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Instrumental Variables


Lecture number:
Lecture Note
Cornell University
Econ 3120 - Applied Econometrics

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Econ 3120 1st Edition Lecture 27 Outline of Last Lecture I. Counterfactual in Policies Outline of Current Lecture II. Instrumental Variables Current Lecture Review: Omitted Variable Bias Let’s start with the standard education-earnings model that includes ability as a regressor: log(wage) = β0 +β1educ+β2abil +e We have seen previously, that if you instead estimated the model without including ability: log(wage) = β0 +β1educ+u your estimates for β1 will be biased and inconsistent. This is because, in general, Cov(educ,u) 6= 0 since u = β2abil +e. This implies that we need to include a measure of ability in the regression in order to have unbiased estimates. But what if one isn’t available? Instrumental Variables We can get around this problem by using an instrumental variable. One way to think about an instrumental variable is that it allows us to isolate the variation in educ that is unrelated to u, and thus we can back out an estimate of β1. An instrumental variable is a variable that satisfies two properties: 1. It has to be correlated with the variable it is instrumenting for, educ. 2. It has to be uncorrelated with the error term u. In this example, we’ll use the individual’s proximity to college when a teenager as an instrumental variable. Is it likely to satisfy these two properties? 1 Let’s formalize this a bit. Take the general model y = β0 +β1x+u where Cov(x,u) 6= 0 (so MLR.4 is not satisfied). If we can find an instrument z, it needs to satisfy the following two properties: 1. First stage: Cov(z, x) = 0 2. Exclusion restriction: Cov(z,u) = 0 We can now use these properties to back out an IV estimate of β1: y = β0 +β1x+u Cov(y,z) = Cov(β0 +β1x+u,z) = Cov(β0,z) +Cov(β1x,z) +Cov(u,z) = 0+β1Cov(x,z) +0 β1 = Cov(y,z) Cov(x,z) We can then use the sample analogs to get an estimate: ˆβ1,IV = Covd(y,z) Covd(x,z) It can be shown that ˆβ1,IV is a consistent estimator for β1. It turns out that in small samples, IV estimates are STILL a bit biased in finite samples, but in this class we are not going to go into detail as to why.1 Back to our example with schooling and earnings. For our instrument, we will use a dummy variable equal to 1 if the person grew up near a 4-year college. The estimates are: ˆβ1,OLS = 0.052 ˆβ1,IV = 0.188 1The magnitude of the bias depends inversely on the correlation between z and x. If we have a strong correlation between z and x, we can safely ignore finite-sample bias. However, if z and x are only weakly correlated, then this bias can become problematic. 2 The IV estimate is almost 4X as large as the OLS estimate! We’ll hold off on discussing why until we have developed a more complete model. These notes represent a detailed interpretation of the professor’s lecture. GradeBuddy is best used as a supplement to your own notes, not as a substitute.

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