1、Econometrics IProfessor William GreeneStern School of BusinessDepartment of EconomicsEconometrics IPart 13 - EndogeneityCornwell and Rupert DataCornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file areEXP = work experienceWKS = weeks workedOCC = occupation, 1 i

2、f blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA= 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by union contractED = years of educationLWAGE = log of wage = dependent variable in regressionsThese data were analyzed in Cornwell,

3、 C. and Rupert, P., Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators, Journal of Applied Econometrics, 3, 1988, pp. 149-155. See Baltagi, page 122 for further analysis. The data were downloaded from the website for Baltagis text. Specification: Quadra

4、tic Effect of ExperienceThe Effect of Education on LWAGEWhat Influences LWAGE?An Exogenous InfluenceInstrumental VariablesStructureLWAGE (ED,EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION)ED (MS, FEM)Reduced Form: LWAGE ED (MS, FEM), EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION Two Stage Least Squares StrategyReduced F

5、orm: LWAGE ED (MS, FEM,X), EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION Strategy (1) Purge ED of the influence of everything but MS, FEM (and the other variables). Predict ED using all exogenous information in the sample (X and Z).(2) Regress LWAGE on this prediction of ED and everything else.Standard errors

6、 must be adjusted for the predicted EDOLSThe weird results for the coefficient on ED happened because the instruments, MS and FEM are dummy variables. There is not enough variation in these variables.Source of EndogeneityLWAGE = f(ED, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) + ED = f(MS,FEM, EXP,EXPSQ,W

7、KS,OCC, SOUTH,SMSA,UNION) + uRemove the EndogeneityLWAGE = f(ED, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) + u + StrategyEstimate uAdd u to the equation. ED is uncorrelated with when u is in the equation.Auxiliary Regression for ED to Obtain ResidualsOLS with Residual (Control Function) Added2SLSA Warnin

8、g About Control FunctionThe ProblemInstrumental VariablesFramework: y = X + , K variables in X.There exists a set of K variables, Z such that plim(ZX/n) 0 but plim(Z/n) = 0The variables in Z are called instrumental variables.An alternative (to least squares) estimator of is bIV = (ZX)-1ZyWe consider

9、 the following:Why use this estimator?What are its properties compared to least squares?We will also examine an important applicationIV EstimatorsConsistentbIV = (ZX)-1Zy = (ZX/n)-1 (ZX/n)+ (ZX/n)-1Z/n = + (ZX/n)-1Z/n Asymptotically normal (same approach to proof as for OLS)Inefficient to be shown.T

10、he General ResultBy construction, the IV estimator is consistent. So, we have an estimator that is consistent when least squares is not.LS as an IV EstimatorThe least squares estimator is (X X)-1Xy = (X X)-1ixiyi = + (X X)-1ixii If plim(XX/n) = Q nonzero plim(X/n) = 0 Under the usual assumptions LS

11、is an IV estimator X is its own instrument.IV EstimationWhy use an IV estimator? Suppose that X and are not uncorrelated. Then least squares is neither unbiased nor consistent.Recall the proof of consistency of least squares: b = + (XX/n)-1(X/n). Plim b = requires plim(X/n) = 0. If this does not hol

12、d, the estimator is inconsistent.A Popular MisconceptionA popular misconception. If only one variable in X is correlated with , the other coefficients are consistently estimated. False. The problem is “smeared” over the other coefficients.Asymptotic Covariance Matrix of bIVAsymptotic EfficiencyAsymp

13、totic efficiency of the IV estimator. The variance is larger than that of LS. (A large sample type of Gauss-Markov result is at work.)(1) Its a moot point. LS is inconsistent.(2) Mean squared error is uncertain:MSEestimator|=Variance + square of bias.IV may be better or worse. Depends on the dataTwo

14、 Stage Least SquaresHow to use an “excess” of instrumental variables(1) X is K variables. Some (at least one) of the K variables in X are correlated with .(2) Z is M K variables. Some of the variables in Z are also in X, some are not. None of the variables in Z are correlated with .(3) Which K varia

15、bles to use to compute ZX and Zy?Choosing the InstrumentsChoose K randomly?Choose the included Xs and the remainder randomly?Use all of them? How?A theorem: (Brundy and Jorgenson, ca. 1972) There is a most efficient way to construct the IV estimator from this subset:(1) For each column (variable) in

16、 X, compute the predictions of that variable using all the columns of Z.(2) Linearly regress y on these K predictions.This is two stage least squaresAlgebraic EquivalenceTwo stage least squares is equivalent to(1) each variable in X that is also in Z is replaced by itself.(2) Variables in X that are

17、 not in Z are replaced by predictions of that X with all the variables in Z that are not in X.2SLS AlgebraAsymptotic Covariance Matrix for 2SLS2SLS Has Larger Variance than LSEstimating 2Cornwell and Rupert DataCornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the f

18、ile areEXP = work experienceWKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA= 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by union contractED = years of educationLWAGE = log of wage = dep

19、endent variable in regressionsThese data were analyzed in Cornwell, C. and Rupert, P., Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators, Journal of Applied Econometrics, 3, 1988, pp. 149-155. See Baltagi, page 122 for further analysis. The data were d

20、ownloaded from the website for Baltagis text. Endogeneity Test? (Hausman) Exogenous EndogenousOLS Consistent, Efficient Inconsistent 2SLS Consistent, Inefficient Consistent Base a test on d = b2SLS - bOLS Use a Wald statistic, dVar(d)-1d What to use for the variance matrix? Hausman: V2SLS - VOLS Hau

21、sman TestEndogeneity Test: WuConsiderable complication in Hausman test (text, pp. 322-323)Simplification: Wu test.Regress y on X and X estimated for the endogenous part of X. Then use an ordinary Wald test.Wu TestNote: .05544 + .54900 = .60444, which is the 2SLS coefficient on ED.Alternative to Haus

22、mans Formula?H test requires the difference between an efficient and an inefficient estimator.Any way to compare any two competing estimators even if neither is efficient?Bootstrap? (Maybe)Measurement Error y = x* + all of the usual assumptionsx = x* + uthe true x* is not observed (education vs. yea

23、rs of school) What happens when y is regressed on x? Least squares attenutation:Why Is Least Squares Attenuated?y = x* + x = x* + uy = x + ( - u)y = x + v, cov(x,v) = - var(u)Some of the variation in x is not associated with variation in y. The effect of variation in x on y is dampened by the measur

24、ement error.Measurement Error in Multiple RegressionTwins Application from the literature: Ashenfelter/Kreuger: A wage equation for twins that includes “schooling.”O(jiān)rthodoxyA proxy is not an instrumental variableInstrument is a noun, not a verbAre you sure that the instrument really exogenous? The “

25、natural experiment.”The First IV Study(Snow, J., On the Mode of Communication of Cholera, 1855)London Cholera epidemic, ca 1853-4Cholera = f(Water Purity,u)+.Effect of water purity on cholera?Purity=f(cholera prone environment (poor, garbage in streets, rodents, etc.). Regression does not work. Two

26、London water companies Lambeth Southwark=|= Main sewage dischargePaul Grootendorst: A Review of Instrumental Variables Estimation of Treatment Effects IV EstimationCholera=f(Purity,u)+Z = water companyCov(Cholera,Z)=Cov(Purity,Z)Z is randomly mixed in the population (two full sets of pipes) and unco

27、rrelated with behavioral unobservables, u)Cholera=+Purity+u+Purity = Mean+random variation+uCov(Cholera,Z)= Cov(Purity,Z)Autism: Natural ExperimentAutism Television watchingWhich way does the causation go?We need an instrument: RainfallRainfall effects staying indoors which influences TV watchingRai

28、nfall is definitely absolutely truly exogenous, so it is a perfect instrument.The correlation survives, so TV “causes” autism.Two Problems with 2SLSZX/n may not be sufficiently large. The covariance matrix for the IV estimator is Asy.Cov(b ) = 2(ZX)(ZZ)-1(XZ)-1If ZX/n - 0, the variance explodes.Addi

29、tional problems:2SLS biased toward plim OLSAsymptotic results for inference fall apart.When there are many instruments, is too close to X; 2SLS es OLS.Weak InstrumentsSymptom: The relevance condition, plim ZX/n not zero, is close to being violated.Detection: Standard F test in the regression of xk o

30、n Z. F 0) HOSPITAL= 1(Number of hospital visits 0) HSAT = health satisfaction, coded 0 (low) - 10 (high) DOCVIS = number of doctor visits in last three months HOSPVIS = number of hospital visits in last calendar year PUBLIC = insured in public health insurance = 1; otherwise = 0 ADDON = insured by a

31、dd-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net e in German marks / 10000. (4 observations with e=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = 0 EDUC = years of schooling AGE = age in years MARRIED = marital status EDUC = years of educationEvidence of Moral Hazard?Regression StudyEndogenous Dummy VariableDoctor Visits = f(Age, Educ, Health,