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Stochastic Production
Functions II: Maximum
Likelihood
Lecture XI
Normal-Half Normal Model
 Assumptions about
Errors:
2
v
vi ~ N  0,

ui ~ N  0,
2
u
*

vi and ui independent
Distribution functions:
 The distribution function of v follows
the standard zero-mean normal
distribution function
 v 
f v 
exp   2 
2 v
 2 v 
1
2
 The half-normal distribution is
represented by
2

2
u 
g u  
exp   2 
2 v
 2 u 
 Assuming independence
2
2

2
u
v 
f  u, v   f  v  g  u  
exp   2  2 
2 u v
 2 u 2 v 
 Since ε = v - u , or by definition of
the composed error term
2
2

  u

2
u
f  u,   
exp   2 
2

2 u v
2

2

u
v





 Integrating u out, we obtain the
marginal distribution function for ε
 From Weinstein
X  X 

x
X
Y  Y 

y
Y
x is distributed normal, while y is
distributed half-normal
 x2 
1
f  x 
exp   
2
 2 
y

f


y  a

g  y    1  F  a 
 
 

0 y  a

 


where

  Y
X
F u  
u


f  x  dx
 Note that if a = 0
g  y 

 
f y
 1  F  0  
 
f y
 1  0.5

  
2f y


y2 

exp  
2
   
2 X 
X


2
 Substituting in

  Y
X


2
y2
g  y 
exp  

 Y
2 X Y
X

X

X



2


 y2 
2

exp   2 
2 Y

 Y 


 By integration
Q t  

t



 dz  g  z  x  f  x  dx
t
1


za
dz 


 1  F  a  


t a

zx
f
 f  x  dx
  
 tx
1
 a  

F
  F    f  x  dx


1  F a      
  
 



 Again note that if a = 0
 tx

1
Q t  
F
  F  0   f  x  dx


1  F  0       

t
 t x

 2  F 
  0.5 f  x  dx
  

 
t
1 tx 
 2  2F 
  1 f  x  dx
2    

t
 Thus,

f      f  u ,   du
0
2

2
 2 

   
1       exp   2 2 
 



2 
 
 
   
  
   
   
2
u

 u
2
v
v
 Note that as λ-> 0 , either σv2 -> ∞ or
σu2 -> 0 or the symmetric error
dominates the one-sided component.
 Note that as λ-> ∞ , either σu2 -> ∞ or
σv2 -> 0 or the symmetric error
dominates the one-sided component.
Maximum Likelihood
 The parameters of the model can be
estimated by maximizing
  i  
1
ln  L   N ln     ln    


2
i 1
     2
N
N
2

i
i 1
Results
Half-Normal Estimates
Parameter
A_0
Estimate
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)
 More mess

 u
v
  u   v
   
2
u
2
2
v
    
2
2

2
v

  
2
v
2
v
2
2
 1

2
    
2
u
2
2
v
 1
2
v
 v  4.98
 u  1.04
Comparison
Stochastic Frontier
Ordinary Least Squares
Parameter
Estimate
Parameter
Estimate
A_0
4.99564
A_0
4.58582
(0.03574)
A_1
0.00903
(0.05570)
A_1
(0.00706)
A_2
0.00504
(0.01171)
A_2
(0.00500)
A_3
0.00452
(0.00424)
0.01265
0.01677
(0.00728)
A_3
0.01322
(0.00625)