2010 ECON 7710

5.1

Hypothesis Testing 2: Joint Restrictions

• Testing joint hypotheses

• Chow test

Objectives

2010 ECON 7710

5.2

Yi = 0 + 1X1i + 2X1i2 + 3X2i + i

Can the variation in X1 explain the variation in Y significantly?

Yi = 0 + 1X1i + 2X2i + 3X3i + i

Can X2 and X3 explain Y significantly simultaneously?

1. Adding or Dropping Variables

2010 ECON 7710

5.3

Regression model: Yi = 0 + 1X1i + 2X2i + … + KXKi + i

Ho: 1 =0, 2 = 0, , M = 0HA: i 0 for at least one i = 1, 2, , M.Unconstrained model: Yi = 0 + 1X1i + 2X2i + … + KXKi + i

ESSU, RSSU.

Constrained model: (delete M regressors) Yi = 0 + M+1XM+1,i + … + KXKi + i

ESSC ESSU, RSSC RSSU .

2010 ECON 7710

5.4

Unconstrained model:

Yi = 0 + 1X1i + 2X2i + … + KXKi + i

ESSU, RSSU.

Constrained model: (delete M regressors)

Yi = 0 + M+1XM+1,i + … + KXKi + i

ESSC ESSU, RSSC RSSU .

2010 ECON 7710

5.5

If is normally distributed, then

RSSC – RSSU has a chi-square distribution.

Total variation in Y: 2

iTSS Y Y Unexplained variation in the unconstrained model: RSSU = eU

2

Increased unexplained variation in the constrained model:

RSSC – RSSU

= eC2 – eU

2

2010 ECON 7710

5.6

F

0

f(F)

F Distribution

1-

Fc

1,~

1/

/

KNMFKNRSS

MRSSRSS

U

UC

Ho: 1 =0, 2 = 0, , M = 0

HA: i 0 for at least one i = 1, 2, , M.

2010 ECON 7710

5.7

• Yi = 0 + 1X1i + … + KXKi + i

Unconstrained model RSSU with N – K – 1 degrees of freedom

• Constrained model with M restrictions RSSC with N – K – 1 + M degrees of freedom

C U

U

RSS RSS / M Test statistic: F

RSS /(N K 1)

• Critical value: Fc = FM,N-K-1,

• Reject Ho if F > Fc.

Testing Procedures

2010 ECON 7710

5.8

Example 1: Consider the following regression model:

Yi = 0 + 1X1i + 2X2i + 3X3i + i.

What are the unconstrained and constrained models that are used to test the following null hypotheses?

. 1 = 0

. 2 = 0 and 3 = 0

c. k = 0 for k = 1, 2, 3

2010 ECON 7710

5.9

Exclusion Restriction: 1 variable

Unconstrained model:

uniGPA = 0 + 1hsGPA + 2HKAL + 3skipped +

Restriction: 3=0

Constrained model:

uniGPA = 0 + 1hsGPA + 2HKAL +

Example 2

2010 ECON 7710

5.10

Example 2 (Cont’d): Regression results

Unconstrained model

uniGPA’ = 1.390 + .412hsGPA + .015HKAL - .083skipped

se (0.332) (0.094) (0.011) (0.026)

R2 = 0.2336, N = 141, RSSU = 14.87297

Constrained model

uniGPA’ = 1.286 + .453hsGPA + .009HKAL

se (0.341) (0.096) (0.011)

R2 = 0.1764, N = 141, RSSC = 15.98244

2010 ECON 7710

5.11

Ho: 3 = 0; HA: 3 0

(RSSC RSSU)/M

RSSU/(N K – 1)F =

(15.98244 14.87297)/1

14.87297/(141 - 4)=

= 10.2197

-0.083

0.026t =

= 3.1923

Note that when there is only one restriction, F = t2.

Example 2 (Cont’d): Hypothesis testing

2010 ECON 7710

5.12

Exclusion Restrictions: 2 variables

TRi = 0 + 1Pi + 2Ai + 3A2i + i

Example 3: A general functional form

H0: = 0, 3 = 0

HA: H0 not true

Testing the significance of advertising expenses on revenue.

Constrained model: TRi = 0 + 1Pi + i

2010 ECON 7710

5.13

Next run the constrained regression by dropping Ai and Ai

2 to get RSSC.

First run unconstrained regression to get RSSU.

(RSSC RSSU)/M

RSSU/(N K – 1)F = =

TR = 110.46*** – 10.20***P + 3.36***A – 0.027*A2

se (3.74) (1.58) (0.42) (0.016)R2 = 0.878, N = 78, RSSU = 2,592.30

^

TR = 111.71*** + 5.06P se (8.85) (4.01)R2 = 0.0205, N = 78, RSSC = 20,907.33

^

2010 ECON 7710

5.14

Remark: Relation between F and R2

RSSC = TSS(1 - RC2)

RSSU = TSS(1 - RU2)

C U

U

(RSS RSS ) MF

RSS (N K 1)

2010 ECON 7710

5.15

To test the overall significance of the regression equation, the null and alternative hypotheses are

A Special Case: Testing the Overall Significance of the Regression Equation

Yi = 0 + 1X1i + 2X2i + … + KXKi + i

H0: 1 = 2 = = K = 0

HA: H0 not true

2010 ECON 7710

5.16

The test statistic is

ESS / K = average explained sum of squares

RSS / (N – K – 1) = average unexplained sum of squares

Larger F means higher explanatory power.

Degrees of freedom: 1 = K, 2 = N – K – 1

.

//

//

111 2

2

KNR

KR

KNRSS

KESSF

U

U

Reject Ho if F > Fc = F1,2,

2010 ECON 7710

5.17

Example 4: Picking restaurant locations pp. 75 – 78)

Yi = 0 + 1Ni + 2Pi + 3Ii + i

N: Competition

P: Population

I: Income

If the model cannot explain the variation of Y:

1 = 2 = 3 = 0.

2010 ECON 7710

5.18

Example 4: Picking Restaurant Locations (Table 3.1)

Yhat = 102192 – 9075N + 0.3547P + 1.2879I

se (12800) (2053) (0.0727) (0.5433)

R2 = 0.6182, N = 33,

RSS = 6133282062, TSS = 16062183882

F =ESS/K

RSS/(N-K-1)

6489.15

1333/6133282062

3/613328206221606218388

2010 ECON 7710

5.19

2010 ECON 7710

5.20

Example 6: Consider the following estimated saving function:

Shat = -42.75 + 0.015Y + 0.007W + 7.67r se (-3.30) (2.09) (1.75) (3.81) Adj.R2 = 0.962, F = 251.5, RSS = 2470.8, N = 31

Another regression has been run with the same data set, Shat = 9.4 + 0.06Y se (2.1) (17.2) Adj.R2 = 0.908, F = 296.4, RSS = 6372.8, N = 31.

Are the coefficients for wealth and interest rate jointly significant at 1% level?

2010 ECON 7710

5.21

2. Are Two Equations Equal?

45

50

55

60

65

70

75

80

85

155 160 165 170 175 180 185 190

HEIGHT

WE

IGH

T

Scatter DiagramWeight vs. Height for Male Students in 2004

35

40

45

50

55

60

65

70

75

140 150 160 170 180 190 200

HEIGHT

WE

IGH

T

Scatter DiagramWeight vs. Height for Female Students in 2004

ˆM & F 2004: 66.631 0.729weight height

2010 ECON 7710

5.22

Suppose there are 2 groups of data:

Group A: (Yi, X2i,…,XKi), i = 1,…,N1.

Group B: (Yi, X2i ,…,XKi), i = N1+1,…,N

If the relation between X & Y is different, then

(1) Yi = 0 + 1X1i +…+ KXKi + 1i, i = 1,…,N1

(2) Yi = 0 + 1X1i +…+ KXKi + 2i, i = N1+1,…,N

If the relation between is identical for both groups,

(3) Yi = 0 + 1X1i +…+ KXKi + i, i = 1,…,N

2010 ECON 7710

5.23

Should the two groups be treated as one group?

1. Ho: k= k, k= 0,1,…,K;

H1: k k for at least one k

2. Estimate equations (1) and (2) to get RSS1 and RSS2.

RSSU = RSS1 + RSS2.

3. Estimate equation (3) to get RSSC.

2010 ECON 7710

5.24

)2K2N(,1K

U

UC F~)2K2N/(RSS

)1K(RSSRSSF

Reject Ho if F > FK+1,(N-2K-2),.

Remarks:

a. This method is called the Chow test.

b. It is assumed that the variances of the two groups are equal.

c. One can use dummy variables to test for this change of equation structure.

2010 ECON 7710

5.25

Example 7: Structural change in the US saving function (BE4_Tab0809)

savings = 0 + 1Income +

40

80

120

160

200

240

280

0 1,000 2,000 3,000 4,000 5,000 6,000

INCOME

SA

VIN

GS

D1982=0D1982=1

Two Separate Regression Lines

40

80

120

160

200

240

280

0 1,000 2,000 3,000 4,000 5,000 6,000

INCOME

SA

VIN

GS

One Regression Line

2010 ECON 7710

5.26Example 7 (Cont’d): Empirical results of different periods

Dep.variable

Constant Indep. VX

R2 SEE RSS N

Y(70-81)

1.0161(11.6377)

0.0803(0.00837)

0.9021 0.1412 1785 12

Y(82-95)

153.4947(32.7123)

0.0148(0.00839)

0.2971 0.1660 10005 14

Y(70-95)

62.4226(12.7608)

0.0376(0.0424)

0.7672 0.1891 23248 26

F = (RSSC - RSSU)/(K+1)

RSSU / (N-2K-2)=

(23248.30 – 1785.032 – 10005.22)/ 2

(1785.032 + 10005.22) / (26-2*2)

F = 10.69 Fc = F 2,22, 0.01

= 5.72>

2010 ECON 7710

5.27

Exercises:

1. Find the following critical values

. = 5%, 1 = 3, 2 = 10;

. = 5%, 1 = 12, 2 = 5;

c. = 1%, 1 = 2, 2 = 19;

2010 ECON 7710

5.28

2. Testing the joint significance of the estimated coefficients of X3, X4 and X5. ( = 5%)

*** *** ***

1 2 3 4 5 60.65 0.09 0.14 0.08 0.08 0.03 0.07

2

ˆ 4.48 0.37 0.16 0.05 0.09 0.05 0.26

0.8975, N = 63, RSS = 3.3324.

seY X X X X X X

R

*** *** *** ***

1 2 60.55 0.09 0.10 0.07

2

ˆ 4.33 0.41 0.28 0.29

0.8971, N = 63, RSS = 3.5245.

seY X X X

R

2010 ECON 7710

5.29

3. Consider the following regression results using the weight-height data of some students in 2004. Test whether the weight-height relation is different for male and female.

All Female Male

Intercept -66.63***

(18.7198)

-39.00*

(21.0006)

-43.99

(51.6617)

Height 0.7287***

(0.1110)

0.5491***

(0.1291)

0.6097*

(0.2946)

R2 0.6238 0.5818 0.2802

N 28 15 13

RSS 960.8552 274.6812 559.7543