# Part 13: Endogeneity 13-1/62 Econometrics I Professor William Greene Stern School of Business Department of Economics.

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Part 13: Endogeneity 13-1/62 Econometrics I Professor William Greene Stern School of Business Department of Economics

Part 13: Endogeneity 13-2/62 Econometrics I Part 13 - Endogeneity

Part 13: Endogeneity 13-3/62 I am here to ask a little help for endogeneity. I have a main regression, in which the independent variabels are lagged 1 year (this is an unbalanced panel dataset); I use fixed effect, xtreg: Main Regression: Yt = Xt-1 + Qt-1 + Z3t-1 I suspect endogeneity: variable X may be itself determined by prior-year Y. As a solution, I read this strategy: regress the endogenous variable Xt-1 on the dependent variable (Yt-2) and other independent variables (i.e., Qt-2 and Zt-2); these Y Q and Z are all in year t-2, while X is in t-1. Then, from this regression, calculate the “predicted” values for X, and include them as a control-for- endogeneity (e.g., a variable named “Endogeneity-control”) in the main regression above. Question 1: in the Main Regression above, when including the control for endogeneity (i.e., the variable “Endogeneity-control”), do I have to lag its value? That is, do I have to include Endogeneity-control in t-1? or just the predicted values, without lagging?

Part 13: Endogeneity 13-4/62 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.

Part 13: Endogeneity 13-5/62 Specification: Quadratic Effect of Experience

Part 13: Endogeneity 13-6/62 The Effect of Education on LWAGE

Part 13: Endogeneity 13-7/62 What Influences LWAGE?

Part 13: Endogeneity 13-8/62 An Exogenous Influence

Part 13: Endogeneity 13-9/62 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 ]

Part 13: Endogeneity 13-10/62 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

Part 13: Endogeneity 13-11/62 OLS

Part 13: Endogeneity 13-12/62 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.

Part 13: Endogeneity 13-13/62 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

Part 13: Endogeneity 13-14/62 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.

Part 13: Endogeneity 13-15/62 Auxiliary Regression for ED to Obtain Residuals

Part 13: Endogeneity 13-16/62 OLS with Residual (Control Function) Added 2SLS

Part 13: Endogeneity 13-17/62 A Warning About Control Function

Part 13: Endogeneity 13-18/62 The Problem

Part 13: Endogeneity 13-19/62 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

Part 13: Endogeneity 13-20/62 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 – to be shown.

Part 13: Endogeneity 13-21/62 The General Result By construction, the IV estimator is consistent. So, we have an estimator that is consistent when least squares is not.

Part 13: Endogeneity 13-22/62 LS as an IV Estimator The least squares estimator is (X X) -1 Xy = (X X) -1  i x i y i =  + (X X) -1  i x i ε i 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.

Part 13: Endogeneity 13-23/62 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.

Part 13: Endogeneity 13-24/62 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.

Part 13: Endogeneity 13-25/62 Asymptotic Covariance Matrix of b IV

Part 13: Endogeneity 13-26/62 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

Part 13: Endogeneity 13-27/62 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?

Part 13: Endogeneity 13-28/62 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

Part 13: Endogeneity 13-29/62 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.

Part 13: Endogeneity 13-30/62 2SLS Algebra

Part 13: Endogeneity 13-31/62 Asymptotic Covariance Matrix for 2SLS

Part 13: Endogeneity 13-32/62 2SLS Has Larger Variance than LS

Part 13: Endogeneity 13-33/62 Estimating σ 2

Part 13: Endogeneity 13-34/62 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.

Part 13: Endogeneity 13-35/62 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

Part 13: Endogeneity 13-36/62 Hausman Test

Part 13: Endogeneity 13-37/62 Hausman Test: One at a Time?

Part 13: Endogeneity 13-38/62 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.

Part 13: Endogeneity 13-39/62 Wu Test Note:.05544 +.54900 =.60444, which is the 2SLS coefficient on ED.

Part 13: Endogeneity 13-40/62 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)

Part 13: Endogeneity 13-41/62

Part 13: Endogeneity 13-42/62 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:

Part 13: Endogeneity 13-43/62 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.

Part 13: Endogeneity 13-44/62 Measurement Error in Multiple Regression

Part 13: Endogeneity 13-45/62 Twins Application from the literature: Ashenfelter/Kreuger: A wage equation for twins that includes “schooling.”

Part 13: Endogeneity 13-46/62 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.”

Part 13: Endogeneity 13-47/62 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

Part 13: Endogeneity 13-48/62 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)

Part 13: Endogeneity 13-49/62 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.

Part 13: Endogeneity 13-50/62 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.

Part 13: Endogeneity 13-51/62 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.

Part 13: Endogeneity 13-52/62 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? For a simple example, we examine the PUBLIC insurance (89%) instead of ADDON insurance (2%).

Part 13: Endogeneity 13-53/62 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

Part 13: Endogeneity 13-54/62 Evidence of Moral Hazard?

Part 13: Endogeneity 13-55/62 Regression Study

Part 13: Endogeneity 13-56/62 Endogenous Dummy Variable  Doctor Visits = f(Age, Educ, Health, Presence of Insurance, Other unobservables)  Insurance = f(Expected Doctor Visits, Other unobservables)

Part 13: Endogeneity 13-57/62 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)  (?) Propensity Score Matching (Heckman et al., Becker/Ichino, Many recent researchers)

Part 13: Endogeneity 13-58/62 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 Magnitude = 11.1200 is nonsensical in this context.

Part 13: Endogeneity 13-59/62 Instrumental Variable Approach Construct a prediction for T using only the exogenous information Use 2SLS using this instrumental variable. Magnitude = 23.9012 is also nonsensical in this context.

Part 13: Endogeneity 13-60/62 Propensity Score Matching  Create a model for T that produces probabilities for T=1: “Propensity Scores”  Find people with the same propensity score – some with T=1, some with T=0  Compare number of doctor visits of those with T=1 to those with T=0.

Part 13: Endogeneity 13-61/62 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

Part 13: Endogeneity 13-62/62 Treatment Effects and Natural Experiments

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