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Probability behind Probability behind Spatial Stochastic Frontier modelSpatial Stochastic Frontier modelSpatial Stochastic Frontier modelSpatial Stochastic Frontier model
Dmitry Pavlyuk
The Mathematical Seminar,The Mathematical Seminar,
Transport and Telecommunication Institute, Riga, 08.05.2015
Outline
• Classical stochastic frontier model
• A case of truncated normal inefficiency
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
• A case of truncated normal inefficiency
• Spatial stochastic frontier model
• Moment generating and characteristic functions
• Closed/Unified skew-normal distribution
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 2
Economic background of stochastic
frontier model
We consider a company, which uses K inputs, indexed k
= 1, 2, …, K, to produce M outputs, indexed m = 1, 2, …,
M:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
= 1, 2, …, K, to produce M outputs, indexed m = 1, 2, …,
M:
Than production possibility set is defined as:
( )( ).,...,,
,,...,,
21
21
M
K
yyyy
xxxx
==
{ }yxyxPPS producecan :,=
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
The set of feasible outputs for an input vector:
3
{ }yxyxPPS producecan :,=
( ) ( ){ }PPSyxyxP ∈= ,:
Production possibility frontier
A production possibility frontier is defined as a
function:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( ) ( ) ( ){ }xPyxPyyxf yy ∉∀∈= > ',: '
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 4
Technical efficiency
Debreu-Farrell definition of technical efficiency of
output vector y is:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
output vector is:
can be presented in a form
of equation:
( ) ( ){ }1
:sup,−
≤= xfyyxTE θθθ
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
of equation:
5
( ) ( )yxTExfy ,⋅=
Classical stochastic frontier
For estimation purposes the technical efficiency term is
usually transformed as:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
usually transformed as:
Introducing the random disturbances v into the
formula, we consider a classical stochastic frontier model:
( ) ( ) .0,exp, ≥−= uuyxTE
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
where β is a vector of unknown coefficients.
6
( ) ( ) ( )iiii uvxfy −⋅⋅= expexp, β
Classical stochastic frontier
The model is frequently presented in the logarithmic
form:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
form:
So the only probabilistic feature of the stochastic
frontier model is a composed error term:
( ) iiii uvxfy −+= β,lnln
uv −=ε
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 7
iii uv −=ε
Outline
• Classical stochastic frontier model
• A case of truncated normal inefficiency
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
• A case of truncated normal inefficiency
• Spatial stochastic frontier model
• Moment generating and characteristic functions
• Closed/Unified skew-normal distribution
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 8
Stochastic frontier: a case of
truncated normal inefficiency
The distribution of the random disturbances v is
usually set to independent identically distributed normal
with zero mean and constant deviation σ :
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
usually set to independent identically distributed normal
with zero mean and constant deviation σv:
The inefficiency term u can be modelled with different
distributions, e.g. truncated normal:
( )2,0~ vi Nv σ
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
distributions, e.g. truncated normal:
9
( )2,0 ,~ ui TNu σµ+∞
Stochastic frontier: a case of
truncated normal inefficiency
The probability density function for truncated normal
distribution is (truncation limits are set to 0 and +∞ to
match the non-negativity requirement of ):
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
distribution is (truncation limits are set to 0 and +∞ to
match the non-negativity requirement of u):
( )( )
<
≥
−−
Φ
=
−
0,0
0,2
exp2
12
21
i
iu
i
uui
u
uu
uf σµ
σπσµ
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 10
Stochastic frontier: a case of
truncated normal inefficiency
According to the convolution formula, if u and v are
independent, then:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( ) ( ) ( )∫∞+
∞−
+=
−=
iiuiivi
iii
duufuff
uv
εε
ε
ε
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 11
Stochastic frontier: a case of
truncated normal inefficiency
Transformations:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( ) ( ) ( ) =+=+∞
∫ iiuiivi duufuff εεε
( ) ( )
( ) ( )
=
+−−
+−
Φ=
=
−−+−
Φ=
=
−−
Φ
+−=
∞+−
∞+−
∞+ −
∞−
∫
∫
∫
∫
221
02
2
2
21
02
21
2
2
/1exp
1exp
1
22exp
2
1
2exp
2
1
2exp
2
1
vuii
iu
i
v
ii
uvu
iu
i
uuv
ii
v
iiuiivi
du
duuu
duuu
µσσεµεµ
σµ
σε
σπσσµ
σµ
σπσµ
σε
σπ
ε
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 12
( )
( )
++
+−Φ
++
Φ
+=
=
++
+−−
+−
Φ=
−
∫
222222
1
22
0222222
/
/1
/2exp
2exp
2
uvvuuv
vui
uv
i
uuv
i
uvvuuv
vui
uv
i
uvu
du
σσσσµ
σσσσε
σσµεϕ
σµ
σσ
σσσσσσσσσπσσ
Univariate Extended skew normal
distribution
So
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( )( )
++
+−Φ
++
Φ
+=
−
222222
1
22
/1 vuiiif
σσσσµ
σσσσε
σσµεϕ
σµ
σσεε
This density function is
known as an
univariate
extended skew normal
( )( )
+
++
−Φ
+
Φ+
=22222222 / uvvuuvuvuuv
ifσσσσσσσσ
ϕσσσ
εε
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
extended skew normal
distribution function,
introduced by Azzalini (1985).
13
Outline
• Classical stochastic frontier model
• A case of truncated normal inefficiency
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
• A case of truncated normal inefficiency
• Spatial stochastic frontier model
• Moment generating and characteristic functions
• Closed/Unified skew-normal distribution
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 14
Spatial stochastic frontier model
A basic assumption of the classical stochastic frontier:
Observations are independent!
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
Observations are independent!
( )( )nu
nvn
IMVTNu
IMVNv
uvXy
2~,0
2~
,~
,,0~
,
σµσ
β
+∞
−+=
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
In practice, there are a lot of theoretical and empirical
evidences of relationships between observations.
15
Spatial stochastic frontier model
One of the simplest forms of this relationship is a linear
combination:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
combination:
In spatial econometrics this is assumed that
coefficients wij can be explained with a distance between
objects i and j, e.g.
∑≠ ji
jiji ywy on depends
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
objects i and j, e.g.
16
( )jiij objectobject
w,distance
1=
Spatial stochastic frontier model
Spatial contiguity matrix:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
...0 112 nww
Matrix form of dependence (spatial lags):
=
0...
............
...0
...0
21
221
112
nn
n
n
ww
ww
ww
W
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
Matrix form of dependence (spatial lags):
17
Wyyywyji
jijini on depends on depends ..1 ⇔∀ ∑≠
=
Spatial stochastic frontier model
The spatial stochastic frontier model specification:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
,)( uvXWXβYWρY s −+++= β
where
.~,~
,)(
uuWρu
vvWρv
uvXWXβYWρY
uu
vv
sXYY
+=+=
−+++= β
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
• WY and ρY are contiguity matrix and coefficient for endogenous spatial effects (spatial
dependency),
• WX and β(s) are contiguity matrix and coefficients for exogenous spatial effects,
• Wv and ρv are contiguity matrix and coefficients for spatially correlated random
disturbances (spatial heterogeneity),
• Wu and ρu are contiguity matrix and coefficients for spatially related inefficiency.
18
Spatial stochastic frontier model
Assuming that
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( )( )nu
nvn
IMVTNu
IMVNv2~,0
2~
,~~,,0~~
σµσ
+∞
and using a simple transformation
we obtain random component distributions:( )MVNv ,,0~ Σ
( )nu IMVTNu ~,0 ,~ σµ+∞
( )( ) ,~
,~
1
1
uWρIu
vWρIv
uu
vv
−
−
−=
−=
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 19
( )( ) ( )[ ]
( )( ) ( )[ ]Tuuuuuu
u
T
vvvvvv
vn
WρIWρI
MVTNu
WρIWρI
MVNv
112~
,0
112~
,,~
,,0~
−−
+∞
−−
−−=Σ
Σ−−=Σ
Σ
σ
µσ
Spatial stochastic frontier model
The main problem is a distribution of the composed
error term:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
error term:
Using of the convolution formula for derivation of the
probability density function for ε is quite complicated
( ) ( ).,~,,0~ ,0 uvn MVTNuMVNv
uv
ΣΣ−=
+∞ µε
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
probability density function for ε is quite complicated
(see in my PhD thesis).
20
Outline
• Classical stochastic frontier model
• A case of truncated normal inefficiency
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
• A case of truncated normal inefficiency
• Spatial stochastic frontier model
• Moment generating and characteristic functions
• Closed/Unified skew-normal distribution
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 21
Moment generating and
characteristic functions
Let f(x) is a continuous probability density function of a
random variable X.
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
random variable .
Moment generating function for a continuous
probability density function:
( ) ( )∫+∞
∞−
= dxxfetMGF txX
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
Characteristic function for a continuous probability
density function:
22
( ) ( ) ( )12 −== ∫+∞
∞−
idxxfetCF itxX
Moment generating and
characteristic functions
Note that
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
MGFX(-t) is the two-sided Laplace transformation of
the probability density function,
CFX (t) is an inverse Fourier transformation of the
probability density function.
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
probability density function.
23
Moment generating and
characteristic functions
Very useful properties of MGF:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
1. Uniqueness theorem: if two distributions have the
same moment-generating function, then they are
identical at almost all points.
2. Convolution theorem: if random variables X and Y are
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
2. Convolution theorem: if random variables X and Y are
independent and Z=X+Y, then:
MGFZ(t)=MGFX(t)MGFY(t)
24
Moment generating and
characteristic functions
Classical stochastic frontier:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( )
=2
exp22t
tMGF vv
σ( )2,0~ Nv σ
So the composed error term:
( )
=2
exptMGFvi( ),0~ vi Nv σ
( )2,0 ,~ ui TNu σµ+∞
iii uv −=ε
( )
+Φ
Φ
+=
−
tt
ttMGF uuu
uui
σσµ
σµσµ
122
2exp
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 25
( ) ( ) ( )
( )
−Φ
Φ
++−=
=−−=−
tt
t
tMGFtMGFtMGF
uuu
uv
uv iii
σσµ
σµσσµ
ε
1222
2exp
iii uv −=ε
Moment generating and
characteristic functions
Similarly for the spatial stochastic frontier model
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( )vnMVNv Σ,0~
( )
Σ= tttMGF vT
v 2
1exp
( )vnMVNv Σ,0~
( ).,~ ,0 uMVTNu Σ+∞ µ
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 26
( ) ( )[ ] ( )uunuTT
unnu tttttMGF Σ−ΣΦ
Σ+Σ−Φ= − ,,2
1exp,,0 1 µµ
Moment generating and
characteristic functions
MGF of the composed error term:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
uv −=ε
which is a specific case of the Closed Skew Normal
( ) ( ) ( )
( )[ ] ( ) ( )uunuvTT
un
uv
tttt
tMGFtMGFtMGFiii
Σ−Σ−Φ
Σ+Σ+−ΣΦ=
=−=
− ,,2
1exp,,0 1 µµ
ε
iii uv −=ε
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
which is a specific case of the Closed Skew Normal
distribution.
27
Outline
• Classical stochastic frontier model
• A case of truncated normal inefficiency
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
• A case of truncated normal inefficiency
• Spatial stochastic frontier model
• Moment generating and characteristic functions
• Closed/Unified skew-normal distribution
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 28
Closed Skew Normal distribution
The Closed Skew Normal distribution:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( ), ,,,,,~ ∆′′Γ′Σ′′nnCSNx νµ( )
( )
( ) 111
1
,
,
,
,
,
,,,,,~
−−−
−
Σ+Σ=∆′
−=′Σ+ΣΣ−=Γ′
Σ+Σ=Σ′−=′
∆′′Γ′Σ′′
uv
uvu
uv
nnCSNx
µν
µµνµ
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
Introduced by González-Farías, Domínguez-Molina, and Gupta
(2004).
29
( )Σ+Σ=∆′ uv
Unified Skew Normal distribution
There are a lot of different variants of multivariate
skewed distributions:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
skewed distributions:
1. Closed Skew Normal
2. Hierarchical Skew Normal (Liseo, Loperfido, 2003)
3. Fundamental Skew Normal (Arellano-Valle, Genton,
2005)
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
Arellano-Valle and Azzalini (2006) introduced a Unified
Skew Normal distribution, which cover many private
cases, including CSN.
30
Estimation of the spatial stochastic
frontier model
Maximum likelihood estimator:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
( ) ( ),,0ln,,,,,,,ln 22)( +Σ−Φ−=s ρρρβL µµσσβ( ) ( )( ) ( ) ( )( ) ( )
( )( ) ( )( )
,
,
,,,ln,,ln
,,0ln,,,,,,,ln
112~
)(
1111
2~
2~
)(
−−
−−−−
−−=Σ
−−−=
Σ+Σ−+Σ+Σ−+Σ+ΣΣ−Φ+
+Σ−Φ−=
vvn
T
vvnvv
sXYY
uvnuvuvun
unuvYuvs
WρIWρI
XβWXβYWρYe
ee
ρρρβL
σ
µϕµµ
µµσσβ
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 31
( )( ) ( ) ., 112~
−− −−=Σ uun
T
uunuu
vvnvvnvv
WρIWρIσ
Estimation of the spatial stochastic
frontier model
Inefficiency component u can be estimated via
conditional distribution, which is multivariate truncated
normal:
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
conditional distribution, which is multivariate truncated
normal:
( )
( ) ( )( ) 1
1
,0
where
,,~
−−−
−
+∞
Σ+Σ=Σ
+Σ+ΣΣ−=
Σ
uvuu
uuMVTNu
ε
εε
µεµµ
µε
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
Corresponding theoretical moments of the multivariate
truncated normal distribution are well-known.
32
( ) 111 −−− Σ+Σ=Σ uvu ε
Alternative research paths• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
Sum of normal and
truncated normal
Convolution
Obtained probability
density of CSN
Moment generating
functions
Probability density
functions
Moments of CSN
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015 33
density of CSN
Maximum Likelihood
estimator
Method of Moments
estimator
Related literature
1. González-Farías, G., Domínguez-Molina, J.A., Gupta, A. (2004). The closed
skew-normal distribution, in Skew-elliptical distributions and their applications: a
journey beyond normality, Chapman & Hall/CRC, Boca Raton, FL, pp. 25–42.
• Classical stochastic frontier
• Normal-Truncated Normal Case
• Spatial stochastic frontier
• MGF and CF
• Closed/Unified skew normal
journey beyond normality, Chapman & Hall/CRC, Boca Raton, FL, pp. 25–42.
2. Domínguez-Molina, J.A., González-Farías, G., Ramos-Quiroga, R. (2004). Skew-
normality in stochastic frontier analysis, Skew-elliptical distributions and their
applications: A journey beyond normality, pp. 223–241.
3. Arellano-Valle, R. B. and Azzalini, A. (2006). On the unification of families of
skew-normal distributions. Scand. J. Statist., 33, 561-574.
4. Aziz M. (2011) Study of Unified Multivariate Skew Normal Distribution with
Applications in Finance and Actuarial Science, PhD thesis.
The Mathematical Seminar, Transport and Telecommunication Institute, Riga, 08.05.2015
Applications in Finance and Actuarial Science, PhD thesis.
34
Thank you for your attention!Thank you for your attention!Thank you for your attention!Thank you for your attention!
Questions are very appreciated
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