1 multiple restrictions and zero restrictions the f test approach to testing a restriction may be...

1

MULTIPLE RESTRICTIONS AND ZERO RESTRICTIONS

The F test approach to testing a restriction may be extended to cover the case where we wish to test whether several restrictions are valid simultaneously.

knRSS

pRSSRSSknpF

U

UR

/

/,

Suppose that there are p restrictions. Let RSSU be RSS for the fully unrestricted model and RSSR be RSS for the model where all p restrictions have been imposed. The test statistic is then as shown.

2


knRSS

pRSSRSSknpF

U

UR

/

/,

The numerator is the reduction in RSS comparing the fully restricted model with the unrestricted model, divided by the number of degrees of freedom lost when the restrictions are relaxed.

3


knRSS

pRSSRSSknpF

U

UR

/

/,

The denominator is the RSS for the unrestricted model, divided by the number of degrees of freedom remaining when that model is fitted. k is the number of parameters in the unrestricted model.

4


knRSS

pRSSRSSknpF

U

UR

/

/,

The t test approach can be used, as before, to test individual restrictions in isolation.

knRSS

pRSSRSSknpF

U

UR

/

/,

5


You will often encounter references to zero restrictions. This just means that a particular parameter is hypothesized to be equal to zero, for example, b5 in the model above. Taken in isolation, the appropriate test is of course the t test.

6


uXXXXY 554433221

05

uXXXXY 554433221

05

It can be considered to be a special case of the t test of a restriction discussed above where there is no need for reparameterization. The test statistic is just the t statistic for the parameter in question.

7


Likewise the testing of multiple zero restrictions can be thought of as a special case of the testing of multiple restrictions. The example shown is for a model where there are two zero restrictions.

8


knRSS

pRSSRSSknpF

U

UR

/

/,

uXXXY 4433221 uXY 221

URSS

RRSS

knRSSRSSRSS

knFU

UR

/

2/,2

0: 430 H

0 4 0 : 31 H or 0 4 or both

and 3

The F test of the joint explanatory power of a group of explanatory variables discussed in Section 3.5 in the text can be thought of in this way.

9


knRSS

pRSSRSSknpF

U

UR

/

/,

uXXXY 4433221 uXY 221

URSS

RRSS

knRSSRSSRSS

knFU

UR

/

2/,2

0: 430 H

0 4 0 : 31 H or 0 4 or both

and 3

Even the F statistic for the equation as a whole can be treated as a special case. Here the unrestricted and restricted models are as shown.

10


uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:

Restrictions: 0...32 k

RSSU

RSSR

Hypotheses:

00...: 320 kH

:1H at least one of the slope coefficients

When we fit the restricted model, we find that the OLS estimator of b1 is the sample mean of Y (see Exercise 1.3).

11


Fitting restricted model: Yb 1

uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH


Hence the fitted value of Y in all observations is equal to the sample mean of Y.

12



YbYi 1ˆ for all i

uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH


Now we know that for any OLS regression, TSS = ESS + RSS.

13



YbYi 1ˆ

RSSESSTSS for all i

uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH


Hence TSS = RSS for the restricted regression.

14



YbYi 1ˆ

RSSESSTSS

2222 0ˆiiii eeYYYY

for all i

uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH



YbYi 1ˆ

RSSESSTSS

2222 0ˆiiii eeYYYY

for all i

Obviously, if there are no explanatory variables, none of the variation in Y is explained by the model and so RSS is equal to TSS.

15


uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH


knRSSkESS

knRSSkRSSTSS

knRSSkRSSRSS

knkF

U

U

U

U

U

UR

/1/

/1/

/1/

,1

uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH


Here is the F statistic for the comparison of the unrestricted model with all of the X variables and the restricted model with only the intercept.

16


knRSSkESS

knRSSkRSSTSS

knRSSkRSSRSS

knkF

U

U

U

U

U

UR

/1/

/1/

/1/

,1

uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH


We have just seen that RSS from the restricted version is equal to TSS.

17


knRSSkESS

knRSSkRSSTSS

knRSSkRSSRSS

knkF

U

U

U

U

U

UR

/1/

/1/

/1/

,1

UU RSSESSTSS

uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH


Now we refer to the decomposition of TSS in the case of the unrestricted regression. This is similar to the decomposition for the restricted model, with the difference that RSSU will in be smaller than RSSR and ESSU will be positive, instead of zero.

18


Given the decomposition for the unrestricted version, we can rewrite the F statistic as shown. This is the expression for the F statistic for the equation as a whole that was given in Section 3.5.

knRSSkESS

knRSSkRSSTSS

knRSSkRSSRSS

knkF

U

U

U

U

U

UR

/1/

/1/

/1/

,1

UU RSSESSTSS

19


uXYk

jjj

21

uY 1

Unrestricted model:

Restricted model:


RSSU

RSSR

Hypotheses:

00...: 320 kH


Copyright Christopher Dougherty 2012.

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Introduction to Econometrics, fourth edition 2011, Oxford University Press.

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