Instrumental Variables I IVs are not magic 1. Review: Problems that extra variables and experiments don’t solve. 1a. Discussing omitted variable bias in VSPs 2. IV vs. Measurement Error 3. IV vs. Selection Bias: Selection in Military Service & Draft Lottery – Angrist (1990) 4. Wald Estimator 5. IV and Overidentification1. Review: Pop. Parameters - What did your sample regression aspire to estimate?  Sample Population  1. CEF  y = Xb + e, x’e=0 2. BLP  3. Causal Effect  4. Linear Causal Effect  5. Perfectly specified equation model including all relevant variables  In principle #4 and #5 yield identical population parameters for β1 if Cov (x1, ε | β2’x2) = 0, i.e., no omitted variable bias.1. Review: Problems that extra variables and experiments don’t solve. Solution Problem Add the omitted var. experiment instrument 1. Forgot X2 2. Selection 3. Meas. Err. 4. Misspecification 5. Heterogeneity 6. Endogeneity/ Simultaneity Good omitted variables, experimental data and instruments are all hard to find. Where do control functions and matching fit?1a. Discussing omitted variable bias in VSP e.g. selection bias & reconstruction spending (CERP, JPE 2011) -30 3 6Violence-100 -50 0 50 100CERPcoef = .015, (robust) se = .004, t = 3.9CERP w/controls-30 3 6Violence-100 -50 0 50 100CERPcoef = -.009, (robust) se = .004, t = -2.2CERP in FD, 2004-08-30 3 6Violence-100 -50 0 50 100CERPcoef = -.018, (robust) se = .006, t = -3.0CERP in FD, 2007-08Discussing omitted variable bias: reporting results ------ 2004-2008 ------ Incidents per 1000 (1) (2) (3) (4) (5) (6) Basic controls Y Y Time controls Y Y Y Y First differences Y Y Y Pre-existing trend (∆vt-1) Y Y District specific trends Y CERP per Capita 0.0213*** 0.0147*** 0.0115*** -0.00945** -0.0111** -0.0110** (0.004) (0.0038) (0.0040) (0.0043) (0.0043) (0.0046) Pre-existing trend (∆vt-1) 0.195** 0.192** (0.080) (0.087) Constant 0.361*** 0.306** 0.262** 0.217*** -0.124*** 0.0890** (0.085) (0.13) (0.10) (0.046) (0.041) (0.042) Observations 1040 1000 1000 936 832 832 R-squared 0.08 0.25 0.33 0.17 0.21 0.21 MSPE (10-fold CV) 3.52 3.05 2.81 4.77 4.95 5.25 TABLE 4 Violent Incidents on CERP Spending (Berman, Shapiro, Felter JPE, August 2011) Robust standard errors in parentheses, clustered by district. Results are robust to clustering by governorate instead. Regressions weighted by estimated population. Basic controls include sect, unemployment, and income variables (as in Table 3). Time controls include year indicators and their interaction with Sunni vote share (as in Table 3). District specific trends are district effects in a differenced specification. Basic controls are dropped from first-differenced specifications as they do not vary on a semi-annual basis. *** significant at 1% level; ** significant at 5% level; * significant at 10% levelThe limits of adding more variables and of experiments  Adding the extra variables is expensive or impossible  Experiments are often expensive, unethical or impossible - even when possible they are a lot of work (see McIntosh course)2. Instrumental Variables vs. Measurement Error  yi = β0 + β1x*i + εi  x*i not observed. The best we can do is observe a noisy measure of x*.. xi = x*i + vi , Cov (v,x*)=0, Cov(ε,x*)=0, Cov(v, ε)=0 “classical” measurement error assumptions  Rewrite as an omitted variable.. yi = β0 + β1 (xi – v) + εi = β0 + β1xi - β1v + εi , (L) yi = β0 + β1 xi + ui (S), Cov(x,u) ≠ 0  ..and use OVB formula to solve b1s = b1L + b21 b2L2. IV. vs. Meas. Error: Solution  Solution:  A. Find another noisy measure of x* z1i = x*1i + wi , Cov (w,x*)=0, Cov(w,v)=0, Cov (w,ε)=0  B. Note that Cov(z, ε) = 0 (valid) and Cov(z,x) > 0 (relevant)  C.  Why does this work? - Uses only the variance in signal to estimate, IV removes the variance in noise2. IV. Vs. Meas. Error – Returns to Schooling in Twinsville Sample  Orley Ashenfelter and Alan Krueger set up a booth at the annual Twinsville Twins festival in 1991 and surveyed identical and fraternal twins.  They were really looking for fixed-effects estimates but they happened to ask the twins to report each other’s schooling2. IV. Vs. Meas. Error – Returns to Schooling, Twinsville Data, AER 19942. IV. Vs. Meas. Error – Returns to Schooling Note: GLS is Seemingly Unrelated Regression (SUR)3. IV vs. Selection Bias: Military Service and the Draft Lottery  “Be all that you can be!” What’s the effect of military service on lifetime outcomes?  Imagine y = α + β x + ε, where x is an indicator of service, y is some outcome measure, β is the “linear causal effect” we would get if x were randomly assigned and we ran a regression in the population. β = ζXY / ζxx. But (x,y) pairs are drawn from nonexperimental data, so Cov(x,ε) ≠ 0  Selection Bias would be a problem: selection in volunteer military by both individuals and military selections in conscripted military by both individuals and military - Hard to sign the bias. In WWII there was apparently positive selection into the military.3. IV vs. Selection Bias: Draft Lottery  Draft lottery over birthdates instituted in 1967 for 1950 birth cohort.  Someone pulled a ball with a birthdate on it from a rotating bin with 365 balls, on national TV.  z є (0,1) Cov(z, ε) = 0, Cov(z,x) > 0  Everyone with that birthdate was draft eligible but not all eligibles ended up serving (e.g., Bill Clinton was z=1 but x=0.)3. IV vs. Selection: Draft LotteryDifferences in Earnings w/ trend removed [Reduced Form]4. Wald Estimator  For a simple regression with a binary instrument the IV estimator simplifies to a Wald (1940) estimator.  IV estimator can also be interpreted as the ratio of reduced form to first stage, in this simple regression case.  Which variance (subpopulation) provides identifying information? (Forest Gump, Bill Clinton, groaners)Effect of Eligibility on Service [1st Stage]Effect of Service on Earnings [2nd Stage] 487.8/.1594= 3060.25. Efficient IV estimates and over-identification statistics • Table 3 has multiple consistent estimates, so 1) you can estimate more efficiently by combining estimates, and 2) you can test a necessary condition for validity.