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Instrumental Variables

# Instrumental Variables - Instrumental Variables PAM 3100...

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Instrumental Variables PAM 3100 Professor Michael Lovenheim Fall 2010 PAM 3100 Instrumental Variables

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Introduction In the last section, we learned about many ways in which OLS fails to identify the population parameters of interest due to endogeneity. In this section, we will learn about a potential fix to this problem: instrumental variables (IV). The idea behind IV is to find a variable that is correlated with the independent variable of interest but uncorrelated with u, the population error term. Then, we use this instrumental variable to create exogenous variation in our explanatory variable. We know this variation is exogenous because the variable that is generating it is exogenous. IV can fix biases stemming from omitted variables bias, simultaneous equations bias and measurement error. But, finding a good instrument is very hard. PAM 3100 Instrumental Variables
IV in a Bivariate Model Say we have a population model y = β 0 + β 1 x + u , where x is endogenous (i.e., cov(x,u) negationslash =0). Now take an instrument, z, that has two properties: 1 Cov(z,u)=0 2 Cov(z,x) negationslash =0 The first condition is usually called the “exclusion restriction.” It means that z is correctly excluded from the population model. The second condition requires that z and x actually are correlated - if not, z will not wiggle x. Because we can easily test assumption 2 but cannot test assumption 1, most of the work in justifying a given instrument goes towards trying to convince people the exclusion restriction is valid. PAM 3100 Instrumental Variables

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IV in a Bivariate Model Under the IV assumptions (1 and 2 above), Cov(z,y)=Cov(z, β 0 + β 1 x + u )= β 1 Cov(z,x)+Cov(z,u). Because Cov(z,u)=0 by assumption and Cov(z,x) negationslash =0 by assumption, β 1 = Cov ( z , y ) Cov ( x , z ) . Thus ˆ β 1 = n i =1 ( z i - ¯ z )( y i - ¯ y ) n i =1 ( z i - ¯ z )( x i - ¯ x ) Note that if you replace z with x, this gives you the familiar OLS estimator. Though beyond our scope, one can show that the IV estimator is consistent (it converges in probability to β 1 ) but in small samples it is biased. Thus, we prefer IV on bigger samples. Also, the asymptotic variance of the IV estimator is larger than that of OLS. Thus, our standard errors usually go up (sometimes by a lot) when we use IV. PAM 3100 Instrumental Variables
Some Famous Instruments Angrist and Krueger (QJE, 1991): They study the effect of education on earnings. Endogeneity: Those who obtain more education likely have other attributes that are unobserved and are rewarded by the labor force. Instrument: Quarter of birth combined with compulsory schooling laws (you must enroll from the fall you turn 6 until you are 16). Exclusion restriction: quarter of birth is uncorrelated with all other factors aside from schooling that affect wages. Strength of Instrument: Low - it turns out, particulary since the 1980s, compulsory schooling laws have only a weak association with amount of schooling.

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Instrumental Variables - Instrumental Variables PAM 3100...

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