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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 ); errors-in-variables 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 errors-in-variable 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: 10-1 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