Summary Chapter6-Econometrics-RegressionAnalysisUnderLinearRestrictions

Chapter 6
Regression Analysis Under Linear Restrictions and Preliminary Test Estimation


One of the basic objectives in any statistical modeling is to find good estimators of the parameters. In the
context of multiple linear y X β + ε , the ordinary least squares estimator
regression model =

b = ( X ' X ) X ' y is the best linear unbiased estimator of β . Several approaches have been attempted in the
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literature to improve further the OLSE. One approach to improve the estimators is the use of extraneous
information or prior information. In applied work, such prior information may be available about the
regression coefficients. For example, in economics, the constant returns to scale imply that the exponents
in a Cobb-Douglas production function should sum to unity. In another example, absence of money illusion
on the part of consumers implies that the sum of money income and price elasticities in a demand function
should be zero. These types of constraints or the prior information may be available from
(i) some theoretical considerations.
(ii) past experience of the experimenter.
(iii) empirical investigations.
(iv) some extraneous sources etc.


To utilize such information in improving the estimation of regression coefficients, it can be expressed in the
form of
(i) exact linear restrictions
(ii) stochastic linear restrictions
(iii) inequality restrictions.


We consider the use of prior information in the form of exact and stochastic linear restrictions in the model
y X β + ε where y is a (n × 1) vector of observations on study variable, X is a (n × k ) matrix of
=
observations on explanatory variables X 1 , X 2 ,..., X k , β is a (k ×1) vector of regression coefficients and ε is

a (n ×1) vector of disturbance terms.




Econometrics | Chapter 6 | Linear Restrictions and Preliminary Test Estimation | Shalabh, IIT Kanpur
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,Exact linear restrictions:
Suppose the prior information binding the regression coefficients is available from some extraneous sources
which can be expressed in the form of exact linear restrictions as
r = Rβ
where r is a (q ×1) vector and R is a (q × k ) matrix with rank=
( R) q (q < k ). The elements in
r and R are known.


Some examples of exact linear restriction r = Rβ are as follows:
(i) If there are two restrictions with k = 6 like
β2 = β4
β3 + 2β 4 + β5 =
1
then


0  0 1 0 − 1 0 0 0 
=r =  , R  .
1  0 0 1 2 1 0 0 


(ii) If k = 3 and suppose β 2 = 3, then

=r [3=
] , R [ 0 1 0]
(iii) If k = 3 and suppose β1 : β 2 : β3 :: ab : b :1

0  1 − a 0 
then r =
= 0  , R 0 1
 −b  .
0  1 0 −ab 

The ordinary least squares estimator b = ( X ' X ) −1 X ' y does not uses the prior information. It does not obey
the restrictions in the sense that r ≠ Rb. So the issue is how to use the sample information and prior
information together in finding an improved estimator of β .




Econometrics | Chapter 6 | Linear Restrictions and Preliminary Test Estimation | Shalabh, IIT Kanpur
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, Restricted least squares estimation
The restricted least squares estimation method enables the use of sample information and prior information
simultaneously. In this method, choose β such that the error sum of squares is minimized subject to linear
restrictions r = Rβ . This can be achieved using the Lagrangian multiplier technique. Define the Lagrangian
function
S (β , λ ) =( y − X β ) '( y − X β ) − 2λ '( R β − r )
where λ is a (k ×1) vector of Lagrangian multiplier.


Using the result that if a and b are vectors and A is a suitably defined matrix, then
∂
= ( A + A ')a
a ' Aa
∂a
∂
a ' b = b,
∂a
we have
∂S ( β , λ )
= 2 X ' X β − 2 X ' y − 2 R ' λ =' 0 (*)
∂β
∂S ( β , λ )
= R β − r= 0.
∂λ
Pre-multiplying equation (*) by R( X ' X ) −1 , we have

2 Rβ − 2 R( X ' X ) −1 X ' y − 2 R( X ' X ) −1 R ' λ ' =
0

or Rβ − Rb − R( X ' X ) −1 R ' λ ' =
0
−1
⇒λ'=
−  R( X ' X ) −1 R ' ( Rb − r )

using R ( X ' X ) −1 R ' > 0.
Substituting λ in equation (*), we get
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2 X ' X β − 2 X ' y + 2 R '  R( X ' X ) −1 R ' ( Rb − r ) =
0

X ' y − R ' ( R( X ' X ) −1 R ') ( Rb − r ) .
−1
or X 'Xβ =

Pre-multiplying by ( X ' X ) yields
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−1
( X ' X ) X ' y + ( X ' X ) R '  R( X ' X )−1 R ' ( r − Rb )
−1 −1
βˆR =
−1
b − ( X ' X ) R '  R ( X ' X ) R ' ( Rb − r ) .
−1 −1
=
 
This estimation is termed as restricted regression estimator of β .
Econometrics | Chapter 6 | Linear Restrictions and Preliminary Test Estimation | Shalabh, IIT Kanpur
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