probability behind spatial stochastic frontier model · probability behind spatial stochastic...

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


• 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:




• MGF and CF


= 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:




• MGF and CF


( ) ( ) ( ){ }xPyxPyyxf yy ∉∀∈= > ',: '


Technical efficiency

Debreu-Farrell definition of technical efficiency of

output vector y is:




• MGF and CF


output vector is:

can be presented in a form

of equation:

( ) ( ){ }1

:sup,−

≤= xfyyxTE θθθ


of equation:

5

( ) ( )yxTExfy ,⋅=

Classical stochastic frontier

For estimation purposes the technical efficiency term is

usually transformed as:




• MGF and CF


usually transformed as:

Introducing the random disturbances v into the

formula, we consider a classical stochastic frontier model:

( ) ( ) .0,exp, ≥−= uuyxTE


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:




• MGF and CF


form:

So the only probabilistic feature of the stochastic

frontier model is a composed error term:

( ) iiii uvxfy −+= β,lnln

uv −=ε


iii uv −=ε

Outline






• MGF and CF







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 :




• MGF and CF


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


distributions, e.g. truncated normal:

9

( )2,0 ,~ ui TNu σµ+∞



The probability density function for truncated normal

distribution is (truncation limits are set to 0 and +∞ to

match the non-negativity requirement of ):




• MGF and CF


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




According to the convolution formula, if u and v are

independent, then:




• MGF and CF


( ) ( ) ( )∫∞+

∞−

+=

−=

iiuiivi

iii

duufuff

uv

εε

ε

ε




Transformations:




• MGF and CF


( ) ( ) ( ) =+=+∞

∫ 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


( )

( )

++

+−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




• MGF and CF


( )( )

++

+−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


extended skew normal

distribution function,

introduced by Azzalini (1985).

13

Outline






• MGF and CF







Spatial stochastic frontier model

A basic assumption of the classical stochastic frontier:

Observations are independent!




• MGF and CF


Observations are independent!

( )( )nu

nvn

IMVTNu

IMVNv

uvXy

2~,0

2~

,~

,,0~

,

σµσ

β

+∞

−+=


In practice, there are a lot of theoretical and empirical

evidences of relationships between observations.

15


One of the simplest forms of this relationship is a linear

combination:




• MGF and CF


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


objects i and j, e.g.

16

( )jiij objectobject

w,distance

1=


Spatial contiguity matrix:




• MGF and CF


...0 112 nww

Matrix form of dependence (spatial lags):

=

0...

............

...0

...0

21

221

112

nn

n

n

ww

ww

ww

W


Matrix form of dependence (spatial lags):

17

Wyyywyji

jijini on depends on depends ..1 ⇔∀ ∑≠

=


The spatial stochastic frontier model specification:




• MGF and CF


,)( uvXWXβYWρY s −+++= β

where

.~,~

,)(

uuWρu

vvWρv

uvXWXβYWρY

uu

vv

sXYY

+=+=

−+++= β


• 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


Assuming that




• MGF and CF


( )( )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

−

−

−=

−=


( )( ) ( )[ ]

( )( ) ( )[ ]Tuuuuuu

u

T

vvvvvv

vn

WρIWρI

MVTNu

WρIWρI

MVNv

112~

,0

112~

,,~

,,0~

−−

+∞

−−

−−=Σ

Σ−−=Σ

Σ

σ

µσ


The main problem is a distribution of the composed

error term:




• MGF and CF


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


probability density function for ε is quite complicated

(see in my PhD thesis).

20

Outline






• MGF and CF







Moment generating and

characteristic functions

Let f(x) is a continuous probability density function of a

random variable X.




• MGF and CF


random variable .

Moment generating function for a continuous

probability density function:

( ) ( )∫+∞

∞−

= dxxfetMGF txX


Characteristic function for a continuous probability

density function:

22

( ) ( ) ( )12 −== ∫+∞

∞−

idxxfetCF itxX



Note that




• MGF and CF


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.


probability density function.

23



Very useful properties of MGF:




• MGF and CF


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


2. Convolution theorem: if random variables X and Y are

independent and Z=X+Y, then:

MGFZ(t)=MGFX(t)MGFY(t)

24



Classical stochastic frontier:




• MGF and CF


( )

=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


( ) ( ) ( )

( )

−Φ

Φ

++−=

=−−=−

tt

t

tMGFtMGFtMGF

uuu

uv

uv iii

σσµ

σµσσµ

ε

1222

2exp

iii uv −=ε



Similarly for the spatial stochastic frontier model




• MGF and CF


( )vnMVNv Σ,0~

( )

Σ= tttMGF vT

v 2

1exp

( )vnMVNv Σ,0~

( ).,~ ,0 uMVTNu Σ+∞ µ


( ) ( )[ ] ( )uunuTT

unnu tttttMGF Σ−ΣΦ

Σ+Σ−Φ= − ,,2

1exp,,0 1 µµ



MGF of the composed error term:




• MGF and CF


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


which is a specific case of the Closed Skew Normal

distribution.

27

Outline






• MGF and CF







Closed Skew Normal distribution

The Closed Skew Normal distribution:




• MGF and CF


( ), ,,,,,~ ∆′′Γ′Σ′′nnCSNx νµ( )

( )

( ) 111

1

,

,

,

,

,

,,,,,~

−−−

−

Σ+Σ=∆′

−=′Σ+ΣΣ−=Γ′

Σ+Σ=Σ′−=′

∆′′Γ′Σ′′

uv

uvu

uv

nnCSNx

µν

µµνµ


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:




• MGF and CF


skewed distributions:

1. Closed Skew Normal

2. Hierarchical Skew Normal (Liseo, Loperfido, 2003)

3. Fundamental Skew Normal (Arellano-Valle, Genton,

2005)


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:




• MGF and CF


( ) ( ),,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


( )( ) ( ) ., 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:




• MGF and CF


conditional distribution, which is multivariate truncated

normal:

( )

( ) ( )( ) 1

1

,0

where

,,~

−−−

−

+∞

Σ+Σ=Σ

+Σ+ΣΣ−=

Σ

uvuu

uuMVTNu

ε

εε

µεµµ

µε


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



• MGF and CF


Sum of normal and

truncated normal

Convolution

Obtained probability

density of CSN

Moment generating

functions

Probability density

functions

Moments of CSN


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.




• MGF and CF


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.


Applications in Finance and Actuarial Science, PhD thesis.

34

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Questions are very appreciated

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