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Instrumental Variables I

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Objective We are trying to learn the effect of education on income We have Card (1993)’s data on years of schooling, wages, proximity to a four year college and various other controls. We will obtain OLS and IV estimates of the returns to education and discuss any problems in this particular context and in general

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OLS Results. reg lwage educ exper expersq black smsa smsa66 south reg66*, robust Linear regression Number of obs = 3010 F( 15, 2994) = Prob > F = R-squared = Root MSE = | Robust lwage | Coef. Std. Err. t P>|t| [95% Conf. Interval] educ | exper | expersq | black | smsa | smsa66 | south | reg661 | reg662 | …… Are you surprised? What is the OLS Identification Assumption? What sources of bias are likely to be present? Which direction are these sources of bias likely to bias our estimates?

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What do we require for an instrument to be valid?

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1.Relevance: cov(z, x) ≠ 0 2.Exogeneity cov(z, e) = 0

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What do we require for an instrument to be valid? 1.Relevance: cov(z, x) ≠ 0 – Important because if the instrument isn’t correlated with the endogenous variable then knowing the value of the instrument doesn’t tell us anything about the endogenous variable. – Do we care about the unconditional correlation or the correlation conditional on the other controls? Why? – Can we test this? How? 2.Exogeneity cov(z, e) = 0

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What do we require for an instrument to be valid? 1.Relevance: cov(z, x) ≠ 0 2.Exogeneity cov(z, e) = 0 – Important because we want the instrument to effect z only through x – Can we test this? If not what do we do instead? – How does this assumption relate to the key OLS identification assumption?

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Testing Relevance How can we test the relevance of an instrument?

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Testing Relevance How can we test the relevance of an instrument? 1.Calculate cor(x,z) – Better than nothing but not ideal. Why? 2.Run the ‘first stage’ regression – What should we include? – What do we look at? – What if we have more than one instrument? – What if we have more than one endogenous variable? 3.Use the post-estimation commands after estimating our main regression. We’ll do (2) today.

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1 st Stage Results reg educ nearc4 exper expersq black smsa smsa66 south reg66*, robust note: reg666 omitted because of collinearity Linear regression Number of obs = 3010 F( 15, 2994) = Prob > F = R-squared = Root MSE = | Robust educ | Coef. Std. Err. t P>|t| [95% Conf. Interval] nearc4 | exper | expersq | Where do we look to test the Relevance condition? Is it satisfied?

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First-Stage F A ‘First Stage F-Statistic’ in excess of 10 is often used as the threshold for satisfaction of the Relevance condition What do we mean by a first stage F Statistic Can we see it on the previous slide? – (we can, but not directly) in general you can use Stata’s ‘test’ command

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How plausible is it that nearc4 is exogenous?

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IV Results ivregress 2sls lwage (educ=nearc4) exper expersq black smsa smsa66 south reg66*, robust note: reg669 omitted because of collinearity Instrumental variables (2SLS) regression Number of obs = 3010 Wald chi2(15) = Prob > chi2 = R-squared = Root MSE = | Robust lwage | Coef. Std. Err. z P>|z| [95% Conf. Interval] educ | exper | expersq | black | smsa | smsa66 | south | reg661 | How have the results changed? Are they what you expect? What explanations could there be for the differences?

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Does the exclusion of IQ break the exogeneity condition?. reg IQ nearc4 Source | SS df MS Number of obs = F( 1, 2059) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = IQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] nearc4 | _cons |

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How about now?. reg IQ nearc4 smsa66 reg662-reg669 Source | SS df MS Number of obs = F( 10, 2050) = Model | Prob > F = Residual | R-squared = Adj R-squared = Total | Root MSE = IQ | Coef. Std. Err. t P>|t| [95% Conf. Interval] nearc4 | smsa66 | reg662 | reg663 | reg664 | reg665 | reg666 | reg667 | reg668 | reg669 | _cons |

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Do we believe the IV results?

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