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Econometrics I

Econometrics I. Professor William Greene Stern School of Business Department of Economics. Econometrics I. Part 13 - Endogeneity. Cornwell and Rupert Data. Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years Variables in the file are

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Econometrics I

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  1. Econometrics I Professor William Greene Stern School of Business Department of Economics

  2. Econometrics I Part 13 - Endogeneity

  3. Cornwell and Rupert Data Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file are EXP = 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 = dependent variable in regressions These 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 downloaded from the website for Baltagi's text.

  4. Specification: Quadratic Effect of Experience

  5. The Effect of Education on LWAGE

  6. What Influences LWAGE?

  7. An Exogenous Influence

  8. Instrumental Variables • Structure • LWAGE (ED,EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) • ED (MS, FEM) • Reduced Form: LWAGE[ ED (MS, FEM),EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION ]

  9. Two Stage Least Squares Strategy • Reduced Form: 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 must be adjusted for the predicted ED

  10. OLS

  11. The 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.

  12. Source of Endogeneity • LWAGE = f(ED, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) +  • ED = f(MS,FEM, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) + u

  13. Remove the Endogeneity • LWAGE = f(ED, EXP,EXPSQ,WKS,OCC, SOUTH,SMSA,UNION) + u +  • Strategy • Estimate u • Add u to the equation. ED is uncorrelated with  when u is in the equation.

  14. Auxiliary Regression for ED to Obtain Residuals

  15. OLS with Residual (Control Function) Added 2SLS

  16. A Warning About Control Function

  17. The Problem

  18. Instrumental Variables • Framework: y = X + , K variables in X. • There exists a set of K variables, Z such that plim(Z’X/n) 0 but plim(Z’/n) = 0 The variables in Z are called instrumental variables. • An alternative (to least squares) estimator of  is bIV = (Z’X)-1Z’y • We consider the following: • Why use this estimator? • What are its properties compared to least squares? • We will also examine an important application

  19. IV Estimators Consistent bIV = (Z’X)-1Z’y = (Z’X/n)-1 (Z’X/n)β+ (Z’X/n)-1Z’ε/n = β+ (Z’X/n)-1Z’ε/n  β Asymptotically normal (same approach to proof as for OLS) Inefficient – to be shown.

  20. The General Result By construction, the IV estimator is consistent. So, we have an estimator that is consistent when least squares is not.

  21. LS as an IV Estimator The least squares estimator is (X X)-1Xy = (X X)-1ixiyi =  + (X X)-1ixiεi If plim(X’X/n) = Q nonzero plim(X’ε/n) = 0 Under the usual assumptions LS is an IV estimator X is its own instrument.

  22. IV Estimation Why 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 =  + (X’X/n)-1(X’/n). Plim b =  requires plim(X’/n) = 0. If this does not hold, the estimator is inconsistent.

  23. A Popular Misconception A 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.

  24. Asymptotic Covariance Matrix of bIV

  25. Asymptotic Efficiency Asymptotic 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) It’s a moot point. LS is inconsistent. (2) Mean squared error is uncertain: MSE[estimator|β]=Variance + square of bias. IV may be better or worse. Depends on the data

  26. Two Stage Least Squares How 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 ε. (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 ε. (3) Which K variables to use to compute Z’X and Z’y?

  27. Choosing the Instruments • Choose 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 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 squares

  28. Algebraic Equivalence • Two 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 not in Z are replaced by predictions of that X with all the variables in Z that are not in X.

  29. 2SLS Algebra

  30. Asymptotic Covariance Matrix for 2SLS

  31. 2SLS Has Larger Variance than LS

  32. Estimating σ2

  33. Cornwell and Rupert Data Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file are EXP = 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 = dependent variable in regressions These 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 downloaded from the website for Baltagi's text.

  34. Endogeneity Test? (Hausman) Exogenous EndogenousOLS Consistent, Efficient Inconsistent2SLS Consistent, Inefficient Consistent Base a test on d = b2SLS - bOLS Use a Wald statistic, d’[Var(d)]-1d What to use for the variance matrix? Hausman: V2SLS - VOLS

  35. Hausman Test

  36. Endogeneity Test: Wu • Considerable 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.

  37. Wu Test Note: .05544 + .54900 = .60444, which is the 2SLS coefficient on ED.

  38. Alternative to Hausman’s 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)

  39. Measurement Error y = x* +  all of the usual assumptions x = x* + u the true x* is not observed (education vs. years of school) What happens when y is regressed on x? Least squares attenutation:

  40. Why Is Least Squares Attenuated? y = x* +  x = x* + u y = 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 measurement error.

  41. Measurement Error in Multiple Regression

  42. Twins Application from the literature: Ashenfelter/Kreuger: A wage equation for twins that includes “schooling.”

  43. Orthodoxy • A proxy is not an instrumental variable • Instrument is a noun, not a verb • Are you sure that the instrument really exogenous? The “natural experiment.”

  44. The First IV Study(Snow, J., On the Mode of Communication of Cholera, 1855) London Cholera epidemic, ca 1853-4 Cholera = f(Water Purity,u)+ε. Effect of water purity on cholera? Purity=f(cholera prone environment (poor, garbage in streets, rodents, etc.). Regression does not work. Two London water companies Lambeth Southwark ======|||||====== Main sewage discharge Paul Grootendorst: A Review of Instrumental Variables Estimation of Treatment Effects…http://individual.utoronto.ca/grootendorst/pdf/IV_Paper_Sept6_2007.pdf

  45. IV Estimation Cholera=f(Purity,u)+ε Z = water company Cov(Cholera,Z)=δCov(Purity,Z) Z is randomly mixed in the population (two full sets of pipes) and uncorrelated with behavioral unobservables, u) Cholera=α+δPurity+u+ε Purity = Mean+random variation+λu Cov(Cholera,Z)= δCov(Purity,Z)

  46. Autism: Natural Experiment • Autism ----- Television watching • Which way does the causation go? • We need an instrument: Rainfall • Rainfall effects staying indoors which influences TV watching • Rainfall is definitely absolutely truly exogenous, so it is a perfect instrument. • The correlation survives, so TV “causes” autism.

  47. Two Problems with 2SLS • Z’X/n may not be sufficiently large. The covariance matrix for the IV estimator is Asy.Cov(b ) = σ2[(Z’X)(Z’Z)-1(X’Z)]-1 • If Z’X/n -> 0, the variance explodes. • Additional problems: • 2SLS biased toward plim OLS • Asymptotic results for inference fall apart. • When there are many instruments, is too close to X; 2SLS becomes OLS.

  48. Weak Instruments • Symptom: The relevance condition, plim Z’X/n not zero, is close to being violated. • Detection: • Standard F test in the regression of xk on Z. F < 10 suggests a problem. • F statistic based on 2SLS – see text p. 351. • Remedy: • Not much – most of the discussion is about the condition, not what to do about it. • Use LIML? Requires a normality assumption. Probably not too restrictive.

  49. A study of moral hazardRiphahn, Wambach, Million: “Incentive Effects in the Demand for Healthcare”Journal of Applied Econometrics, 2003Did the presence of the ADDON insurance influence the demand for health care – doctor visits and hospital visits?For a simple example, we examine the PUBLIC insurance (89%) instead of ADDON insurance (2%).

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