Ch12ivPart1Daria - Introduction to the Instrumental...

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1 Introduction to the Instrumental Variable (IV) Method 1 Instrumental Variables Regression (SW Chapter 12) Three important threats to internal validity are: mitted variable bias from a variable that is correlated with 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) 2 Instrumental variables regression can eliminate bias when E ( u | X ) ¹ 0 – using an instrumental variable , Z IV Regression with One Regressor and One Instrument (SW Section 12.1) Y i = 0 + 1 X i + u i 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 3 uncorrelated with u i . The instrumental variable detects movements in X i that are uncorrelated with u i , and uses these to estimate 1 . Terminology: endogeneity and exogeneity An endogenous variable is one that is correlated with u n ogen us ariable is one that is uncorrelated with 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 salit Ho e er this de nition is narro and IV 4 causality. However, this definition is narrow and IV regression can be used to address OV bias and errors-in- variable bias, not just to simultaneous causality bias.
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2 Three conditions for a valid instrument Y i = 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 ) ¹ 0 2. Instrument exogeneity : corr( Z i , u i ) = 0 3. Exclusion : Z cannot be an omitted variable in model 5 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 ? The IV Estimator, one X and one Z 6 Two Stage Least Squares, ctd. 7 8
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3 Two Stage Least Squares, ctd. Suppose you have a valid instrument, Z i . Stage 1: Regress X i on Z i , obtain the predicted values ˆ i X Stage 2: Regress Y i on ˆ i X ; the coefficient on ˆ i X is the TSLS estimator, 1 ˆ TSLS . 9 1 ˆ TSLS is a consistent estimator of 1 . 10 The IV Estimator, one X and one Z, ctd. Explanation #2: a little algebra… Y i = 0 + 1 X i + u i Thus, cov( Y i , Z i ) = cov(( 0 + 1 X i + u i ) , Z i ) = cov( 0 , Z i ) + cov( 1 X i , Z i ) + cov( u i , Z i ) = 0 + cov( 1 X i , Z i ) + 0 = 1 cov( X i , Z i ) 11 where cov( u i , Z i ) = 0 (instrument exogeneity); thus 1 = cov( , ) cov( , ) ii YZ X Z The IV Estimator, one X and one Z, ctd. 1 = cov( , ) ov( ) Z cov( , XZ The IV estimator replaces these population covariances with sample covariances: 1 ˆ TSLS = YZ s , 12 XZ s s YZ and s XZ are the sample covariances. This is the TSLS estimator – just a different derivation!
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4 Consistency of the TSLS estimator 1 ˆ TSLS = YZ s XZ s The sample covariances are consistent: s YZ p cov( Y
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This note was uploaded on 11/28/2010 for the course ECON Economics taught by Professor Davidbrownstone during the Spring '10 term at UC Irvine.

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Ch12ivPart1Daria - Introduction to the Instrumental...

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