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

2. Regression Extensions
Time Varying Fixed Effects
Measurement Error
Spatial Autoregression and Autocorrelation (Baltagi 10.5)

3. Time Varying Effects Models
Time Varying Fixed Effects: Additive yit = β’xit + ai(t) + εit yit = β’xit + ai + ct + εit ai(t) = ai + ct, t=1,…,T Two way fixed effects model Now standard in fixed effects modeling.

4. Evidence of technical change
LSDV least squares with fixed effects ....
LHS=YIT Mean =
Standard deviation =
No. of observations = DegFreedom Mean square
Regression Sum of Squares =
Residual Sum of Squares =
Total Sum of Squares =
Standard error of e = Root MSE
Fit R-squared = R-bar squared
Estd. Autocorrelation of e(i,t) =
Panel:Groups Empty , Valid data
Smallest 6, Largest
Average group size in panel
Variances Effects a(i) Residuals e(i,t)
Std.Devs
Rho squared: Residual variation due to ai
Within groups variation in YIT D+02
R squared based on within group variation
Between group variation in YIT D+03
| Standard Prob % Confidence
YIT| Coefficient Error z |z|>Z* Interval
X1| ***
X2| ***
X3|
X4| ***
T| Base = 1993
1994 | ***
1995 | ***
1996 | ***
1997 | ***
1998 | ***
Evidence of technical change

5. Time Varying Fixed Effects
911
Rescue

6. Need for Clarification

7. Time Varying Fixed Effects

8. A Partial Fixed Effects Model

9. Time Varying Effects Models
Time Varying Fixed Effects: Additive Polynomial
yit = β’xit + ai(t) + εit
yit = β’xit + ai0 + ai1t + ai2t2+ εit
Let Wi = [1,t,t2]Tx3 Ai = stack of Wi with 0s inserted
Use OLS, Frisch and Waugh. Extend “within” estimator.
Note Ai’Aj = 0 for all i  j.
See Cornwell, Schmidt, Sickles (1990) (Frontiers literature.)

10. Cornwell Schmidt Sickles

13. calc;list;r2=1-(1482-col(x)-3*247)*sst/((n-1)*var(yit))$
[CALC] R =
F[2*247, *247]
= ( )/(2*247) / (( )/(1482 – 4 – 3*247))
= 6.45
Wald = 6.45*494 = Critical chi squared for 494 DF =

14. Time Varying Effects Models
Random Effects yit = β’xit + εit + ai(t) or yit = β’xit + εit + uig(t,) A heteroscedastic random effects model Stochastic frontiers literature – Battese-Coelli (1992)

16. Munnell State Production Model

17. No Effects

18. Quadratic Fixed Effects
Correct DF: (48)=666
Multiply standard errors by sqr(810/666) = 1.103

19. Time Varying Effects Models
Time Varying Fixed Effects: Multiplicative yit = β’xit + ai(t) + εit yit = β’xit + it + εit Not estimable. Needs a normalization. 1 = 1. An EM iteration: (Chen (2015).)

20. EM Algorithm (Chen (2015))

21. Measurement Error

22. General Conclusions About Measurement Error
In the presence of individual effects, inconsistency is in unknown directions
With panel data, different transformations of the data (first differences, group mean deviations) estimate different functions of the parameters – possible method of moments estimators
Model may be estimable by minimum distance or GMM
With panel data, lagged values may provide suitable instruments for IV estimation.
Various applications listed in Baltagi (pp ).

23. Application: A Twins Study

24. Wage Equation

26. Spatial Autocorrelation
Thanks to Luc Anselin, Ag. U. of Ill.

27. Spatially Autocorrelated Data
Per Capita Income in
Monroe County, NY
Thanks Arthur J. Lembo Jr., Geography, Cornell.

28. Hypothesis of Spatial Autocorrelation
Thanks to Luc Anselin, Ag. U. of Ill.

29. Testing for Spatial Autocorrelation
W = Spatial Weight Matrix. Think “Spatial Distance Matrix.” Wii = 0.

30. Modeling Spatial Autocorrelation

31. Spatial Autoregression

32. Generalized Regression
Potentially very large N – GPS data on agriculture plots
Estimation of . There is no natural residual based estimator
Complicated covariance structure – no simple transformations

33. Spatial Autocorrelation in Regression

34. Panel Data Application: Spatial Autocorrelation

36. Spatial Autocorrelation in a Panel

37. Spatial Autocorrelation in a Sample Selection Model
Flores-Lagunes, A. and Schnier, K., “Sample Selection and Spatial Dependence,” Journal of Applied Econometrics, 27, 2, 2012, pp
 Alaska Department of Fish and Game.
 Pacific cod fishing eastern Bering Sea – grid of locations
 Observation = ‘catch per unit effort’ in grid square
 Data reported only if 4+ similar vessels fish in the region
 sample = 320 observations with 207 reported full data

38. Spatial Autocorrelation in a Sample Selection Model
 LHS is catch per unit effort = CPUE
 Site characteristics: MaxDepth, MinDepth, Biomass
 Fleet characteristics:
Catcher vessel (CV = 0/1)
Hook and line (HAL = 0/1)
Nonpelagic trawl gear (NPT = 0/1)
Large (at least 125 feet) (Large = 0/1)

39. Spatial Autocorrelation in a Sample Selection Model

40. Spatial Autocorrelation in a Sample Selection Model

41. Spatial Weights

42. Appendix: Miscellaneous

43. Ordinary Least Squares
Standard results for OLS in a GR model
Consistent
Unbiased
Inefficient
Variance does (we expect) converge to zero;

44. Estimating the Variance for OLS

45. White Estimator for OLS

46. Generalized Least Squares

47. Maximum Likelihood

48. Conclusion Het. in Effects
Choose robust OLS or simple FGLS with moments based variances.
Note the advantage of panel data – individual specific variances
As usual, the payoff is a function of
Variance of the variances
The extent to which variances are correlated with regressors.
MLE and specific models for variances probably don’t pay off much unless the model(s) for the variances is (are) of specific interest.

49. Generalized Regression

50. OLS Estimation

51. Feasible GLS

52. GLS Estimation

53. Heteroscedasticity
Naturally expected in microeconomic data, less so in macroeconomic
Model Platforms
Fixed Effects
Random Effects
Estimation
OLS with (or without) robust covariance matrices
GLS and FGLS
Maximum Likelihood

54. Dear Professor Greene, I have to apply multiplicative heteroscedastic models, that I studied in your book, to the analysis of trade data. Since I have not found any Matlab implementations, I am starting to write the method from scratch. I was wondering if you are aware of reliable implementations in Matlab or any other language, which I can use as a reference. 

55. Baltagi and Griffin’s Gasoline Data
World Gasoline Demand Data, 18 OECD Countries, 19 years Variables in the file are
COUNTRY = name of country
YEAR = year, LGASPCAR = log of consumption per car LINCOMEP = log of per capita income LRPMG = log of real price of gasoline
LCARPCAP = log of per capita number of cars
See Baltagi (2001, p. 24) for analysis of these data. The article on which the analysis is based is Baltagi, B. and Griffin, J., "Gasoline Demand in the OECD: An Application of Pooling and Testing Procedures," European Economic Review, 22, 1983, pp  The data were downloaded from the website for Baltagi's text.

56. Heteroscedastic Gasoline Data

57. Does Teaching Load Affect Faculty Size. Becker, W. , Greene, W
Does Teaching Load Affect Faculty Size? Becker, W., Greene, W., Seigfried, J.
Do Undergraduate Majors or PhD Students Affect Faculty Size? American Economist 56(1): Becker, Jr., W.E., W.H. Greene & J.J. Siegfried. 2011
Mundlak form of Random Effects

58. Random Effects Regressions

59. Modeling the Scedastic Function

60. Two Step Estimation

61. Heteroscedasticity in the RE Model

62. LSDV Residuals

63. Evidence of Country Specific Heteroscedasticity

64. Heteroscedasticity in the FE Model
Ordinary Least Squares
Within groups estimation as usual.
Standard treatment – this is just a (large) linear regression model.
White estimator

65. In order to test robustness two versions of the fixed effects model were run. The first is Ordinary Least Squares, and the second is heteroscedasticity and auto-correlation robust (HAC) standard errors in order to check for heteroscedasticity and autocorrelation.
[Only one version of the model was computed. There was no “check.”]

66. Narrower Assumptions

67. Heteroscedasticity in Gasoline Data
| Least Squares with Group Dummy Variables |
| LHS=LGASPCAR Mean = |
| Fit R-squared = |
| Adjusted R-squared = |
Least Squares - Within
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X|
LINCOMEP
LRPMG
LCARPCAP
White Estimator
LINCOMEP
LRPMG
LCARPCAP
White Estimator using Grouping
LINCOMEP
LRPMG
LCARPCAP

68. Estimating the Variance Components: Baltagi
Invoking Mazodier and Trognon (1978) and Baltagi and Griffin (1988).

69. Estimating the Variance Components: Hsiao
So, who’s right? Hsiao. This is no longer in Baltagi.
Invoking Mazodier and Trognon (1978) and Baltagi and Griffin (1988).

70. Maximum Likelihood

71. OLS and PCSE +--------------------------------------------------+
| Groupwise Regression Models |
| Pooled OLS residual variance (SS/nT) |
| Test statistics for homoscedasticity: |
| Deg.Fr. = 17 C*(.95) = C*(.99) = |
| Lagrange multiplier statistic = |
| Wald statistic = |
| Likelihood ratio statistic = |
| Log-likelihood function = |
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
Constant
LINCOMEP
LRPMG
LCARPCAP
| OLS with Panel Corrected Covariance Matrix |
Constant
LINCOMEP
LRPMG
LCARPCAP

72. FGLS +--------------------------------------------------+
| Groupwise Regression Models |
| Pooled OLS residual variance (SS/nT) |
| Log-likelihood function = |
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
Constant
LINCOMEP
LRPMG
LCARPCAP
| Test statistics against the correlation |
| Deg.Fr. = 153 C*(.95) = C*(.99) = |
| Likelihood ratio statistic = |
Constant
LINCOMEP
LRPMG
LCARPCAP

73. Autocorrelation Source? Already present in RE model – equicorrelated.
Models:
Autoregressive: εi,t = ρεi,t-1 + vit – how to interpret
Unrestricted: (Already considered)
Estimation requires an estimate of ρ

74. FGLS – Fixed Effects

75. FGLS – Random Effects

76. Microeconomic Data - Wages
| Least Squares with Group Dummy Variables |
| LHS=LWAGE Mean = |
| Model size Parameters = |
| Degrees of freedom = |
| Estd. Autocorrelation of e(i,t) |
|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
OCC
SMSA MS
EXP
EXPSQ D

77. Macroeconomic Data – Baltagi/Griffin Gasoline Market
| Least Squares with Group Dummy Variables |
| LHS=LGASPCAR Mean = |
| Estd. Autocorrelation of e(i,t) |
|Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] |
LINCOMEP
LRPMG
LCARPCAP

78. Aggregation Test

79. Baltagi and Griffin’s Gasoline Data
World Gasoline Demand Data, 18 OECD Countries, 19 years Variables in the file are
COUNTRY = name of country
YEAR = year, LGASPCAR = log of consumption per car LINCOMEP = log of per capita income LRPMG = log of real price of gasoline
LCARPCAP = log of per capita number of cars
See Baltagi (2001, p. 24) for analysis of these data. The article on which the analysis is based is Baltagi, B. and Griffin, J., "Gasoline Demand in the OECD: An Application of Pooling and Testing Procedures," European Economic Review, 22, 1983, pp  The data were downloaded from the website for Baltagi's text.

80. A Test Against Aggregation
Log Likelihood from restricted model = Free parameters in  and Σ are (19)/2 = 175.
Log Likelihood from model with separate country dummy variables = Free parameters in  and Σ are = 192
Chi-squared[17]=2( )=442.07
Critical value= Homogeneity hypothesis is rejected a fortiori.