Ec1123 Section 7 Instrumental Variables

Ec1123 Section 7 Instrumental Variables Andrea Passalacqua Harvard University andreapassalacqua@g.harvard.edu November 16th, 2017 Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 1 / 28

Outline 1 Simultaneous Causality 2 Instrumental Variable Regression: Introduction Conditions 3 IV: Examples 4 Two-Stage Least Squares 5 Testing the Validity of Instruments Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 2 / 28

Simultaneous causality: What is that? Y i = β 0 + β 1 X i + β 2 W 1i + β 3 W 2i + u i Simultaneous causality arises when X Y and X Y OLS will fail because conditional mean independence (CMI) is violated. Recall CMI: E[u X, W 1, W 2] = E[u W 1, W 2] conditional on Z 1 and Z 2, X is as good as randomly assigned However, it is not as fixable as OVB (ie. when E[u X ] 0) since adding controls won t address the fact that X Y Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 3 / 28

Examples of simultaneous causality QUANTITY i = β 0 + β 1 PRICE i + u i Changes in price affect quantities supplied and demanded Changes in quantity supplied and demanded affect price Price and quantity are jointly determined by a set of simultaneous equations (ie. the demand and supply curve) Other examples: Democracy and growth Parental involvement and child s performance in school Police and crime Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 4 / 28

Instrumental Variables Instrumental Variables (IV) are useful for estimating models with measurement error with simultaneous causality or with omitted variable bias IV especially useful when we cannot plausibly control for all omitted variables More generally, whenever conditional mean independence on X fails Our estimated coefficient is biased and cannot be interpreted causally IV relies on using a valid instrumental variable Z to recover as-if random assignment of X Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 6 / 28

Instrumental Variables Y i = β 0 + β 1 X i + u i We want to know the causal effect of X on Y W 1i W 2i X i Y i S 1i but are confounded by OVB and simultaneous causality ˆβ 2 is biased Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 7 / 28

Conditions for IV: Intuition Y i = β 0 + β 1 X i + u i Z is an instrumental variable for X if: Condition 1: Relevance Z is related to X Condition 2: Exogeneity of Z Two ways of saying the Exogeneity condition: Z is as-if randomly assigned The only relationship between Z and Y goes through X after conditioning on any control variable W s. Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 8 / 28

Instrumental Variables Y i = β 0 + β 1 X i + u i Z 1i W 1i W 2i X i Y i S 1i Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 9 / 28

Instrumental Variables Y i = β 0 + β 1 X i + u i Z 1i W 1i W 2i X i Y i Z 2i S 1i There can be multiple instruments Z 1 and Z 2 for the same X Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 10 / 28

IV - Instrument Relevance In the single-variable case: Y i = β 0 + β 1 X i + u i Suppose the model fails CMI (Recall CMI: E[u X, W 1, W 2 ] = E[u W 1, W 2 ]), but we have a valid instrument Z: ˆβ IV = Cov(Y, Z) Cov(X, Z) Notice that clearly we need Cov(X, Z) 0. In fact, in our dataset, we really want Cov(X, Z) to be far away from zero. An instrument where Cov(X, Z) (ie. the relevance of Z) is close to zero is known as a weak instrument. Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 11 / 28

IV - Exogeneity Condition Y i = β 0 + β 1 X i + u i All of these would violate Condition 2 (exogeneity): Z i Y i Z i S 1i Z i W 1i and Z W 2i But we can control for W 1i or W 2i In other words, if Z is related to Y (causally or not) through any relationship other than through X. In the figure, this would be any path from Z to Y that does not go through X. Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 12 / 28

Instrumental Variables RECAP Consider Y i = β 0 + β 1 X i + u i where OLS yields a biased ˆβ 1 Conditions for IV Z is an instrumental variable for X in this model if: 1 Relevance: Z is related to X Corr(Z, X ) 0 2 Exogeneity: The only relationship between Z and Y goes through X Corr(Z, u) = 0 Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 13 / 28

Instrumental Variables Example: Roommate Assignment Suppose we analyzing among first semester college freshmen: GPA i = β 0 + β 1 (Hours Studying) i + u i E[u X ] 0 because of omitted variable bias. Proposed instrumental variable: Z = whether randomly assigned roommate brought a video game to college Relevant? Exogeneity? Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 15 / 28

Instrumental Variables Example: Roommate Assignment GPA i = β 0 + β 1 (Hours Studying) i + u i Condition 1: Relevance Z is related to X. i.e. Corr(Z, X ) 0 How to check? Examine the first-stage relationship between Z and X : First-stage: (Hours Studying) i = γ 0 + γ 1 Z i + v i regress hours videogame From lecture, ˆγ 1 = 0.668 (large and significant) (later) Conduct a F -test to confirm Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 16 / 28

Instrumental Variables Example: Roommate Assignment GPA i = β 0 + β 1 (Hours Studying) i + u i Condition 2: Exogeneity The only relationship between Z and Y goes through X : Corr(Z, u) = 0 Condition 2 holds if roommates having video games (Z) only affects GPA (Y ) through affecting hours studying (X ) Plausible? Are there other channels through which Z is related to Y? What if being assigned a roommate with video games made students sleep less? What if male students are more likely to bring video games and males have lower grades on average? (for argument s sake) Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 17 / 28

Instrumental Variables Example: Roommate Assignment The Z to Y alternate channel does NOT have to be causal to violate Condition 2 More mathematically: hours of sleep and gender lie in the error term u, so we have Corr(Z, u) 0 So control for hours of sleep (W 1i ) and gender (W 2i ): GPA i = β 0 + β 1 (Hours Studying) i + β 2 W 1i + β 3 W 2i + u i Similar to OVB, we can never directly test whether the exogeneity condition is violated or satisfied since u is unobserved. We can only provide arguments (using theory or institutional knowledge) Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 18 / 28

Examples of Instrumental Variables Z X Y How does prenatal health affect a child s long-run development? In womb during Ramadan Prenatal health Adult health & income What effect does serving in the military have on future wages? Military draft lottery # Military service Income What is the effect of rioting on community development? Rainfall on day Number of Long-run of MLK assassination riots property values Each of these examples also requires some control variables W s for the exogeneity condition to hold. In general, arguing the exogeneity condition can be very difficult. Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 19 / 28

IV in STATA IV regression of Y on X using instrument Z: ivregress 2sls y (x = z), robust IV regression of Y on X using instrument Z and controls W 1 and W 2 : ivregress 2sls y w1 w2 (x = z), robust IV regression of Y on X using instruments Z 1 and Z 2 and controls W 1 and W 2 : ivregress 2sls y w1 w2 (x = z1 z2), robust where 2sls stands for Two-Stage Least Squares Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 21 / 28

Two-Stage Least Squares (TSLS or 2SLS) Goal of IV: estimate the causal relationship of X on Y using instrument Z Y i = β 0 + β 1 X i + u i 2SLS estimates ˆβ 1 in two stages : Stage 1: Regress X on Z and calculate predicted values ˆX i ˆX i = ˆγ 0 + ˆγ 1 Z i Stage 2: Regress Y on ˆX to get ˆβ 1 = ˆβ 2SLS = ˆβ IV Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 22 / 28

2SLS Intuition Y i = β 0 + β 1 X i + u i We cannot causally interpret β 1 since X is not randomly assigned Think of the variation (not variance) in X as coming from two separate sources: Variation in X = As-if random part + Non-random part The non-random part is giving us problems (OVB or simultaneous causality) Stage 1 of 2SLS isolates the as-if random part of X, which is ˆX. Since Z is a valid instrument for X in this model, Z is as-if randomly assigned by Condition 2 The variation in X related to Z is also as-if randomly assigned Stage 2 uses the as-if random part to estimate the causal relationship of X on Y. Regressing Y on ˆX is like regressing Y on the random variation in X. Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 23 / 28

Testing the Validity of Instruments Y i = β 0 + β 1 X i + β 2 W 1i + β 3 W 2i + u i Conditions for IV Z is an instrumental variable for X in this model if controlling for W s: 1 Relevance: Corr(Z, X ) 0 2 Exogeneity: Corr(Z, u) = 0 Testing Condition 1 is straightforward, since we have data on both Z and X Testing Condition 2 is trickier, because we never observe u. In fact, we can only test Condition 2 when we have more instruments Zs than X s Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 25 / 28

Testing Condition 1: Relevance Condition 1: Relevance Z must be related to X. i.e. Corr(Z, X ) 0 We need the relationship between X and Z to be meaningfully large How to check? Run first-stage regression with OLS X i = α 0 + α 1 Z 1i + α 2 Z 2i + α 3 W 1i + α 4 W 2i + + v i Check the F-test on all the coefficients on the instruments H 0 : α 1 = α 2 = 0 If ˆF > 10, we claim that Z is a strong instrument If ˆF 10, we have a weak instruments problem Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 26 / 28

Testing Condition 2: Exogeneity Condition 2: Exogeneity of Z Z is as-if randomly assigned. i.e. Corr(Z, u) = 0 To check exogeneity, we need more instruments Zs than endogenous X s (ie., our model is overidentified) Suppose there is one treatment variable of interest X, multiple Zs, potentially, multiple control variables W s. Y i = β 0 + β 1 X i + β 2 W 1i + β 3 W 2i + u i Use Z 1 to estimate ˆβ 1 and predict û i. If Z 1 and Z 2 are both valid instruments, then: Corr(Z 2, û) = 0 This is the basic idea of a J-test for a model with just one instrument and one endogenous variable there is no formal test Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 27 / 28

Testing Condition 2: Exogeneity J-test for overidentifying restrictions: H 0 : H a : Both Z 1 and Z 2 satisfy the exogeneity condition Either Z 1, Z 2, or both are invalid instruments In STATA: ivregress 2sls y w1 w2 (x = z1 z2), robust estat overid display "J-test = " r(score) " p-value = " r(p score) If the p-value < 0.05, then we reject the null hypothesis that all our instruments are valid But just like an F -test, rejecting the test does not reveal which instrument is invalid, only that at least one fails the exogeneity condition Andrea Passalacqua (Harvard) Ec1123 Section 7 Instrumental Variables November 16th, 2017 28 / 28