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

2. 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 (?)

3. 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.

4. 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)
London Cholera epidemic, ca
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…

5. 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)

6. 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  See Baltagi, page 122 for further analysis.  The data were downloaded from the website for Baltagi's text. 6

8. Specification: Quadratic Effect of Experience

9. The Effect of Education on LWAGE

10. What Influences LWAGE?

11. An Exogenous Influence

12. 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 ]

13. 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

14. OLS

15. 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.

16. 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. 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. Auxiliary Regression for ED to Obtain Residuals

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

20. 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.

21. 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?

25. The General Problem

26. 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

27. 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.

28. 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.

29. 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.

30. Consistency and Asymptotic Normality of the IV Estimator

31. Asymptotic Covariance Matrix of bIV

32. 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

33. 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?

34. 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

35. 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.

36. 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.

37. 2SLS Algebra

38. Asymptotic Covariance Matrix for 2SLS

39. 2SLS Has Larger Variance than LS

40. Estimating σ2

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

42. 2SLS vs. Robust Standard Errors
| Robust Standard Errors |
|Variable | Coefficient | Standard Error |b/St.Er.| B_ B_ B_ B_ B_ B_ B_
| 2SLS Standard Errors | B_ B_ B_ B_ B_ B_ B_

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

44. Hausman Test

45. Hausman Test: One at a Time?

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

47. Wu Test

48. Regression Based Endogeneity Test

49. 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
EXP
EXPSQ D D
OCC
SOUTH
SMSA
WKS
U D-14
WKS
WKSHAT

50. General Test for Endogeneity

51. 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)

53. 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.

54. Weak Instruments (cont.)

55. Weak Instruments

56. A study of moral hazard Riphahn, Wambach, Million: “Incentive Effects in the Demand for Healthcare” Journal of Applied Econometrics, 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%).

57. 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 = ADDON =  insured by add-on insurance = 1; otherswise = 0
HHNINC =  household nominal monthly net income in German marks /
(4 observations with income=0 were dropped) HHKIDS = children under age 16 in the household = 1; otherwise = EDUC =  years of schooling
AGE = age in years
MARRIED = marital status
EDUC = years of education 57

58. Evidence of Moral Hazard?

59. Regression Study

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

61. 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)

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 = is nonsensical in this context.

63. Instrumental Variable Approach
Construct a prediction for T using only the exogenous information Use 2SLS using this instrumental variable.
Magnitude = is also nonsensical in this context.

64. 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.

65. 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,

66. Treatment Effects and Natural Experiments

67. 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.

68. Appendix: Structure and Regression

69. Least Squares Revisited

70. Inference with IV Estimators

71. Comparing OLS and IV

72. Testing for Endogeneity(?)

73. 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.

74. 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.

75. 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

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