1. Microeconometric Modeling
William Greene
Stern School of Business
New York University
New York NY USA
1.2 Extensions of the Linear Regression Model

2. Concepts Models Multiple Imputation Robust Covariance Matrices
Bootstrap
Maximum Likelihood
Method of Moments
Estimating Individual Outcomes
Linear Regression Model
Quantile Regression
Stochastic Frontier

3. Multiple Imputation for Missing Data

4. Imputed Covariance Matrix

5. Implementation
SAS, Stata: Create full data sets with imputed values inserted. M = 5 is the familiar standard number of imputed data sets. Data are replicated and redistributed
SAS: Standard procedure and code distributed.
Stata: Elaborate imputation equations, M=5
NLOGIT
Create an internal map of the missing values and a set of engines for filling missing values
Loop through imputed data sets during estimation.
M may be arbitrary – memory usage and data storage are independent of M. Data may be replicated

6. Regression with Conventional Standard Errors

7. Robust Covariance Matrices
Robust standard errors, not estimates
Robust to: Heteroscedasticty
Not robust to: (all considered later)
Correlation across observations
Individual unobserved heterogeneity
Incorrect model specification
‘Robust inference’ means hypothesis tests and confidence intervals using robust covariance matrices

8. A Robust Covariance Matrix
Uncorrected

9. Bootstrap Estimation of the Asymptotic Variance of an Estimator
Known form of asymptotic variance: Compute from known results
Unknown form, known generalities about properties: Use bootstrapping
Root N consistency
Sampling conditions amenable to central limit theorems
Compute by resampling mechanism within the sample.

10. Bootstrapping Algorithm
1. Estimate parameters using full sample:  b 2. Repeat R times: Draw n observations from the n, with replacement Estimate  with b(r). 3. Estimate variance with V = (1/R)r [b(r) - b][b(r) - b]’ (Some use mean of replications instead of b. Advocated (without motivation) by original designers of the method.)

11. Application: Correlation between Age and Education

12. Bootstrapped Regression

13. Bootstrap Replications

14. Bootstrapped Confidence Intervals Estimate Norm()=(12 + 22 + 32 + 42)1/2

16. Quantile Regression Q(y|x,) = x,  = quantile
Estimated by linear programming
Q(y|x,.50) = x, .50  median regression
Median regression estimated by LAD (estimates same parameters as mean regression if symmetric conditional distribution)
Why use quantile (median) regression?
Semiparametric
Robust to some extensions (heteroscedasticity?)
Complete characterization of conditional distribution

17. Estimated Variance for Quantile Regression

18. Quantile Regressions
 = .25
 = .50
 = .75

19. OLS vs. Least Absolute Deviations

21. Coefficient on MALE dummy variable in quantile regressions

22. A Production Function Model with Inefficiency The Stochastic Frontier Model

23. Inefficiency in Production

24. Cost Inefficiency
y* = f(x)  C* = g(y*,w) (Samuelson – Shephard duality results) Cost inefficiency: If y < f(x), then C must be greater than g(y,w). Implies the idea of a cost frontier. lnC = lng(y,w) + u, u > 0.

25. Corrected Ordinary Least Squares

26. COLS Cost Frontier

27. Stochastic Frontier Models
Motivation:
Factors not under control of the firm
Measurement error
Differential rates of adoption of technology
Frontier is randomly placed by the whole collection of stochastic elements which might enter the model outside the control of the firm.
Aigner, Lovell, Schmidt (1977),
Meeusen, van den Broeck (1977),
Battese, Corra (1977)

28. The Stochastic Frontier Model
ui > 0, but vi may take any value. A symmetric distribution, such as the normal distribution, is usually assumed for vi. Thus, the stochastic frontier is
+’xi+vi
and, as before, ui represents the inefficiency.

29. Least Squares Estimation
Average inefficiency is embodied in the third moment of the disturbance εi = vi - ui. So long as E[vi - ui] is constant, the OLS estimates of the slope parameters of the frontier function are unbiased and consistent. (The constant term estimates α-E[ui]. The average inefficiency present in the distribution is reflected in the asymmetry of the distribution, which can be estimated using the OLS residuals:

30. Application to Spanish Dairy Farms
N = 247 farms, T = 6 years ( )
Input
Units
Mean
Std. Dev.
Minimum
Maximum
Milk
Milk production (liters)
131,108
92,539
14,110
727,281
Cows
# of milking cows
2.12
11.27
4.5
82.3
Labor
# man-equivalent units
1.67
0.55
1.0
4.0
Land
Hectares of land devoted to pasture and crops.
12.99
6.17
2.0
45.1
Feed
Total amount of feedstuffs fed to dairy cows (tons)
57,941
47,981
3,924.14
376,732

31. Example: Dairy Farms

32. The Normal-Half Normal Model

33. Skew Normal Variable

34. Estimation: Least Squares/MoM
OLS estimator of β is consistent
E[ui] = (2/π)1/2σu, so OLS constant estimates α+ (2/π)1/2σu
Second and third moments of OLS residuals estimate
Method of Moments: Use [a,b,m2,m3] to estimate [,,u, v]

35. Standard Form: The Skew Normal Distribution

36. Log Likelihood Function
Waldman (1982) result on skewness of OLS residuals: If the OLS residuals are positively skewed, rather than negative, then OLS maximizes the log likelihood, and there is no evidence of inefficiency in the data.

37. Airlines Data – 256 Observations

38. Least Squares Regression

40. Alternative Models: Half Normal and Exponential

41. Normal-Exponential Likelihood
Other Models
Many other parametric models
Semiparametric and nonparametric – the recent outer reaches of the theoretical literature
Other variations including heterogeneity in the frontier function and in the distribution of inefficiency

42. A Test for Inefficiency?
Base test on u = 0 <=>  = 0
Standard test procedures
Likelihood ratio
Wald
Lagrange Multiplier
Nonstandard testing situation:
Variance = 0 on the boundary of the parameter space
Standard chi squared distribution does not apply.

44. Estimating ui No direct estimate of ui
Data permit estimation of yi – β’xi. Can this be used?
εi = yi – β’xi = vi – ui
Indirect estimate of ui, using E[ui|vi – ui]
This is E[ui|yi, xi]
vi – ui is estimable with ei = yi – b’xi.

45. Fundamental Tool - JLMS
We can insert our maximum likelihood estimates of all parameters.
Note: This estimates E[u|vi – ui], not ui.

46. Application: Electricity Generation

47. Estimated Translog Production Frontiers

48. Inefficiency Estimates

49. Estimated Inefficiency Distribution

50. Estimated Efficiency

51. A Semiparametric Approach
Y = g(x,z) + v - u [Normal-Half Normal] (1) Locally linear nonparametric regression estimates g(x,z) (2) Use residuals from nonparametric regression to estimate variance parameters using MLE (3) Use estimated variance parameters and residuals to estimate technical efficiency.

52. Airlines Application

53. Efficiency Distributions

54. Nonparametric Methods - DEA

55. DEA is done using linear programming

57. Methodological Problems with DEA
Measurement error
Outliers
Specification errors
The overall problem with the deterministic frontier approach

58. DEA and SFA: Same Answer?
Christensen and Greene data
N=123 minus 6 tiny firms
X = capital, labor, fuel
Y = millions of KWH
Cobb-Douglas Production Function vs. DEA

60. Comparing the Two Methods.