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

2. The Incidental Parameters Problem
The model is correctly specified
The log likelihood is correctly specified and maximized
The estimator of 2 is inconsistent
The number of parameters grows with N
The “bias” in the MLE gets smaller as T grows
At infinite T, the estimator is consistent in N
In the linear FEM, the MLE of 2 is affected by this problem.

3. FE Log Likelihood for Normal 

5. The Incidental Parameters Problem
The true value of 2 is The estimator converges on 4 from below.

6. Chamberlain’s Approach and Minimum Distance Estimation
Chamberlain (1984) “Panel Data,” Handbook of Econometrics
Innovation: treat the panel as a system of equations: SUR Models, See Wooldridge, Ch. 7 through p. 172.
Assumptions:
Balanced panel
Minimal restrictions on variances and covariances of disturbances (zero means, finite fourth moments)
Model the correlation between effects and regressors

7. Chamberlain

8. Chamberlain - Data

9. Chamberlain Model

10. Chamberlain SUR Model

11. Chamberlain – Implied Model

12. Chamberlain Estimation of Π
FGLS. Use the usual two step GLS estimator.
OLS. System has an unrestricted covariance matrix and the same regressors in every equation. GLS = FGLS = equation by equation OLS. Denote the T OLS coefficient vectors as
P = [p1, p2, p3 …, pT].
Unconstrained OLS will be consistent. Plim pt = πt, t=1,…,T
OLS is inefficient. There are T(T-1) different estimates of  in P and T-1 estimates of each δt.

13. Chamberlain Estimation of Σ

14. Chamberlain Estimator: Application
Cornwell and Rupert:
Lwageit = αi + β1Expit + β2Expit2 + β3Wksit + εit
αi projected onto all 7 periods of Exp, Exp2 and Wks.
For each of the 7 years, we regress Lwageit on a constant
and the three variables for all 7 years.
Each regression has 22 coefficients.

15. Chamberlain Approach Least Squares Estimates
What They Estimate
There are 7 estimates of 
There are potentially 42 estimates of 
There are potentially 6 estimates of each t.
How do we “average” the different estimates to get a single estimate?

16. Efficient Estimation of Π
Minimum Distance Estimation: Chamberlain (1984). (See Wooldridge, pp )
Asymptotically efficient
Assumes only finite fourth moments of vit
Maximum likelihood Estimation: Joreskog (1981), Greene (1981,2008)
Add normality assumption
Same asymptotic properties as MDE (!)

17. MDE-1 Cornwell and Rupert. Pooled, all 7 years
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
Constant|
EXP |
EXPSQ | D
WKS |
OCC |
IND |
SOUTH |
SMSA |
MS |
FEM |
UNION |
ED |

18. MDE-2 Cornwell and Rupert. Year 1
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
Constant|
EXP |
EXPSQ |
WKS |
OCC |
IND |
SOUTH |
SMSA |
MS |
FEM |
UNION |
ED |

19. MDE-3 Cornwell and Rupert. Year 7
|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|
Constant|
EXP |
EXPSQ |
WKS |
OCC |
IND |
SOUTH |
SMSA |
MS |
FEM |
UNION |
ED |

20. MDE-4

21. MDE-5

22. MDE-6

23. MDE-7
S11 S21
S12 S22

24. MDE-8

25. MDE-9

26. Carey Hospital Cost Model

27. Multiple Estimates (25) of 10 Structural Parameters

31. Appendix I. Chamberlain Model Algebra

32. Minimum Distance Estimation

33. MDE (2)

34. MDE (3)

35. Maximum Likelihood Estimation

36. MLE (2)

37. Rearrange the Panel Data

38. Generalized Regression Model

39. Least Squares

40. GLS and FGLS