Econometrics, 3rd Edition
Slide 7-4
AC  1 2Q 3Q2  e
(7.1)
TC  1  2Q  3Q2  4Q3  e
(7.2)
dE ( AC )
 2  23Q
dQ
(7.3)
dE (TC )
  2  2 3Q  3 4Q 2
dQ
(7.4)
Principles of Econometrics, 3rd Edition
Slide 7-5
WAGE  1 2 EDUC 3 EXPER+4EXPER2  e
E (WAGE )
 3  24 EXPER
EXPER
(7.5)
(7.6)
If Deriv < 0 => Exper < -β3/2β4 ,
or Exper > β3/2β4 .
Principles of Econometrics, 3rd Edition
Slide 7-6
Principles of Econometrics, 3rd Edition
Slide 7-7
E (WAGE )
EXPER
 .3409  2(.0051)18  .1576
EXPER 18
EXPER  3 24  .3409 / 2(.0051)  33.47
Where 33.47 is the ‘turning point’
Principles of Econometrics, 3rd Edition
Slide 7-8
PRICE  1  2 SQFT  e
(7.7)
1 if characteristic is present
D
0 if characteristic is not present
(7.8)
1 if property is in the desirable neighborhood
D
0 if property is not in the desirable neighborhood
Principles of Econometrics, 3rd Edition
Slide 7-9
PRICE  1  D  2 SQFT  e
(1  )  2 SQFT
E ( PRICE )  
 1  2 SQFT
Principles of Econometrics, 3rd Edition
when D  1
when D  0
(7.9)
(7.10)
Slide 7-10
Figure 7.3 An intercept dummy variable
Principles of Econometrics, 3rd Edition
Slide 7-11
1 if property is not in the desirable neighborhood
LD  
0 if property is in the desirable neighborhood
PRICE  1  LD  2 SQFT  e
PRICE  1  D  LD  2 SQFT  e
Principles of Econometrics, 3rd Edition
Slide 7-12
PRICE  1  2 SQFT   ( SQFT  D)  e
1  (2   ) SQFT
E ( PRICE )  1  2 SQFT    SQFT  D   
 1  2 SQFT
E ( PRICE )  2  

SQFT
2
Principles of Econometrics, 3rd Edition
(7.11)
when D  1
when D  0
when D  1
when D  0
Slide 7-13
Figure 7.4 (a) A slope dummy variable. (b) A slope and intercept dummy variable
Principles of Econometrics, 3rd Edition
Slide 7-14
PRICE  1  D  2 SQFT   ( SQFT  D)  e
(1  )  (2   ) SQFT
E ( PRICE )  
 1  2 SQFT
Principles of Econometrics, 3rd Edition
(7.12)
when D  1
when D  0
Slide 7-15
Principles of Econometrics, 3rd Edition
Slide 7-16
PRICE  1  1UTOWN  2 SQFT    SQFT  UTOWN 
+ 3 AGE   2 POOL  3 FPLACE  e
Principles of Econometrics, 3rd Edition
(7.13)
Slide 7-17
Principles of Econometrics, 3rd Edition
Slide 7-18
PRICE  (24.5  27.453)  (7.6122  1.2994) SQFT  .1901AGE
 4.3772 POOL  1.6492 FPLACE
 51.953+8.9116SQFT  .1901AGE  4.3772 POOL  1.6492 FPLACE
PRICE  24.5  7.6122SQFT  .1901AGE  4.3772POOL  1.6492FPLACE
Principles of Econometrics, 3rd Edition
Slide 7-19
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
Principles of Econometrics, 3rd Edition
Slide 7-20

7.3.1 Interactions Between Qualitative Factors
WAGE  1  2 EDUC  1BLACK  2 FEMALE    BLACK  FEMALE   e
(7.14)
WHITE  MALE
1  2 EDUC
      EDUC
BLACK  MALE
 1 1  2
E (WAGE )  
WHITE  FEMALE
 1  2   2 EDUC

 1  1  2     2 EDUC BLACK  FEMALE
Principles of Econometrics, 3rd Edition
Slide 7-21
Principles of Econometrics, 3rd Edition
Slide 7-22
(SSER  SSEU ) / J
F
SSEU / ( N  K )
WAGE  4.9122  1.1385EDUC
(se)
(.9668) (.0716)
(SSER  SSEU ) / J (31093  29308) / 3
F

 20.20
SSEU / ( N  K )
29308 / 995
Principles of Econometrics, 3rd Edition
Slide 7-23
WAGE  1  2 EDUC + 1SOUTH  2 MIDWEST  3WEST  e
(7.15)
 1  3   2 EDUC WEST

 1  2   2 EDUC MIDWEST
E (WAGE )  
 1  1   2 EDUC SOUTH
1  2 EDUC
NORTHEAST
Principles of Econometrics, 3rd Edition
Slide 7-24
Principles of Econometrics, 3rd Edition
Slide 7-25
WAGE  1  2 EDUC  1BLACK  2 FEMALE    BLACK  FEMALE  
1SOUTH  2  EDUC  SOUTH   3  BLACK  SOUTH  
(7.16)
4  FEMALE  SOUTH   5  BLACK  FEMALE  SOUTH   e
1  2 EDUC  1BLACK   2 FEMALE 
SOUTH  0


BLACK

FEMALE




E WAGE   
(   )  (   ) EDUC  (   ) BLACK 
SOUTH  1
2
2
1
3
 1 1
(2  4 ) FEMALE  (   5 )  BLACK  FEMALE 
Principles of Econometrics, 3rd Edition
Slide 7-26
Principles of Econometrics, 3rd Edition
Slide 7-27
H 0 : 1  2  3  4  5  0
(SSER  SSEU ) / J (29307.7  29012.7) / 5
F

 2.0132
SSEU / ( N  K )
29012.7 / 990
Principles of Econometrics, 3rd Edition
Slide 7-28
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.
Principles of Econometrics, 3rd Edition
Slide 7-29
7.3.4a Seasonal Dummies
7.3.4b Annual Dummies
7.3.4c Regime Effects
1 1962  1965,1970  1986
ITC  
0 otherwise
INVt  1  ITCt  2GNPt  3GNPt 1  et
Principles of Econometrics, 3rd Edition
Slide 7-30
Principles of Econometrics, 3rd Edition
Slide 7-31
PIZZA  1  2 AGE  3 INCOME  e
(7.17)
PIZZA  1  2 AGE  3 INCOME  e
E ( PIZZA) INCOME  3
PIZZA  342.88  7.58 AGE  .0024INCOME
(t )
(  3.27)
(3.95)
Principles of Econometrics, 3rd Edition
Slide 7-32
PIZZA  1  2 AGE  3 INCOME  4 ( AGE  INCOME )  e (7.18)
E ( PIZZA) AGE  2  4 INCOME
E ( PIZZA) INCOME  3  4 AGE
PIZZA  161.47  2.98 AGE  .009INCOME  .00016( AGE  INCOME )
(t )
(  .89)
(2.47)
(  1.85)
Principles of Econometrics, 3rd Edition
Slide 7-33
E ( PIZZA)
 b2  b4 INCOME
AGE
 2.98  .00016 INCOME
  6.98 for INCOME  $25,000

17.40 for INCOME  $90,000
Principles of Econometrics, 3rd Edition
Slide 7-34
7.5.1 Dummy Variables
ln(WAGE )  1  2 EDUC  FEMALE
(7.19)
MALES
1  2 EDUC
ln(WAGE )  
1     2 EDUC FEMALES
Principles of Econometrics, 3rd Edition
Slide 7-35
ln(WAGE ) FEMALES  ln(WAGE )MALES   ln WAGE   
ln(WAGE )  .9290  .1026 EDUC  .2526 FEMALE
(se)
(.0837) (.0061)
(.0300)
Principles of Econometrics, 3rd Edition
Slide 6-36
ln(WAGE ) FEMALES  ln(WAGE )MALES
 WAGEFEMALES
 ln 
 WAGEMALES



WAGEFEMALES
 e
WAGEMALES
WAGEFEMALES WAGEMALES WAGEFEMALES  WAGEMALES


 e  1
WAGEMALES WAGEMALES
WAGEMALES


100 e  1 %  100  e.2526  1 %  22.32%
ˆ
Principles of Econometrics, 3rd Edition
Slide 7-37
ln(WAGE )  1  2 EDUC  3 EXPER   ( EDUC  EXPER) (7.20)
 ln(WAGE )
 3  EDUC
EXPER EDUC fixed
ln(WAGE )  .1528 +.1341EDUC  .0249EXPER  .000962(EDUC  EXPER)
(se)
(.1722) (.0127)
(.0071)
(.00054)
Principles of Econometrics, 3rd Edition
Slide 7-38
ln(WAGE )  1  2 EDUC  3 EXPER  4 EXPER 2    EDUC  EXPER 
%WAGE  100 3  24 EXPER  EDUC  %
Principles of Econometrics, 3rd Edition
Slide 7-39














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
Principles of Econometrics, 3rd Edition




reference group
regional dummy variable
seasonal dummy variables
slope dummy variable
Slide 7-40
Principles of Econometrics, 3rd Edition
Slide 7-41
E ( y)  exp(1  2 x  2 2)  exp 1  2 x   exp  2 2 
E ( y)  exp 1  2 x  D   exp  2 2 
Principles of Econometrics, 3rd Edition
Slide 7-42
 E  y1   E  y0  
%E ( y )  100 
%
E
y
 0 


 

 exp  1  2 x     exp   2 2   exp  1  2 x   exp   2 2  
%
 100 
2


exp  1  2 x   exp   2 




 exp  1  2 x  exp     exp  1  2 x  
 100 
%
exp



x
 1 2 


 100 exp     1 %
Principles of Econometrics, 3rd Edition
Slide 7-43
E ( y)  exp 1  2 x  3 z   ( xz )   exp  2 2 
E ( y)
 exp 1  2 x  3 z   ( xz )   exp  2 2   3  x) 
z
 E ( y) E ( y) 
100 
 100 3  x  %

z


Principles of Econometrics, 3rd Edition
Slide 7-44