Stochastic Production Functions II: Maximum Likelihood

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Stochastic Production Functions II Maximum
Likelihood

• Lecture XI

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Normal-Half Normal Model

• Assumptions about Errors

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Distribution functions

• The distribution function of v follows the
  standard zero-mean normal distribution function

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• The half-normal distribution is represented by

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• Assuming independence

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• Since e v - u , or by definition of the
  composed error term

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• Integrating u out, we obtain the marginal
  distribution function for e
• From Weinstein

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• x is distributed normal, while y is distributed
  half-normal

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• where

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• Note that if a 0

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• Substituting in

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• By integration

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• Again note that if a 0

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• Thus,

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• Note that as ?-gt 0 , either sv2 -gt 8 or su2 -gt 0
  or the symmetric error dominates the one-sided
  component.

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• Note that as ?-gt 8 , either su2 -gt 8 or sv2 -gt 0
  or the symmetric error dominates the one-sided
  component.

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Maximum Likelihood

• The parameters of the model can be estimated by
  maximizing

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Results
Half-Normal Estimates Half-Normal Estimates
Parameter Estimate
A_0 4.99564
(0.03574)
A_1 0.00903
(0.00706)
A_2 0.00504
(0.00500)
A_3 0.00452
(0.00424)
sigma 0.45639
(0.02126)
lambda 5.08765
(0.77545)
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• More mess

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Comparison Comparison Comparison Comparison
Stochastic Frontier Stochastic Frontier Ordinary Least Squares Ordinary Least Squares
Parameter Estimate Parameter Estimate
A_0 4.99564 A_0 4.58582
(0.03574) (0.05570)
A_1 0.00903 A_1 0.01265
(0.00706) (0.01171)
A_2 0.00504 A_2 0.01677
(0.00500) (0.00728)
A_3 0.00452 A_3 0.01322
(0.00424) (0.00625)