1Spring 02

Non-Linear Relationships

If a relationship is not linear, how can we deal with it.

For example:

PyM

or

XXY

d

32321

2Spring 02

Non-Linear Relationships

One possibility is to take the natural logarithm of both sides.

The natural logarithm is the inverse of the natural exponential function. Natural exponent:

YYX

eY

nne

e

X

n

lnlog

71828.2)1

1(lim

3Spring 02

Graph of ex

-5

0

5

10

15

20

25

-15 -10 -5 0 5

4Spring 02

Graph of ln x

0

2

4

6

8

0 20 40 60 80 100

5Spring 02

Non-Linear Relationships

Natural logs and exponential functions: e0 =1 ln 1 = 0

Logarithms have certain properties: ln (XY) = ln X+ ln Y ln(X/Y) = ln X – ln Y

ln(1/X) = ln 1 – ln X = - ln X ln ax = x lnA

6Spring 02

Non-Linear Relationships

*33

*22

*1

33221

33221

321

*

)(ln)(ln)(ln)(ln

lnlnlnln

32

XXY

XXY

XXY

XXY

7Spring 02

Non-Linear Relationships

)(ln)(ln)(ln)(ln

lnlnlnln

YPM

YPM

PYM

d

d

d

8Spring 02

Elasticities

When you take the natural logs of a non-linear relationship and estimate the equation, the estimated coefficients are also the elasticities.

vex

xv

yu

xfy

ln

ln

)(

yx

v

Edx

dy

y

x

xdx

dy

ye

dx

dy

y

dv

dx

dx

dy

y

dv

dx

dx

dy

dy

du

dv

du

xd

yd

,

11

1

)(ln

)(ln

9Spring 02

Dummy Variables

Used to quantify qualitative differences.

For example, Consumption as a function of income may be

affected by wartime/peacetime Income as a function of education may be affected

by gender Quantity demanded of ice cream as a function of

price may be affected by season.

10Spring 02

Dummy Variables

How does consumption differ in wartime from peacetime?

Yd

C Cp

Cw

C Cp

Yd

Cw

C Cp

Yd

Cw

11Spring 02

Dummy Variables

)(

)(

4321

321

321

21

dd

dd

d

d

DYDYC

DYYC

DYC

YC

12Spring 02

Example

Salvatore: Below is the quantity supplied of milk by a dairy for 14 months but for 3 of those months there was a strike

Month 1 2 3 4 5 6 7 8 9 10 11 12 13 14QS 98 100 103 105 80 87 94 113 116 118 121 123 126 128P $0.79 $0.80 $0.82 $0.82 $0.93 $0.95 $0.96 $0.88 $0.88 $0.90 $0.90 $0.94 $0.96 $0.97Strike 0 0 0 0 1 1 1 0 0 0 0 0 0 0

13Spring 02

Example

If we estimate quantity supplied as a function of price without taking into consideration strike, then we would get these results:

If we estimate quantity supplied as a function of price taking into consideration strike (just affecting the constant), then we would get these results:

If strike: If no strike:

PQ s 5.509.62ˆ (1.05) (0.76)

DPQ s 3.381609.34ˆ (-3.26) (13.9) (-21.2)

PQ s 1602.73ˆ

PQ s 1609.34ˆ

14Spring 02

Example

If we estimate quantity supplied as a function of price taking into consideration strike (just affecting slope), then we would get these results:

If strike: If no strike:

If we estimate quantity supplied as a function of price taking into consideration strike (affecting both the constant and slope), then we would get these results:

If strike: If no strike:

)(*0.284*9.306*0.166*1.32ˆ DYDPQ s

)(*4.40*0.169*1.35ˆ DYPQ s PQ s 6.1281.35ˆ

PQ s 0.1661.32ˆ PQ s 4050.339ˆ

PQ s 6.1281.35ˆ

15Spring 02

Wald Test Revisited

:

0...:

...

......

210

221

11221

a

kmm

mm

kkmmmm

H

H

uXXY

XXXXY

At least one of the above betas is not zero.

)(

)()(

knESS

mkESSESS

FU

UR

16Spring 02

Example

We can test whether the explanatory variables including strike as a set are significant in explaining price by using a Wald Test:

46796.2

1382

10/6.292

)6.292793(

)(

)()(

knESS

mkESSESS

FU

UR

17Spring 02

Test for Structural Change

Data 7-19 contains data from 1960-1988 on the demand for cigarettes in Turkey and its determinants. Estimate the whole equation and also estimate whether two anti-smoking campaigns had their desired effect. In late 1981, health warning were issued in Turkey

regarding the hazards of cigarette smoking. In 1986, one of the national newspapers launched an

antismoking campaign.

18Spring 02

Smoking Problem

Coefficientsa

-4.585 .752 -6.094 .000

-.484 .106 -1.055 -4.581 .000

.688 .098 1.610 6.990 .000

(Constant)

LNP

LNY

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: LNQa.

Coefficientsa

1.656 .126 13.178 .000

3.423E-04 .000 1.387 6.326 .000

-.419 .100 -.922 -4.204 .000

(Constant)

Y

P

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: Qa.

On levels

On natural logs

19Spring 02

Smoking Problem

Coefficientsa

-4.186 .535 -7.825 .000

-.201 .090 -.440 -2.248 .034

.621 .070 1.452 8.804 .000

-.103 .026 -.413 -3.931 .001

-.103 .037 -.295 -2.802 .010

(Constant)

LNP

LNY

D82

D86

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: LNQa.

Regression with dummy variables affecting constant for after 1982 and 1986:

20Spring 02

Smoking Problem

Coefficientsa

-4.800 .677 -7.095 .000

-.337 .129 -.735 -2.608 .016

.705 .091 1.651 7.761 .000

-.108 .207 -.433 -.521 .608

-.406 .236 -1.161 -1.725 .099

1.637E-02 .246 .066 .067 .947

.288 .250 .944 1.151 .262

(Constant)

LNP

LNY

D82

D86

D82P

D86P

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: LNQa.

Regression with dummy variables affecting constant and slope for after 1982 and 1986:

21Spring 02

Smoking Problem

Coefficientsa

-4.824 .677 -7.127 .000

-.342 .129 -.745 -2.644 .015

.708 .091 1.658 7.793 .000

-1.50E-02 .025 -.516 -.608 .549

3.961E-02 .249 .159 .159 .875

-4.58E-02 .028 -1.129 -1.643 .115

.275 .253 .902 1.087 .289

(Constant)

LNP

LNY

D82Y

D82P

D86Y

D86P

Model1

B Std. Error

UnstandardizedCoefficients

Beta

Standardized

Coefficients

t Sig.

Dependent Variable: LNQa.

Regression with dummy variables affecting just the slope for after 1982 and 1986:

22Spring 02

Coefficients of Different Regressions

To test whether the assumptions of two different regressions is correct, we start with the null hypothesis that the regressions are identical and see whether or not we can reject the null hypothesis. Test whether the stock market has changed the

relationship between consumption and wealth. Test whether the relationship between years of

education and income is different for women and men or for different regions of the country.

23Spring 02

Coefficients of Different Regressions

jkjkjjj

ikikiii

XXXY

XXXY

...

...

33221

33221

To test the null hypothesis:

kk ,...,, 2211

Run a regression on the whole model, N+M observations. Then run two separate regressions.

Ni ,1

Mj ,1

kMNESS

kESSESS

FU

UR

kMNk

2

)(

2,