7.1 Lecture #7 Studenmund(2006) Chapter 7 Objective: Applications of Dummy Independent Variables.

7.1

All right reserved by Dr.Bill Wan Sing Hung - HKBU

Lecture #7

Studenmund(2006) Chapter 7

Objective:

Applications of Dummy Independent Variables

7.2


Qualitative information

Gender: male and female

Regional: HK Island, Kowloon & NT

Zone: East, South, West, North, Center

Time/period: peace and war, before & after crisis

Age: young, middle, elder

Education: Post-graduate, College, High, Element

Others:

7.3


obs Male Dummy Female Dummy Salary(K) Years ofteaching

1 1 0 23 12 0 1 19.5 13 1 0 24 24 0 1 21 25 1 0 25 36 0 1 22 37 1 0 26.5 48 0 1 23.1 49 0 1 25 5

10 1 0 28 511 1 0 29.5 612 0 1 26 613 0 1 27.5 714 1 0 31.5 715 0 1 29 616 1 0 22 517 0 1 19 218 1 0 18 219 0 1 21.7 520 0 1 18.5 221 1 0 21 422 1 0 20.5 423 0 1 17 124 0 1 17.5 125 1 0 21.2 5

Example: Gender issue of whether discrimination is existing for salary

7.4


Separate sample of male

Male Sample: (Gujarati-1995, Table 15.1 & 15.5)

obs Starting salary, Y Years of teaching, X21 23 13 24 25 25 37 26.5 4

10 28 511 29.5 614 31.5 716 22 518 21.7 521 21 422 20.5 425 21.2 5

Total # obs: 1212

7.5


obs Staring salary, Y Years of teaching, X22 19.5 14 21 26 22 38 23.1 49 25 5

12 26 613 27.5 715 29 6

17 19 219 18 220 18.5 223 17 124 17.5 1

Femalesample: (Gujarati-1995, Table15.1 & 15.5)

Separate sample of female

Total # obs: 1313

7.6


10

15

20

25

30

35

0 1 2 3 4 5 6 7 8

Male

Linear (Male)

Salary Y

X teaching years

Y = 0 + 1 X (male)^ ^ ^

Linear (Female)

Female

Y = *0+ 2X (female)^ ^ ^

Two separate models: Yi = 0 + 1 Xi + i

Yj = *0 + 2 Xj + j

(male)

(female)

7.7


Assuming 1 = 2, same slope but different constant between Y and X.

1st model: Yi = 0 + ’0 Di + 1 Xi + i

Yi = 0 + ’0 Di + 1 Xi + ’1 DiXi + i

Yi = annual salary

Xi = years of teaching experience

Di = 1 if male

= 0 otherwise (female)control variable

Assuming 1 2, different slope and different constant between Y and X.

2nd model:

7.8


Salary Y

X teaching years

Y = *0+ 2X (female)^ ^ ^

Y = 0 + 1X (male)^ ^ ^

0 1 2 3 4 5 6 7 8

MaleFemaleLinear (Male)Linear (Female)

15

20

25

30

35

10

Y = *0 + * 1X (whole)^ ^ ^

Two separate models: Yi = 0 + 1 Xi + i

Yj = *0 + 2 Xj + j

(male)

(female)

7.9


D1 + D2 = 1

D1 = 1 - D2

male femaleannualSalary

years ofteaching

obs D2 D1 Y X1 0 1 23 1

2 1 0 19.5 1

3 0 1 24 2

4 1 0 21 2

5 0 1 25 3

6 1 0 22 3

7 0 1 26.5 4

8 1 0 23.1 4

9 1 0 25 5

10 0 1 28 5

11 0 1 29.5 6

12 1 0 26 6

13 1 0 27.5 7

14 0 1 31.5 7

15 1 0 29 6

16 0 1 22 5

17 1 0 19 2

18 0 1 18 2

19 1 0 21.7 5

20 1 0 18.5 2

21 0 1 21 4

22 0 1 20.5 4

23 1 0 17 1

24 1 0 17.5 1

25 0 1 21.2 5

Each dummy identify two

different categories, but when

sum up two dummiesit cannotidentifywhich is

male or female

7.10


(Dummy variable trap)

If we introduce two dummy variables in one model to identify two categories of one qualitative variable such as

Yi = 0 + ’0 D1i + ’’0 D2i + 1 Xi + i

where D1i = 1 if male = 0 otherwise

where D2i = 1 if female = 0 otherwise

This model cannot be estimated because of perfect collinearity between D1 and D2

D1 = 1 - D2or D2 = 1 - D1or D1 + D2 = 1 ( Perfect collinearity )

7.11


Use two dummy variables to identify two different qualitative categories in one model will be fall into the trap of perfect multicollinearity.

General rule : To avoid the perfect multicollinearity

If a qualitative variable has “m” categories, introduce only “m-1” dummy variables.

1

D1 D2 D3 D4 D5 … Dm-1

age1 10 20 30 40 m

Categories dummy =>

Qualitative variable

7.12


Measure the estimated result for two groups:

Male: ==> Yi = (0 + ’0 Di)+ 1Xi Di = 1 ^ ^ ^ ^

Female: ==> Yi = 0 + 1Xi Di = 0^ ^ ^

Now consider different intercepts of two groups:

Model: Yi = 0 + ’0 Di + 1Xi + i

Di = 1 if male

= 0 otherwise, (i.e. female)

When a category is assigned the value of zero, this category is called a control category (or omitted group).

2

7.13


In order to test whether there is any difference in the relationships between two categories

Compare: Yi = 0 + 1Xi

^ ^ ^

Yi = (0 + ’0 D)+ 1 Xi

^ ^ ^ ^

If t-statistics is significant in ’0, there is different in constant term.

=>same 1 means two categories of X have the same relationship with Y

^

^

Check the t-value

7.14


H0 : ’0 = 0

H1 : ’0 > 0 or H1 : ’0 0

Appropriate test is the t-test on ’0^

Compare tc and t*, N-K

2

If t* > tc ==> reject H0 : ’0 = 0

Y = 0 + ’0 D+ 1 Xi + ’1DX^ ^ ^ ^^

Check t-statistics

=

This part is testing the

difference of intercept

This part is testingThe difference of slope in two categoriesCheck

t-statistics

=

7.15


Separate Examples for female and male:

Female Male

The two regression results performed differently in slope and intercept. But are they really statistically different?We cannot answer from these two separate regression resultsunless you test with the F*.

7.16


Yi = (0 + ’0 D)+ Xi

^ ^ ^ ^

= (17.937-1.2810) + 1.561X

D1:Female =1others = 0

D2:Male =1others = 0

= (16.656+1.2810) + 1.561X

Yi = (0 + ’0 D)+ Xi

^ ^ ^ ^

=17.937=16.656 If the dummy were significant

Set two different dummies for the Example

7.17


Yi = 0 + 1Xi

^ ^ ^

= 17.095+1.608X

Whole Sample

7.18


D1: Female =1

Male: Y = 0 + 1 Xi ^ ^ = 18.689 + 1.373 X

Female: Y = (0 + ’0 D)+ (1 + ’1D)X^ ^ ^ ^= 16.255 +1.677 X

=0 =0= 18.689 + 1.373 X

7.19


D2: Male =1

Female: Y = 0 + 1 Xi ^^ =16.255 + 1.677 X

Male: Y = (0 + ’0 D)+(1 + ’1D)X^ ^ ^ ^ =18.689 + 1.373 X=0 =0

=16.255 + 1.677 X

7.20


One qualitative variable with more than two categories

(Health care) = 0 + ’0 D2 + ’’0 D3 + Income + (Y) (X)

D2 = 1 if high school education = 0 otherwise

D3 = 1 if college education = 0 otherwise

2

7.21


Health care

income

Less than high school education

Y = 0 + X^ ^ ^

0^

High school education

Y = (0 + ’0 D2)+ X^ ^ ^ ^

D2 = 1

’0^

D3 = 1College education

Y = (0 + 0 D’’3)+ X^ ^ ^ ^

’’0^

7.22


D2 = 1 High school = 0 otherwise

D3 = 1 College = 0 otherwise

========================================= obs Y X D2 D3=========================================

1 6.000000 40.00000 0.000000 1.0000002 3.900000 31.00000 1.000000 0.0000003 1.800000 18.00000 0.000000 0.0000004 1.900000 19.00000 0.000000 0.0000005 7.200000 47.00000 0.000000 1.0000006 3.300000 27.00000 1.000000 0.0000007 3.100000 26.00000 1.000000 0.0000008 1.700000 17.00000 0.000000 0.0000009 6.400000 43.00000 0.000000 1.000000

10 7.900000 49.00000 0.000000 1.00000011 1.500000 15.00000 0.000000 0.00000012 3.100000 25.00000 1.000000 0.00000013 3.600000 29.00000 1.000000 0.00000014 2.000000 20.00000 0.000000 0.00000015 6.200000 41.00000 0.000000 1.000000

=========================================

7.23


7.24


Less than high school: Yi = -1.2859 + 0.1722 Xi^

Yi = (-1.2859 - 0.068 ) + 0.1722 Xi^

= -1.3539 + 0.1722 X

High school:

When t-value of D2 is statistically significant

Yi = (-1.2859 + 0.447 ) + 0.1722 Xi^

= -0.8389 + 0.1722 Xi

College:

When t-value of D3 is statistically significant

= -1.2859 + 0.1722 X

= -1.2859 + 0.1722 X

When t-value is not statistically significant

7.25


One Qualitative variable with many categories :Example : An estimate model on three different

age’s medical care expenditure

Yi = 0 + ’0 A1 + ’’0 A2 + Xi + i

(t-value) (t-value)

where A1 = 1 if 55 > age > 25 = 0 otherwise

A2 = 1 if age > 55 = 0 otherwise

A1 + A2 1

A2 =1A1 =10

25 55

7.26


Qualitative variable with many categories :(Cont.)then the estimated models are :

age below 25 Y = 0 + X^ ^ ^

Y = (0 + ’0A1)+ X^ ^ ^ ^25 < age < 55

age > 55 Y = (0 + ’’0A2)+ X^ ^ ^ ^

H0 : ’0 = 0, ’’0 = 0 t1*

H1 : ’0 0, ’’0 0 t2*

Compare to tcp, n-k

7.27


’0^

25 < age < 55Y = ( 0 + ’0)+ X^ ^ ^ ^

^

age > 55

Y = ( 0+ ’’0)+ X^ ^ ^ ^

’’0

In scatter diagram :

0^

Y

X

age < 25

Y = ( 0 ) + X^ ^ ^

7.28


One Qualitative variable with many categories :

Example : An estimate model on four different age’s medical care expenditure

Y = 0 + ’0 A1 + ’’0 A2 + ’’’0 A3 + 1 X +

where A1 = 1 if age > 55

= 0 otherwiseA2 = 1 if 35 < age 55

= 0 otherwiseA3 = 1 if 15 < age 35

= 0 otherwise

7.29


Qualitative variable with many categories :(Cont.)

The estimated models are : The estimated models are :

age age 15 15 Y = Y = 00 + + 11 X X^̂ ^̂ ^̂

15 < age 15 < age 35 35 Y = (Y = (00 + + ’’00AA33) + ) + 11 X X^̂ ^̂ ^̂ ^̂

35 < age 35 < age 55 55 Y = (Y = (00 + + ’’’’00AA22)+ )+ 11 X X^̂ ^̂ ^̂ ^̂

age > 55age > 55 Y = (Y = (00 + + ’’’’’’00AA11)+ )+ 11 X X^̂ ^̂ ^̂ ^̂

7.30


Two qualitative variables

(Y) Salary = 0 + ’0D1 + ’’0 D2 + 1 X +

or Y = 0+’0D1+ ’’0D2 + 1 X + ’1D1*X + ’’1D2*X + ’

D1 = 1 if male = 0 otherwise

sex

D2 = 1 if white = 0 otherwise

race

(1) Mean salary for “black” female teacher:

Y = 0 + 1 X that are D1 = 0, D2 = 0^ ^ ^

(2) Mean salary for “black” male teacher:

Y = (0 + ’0 D1) + (1+ 1D1)X that are D1 = 1, D2 = 0^ ^ ^ ^ ^

7.31


(3) Mean salary for “white” female teacher:

Y = (0 + ’’0 D2) + 1 X + 1D2X that are D1 = 0, D2 = 1^ ^ ^

(4) Mean salary for “white” male teacher:

^

Y = (0 + ’0 D0 +’’0D2)+ (1+ ’1D1 + ’’1D2 )X

that are D1 = 1, D2 = 1

^ ^ ^ ^ ^ ^ ^

7.32


D = 1 if 1946-1954 = 0 otherwise (1955-1963)

1. Identical regression:

Y = 0 + 1 X + ’0D + ’1D*X

H0 : ’0 = 0 and ’1 = 0

2. Parallel regression:

Y = 0 + 1 X + ’0 D + ’1D*X

H0 : ’1 = 0

4. Dissimilar regression:

Y = 0 + 1 X + ’0D + ’1D*X

H0 : ’0 0 and ’1 0

3. Concurrent regression:

Y = 0 + 1 X + ’0 D + ’1D*X

H0 : ’0 = 0

Different types of dummy regression:

7.33


Reconstruction (46-54): Yt = A0 + A1 Xt + 1t

Pastreconstruction (55-63): Yt = B0 + B1 Xt +2t

Y

X

A0 = B0

1

A1 = B1

Identical regressions

Y

XA0

1A1

Parallel regressions

A0 B0, A1 = B1

B1

1

B0

7.34


Y

X

A0 = B0

1

B1

Concurrent regressions

A1

1

A0 = B0, A1 B1

Y

X

A0

1

A1

dissimilar regressions

A0 B0, A1 B1

B0

1

B1

7.35


Interactive effects between the two qualitative variables

Spending(Y) = 0 + ’0 D1 + ’’0 D2 + 1 income(X) +

D1 = 1 if female = 0 otherwise

sex

D2 = 1 if college graduate = 0 otherwise

education

Spending(Y) = 0 + ’0D1 + ’’0D2 + ’’’0D1*D2 + 1income(X) +

Interaction effect:

’0 = different effect of being a female’’0= different effect of being a college graduate’’’0 = different effect of being a female with college graduate

7.36


Concurrent model (or Covariance, or Slope shift model)

Example : how can we test the hypothesis that the gasoline spending is different between a new car and a used car ?

Let us assume that at the begin mile, there is no different between used car and new car.

gas spending

miles running

Y

X

0^

* * *

* *

*

* * *

* *

New car Y = 0 + 1 X^ ^ ^

o

o

o

o

o

o

o

o

o

o used car Y = 0+ 1X^

Y = 0+ (1 + ’1)X^ ^ ^ ^

^ ^

7.37


The estimated relations are :

used car : Yi = 0 + (1 + ’1D) Xi where D = 1

^ ^ ^ ^

new car : Yi = 0 + 1 Xi^ ^ ^

==

Yi = 0 + 1 Xi== ^ ^ ^or

If ’1 0, means the estimated slopes for cars is different.^

Let 1= 1 + ’1 D where D = 1 if used car = 0 otherwiseNow in one model :

multiplicative dummy variable

Yi = 0 + (1 + ’1 D) Xi +i

= 0 + 1 Xi + ’1 D*Xi +i

= 0 + 1 Xi + ’1 Zi +i

7.38


Test whether ’1 = 0 or not ?^

(i) Compare : (a) Y = 0 + 1 X^ ^ ^

(b) Y = 0 + 1X^ ^ ^Two separate

models

(ii) use t-test on ’1:Y = 0 +1 Xi + ’1 Z^^ ^ ^ ^

compare tcP, N-3 and t*

H0 : ’1 = 0^

H1 : ’1 > 0^ If t* > tcP, N-3

=> reject H0or (’1 0)

7.39


Check the t-value

…...

…...

…...

…...

…...

Y = 0 + 1 Xi + ’1 Zi

obs Yi Xi Di (Di Xi) = Zi

1 210 100 0 0

2 250 110 1 1103 340 150 1 150

4 305 120 1 120

^ ^ ^ ^

7.40


Shifts in both intercept and slope

Example: Estimating Seasonal effects :

E = 0 + 1 T +

E : electricity consumptionT : temperature

To capture effect of seasonal factors

E = 0 + ’0D1 +’’0D2 + ’’’0 D3 + 1T +

where D1 = 1 if winter 0 otherwise

D2 = 1 if spring 0 otherwise

D3 = 1 if summer 0 otherwise

spring summer fall writerQ1 Q2 Q3 Q4

Control group

7.41


The estimated models :

Fall E = 0 + 1 T ^ ^ ^

Spring E = (0 + ’’0)+(1 + ’’1) T^ ^ ^ ^ ^

Winter E = (0 + ’0)+ (1 + ’1) T^ ^ ^ ^ ^

Summer E = (0 ’’’0)+(1 + ’’’1) T^ ^ ^ ^^

0^

T

E

E = 0 + 1T (Fall)^ ^^

E = (0 + ’0)+(1 +’1)T(winter)^ ^ ^ ^ ^

’0^

E = (0 + ’’0)+(1 + ’’1)T (Spring)^^ ^^

’’0^

’’’0

E=(0+ ’’’0)+(1+ ’’’1)T(Summer)^^ ^^^

^

7.42


Estimating Seasonal effects :(Cont.)

Also consider the slope in different seasons

Let = 0 + ’0D1 + ’’0 D2 + ’’’0 D3

Thus, the full general specification is

E = [0+ ’0D1 + ’’0D2+’’’0D3]+1T + ’1D1 T+’’1D2 T

+ ’’’1D3 T + Z1 Z2

Z3

7.43


Quarterly effect is same as seasonal effect

D1 = 1 1st Quarter

= 0 otherwise

D2 = 1 2nd Quarter

= 0 otherwise

D3 = 1 3rd Quarter

= 0 otherwise

Control quarter is the 4th quarter

7.44


1. Set the seasonal dummy = 1 if there is the 1st quarter = 0 otherwise

7.45


E T D1 D2 D3 D41970 :1 1 0 0 0

:2 0 1 0 0:3 0 0 1 0:4 0 0 0 1

1971 :1 1 0 0 0:2 0 1 0 0:3 0 0 1 0:4 0 0 0 1

1972 :1 1 0 0 0:2 0 1 0 0:3 0 0 1 0:4 0 0 0 1

1973 :1 1 0 0 0:2 0 1 0 0:3 0 0 1 0:4 0 0 0 1

Howdoes the quarterlydummyvariable

look like?

7.46


(2) Structural Test based on Dummy variablesBasic model

YT = 0 + 1 XT + T1974

1960 1989

Define a dummy variable : D = 1 for the period 1974 onward

= 0 otherwise

To test whether the structures of two periods are different, the specification must assume that

* = 0 + ’0 D* = 1 + ’1 D

Dummy regression:

YT = 0 + ’0 D + 1 XT + ’1D XT + T

7.47


The Chow test on the Unemployment rate-capacity utilization rate

Dependent Var. Constant CAPt R2 F RSS n_

Sample : 60 - 89

unemplt 30.0 -0.293 0.761 93.6 17.15 30

(12.1) (9.7) RSSR

^

Sample : 60 - 73

unemplt 19.64 -0.175 0.59 19.7 4.69 14

(5.9) (4.4) RSS1

^

Sample : 74 - 89

unemplt 30.63 -0.296 0.871 102.1 3.29 16

(13.1) (10.1) RSS2

^

Note : t-values are in parentheses

7.48


H0: No structural changeNo structural changeH1: YesYes

For the unrestricted model :

RSSu = RSS1 + RSS2 = 4.69 + 3.29

= 7.98

F* = (RSSR - RSSu) / k+1

RSSu / (N - 2k-2)=

(17.15 - 7.98) / 2

7.98 / (30 - 4)= 14.9

F* > Fc ==> reject H0

Fc 0.01, k, T -2k = Fc

0.01 = 5.530.05 0.05, 2, 26 = 3.37

Restriction F-test procedures:

7.49


The unemployment rate - capacity utilization rateSample : 1960 - 1989

Dt = 1 1974 to 1980 = 0 prior to 1974

unempl = 19.6 + 11.0 Dt - 0.175 CAPt - 0.121 (Dt*CAPt)^(6.7) (2.7) (5.0) (2.5)

R2 = 0.88 SEE = 0.554 F = 72.2 n = 30_

The estimated of 1974-1980:

unempl = (19.6+11.0) - (0.175+0.121)CAP

= 30.6 - 0.296 CAP

^

^The estimated of 1960-1973:

unempl = 19.6 - 0.175 CAP

Using the dummy variable to identify the structural change

7.50


D = 1 if t 74 = 0 otherwise

Observed data Year Ut CAPt Dt Dt*CAPt

60 4.20 5.70 0 061 0 062 0 063 0 0… … ...

68 0 069 0 070 0 071 0 072 0 073 0 074 175 176 177 1

... 1 ... 1 ... 1

89 1

……

……

….…

…

……

...….

……

....

10.511.2

10.511.2

Ut = 0 + 1 CAPt + ’0Dt + 2 Dt*CAPt

7.51


(2) Structural stability test based on dummy variables

The estimated models are :

1974 : 1 and onward Y = 0 + 1 X^ ^ ^

Now the basic model becomes

YT = 0 + ’0 D + 1 XT + ’1 D XT + T

YT = 0 + ’0 D + 1 XT + ’1 X*T + T==>

=== ==t t-test on ’1 = 0 ^

1974 : 11950 1995

Prior to 1974 : 1 Y = ( 0 + ’0D)+(1 + ’1D) X^ ^ ^ ^ ^

* *

7.52


GENR DUMMY = 1 (sample 1970 - 1980)GENR DUMMY = 0 (sample 1981 - 1991)

=================================================obs SAVINGS INCOME DUMMY D*INCOME

=================================================1970 57.50000 831.0000 1.000000 831.00001971 65.40000 893.5000 1.000000 893.50001972 59.70000 980.5000 1.000000 980.50001973 86.10000 1098.700 1.000000 1098.7001974 93.40000 1205.700 1.000000 1205.7001975 100.3000 1307.300 1.000000 1307.3001976 93.00000 1446.300 1.000000 1446.3001977 87.90000 1601.300 1.000000 1601.3001978 107.8000 1807.900 1.000000 1807.9001979 123.3000 2033.100 1.000000 2033.1001980 153.8000 2265.400 1.000000 2265.4001981 191.8000 2534.700 0.000000 0.0000001982 199.5000 2690.900 0.000000 0.0000001983 168.7000 2862.500 0.000000 0.0000001984 222.0000 3154.600 0.000000 0.0000001985 189.3000 3379.800 0.000000 0.0000001986 187.5000 3590.400 0.000000 0.0000001987 142.0000 3802.000 0.000000 0.0000001988 155.7000 4075.900 0.000000 0.0000001989 175.6000 4664.200 0.000000 0.0000001990 175.6000 4664.200 0.000000 0.0000001991 199.6000 4828.300 0.000000 0.000000

=================================================

7.53


Savings = 0 + 1 Income + ’0D + ’1D*Income +

D = 1 1970--1980 = 0 1981--1991

Estimated for 1970 - 1980 : D = 1

Savings = (0 + ’0) +(1 + ’1) Income^ ^ ^ ^

1

Estimated for 1981 - 1991 : D = 0

Savings = 0 + 1 Income^ ^

2

Dummy Regression Results: 1970 - 1991 :

Savings = 217.81 - 203.19 D - 0.010 Income + 0.066 D*Income(7.96) (-6.19) (-1.39) (4.63)

7.54


70 - 80:Savings = (217.81 - 203.19) + (-0.010 + 0.066) Income

= 14.62 + 0.056 Income

81 - 91:Savings = 217.81 - 0.010 Income

1970 - 1991 :Savings = 57.63 + 0.031 Income

(3.86) (5.95)

1970 - 1980 :Savings = 14.61 + 0.056 Income

(1.40) (7.93)

1981 - 1991 :Savings = 217.81 + 0.010 Income

(6.16) (-1.08)

7.55


Savings = -1.250 + 0.091 Dummy + 0.125 Income

(-3.42) (0.506) (7.04)

LS // Dependent Variable is SAVINGSDate: 03/02/99 Time: 22:23Sample: 1946 1963Number of observations: 18 ===================================================== Variable Coefficient Std. Error t-Statistic. Prob.=====================================================

C -1.250957 0.364879 -3.428419 0.0037 DUMMY 0.091857 0.181244 0.506816 0.6197 INCOME -0.125655 0.017837 -7.044517 0.0000=====================================================R-squared 0.919909 Mean dependent var 0.773333Adjusted R-squared 0.909230 S.D. dependent var 0.642806S.E. of likelihood 0.193665 Akaike info criterion -3.132238Sum squared resid 0.562593 Schwarz criterion -2.983843 Log likelihood 5.649250 F-statistic 86.14326Durbin-Watson stat 0.976197 Prob(F-statistic) 0.000000=====================================================

Only consider the difference in intercept

7.56


LS // Dependent Variable is SAVINGSDate: 03/02/99 Time: 22:23Sample: 1946 1963Number of observations: 18 ===================================================== Variable Coefficient Std. Error t-Statistic. Prob.=====================================================

C -1.750172 0.331888 -5.273377 0.0001 DUMMY 1.483923 0.470362 3.154852 0.0070 INCOME 0.150450 0.016286 9.238172 0.0000 DINCOME -0.103422 0.033260 -3.109471 0.0077 =====================================================R-squared 0.952626 Mean dependent var 0.773333Adjusted R-squared 0.942475 S.D. dependent var 0.642806S.E. of likelihood 0.154173 Akaike info criterion -3.546228Sum squared resid 0.332771 Schwarz criterion -3.348367 Log likelihood 10.37516 F-statistic 93.84109Durbin-Watson stat 1.468099 Prob(F-statistic) 0.000000=====================================================

Whether intercept and slope change?

7.57


1946 - 1954 : = -0.2662 + 0.047 Income D1 = 1

1955 - 1963 : = -1.750 + 0.150 Income D1 = 0

Savings = -1.750 + 1.483 D + 0.150 Income - 0.103 (Income*D)

(-5.273) (3.154) (9.238) (-3.109)

^

7.58


The use of Dummy variables in the Pooled data

2. Dummy variable method :

(i) Y = 0 + 1 X1 + 2 X2 + 3 D + H0 : 0 = 0’

H1 : 0 0’

D = 1 for GM = 0 otherwise

==

if t3 > tc ==> reject H0 ^

Panel =time-series

+cross-section

1. For each firm, run the separated regression :

GM : Y = 0 + 1 X1 + 2 X2 + Y = 0’ + 1’ X1 + 2’ X2 + ’Westinghouse :

H0 : 1 = 1’, 2 = 2’, 0 = 0’

7.59


(ii) Y = 0 + 1 X1 + 2 X2 + 3 D + 4 D X2 + 5 D X3 +

H0 : 1 = 1’

H1 : 1 1’

==


H0 : 2 = 2’

H1 : 2 2’

==


7.60


“Chow Test” - structural stability Test Using dummy variables approach

H0 : no structural change

H1 : yes

Procedures:

Generate dummy variableGenerate dummy variable :D1 = 0 for 1946-1954D1 = 1 for 1955-1963 or

D1D1 = 1 for 1946-1954D2D2 = 0 for 1955-1963

7.61


GENRdummy = 0 for 1946-1954dummy = 1 for 1955-1963

7.62


year Dummy1946 01947 01948 01949 01950 01951 01952 01953 01954 01956 11957 11958 11959 11960 11961 11962 11963 1

7.63


Using the dummy variable to identify the structural instability

Check the t-statistics

Generate a dummy series “DUMMY”

7.64


7.65


Slope change?

Intercept change?

7.66


Read the estimated results from the dummy regressions:

For the period of 1955-19631955-1963:savings = (-0.2662 - 1.4839) + (0.0470 + 0.1034) = -1.7501 + 0.1504 Income

For the period of 1946-19541946-1954:Savings = -0.2662 + 0.0470 Income

7.1 Lecture #7 Studenmund(2006) Chapter 7 Objective: Applications of Dummy Independent Variables.

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Transcript of 7.1 Lecture #7 Studenmund(2006) Chapter 7 Objective: Applications of Dummy Independent Variables.