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Unformatted text preview: Instrumental Variables Regression (SW Ch. 10) Three important threats to internal validity are: omitted variable bias from a variable that is correlated with X but is unobserved, so cannot be included in the regression; simultaneous causality bias ( X causes Y , Y causes X ); errorsinvariables bias ( X is measured with error) Instrumental variables regression can eliminate bias from these three sources. The IV Estimator with a Single Regressor and a Single Instrument (SW Section 10.1) Y i = + 1 X i + u i Loosely, IV regression breaks X into two parts: a part that might be correlated with u , and a part that is not. By isolating the part that is not correlated with u , it is possible to estimate 1 . This is done using an instrumental variable , Z i , which is uncorrelated with u i . The instrumental variable detects movements in X i that are uncorrelated with u i , and use these to estimate 1 . Terminology: endogeneity and exogeneity An endogenous variable is one that is correlated with u An exogenous variable is one that is uncorrelated with u Historical note: Endogenous literally means determined within the system, that is, a variable that is jointly determined with Y, that is, a variable subject to simultaneous causality. However, this definition is narrow and IV regression can be used to address omitted variable bias and errorsinvariable bias, not just simultaneous causality bias. Two conditions for a valid instrument Y i = + 1 X i + u i For an instrumental variable (an instrument ) Z to be valid, it must satisfy two conditions: 101 1. Instrument relevance : corr( Z i , X i ) 2. Instrument exogeneity : corr( Z i , u i ) = 0 Suppose for now that you have such a Z i (well discuss how to find instrumental variables later). How can you use Z i to estimate 1 ? The IV Estimator, one X and one Z Explanation #1: Two Stage Least Squares (TSLS) As it sounds, TSLS has two stages two regressions: (1) First isolates the part of X that is uncorrelated with u : regress X i on Z i , i = 1, ,N, using OLS X i = + 1 Z i + v i (1) Because Z i is uncorrelated with u i , + 1 Z i is uncorrelated with u i . We dont know or 1 but we have estimated them, so Compute the predicted values of X i , i X , where i X = + 1 Z i , i = 1,, n . (2) Replace X i by i X in the regression of interest: regress Y i on i X using OLS: Y i = + 1 i X + u i (2) Because i X is uncorrelated with u i in large samples, so the first least squares assumption holds Thus 1 can be estimated by OLS using regression (2) This argument relies on large samples (so and 1 are well estimated using regression (1)) The resulting estimator is called the Two Stage Least Squares (TSLS) estimator, 1 TSLS ....
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This note was uploaded on 03/01/2012 for the course EC 408 taught by Professor Rogerperman during the Fall '07 term at Uni. Strathclyde.
 Fall '07
 RogerPerman

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