1. Part 12: Endogeneity 12-1/71 Econometrics I Professor William Greene Stern School of Business Department of Economics

2. Part 12: Endogeneity 12-2/71 Econometrics I Part 12 - Endogeneity

3. Part 12: Endogeneity 12-3/71 Sources of Endogeneity  One or more variables is correlated with the disturbance.  Sources: Omitted variables Nonrandom sampling Nonrandom attrition from the sample Voluntary participation in the program (Not so) old fashioned simultaneous equations Dynamics in panel data Measurement error

4. Part 12: Endogeneity 12-4/71 The First IV Study – Omitted Variable (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 (waste in streets, rodents, etc.)). Regression does not work. Two London water companies Lambeth Vauxhall/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

5. Part 12: Endogeneity 12-5/71 IV Estimation – The Wald Estimator  Cholera=f(Water Purity,u)+ε  Z = water company Z = 1(Lambeth Water)  Cov(Cholera,Z)=δCov(Water Purity,Z); δ = Cov(C,Z)/Cov(WP,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)

6. Part 12: Endogeneity 12-6/71 Generalities

7. Part 12: Endogeneity 12-7/71 The Standard Example The Effect of Education on Earnings

8. Part 12: Endogeneity 12-8/71 What Influences LWAGE?

9. Part 12: Endogeneity 12-9/71 An Exogenous Influence

10. Part 12: Endogeneity 12-10/71 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 ]

11. Part 12: Endogeneity 12-11/71 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 X 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

12. Part 12: Endogeneity 12-12/71 Cornwell and Rupert Data Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 Years Variables in the file are EXP = work experience WKS = weeks worked OCC = occupation, 1 if blue collar, IND = 1 if manufacturing industry SOUTH = 1 if resides in south SMSA= 1 if resides in a city (SMSA) MS = 1 if married FEM = 1 if female UNION = 1 if wage set by union contract ED = years of education LWAGE = 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.

13. Part 12: Endogeneity 12-13/71 Specification: Quadratic Effect of Experience

14. Part 12: Endogeneity 12-14/71 OLS

15. Part 12: Endogeneity 12-15/71 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. 2 Stage Least Squares

16. Part 12: Endogeneity 12-16/71 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

17. Part 12: Endogeneity 12-17/71 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.

18. Part 12: Endogeneity 12-18/71 Auxiliary Regression for ED to Obtain Residuals

19. Part 12: Endogeneity 12-19/71 OLS with Residual (Control Function) Added 2SLS

20. Part 12: Endogeneity 12-20/71 A Warning About Control Function

21. Part 12: Endogeneity 12-21/71 The two stage LS strategy: (The two stage button in your software.) The software regresses EDUC on all independent variables plus the two instrumental variables (stage 1), then takes the predicted value on education and regresses lwage on that predicted value plus the original independent variables (stage 2). Is this correct? Then the second method you showed is the same except the predicted residuals are included in the second stage OLS. Is one method preferred over another? They produce the same results.

22. Part 12: Endogeneity 12-22/71 General Results By construction, the IV estimator is consistent. So, we have an estimator that is consistent when least squares is not.

23. Part 12: Endogeneity 12-23/71 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 b IV = (Z’X) -1 Z’y  We consider the following: Why use this estimator? What are its properties compared to least squares?  We will also examine an important application

24. Part 12: Endogeneity 12-24/71 IV Estimators Consistent b IV = (Z’X) -1 Z’y = (Z’X/n) -1 (Z’X/n)β+ (Z’X/n) -1 Z’ε/n = β+ (Z’X/n) -1 Z’ε/n  β Asymptotically normal (same approach to proof as for OLS) Inefficient – (relative to what?) to be shown.

25. Part 12: Endogeneity 12-25/71 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.

26. Part 12: Endogeneity 12-26/71 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.

27. Part 12: Endogeneity 12-27/71 Asymptotic Covariance Matrix of b IV

28. Part 12: Endogeneity 12-28/71 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

29. Part 12: Endogeneity 12-29/71 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?

30. Part 12: Endogeneity 12-30/71 Choosing the Instruments  Choose K randomly?  Choose the included Xs and the remainder randomly?  Use all of them? How?  A theorem: 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

31. Part 12: Endogeneity 12-31/71 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.

32. Part 12: Endogeneity 12-32/71 2SLS Algebra

33. Part 12: Endogeneity 12-33/71 Asymptotic Covariance Matrix for 2SLS

34. Part 12: Endogeneity 12-34/71 2SLS Has Larger Variance than LS

35. Part 12: Endogeneity 12-35/71 Estimating σ 2

36. Part 12: Endogeneity 12-36/71 Robust Covariance Matrix for 2SLS Actual x ct Predicted x ct

37. Part 12: Endogeneity 12-37/71 Endogeneity Test? (Hausman) Exogenous Endogenous OLS Consistent, Efficient Inconsistent 2SLS Consistent, Inefficient Consistent Base a test on d = b 2SLS - b OLS Use a Wald statistic, d’[Var(d)] -1 d What to use for the variance matrix? Hausman: V 2SLS - V OLS

38. Part 12: Endogeneity 12-38/71 An Endogenous Dummy Variable  HospitalVisits = f(Age,Health,Insurance, Other unobservables)  Insurance = f(Expected Doctor Visits, Age,Health,Gender,Income,Kids, Education, Other unobservables)

39. Part 12: Endogeneity 12-39/71 Hausman Test

40. Part 12: Endogeneity 12-40/71

41. Part 12: Endogeneity 12-41/71 Rank = 1. Use a generalized inverse. Still not significant

42. Part 12: Endogeneity 12-42/71 Endogeneity Test: Wu  Considerable complication in Hausman test (text, pp. 235-236)  Simplification: Wu test.  Regress y on X and X^ estimated for the endogenous part of X. Then use an ordinary Wald test.

43. Part 12: Endogeneity 12-43/71 Wu Test in C&R Wage Equation

44. Part 12: Endogeneity 12-44/71 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)

45. Part 12: Endogeneity 12-45/71

46. Part 12: Endogeneity 12-46/71 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 x k 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.

47. Part 12: Endogeneity 12-47/71 Two Problems with 2SLS  Z’X/n may not be sufficiently large. The covariance matrix for the IV estimator is Asy.Cov(b ) = σ 2 [(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. Part 12: Endogeneity 12-48/71 Orthodoxy  A proxy is not an instrumental variable  Instrument is a noun, not a verb  Are you sure that the instrument is really exogenous? The “natural experiment.”

49. Part 12: Endogeneity 12-49/71 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.

50. Part 12: Endogeneity 12-50/71 Whitehouse (WJS) Commend on Waldman When economists use one variable as a proxy for another—rainfall patterns instead of TV viewing, for example—it’s not always clear what the results actually measure. Prof. Waldman’s willingness to hazard an opinion on a delicate matter of science reflects the growing ambition of economists—and also their growing hubris, in the view of critics. Academic economists are increasingly venturing beyond their traditional stomping ground, a wanderlust that has produced some powerful results but also has raised concerns about whether they’re sometimes going too far.

51. Part 12: Endogeneity 12-51/71 Treatment Effect  Earnings and Education: Effect of an additional year of schooling  Estimating Average and Local Average Treatment Effects of Education when Compulsory Schooling Laws Really Matter Philip Oreopoulos AER, 96,1, 2006, 152-175

52. Part 12: Endogeneity 12-52/71 Treatment Effects and Natural Experiments

53. Part 12: Endogeneity 12-53/71 Treatment Effects  Outcome Y and treatment T = 0/1  Y = T*Y1 + (1-T)*Y0; Potential outcomes Y1 = outcome if treated Y0 = outcome if not treated  Y = x’b + dT + e d = endogenous (causal) treatment effect How to estimate?  2SLS  Sample Selection  Propensity Score Matching

54. Part 12: Endogeneity 12-54/71 Union Effect in LWAGE

55. Part 12: Endogeneity 12-55/71 Union Dummy

56. Part 12: Endogeneity 12-56/71 2SLS

57. Part 12: Endogeneity 12-57/71 Sample Selection

58. Part 12: Endogeneity 12-58/71 Sample Selection MLE

59. Part 12: Endogeneity 12-59/71 Matching Estimates ATT = E[lnWage(1)|X,Union=1] – E[lnWage(1)|X,Union=0]

60. Part 12: Endogeneity 12-60/71 Measurement Error y =  x* +  all of the usual assumptions x = x* + uthe true x* is not observed (education vs. years of school) What happens when y is regressed on x? Least squares attenutation:

61. Part 12: Endogeneity 12-61/71 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.

62. Part 12: Endogeneity 12-62/71 Measurement Error in Multiple Regression

63. Part 12: Endogeneity 12-63/71 Twins Application from the literature: Ashenfelter/Kreuger: A wage equation for twins that includes “schooling.”

64. Part 12: Endogeneity 12-64/71 A study of moral hazard Riphahn, Wambach, Million: “Incentive Effects in the Demand for Healthcare” Journal of Applied Econometrics, 2003 Did the presence of the ADDON insurance influence the demand for health care – doctor visits and hospital visits?

65. Part 12: Endogeneity 12-65/71 Application: Health Care Panel Data German Health Care Usage Data, 7,293 Individuals, Varying Numbers of Periods Variables in the file are Data downloaded from Journal of Applied Econometrics Archive. This is an unbalanced panel with 7,293 individuals. They can be used for regression, count models, binary choice, ordered choice, and bivariate binary choice. This is a large data set. There are altogether 27,326 observations. The number of observations ranges from 1 to 7. (Frequencies are: 1=1525, 2=2158, 3=825, 4=926, 5=1051, 6=1000, 7=987). Note, the variable NUMOBS below tells how many observations there are for each person. This variable is repeated in each row of the data for the person. (Downloaded from the JAE Archive) DOCTOR = 1(Number of doctor visits > 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 add-on insurance = 1; otherswise = 0 HHNINC = household nominal monthly net income in German marks / 10000. (4 observations with income=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 education

66. Part 12: Endogeneity 12-66/71 Evidence of Moral Hazard?

67. Part 12: Endogeneity 12-67/71 Regression Study

68. Part 12: Endogeneity 12-68/71 Endogenous Dummy Variable  HospitalVisits = f(Age,Health,Insurance, Other unobservables)  Insurance = f(Expected Doctor Visits, Age,Health,Gender,Income,Kids, Education, Other unobservables)

69. Part 12: Endogeneity 12-69/71 Approaches  (Parametric) Control Function: Build a structural model for the two variables (Heckman)  (Semiparametric) Instrumental Variable: Create an instrumental variable for the dummy variable (Barnow/Cain/ Goldberger, Angrist, Current generation of researchers)

70. Part 12: Endogeneity 12-70/71 Heckman’s Control Function Approach  Y = xβ + δT + E[ε|T] + {ε - E[ε|T]}  λ = E[ε|T], computed from a model for whether T = 0 or 1 Result is insignificant and goes in the wrong direction.

71. Part 12: Endogeneity 12-71/71 Instrumental Variable Approach Construct a prediction for AddOn using only the exogenous information. Use 2SLS using this instrumental variable. Same non-result.