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# Definition of Instrumental Variables

Definition: Instrumental Variables either refers to:
1. An estimation technique, often abbreviated IV
2. The exogenous variables used in the estimation technique.
Suppose one has a model:

y = Xb + e

Here y is a T x 1 vector of dependent variables, X is a T x k matrix of independent variables, b is a k x 1 vector of parameters to estimate, and e is a k x 1 vector of errors. OLS can be imagined, but suppose in the environment being modelled that the matrix of independent variables X may be correlated to the e's. Then using a T x k matrix of independent variables Z, correlated to the X's but uncorrelated to the e's one can construct an IV estimator that will be consistent:

bIV = (Z'X)-1Z'y

The two stage least squares estimator is an important extension of this idea.

In that discussion above, the exogenous variables Z are called instrumental variables and the instruments (Z'Z)-1(Z'X) are estimates of the part of X that is not correlated to the e's.

Terms related to Instrumental Variable / Instrumental Variables Estimation:

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