Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

Endogeneity y = X+ε, Definition: E[ε|x]≠0 Why not? Omitted variables Unobserved heterogeneity (equivalent to omitted variables) Measurement error on the RHS (equivalent to omitted variables) Structural aspects of the model Endogenous sampling and attrition Simultaneity (?)

Instrumental Variable Estimation One “problem” variable – the “last” one yit = 1x1it + 2x2it + … + KxKit + εit E[εit|xKit] ≠ 0. (0 for all others) There exists a variable zit such that E[xKit| x1it, x2it,…, xK-1,it,zit] = g(x1it, x2it,…, xK-1,it,zit) In the presence of the other variables, zit “explains” xit E[εit| x1it, x2it,…, xK-1,it,zit] = 0 In the presence of the other variables, zit and εit are uncorrelated. A projection interpretation: In the projection XKt =θ1x1it,+ θ2x2it + … + θk-1xK-1,it + θK zit, θK ≠ 0.

London Cholera epidemic, ca 1853-4 Cholera = f(Water Purity,u)+ε. The First IV Study: Natural Experiment (Snow, J., On the Mode of Communication of Cholera, 1855) http://www.ph.ucla.edu/epi/snow/snowbook3.html London Cholera epidemic, ca 1853-4 Cholera = f(Water Purity,u)+ε. ‘Causal’ 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 River Thames Paul Grootendorst: A Review of Instrumental Variables Estimation of Treatment Effects… http://individual.utoronto.ca/grootendorst/pdf/IV_Paper_Sept6_2007.pdf

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)

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. 6

Specification: Quadratic Effect of Experience

The Effect of Education on LWAGE

What Influences LWAGE?

An Exogenous Influence

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 ]

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

OLS

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.

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

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.

Auxiliary Regression for ED to Obtain Residuals

OLS with Residual (Control Function) Added 2SLS

A Warning About Control Functions Sum of squares is not computed correctly because U is in the regression. A general result. Control function estimators usually require a fix to the estimated covariance matrix for the estimator.

On Sat, May 3, 2014 at 4:48 PM, … wrote: Dear Professor Greene, I am giving an Econometrics course in Brazil and we are using your great textbook. I got a question which I think only you can help me. In our last class, I did a formal proof that var(beta_hat_OLS) is lower or equal than var(beta_hat_2SLS), under homoscedasticity.  We know this assertive is also valid under heteroscedasticity, but a graduate student asked me the proof (which is my problem).  Do you know where can I find it?

The General Problem

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

IV Estimators bIV = (Z’X)-1Z’y = (Z’X/n)-1 (Z’X/n)β+ (Z’X/n)-1Z’ε/n * 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. * By construction, the IV estimator is consistent. We have an estimator that is consistent when least squares is not.

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.

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.

Consistency and Asymptotic Normality of the IV Estimator

Asymptotic Covariance Matrix of bIV

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

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?

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

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 using All other variables in X that are not correlated with ε All the variables in Z that are not in X.

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.

2SLS Algebra

Asymptotic Covariance Matrix for 2SLS

2SLS Has Larger Variance than LS

Estimating σ2

Robust estimation of VC “Predicted” X “Actual” X

2SLS vs. Robust Standard Errors +--------------------------------------------------+ | Robust Standard Errors | +---------+--------------+----------------+--------+ |Variable | Coefficient | Standard Error |b/St.Er.| B_1 45.4842872 4.02597121 11.298 B_2 .05354484 .01264923 4.233 B_3 -.00169664 .00029006 -5.849 B_4 .01294854 .05757179 .225 B_5 .38537223 .07065602 5.454 B_6 .36777247 .06472185 5.682 B_7 .95530115 .08681261 11.000 | 2SLS Standard Errors | B_1 45.4842872 .36908158 123.236 B_2 .05354484 .03139904 1.705 B_3 -.00169664 .00069138 -2.454 B_4 .01294854 .16266435 .080 B_5 .38537223 .17645815 2.184 B_6 .36777247 .17284574 2.128 B_7 .95530115 .20846241 4.583

Endogeneity Test? (Hausman) Exogenous Endogenous OLS Consistent, Efficient Inconsistent 2SLS 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

Hausman Test

Hausman Test: One at a Time?

Endogeneity Test: Wu Considerable complication in Hausman test (text, pp. 234-237) Simplification: Wu test. Regress y on X and estimated for the endogenous part of X. Then use an ordinary Wald test.

Wu Test

Regression Based Endogeneity Test

Testing Endogeneity of WKS (1) Regress WKS on 1,EXP,EXPSQ,OCC,SOUTH,SMSA,MS. U=residual, WKSHAT=prediction (2) Regress LWAGE on 1,EXP,EXPSQ,OCC,SOUTH,SMSA,WKS, U or WKSHAT +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| Constant -9.97734299 .75652186 -13.188 .0000 EXP .01833440 .00259373 7.069 .0000 19.8537815 EXPSQ -.799491D-04 .603484D-04 -1.325 .1852 514.405042 OCC -.28885529 .01222533 -23.628 .0000 .51116447 SOUTH -.26279891 .01439561 -18.255 .0000 .29027611 SMSA .03616514 .01369743 2.640 .0083 .65378151 WKS .35314170 .01638709 21.550 .0000 46.8115246 U -.34960141 .01642842 -21.280 .0000 -.341879D-14 WKS .00354028 .00116459 3.040 .0024 46.8115246 WKSHAT .34960141 .01642842 21.280 .0000 46.8115246

General Test for Endogeneity

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)

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.

Weak Instruments (cont.)

Weak Instruments

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%).

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 57

Evidence of Moral Hazard?

Regression Study

Endogenous Dummy Variable Doctor Visits = f(Age, Educ, Health, Presence of Insurance, Other unobservables) Insurance = f(Expected Doctor Visits, Other unobservables)

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)

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.

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.

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.

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

Treatment Effects and Natural Experiments

How do panel data fit into this? We can use the usual models. We can use far more elaborate models We can study effects through time Observations are surely correlated. The same individual is observed more than once Unobserved heterogeneity that appears in the disturbance in a cross section remains persistent across observations (on the same ‘unit’). Procedures must be adjusted. Dynamic effects are likely to be present.

Appendix: Structure and Regression

Least Squares Revisited

Inference with IV Estimators

Comparing OLS and IV

Testing for Endogeneity(?)

Structure vs. Regression Reduced Form vs. Stuctural Model Simultaneous equations origin Q(d) = a0 + a1P + a2I + e(d) (demand) Q(s) = b0 + b1P + b2R + e(s) (supply) Q(.) = Q(d) = Q(s) What is the effect of a change in I on Q(.)? (Not a regression) Reduced form: Q = c0 + c1I + c2R + v. (Regression) Modern concepts of structure vs. regression: The search for causal effects.

Implications The structure is the theory The regression is the conditional mean There is always a conditional mean It may not equal the structure It may be linear in the same variables What is the implication for least squares estimation? LS estimates regressions LS does not necessarily estimate structures Structures may not be estimable – they may not be identified.

Structure and Regression Simultaneity? What if E[ε|x]≠0 y=x+ε, x=δy+u. Cov[x, ε]≠0 x is not the regression? What is the regression? Reduced form: Assume ε and u are uncorrelated. y = [/(1- δ)]u + [1/(1- δ)]ε x= [1/(1- δ)]u + [δ /(1- δ)]ε Cov[x,y]/Var[x] =  The regression is y = x + v, where E[v|x]=0

Structure vs. Regression Supply = a + b*Price + c*Capacity Demand = A + B*Price + C*Income