Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: reparameterization...

Christopher Dougherty

EC220 - Introduction to econometrics (chapter 6)Slideshow: reparameterization of a model and t test of a linear restriction

Original citation:

Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 6). [Teaching Resource]

© 2012 The Author

This version available at: http://learningresources.lse.ac.uk/132/

Available in LSE Learning Resources Online: May 2012

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1

REPARAMETERIZATION OF A MODEL AND t TEST OF A LINEAR RESTRICTION

Linear restrictions can also be tested using a t test. This involves the reparameterization of a regression model and we will start with this.

uXYk

jjj

21

2

Suppose that you have a regression model shown and that the regression model assumptions are valid.

uXYk

jjj

21


3

We fit the model in the usual way. This enables us to obtain estimates of the parameters and their standard errors.

uXYk

jjj

21

k

jjjXbbY

21

ˆ


4

Suppose, however, that we are interested in a linear combination, , of the parameters, where the j are weights.

uXYk

jjj

21

k

jjjXbbY

21

ˆ

k

jjj

1


5

Obviously, the can obtain a point estimate of as the corresponding linear combination of the estimates of the individual parameters. If the regression model assumptions are valid, it can easily be shown that it is unbiased and that it is the most efficient estimator of .

uXYk

jjj

21

k

jjjXbbY

21

ˆ

k

jjj

1

k

jjjb

1

ˆ


6

However you do not have information on its standard error and hence you are not able to construct confidence intervals for or to perform t tests. There are three ways that you might use to obtain such information.

uXYk

jjj

21

k

jjjXbbY

21

ˆ

k

jjj

1

k

jjjb

1

ˆ


7

(1) Some regression applications have a special command that produces it. For example, Stata has the lincom command.

uXYk

jjj

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k

jjjXbbY

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ˆ

k

jjj

1

k

jjjb

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ˆ


8

(2) Given the appropriate command, most regression applications will produce the variance-covariance matrix for the estimates of the parameters. This is the complete list of the estimates of their variances and covariances, for convenience arranged in matrix form.

uXYk

jjj

21

k

jjjXbbY

21

ˆ

k

jjj

1

k

jjjb

1

ˆ


9

The standard errors in the ordinary regression output are the square roots of the variances. The estimate of the variance of the estimate of is given by the expression shown, where s subscript bpbj is the estimate of the covariance between bp and bj.

uXYk

jjj

21

k

jjjXbbY

21

ˆ

k

jjj

1

k

jjjb

1

ˆ

jp

bbjp

k

jbj jpj

sss 21

222ˆ


10

This method is cumbersome and avoided when possible.

uXYk

jjj

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k

jjjXbbY

21

ˆ

k

jjj

1

k

jjjb

1

ˆ

jp

bbjp

k

jbj jpj

sss 21

222ˆ


11

(3) The third method is to reparameterize the model, manipulating it so that and its standard error are estimated directly as part of the regression output.

uXYk

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k

jjjXbbY

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k

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k

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12

To do this, we rewrite the expression for so that one of the parameters is expressed in terms of and the other parameters. This will be illustrated with two simple examples, the general case being left as an additional exercise (see the study guide on the website).

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k

jjjXbbY

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k

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13

Suppose the regression model is as shown and suppose we are interested in the sum of 2 and 3.

uXXY 33221

32


14

We rewrite the relationship as shown, expressing one of the parameters in terms of and the other parameter.

uXXY 33221

32

23


15

Substituting in the original model, we reparameterize it as shown.

uXXY 33221

32

23

32321

33221

32221

where XXZuXZ

uXXX

uXXY


16

Thus if we define a new variable Z = X2 – X3, and regress Y on Z and X3, the coefficient of Z will be an estimate of 2 and that of X3 will be an estimate of .

uXXY 33221

32

23

32321

33221

32221

where XXZuXZ

uXXX

uXXY


17

The estimate of will be exactly the same as that obtained by summing the estimates of 2 and 3 in the original model. The difference is that we obtain its standard error directly from the regression results. The estimate of 2 and its standard error will be the same as those obtained with the original model.

uXXY 33221

32

23

32321

33221

32221

where XXZuXZ

uXXX

uXXY


18

Suppose instead that we were interested in the difference between 2 and 3.

uXXY 33221

23


19

We rewrite this as shown and substitute in the original model. Thus if we define a new variable Z = X2 + X3 and regress Y on Z and X3, the coefficient of Z will be an estimate of 2 and that of X3 will be an estimate of .

uXXY 33221

23

32321

33221

32221

where XXZuXZ

uXXX

uXXY


23

20

The estimate of will be exactly the same as that obtained by taking the difference of the estimates of 2 and 3 in the original model and we obtain its standard error directly from the regression results. The estimate of 2 and its standard error will be the same as those obtained with the original model.

uXXY 33221

23

32321

33221

32221

where XXZuXZ

uXXX

uXXY


23

21

An obvious application of reparameterization is its use in the testing of linear restrictions. Suppose that your hypothetical restriction is as shown, where is a scalar.

uXXY 33221

k

jjj

1


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Define as shown and reparameterize. will become the coefficient of one of the variables in the model, and a t test of H0: = 0 is effectively a t test of the restriction.

uXXY 33221

k

jjj

1

k

jjj

1


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As an illustration. we will use the example discussed in the previous sequence. The model relates years of schooling, S, to the cognitive ability score ASVABC and years of schooling of the mother and the father, SM and SF.

uSFSMASVABCS 4321


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It was hypothesized that mother’s education and father’s education are equally important for educational attainment, implying the restriction 4 = 3.

uSFSMASVABCS 4321

34


25

The restriction may be rewritten as 4 – 3 = 0. Define as shown.

uSFSMASVABCS 4321

34

034

34


26

Hence express one of the parameters in terms of and the other parameter, and substitute in the original model.

uSFSMASVABCS 4321

34

034

34

34

uSFSPASVABC

uSFSFSMASVABC

uSFSMASVABCS

321

321

3321


27

SP is defined as the sum of SM and SF. A t test on the coefficient of SF is a test of the restriction.

uSFSMASVABCS 4321

34

034

34

34

uSFSPASVABC

uSFSFSMASVABC

uSFSMASVABCS

321

321

3321

0:,0: 341340 HH


. reg S ASVABC SP SF

Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943

------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SP | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .0584401 .0617051 0.95 0.344 -.0627734 .1796536 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------

28

Here is the corresponding regression. We see that the coefficient of SF is not significantly different from zero, indicating that the restriction is not rejected.


. reg S ASVABC SP SF

Source | SS df MS Number of obs = 540-------------+------------------------------ F( 3, 536) = 104.30 Model | 1181.36981 3 393.789935 Prob > F = 0.0000 Residual | 2023.61353 536 3.77539837 R-squared = 0.3686-------------+------------------------------ Adj R-squared = 0.3651 Total | 3204.98333 539 5.94616574 Root MSE = 1.943

------------------------------------------------------------------------------ S | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- ASVABC | .1257087 .0098533 12.76 0.000 .1063528 .1450646 SP | .0492424 .0390901 1.26 0.208 -.027546 .1260309 SF | .0584401 .0617051 0.95 0.344 -.0627734 .1796536 _cons | 5.370631 .4882155 11.00 0.000 4.41158 6.329681------------------------------------------------------------------------------

29

It can be shown mathematically that the F test of a restriction and the corresponding t test are equivalent. The F statistic is the square of the t statistic and the critical value of F is the square of the critical value of t.


Copyright Christopher Dougherty 2011.

These slideshows may be downloaded by anyone, anywhere for personal use.

Subject to respect for copyright and, where appropriate, attribution, they may be

used as a resource for teaching an econometrics course. There is no need to

refer to the author.

The content of this slideshow comes from Section 6.5 of C. Dougherty,

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

Additional (free) resources for both students and instructors may be

downloaded from the OUP Online Resource Centre

http://www.oup.com/uk/orc/bin/9780199567089/.

Individuals studying econometrics on their own and who feel that they might

benefit from participation in a formal course should consider the London School

of Economics summer school course

EC212 Introduction to Econometrics

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

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11.07.25

http://www.oup.com/uk/orc/bin/9780199567089/

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

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Christopher Dougherty EC220 - Introduction to econometrics (chapter 6) Slideshow: reparameterization...

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