Summary Chapter15-Econometrics-InstrumentalVariable

Chapter 15
Instrumental Variables Estimation


A basic assumption in analyzing the performance of estimators in multiple regression is that the explanatory
variables and disturbance terms are independently distributed. The violation of such assumption disturbs the
optimal properties of the estimators. The instrumental variable estimation method helps in estimating the
regression coefficients in the multiple linear regression model when such violation occurs.


Consider the multiple linear regression model
y  X 
where y is (n  1) vector of observation on study variable, X is ( n  k ) matrix of observations on
X 1 , X 2 ,..., X k ,  is a (k  1) vector of regression coefficient and  is a (n  1) vector of disturbances.


Suppose one or more explanatory variables is correlated with the disturbances in the limit, then we can write
1 
plim  X '    0.
n 
The consequences of such an assumption on ordinary least squares estimator are as follows:

b X 'X  X 'y
1



  X ' X  X ' X    
1



b     X ' X  X '
1


1
 X ' X   X ' 
   
 n   n 
1
 X 'X   X ' 
plim  b     plim   plim  
 n   n 
0

 X 'X 
assuming plim     XX exists and is nonsingular. Consequently plim b   and thus the OLSE
 n 
becomes an inconsistent estimator of  .


To overcome this problem and to obtain a consistent estimator of  , the instrumental variable estimation can
be used.


Econometrics | Chapter 15 | Instrumental Variables Estimation | Shalabh, IIT Kanpur
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, Consider the model
1 
y  X    with plim  X '    0.
n 
Suppose that it is possible to find a data matrix Z of order  n  k  with the following properties.:



Z'X 
(i) plim     ZX is a finite and nonsingular matrix of full rank. This interprets that the variables in
 n 
Z are correlated with those in X , in the limit.

 Z ' 
(ii) plim    0,
 n 
i.e., the variables in Z are uncorrelated with  , in the limit.
 Z 'Z 
(iii) plim     ZZ exists.
 n 


Thus Z  variables are postulated to be
 uncorrelated with  , in the limit and
 to have a nonzero cross product with X .
Such variables are called instrumental variables.


If some of X variables are likely to be uncorrelated with  , then these can be used to form some of the
columns of Z and extraneous variables are found only for the remaining columns.


First, we understand the role of the term X '  in the OLS estimation. The OLSE b of  is derived by
solving the equation
  y  X   ' y  X  
0

or X ' y  X ' Xb

or X '  y  Xb   0.




Econometrics | Chapter 15 | Instrumental Variables Estimation | Shalabh, IIT Kanpur
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