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# Notes10 - Lecture Notes 10 Econ 410 Introduction to...

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Lecture Notes 10 Econ 410 – Introduction to Econometrics 1 Instrumental Variables Regression Instrumental variables (IV) regression is a way to obtain consistent estimators of the regression coefficients when one or more of the regressors are correlated with the error term. This correlation might arise for different reasons: omitted variables, errors-in- variable, simultaneous causality…. In IV regression, the variables in the regression model are called: endogenous variables, if they are correlated with the error term; exogenous variables, if they are not correlated with the error term. Let X be an endogenous variable. The idea of IV regression is to think of X as composed of two parts: one that is correlated with u and the other one that is not. If we had information that allowed us to isolate this second part, than we could focus on the variations of X that are not correlated with the error term and disregard the part of X that causes problems to the OLS estimator. The information about the movements in X that are uncorrelated with u is obtained from one or more additional variables that are called instrumental variables (or instruments ). IV regression uses these instruments to isolate the movements in X that are uncorrelated with u which in turns permits to estimate consistent OLS coefficients. Regression model with one regressor Consider the model: i i i u X Y + + = 1 0 β β If ( ) 0 , i i u X corr , then ( ) 0 | i i X u E and the OLS estimators 0 ˆ β and 1 ˆ β are biased and inconsistent. Let Z be the instrument that we want to use to isolate the movements in X that are uncorrelated with u . For Z to be a valid instrument, it must satisfy two conditions: 1) Instrument relevance : ( ) 0 , i i X Z corr This condition ensures that the movements in Z are related to the movements in X . 2) Instrument exogeneity : ( ) 0 , = i i u Z corr This condition ensures that the part of the movements in X that is captured by Z is exogenous (that is, not correlated with the error term).

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Lecture Notes 10 Econ 410 – Introduction to Econometrics 2 Two Stage Least Squares Estimator The Two Stage Least Squares (TSLS) estimator is a common IV estimator that can be used when one or more of the regressors are correlated with the error term. The TSLS estimator is obtained in two stages.
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