Econometrics Lecture 9 - Instrumental Variables (25117) PDF

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This document appears to be lecture notes on instrumental variables in econometrics, for a class with the code 25117 at Universitat Pompeu Fabra, on November 18th, 2024. The document covers topics such as instrumental variables, regression models and the two-stage least squares (2SLS) method.

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Lecture 9: An Introduction to Instrumental Variables 25117 - Econometrics Universitat Pompeu Fabra November 18th, 2024 What we learned in the last lesson Statistical studies are evaluated by asking whether the analysis is internally and externally valid....

Lecture 9: An Introduction to Instrumental Variables 25117 - Econometrics Universitat Pompeu Fabra November 18th, 2024 What we learned in the last lesson Statistical studies are evaluated by asking whether the analysis is internally and externally valid. - A study is internally valid if the statistical inferences about causal effects are valid for the population being studied. In regression estimation of causal effects, there are two main types of threats to internal validity: - First, OLS estimators are biased and inconsistent if the regressors and error terms are correlated. - Second, confidence intervals and hypothesis tests are not valid when the standard errors are incorrect. - A study is externally valid if its inferences and conclusions can be generalized from the population and setting studied to other populations and settings. References 2 / 27 Instrumental Variables Consider the following regression model: Yi = β0 + β1 Xi + ui - OVB, Simultaneous causality bias, Errors-in-variables bias... all three result in E(u | X ) ̸= 0 - Instrumental variables regression can eliminate bias when E(u | X ) ̸= 0, using an instrumental variable (IV), Z. - Intuitively, IV regression breaks X into two parts: 1 an endogenous part that might be correlated with u, and 2 an exogenous part that is not. By isolating the part that is not correlated with u, it is possible to estimate β1 References 3 / 27 Instrumental Variables For an instrumental variable (an “instrument”) Z to be valid, it must satisfy two conditions: 1 Instrument relevance: Z is a strong determinant of X (corr (Z , X ) ̸= 0) 2 Instrument exogeneity (exclusion restriction): Z is unrelated to u (corr (Z , u) = 0) Instrumental Variable (IV) u Z E(u | X ) ̸= 0 X Y References 4 / 27 Two Stages Least Squares (2SLS) Assume that Z satifies both conditions. Then the 2SLS estimator of β1 is obtained by - First Stage — Regress Xi on Zi (including the intercept), and obtain the predicted values X̂i. - Second Stage — Regress Yi on X̂i (including the intercept), the coefficient on X̂i is the 2SLS estimator β̂12SLS Formally, suppose that Z satisfies both conditions. As a first step, suppose that we use it as a proxy for X. The reduced-form estimator, β1RF , reads cov (Zi ,Yi ) cov (Zi ,[β0 +β1 Xi +ui ]) β1RF = var (Zi ) = var (Zi ) = β1 cov (Zi ,Xi ) var (Zi ) cov (Zi ,Yi ) cov (Zi ,Xi ) cov (Zi ,Yi ) ⇒ β1 = var (Zi ) / var (Zi ) = cov (Zi ,Xi ) The IV estimator (replacing population covariances with the sample covariances) is a consistent estimator of β1 : ! sZY p cov (Zi ,Yi ) sXY β̂12SLS = sZX → cov (Zi ,Xi ) = β1 Note that β̂1OLS = sXX References 5 / 27 Two Stages Least Squares (2SLS) For another way to see it, consider the system of reduced-form equations: Xi = π0 + π1 Zi + vi Yi = γ0 + γ1 Zi + wi Because Z is exogenous, Z is uncorrelated with both vi and wi. The idea: An exogenous unit change in Z results in a change in X by π1 units and a change in Y by γ1 units. Thus an exogenous change in X by π1 is associated with an exogenous change in Y by γ1 units. So the effect on Y of an exogenous change in X is β12SLS = γ1 /π1 References 6 / 27 A Buttery Example IV regressions were first developed in 1928 by Philip G. Wright to estimate supply and demand elasticities for agricultural goods,in particular, butter. Consider the following demand equation: ln(Qibutter ) = β0 + β1 ln(Pibutter ) + ui Here, β1 captures the price-elasticity of butter, i.e., the percentage change in quantities due to a 1% change in price. Problem: The OLS regression of ln(Qibutter ) on ln(Pibutter ) suffers from simultaneity (reverse causality) bias. Because prices, ln(Pibutter ), are simultaneously determined in equilibrium by the interplay of supply and demand, β1 was unlikely to capture neither the supply or demand elasticity of butter. References 7 / 27 A Buttery Example - In every period, prices are determined in equilibrium by the simultaneous movements of supply and demand - The equilibrium is determined by the intersection of the demand curve Dk and supply curve Sk References 8 / 27 A Buttery Example - As a result, the set of {Pricek , Quantityk } is not informative about the specific demand and supply elasticities (i.e., the curves’ slopes) - In this example, demand and supply elasticities could be anything, from perfectly elastic (a small change in P is associated with a high change in Q) to completely inelastic (a high change in P is associated with a little change in Q). References 9 / 27 A Buttery Example - To determine the elasticity of demand, we need an IV that would shift only supply curves while leaving demand curves unchanged. - What could be such an instrument? References 10 / 27 A Buttery Example A potential IV for the butter prices to estimate the demand elasticity for butter could be drought shocks to dairy regions (Z ). It is a plausibly revelant IV: below-average rainfall in a dairy region (Z ) could impair grazing and thus reduce butter production at a given price, shifting the supply curve to the left and increasing the equilibrium price (X ) Drought shocks are also plausibly unrelated to the quantities of butter demanded (Y )... except through the price increase (X )! Can you think of an instrument to estimate the supply elasticity of butter? In the California Schools example, what could be an instrument for the share of subsidized meals? References 11 / 27 Large-Sample properties 1 var (Zi−µZ )ui - As n → ∞, β̂12SLS ∼ N (β1 , σβ̂2 ) where σβ̂2 = n [cov (Zi ,Xi )]2 1 1 - Statistical inference proceeds in the usual way. - The justification is (as usual) based on large samples - This all assumes that the instruments are valid – we’ll discuss what happens if they aren’t valid shortly. - However, the OLS standard errors from the second stage regression aren’t right because they do not account for the estimation in the first stage - Instead, use a single specialized command that computes the 2SLS estimator and the correct SEs. - As usual, use heteroskedasticity-robust SEs References 12 / 27 The General IV Regression Model We can extend the simple IV framework to multiple endogenous regressors (X s), exogenous regressors or control variables (W s), and exogenous instruments (Z s). We say the coefficients β1 ,... , βk are - underidentified if there are more regressors than instruments (m < k ). - exactly identified if there are as many regressors as instruments (m = k). - overidentified if there are less regressors than instruments (m > k ). E.g., with a single endogenous regressor: Yi = β0 + β1 X1i + β2 W1i + · · · + βr +1 Wri + ui and m instruments, Z1i ,... , Zmi. References 13 / 27 2SLS with a single endogenous regressor With a single endogenous regressor: Yi = β0 + β1 X1i + β2 W1i + · · · + βr +1 Wri + ui and m instruments, Z1i ,... , Zmi. First Stage: - Regress X1 on all exogenous regressors, W1 ,... , Wr , Z1i ,... , Zmi , plus intercept. - Compute predicted values of X̂1 Second Stage: - Regress Y on X̂1 , W1 ,... , Wr and the intercept. - The coefficients from this second stage regression are the 2SLS estimators, but SEs are wrong - To get correct SEs, do this in a single step in your regression software References 14 / 27 Example with m = 1 – Price Elasticity of cigarettes References 15 / 27 Example with m = 1 – Price Elasticity of cigarettes References 16 / 27 Example with m = 1 – Price Elasticity of cigarettes References 17 / 27 Example with m = 1 – Price Elasticity of cigarettes References 18 / 27 The General IV Regression Assumptions Consider the following specification Yi = β0 + β1 X1i +... βk Xki + βk+1 W1i + · · · + βk+r +1 Wri + ui and m instruments, Z1i ,... , Zmi. Under the following assumptions, the 2SLS estimator is consistent and has a sampling distribution that, in large samples, is approximately normal 1. E(ui | W1i ,... , Wri ) = 0 2. (Yi , X1i ,... , Xki , W1i ,... , Wri , Z1i ,... , Zmi ) are i.i.d. 3. Large outliers are unlikely: The X ’s, W ’s, Z ’s, and Y have nonzero finite fourth moments; 4. The instruments Z1i ,... , Zmi are valid: 4.1 Instrument exogeneity: corr (Z1i , ui ) = 0,... , corr (Zmi , ui ) = 0 4.2 Instrument relevance: Suppose the second stage regression could be run using the predicted values from the population first stage regression. Then there is no perfect multicollinearity in this (infeasible) second-stage regression. Intuitively, the instruments must provide enough information about the exogenous movements in these variables to sort out their separate effects on Y. References 19 / 27 Checking Assumption 4.1 – Instrument exogeneity - Instrument exogeneity: All the instruments are uncorrelated with the error term - In other words, Z can only impact Y through X - If the instruments are correlated with the error team, the first stage of 2SLS cannot isolate a component of X that is uncorrelated with the error team, so X̂ is correlated with u and 2SLS inconsistent. - If there are more instruments than endogenous regressors, it is possible to test – partially – for instrument exogeneity. - Intuition: Consider the simplest case: Yi = β0 + β1 Xi + ui Suppose there are two valid instruments: Z1i and Z2i , then you can compute two separate 2SLS estimates. If these 2 2SLS estimates are very different from each other, then something must be wrong: one or the other (or both) of the instruments must be invalid References 20 / 27 Checking Assumption 4.1 – Instrument exogeneity The J-test of overidentifying restrictions (Anderson-Rubin test) makes this comparison in a statistically precise way. Suppose you have m > k. Then, 1 Estimate the residuals ûi2SLS from Yi = β0 + β1 X1i +... βk Xki + βk +1 W1i + · · · + βk+r +1 Wri + ui 2 Estimate the following model ûi2SLS = γ0 + γ1 Z1i +... γm Zmi + γm+1 W1i + · · · + γm+r +1 Wri + ei 3 Recover the homoskedasticity-only F-statistic testing H0 : γ0 = γ1 = · · · = γm = 0. The overidentifying restrictions test statistic is J = mF 4 Under the null hypothesis that all the instruments are exogenous, if ei is homoskedastic, in large samples J ∼ χ2m−k where m − k is the degree of overidentification References 21 / 27 Checking Assumption 4.2 – Instrument Relevance In the first stage, we regress Xi = π0 + π1 Z1i + π2 Z2i + · · · + πm Zmi + πm+1 W1i + πm+2 W2i + · · · + πm+k Wki + ui The instruments are relevant if at least one of π1 ,... , πm is non-zero The instruments are said to be weak if all the π1 ,... , πm are either zero or nearly zero ⇒ Weak instruments explain very little of the variation in X , beyond that explained by the W ’s. If instruments are weak, the sampling distribution of 2SLS and its t-statistic are not (at all) normal, even with n large. sZY p cov (Zi ,Yi ) Remember that β̂12SLS = sZX → cov (Zi ,Xi ) and when instruments are weak, cov (Zi , Xi ) tends to zero... We cannot conduct statistical inference with the usual methods when instruments are weak (inference under weak instruments is beyond the scope of this course). References 22 / 27 Checking Assumption 4.2 – Instrument Relevance X Z is a strong instrument Z Y References 23 / 27 Checking Assumption 4.2 – Instrument Relevance X Z is a weak instrument Z Y References 24 / 27 Checking Assumption 4.2 – Instrument Relevance So... how relevant must the instruments be to draw causal inference in practice? How weak is weak? - Staiger and Stock (1997)’s Rule of thumb (1 endogenous variable!): Reject that your instruments are weak if F ≥ 10 - Stock and Yogo (2005) formalized and generalized Staiger and Stock (1997) : Reject that your instruments are weak if F > J10 (m) where J10 (m) is chosen such that P(F > J10 (m) | 2SLS bias ≤ 10% × OLS bias) = 0.05 The literature on weak instrument testing is still ongoing (technical details are beyond the scope of this class) So if you have weak instruments, - Get better instruments (often easier said than done!) - If you have many instruments, some are probably weaker than others and it’s a good idea to drop the weaker ones (dropping an irrelevant instrument will increase the first-stage F) - If you only have a few instruments, and all are weak, then you need to switch from 2SLS to a method that is robust to weak instruments. References 25 / 27 Summary - A valid instrument lets us isolate a part of X that is uncorrelated with u, and that part can be used to estimate the effect of a change in X on Y - IV regression hinges on having valid instruments: 1 Exogeneity: Test overidentifying restrictions via the J-statistic + exercise your judgment 2 Relevance: Check via first-stage F - A valid instrument isolates variation in X that is “as if” randomly assigned. - The critical requirement of at least m valid instruments cannot be tested – you must use your head! References 26 / 27 Material I – Textbooks: - Introduction to Econometrics, 4th Edition, Global Edition, by Stock and Watson – Chapters 12 - 13 - Introductory Econometrics, 5th Edition, A Modern Approach, by Jeff. Wooldridge – Chapter 15. - Causal Inference, The Mixtape, by Scott Cunningham – Chapter 6 – Papers: - Angrist, J. D., & Krueger, A. B. (1991). Does compulsory school attendance affect schooling and earnings?. The Quarterly Journal of Economics, 106(4), 979-1014. - Hoxby, C. M. (2000). Does competition among public schools benefit students and taxpayers?. American Economic Review, 90(5), 1209-1238. - Gruber, J., & Hungerman, D. M. (2007). Faith-based charity and crowd-out during the great depression. Journal of Public Economics, 91(5-6), 1043-1069. References 27 / 27

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