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Chapter 12 - Instrumental variables regression

Chapter 12 - Instrumental variables regression -...

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10-1 Instrumental Variables Regression (SW Ch. 12) 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 ( IV ) regression can eliminate bias from these three sources. We will mainly focus on simultaneous causality, but be aware that IV may be used in the other cases too.

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10-2 The IV Estimator with a Single Regressor and a Single Instrument (SW Section 12.1) Y i = β 0 + 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 two estimate 1 .
10-3 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 OV bias and errors-in-variable bias, not just simultaneous causality bias.

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10-4 Two conditions for a valid instrument Y i = β 0 + 1 X i + u i For an instrumental variable (an “ instrument ”) Z to be valid, it must satisfy two conditions: 1. Instrument relevance : corr( Z i , X i ) 0 2. Instrument exogeneity : corr( Z i , u i ) = 0 Suppose for now that you have such a Z i (we’ll discuss how to find instrumental variables later). How can you use Z i to estimate 1 ?
10-5 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 on Z using OLS X i = π 0 + 1 Z i + v i ( 1 ) Because Z i is uncorrelated with u i , 0 + 1 Z i is uncorrelated with u i . We don’t know 0 or 1 but we have estimated them, so… Compute the predicted values of X i , ˆ i X , where ˆ i X = 0 ˆ + 1 ˆ Z i , i = 1,…, n .

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10-6 (2) Replace X i by ˆ i X in the regression of interest: regress Y on ˆ i X using OLS: Y i = β 0 + 1 ˆ i X + u i ( 2 ) Because ˆ 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 π 0 and 1 are well estimated using regression (1)) This the resulting estimator is called the “Two Stage Least Squares” (TSLS) estimator, 1 ˆ TSLS .
10-7 Two Stage Least Squares, ctd .

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Chapter 12 - Instrumental variables regression -...

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