Department of Applied Econometrics Working Papers Warsaw School of Economics Al. Niepodleglosci 164 02-554 Warszawa, Poland

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Warsaw School of Economics Institute of Econometrics

Department of Applied Econometrics

Department of Applied Econometrics Working Papers

Warsaw School of Economics Al. Niepodleglosci 164 02-554 Warszawa, Poland

Working Paper No. 1-08

Bayesian analysis of growth using stochastic frontier model

Arkadiusz Wiśniowski

Warsaw School of Economics

This paper is available at the Warsaw School of Economics

Department of Applied Econometrics website at: http://www.sgh.waw.pl/instytuty/zes/wp/

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frontier model

Arkadiusz Wi±niowski

Warsaw S hool of E onomi s

email: awisniowskiop.pl

Abstra t

We employ Bayesian approa h to the analysis of e onomi growth in

Poland.

The results of estimation of a sto hasti frontier modelapplied to pro-

du tion fun tion of Polish voivodships in 2000 - 2004 are presented.

Sto hasti frontier approa h allows to de ompose growth into te hno-

logi al hange,input hange ande ien y hange.In orderto ompute

the posterior hara teristi s of the growth omponents we employ the

GibbsMCMC sampler.

Keywords:Bayesiananalysis,Gibbssampler,e onomi growth,sto has-

ti frontier analysis

JEL odes: C11, C33, O49

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1 Introdu tion

E onomi growth and its sour es belong to the most important issues of

e onomi s [Barro, Sala-i-Martin 1999℄, [Koop, Osiewalski, Steel 2000℄. The

sto hasti frontier model providesa formal framework to de ompose the e o-

nomi growth into three omponents: input hange, te hni al hange and

e ien y hange. The aim of this paper is to present the appli ation of the

Bayesian frameworkinthe analysisof e onomi growth.Wemodelthe growth

by means of a sto hasti produ tion fun tion, thus we use the sold industrial

produ tionasameasure of it.The analysis isperformed for16voivodships in

Poland in the period2000 - 2004.

To our analysis we apply a Bayesian framework with re ently developed

numeri al Markov Chain Monte Carlo methods. Bayesian approa h seems to

betheappropriatetoolsin eitallowsustofo us ofanyquantityofinterestby

derivingitsposteriordistribution(inparti ular the omponentsof theoutput

growth),tointegrateoutallnuisan eparameters,tohandleallrestri tionsand

regularity onditions that result from the e onomi theory, as well as to deal

with alarge numberof parameters in the model.

This resear h is based mainly on the papers by G. Koop, J. Osiewalski

and M.F.J. Steel [1999, 2000℄ and toa lesser extent on[Osiewalski 2001℄ and

[Osiewalski,Steel1998℄. Inthe next se tionwedes ribethe sto hasti frontier

model.Se tion3presentsthede ompositionofgrowth,fourthse tionprovides

the modelina Bayesian framework withthe derivation ofGibbs samplerused

toestimate the unknown parameters. In Se tion5 the resultsfollow, last two

se tionspresent on lusions, modelextensions and a summary.

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2 Sto hasti frontier model

The sto hasti frontier modelwas originallyproposed by W. Meeusen and J.

vandenBroe k[1977℄andD.Aigner,C.A.K.LovellandP.S hmidt[1977℄.The

model onsists of the mi roe onomi produ tion fun tion and two error om-

ponents:one ree tingrandomnessof thefrontier itselfandonethat measures

ine ien y.

In the model we assume that all omparable agents or units (herein we

onsider all 16 voivodships in Poland) produ e a ording to a ommon te h-

nology.Thisassumptionallowsusto onstru ta ommonprodu tionfun tion

and to interpret allsystemati deviations fromit being a result of the under-

usageofinputs.Inotherwords,thevoivodship anoperateeitheronorwithin

afrontier.voivodshipsare territorialunits (provin es) with fullfreedom of in-

formation ow thus the assumption that te hnology used by ompanies, e.g.

in Podkarpa kie voivodship an be opied and used by a plant operating in

Dolno±l¡skie seems tobe reasonable 1

.

The followingpresents the produ tionfun tion under onsideration

Y

it

=f

t (K

it

;L

it )

it z

it

; (2.1)

whereY

it

is anoutput value, f

t (K

it

;L

it

)is aprodu tion fun tion with apital

and labour inputs respe tively,

it

(0 <

it

1) is a random term ree ting

produ tion e ien y (so- allede ien y indi ator) and z

it

is a random term

that aptures the general sto hasti nature of e onomi variables (whi h is a

resultofe.g.measurementerror).Subs ripti=1;:::;N identiestheprodu -

ingunits (inour ase voivodships)intimet=1;:::;T.Therandomtermsare

independent of ea h other, a rosstime and provin es.

Inouranalysisweuse atranslogprodu tionfun tion.This formisadopted

by Koopet al.[Koop, Osiewalski, Steel1999℄ intheir GDP growth analysis of

1

Wearenegle tingallte hnologiesthat areunder patentprote tion.

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aset oftheOECD ountries.Translogformallows ustoree tthevariationof

the data ina better way than the Cobb-Douglasfun tion, whi h is one of the

spe ial ases of the translog form. Moreover, we assume that our produ tion

fun tion is time-dependent and we will onsider two ases, with and without

stru ture imposed.

In a general ase the translogprodu tionfrontier an be written

y

it

=x 0

it

t u

it +v

it

; (2.2)

where

u

it

= log(

it )

is anon-negativerandom variable,

v

it

=log(z

it )

is asymmetri allydistributed random variablewith mean zero,

x

it

=

1; k

it

; l

it

; l

it

k

it

; k 2

it

; l 2

it

;

0

;

and a ve tor of parameters

t

=(

t0

;:::;

t5 )

0

:

Lower ase letters y, l, k indi ate logarithms of the e onomi variables (e.g.

y=log(Y)).Weimposenon-negativity onstraintsonlabour and apitalelas-

ti ities

y

it

l

it

=

t2 +

t3 k

it +2

t5 l

it

0;

y

it

k

it

=

t1 +

t3 l

it +2

t4 k

it

0 (2.3)

foralliandt.Asalo almeasureof e onomiesof s ale(whi hisalsoemployed

by Koop et al. [1999℄) we use the elasti ity of returns to s ale (ER TS, see

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[Varian 1992℄) whi h,for atranslog produ tionfun tion, isgiven by 2

ER TS

it

t1 +

t2 +(

t3 +2

t4 )l

it +(

t3 +2

t5 )k

it

: (2.4)

Thusweobtainthe onstantreturnstos ale by imposingtherestri tion

t1 +

t2

= 1,

t4

=

t5

and

t3

= 2

t4

. The translog fun tion will redu e to the

Cobb-Douglas formfor

t3

=

t4

=

t5

=0.

In this paper we onsider two models originally proposed by Koop et al.

[1999℄.In both of themwe assumenormallydistributed randomerrorv

it with

avarian ethat remains onstanta ross time and voivodships, v

it

N(0; 2

).

Fortherandome ien yindi ator,u

it

= log(

it

),weassumetheexponential

distributionwithitsexpe tedvalue onstanta rosstimeandvoivodships,u

it

Exp (;0) (for justi ation of the exponential form see [Ritter, Simar 1997℄,

[Koop, Osiewalski,Steel 1999℄; for denitions see Appendix A.1).

Asfarastheprodu tionfun tionparametersare on erned(thereareJ =6

of them),we onsider two versions.Wedene the following (NT 1)ve tors:

y = (y 0

1

;:::;y 0

t

;:::;y 0

T )

0

u = (u 0

1

;:::;u 0

t

;:::;u 0

T )

0

(2.5)

v = (v

1

;:::;v 0

t

;:::;v

T )

0

;

wherey

t

=(y

1t

;:::;y

Nt ),u

t

=(u

1t

;:::;u

Nt

)andv

t

=(v

1t

;:::;v

Nt

)are(1N)

ve torsand a (N J) matrix is given by

X

t

=(x

1t

;:::;x

Nt )

0

: (2.6)

2

Given the produ tion fun tion y = f(x) and a s alar t > 0 onsider the fun tion

y(t)=f(tx).Thentheelasti ityofreturnstos aleisdenedas

e(x)= dy(t)

dt

t

y(t)

t=1 :

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In the rst version (indi ated as A version hereafter) we assume that the

parameters are independent of ea h other for every t, hen e our estimation

obje t is a(TJ1) ve tor

A

=( 0

1

;:::; 0

T )

0

: (2.7)

The model an bewritten as

y=X A

A

u+v; (2.8)

wherey,uand v are ve tors dened in2.5, is dened asin2.7and X is the

(NT TJ) matrix

X A

= 2

6

4 X

1

.

X

t

.

X

T 3

7

5

; (2.9)

where X

t

is the matrix given in 2.6.

In the se ond version of our model (indi ated as B) we impose a linear

trend restri tion onthe produ tion fun tionparameters

t

=

+t

: (2.10)

This model an be written asin equation2.8

y=X B

B

u+v; (2.11)

where B

=((

) 0

(

) 0

isa (2J 1)ve tor and

X B

= 2

6

4 X

1 X

1

.

. .

.

X

t tX

t

.

. .

.

X

T TX

T 3

7

5

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is the (NT 2J) matrix. X

t

isgiven in 2.6.

Hen e to des ribe the produ tion frontier in the A version we use (NT +

TJ+2) parameters, whereas the dimension of the version B of the modelis

(NT +2J+2).

3 Growth de omposition

Theprodu tionfrontierallowsustode omposethee onomi growthintothree

omponents: e ien y hange 3

, input hange and te hni al hange. Assume

that the produ tion frontiers for all produ ing units in periods t and t+1

and all inputs in these periods are given. Then the expe ted value of growth

(further abbreviated toexpe ted or predi ted growth) an be writtenas

E(y

i;t+1 y

i;t )=(x

0

i;t+1

t+1 x

0

i;t

t )+(u

ti u

i;t+1

): (3.12)

The rst expression inbra kets indi ates te hni al and input hangewhereas

the se ond one isthe e ien y hange.Let uswrite the rst expression as

x 0

i;t+1

t+1 x

0

i;t

t

= 1

2 (x

i;t+1 x

i;t )

0

(

t+1 +

t )+

1

2 (x

i;t+1 +x

i;t )

0

(

t+1

t

): (3.13)

The rst omponent of equation 3.13 is the hange in inputs for the average

te hnology. The se ondterm indi ateste hni al progress given averageinputs

in two periods onsidered. Te hni al hange for the ith voivodship an be

measuredasexp [x 0

i (

t+1

t

)℄foragiven ve torofinputsx

i

.Duetothefa t

that inputs vary over time, Koop et al. [1999, 2000℄ propose to measure the

impa t of global te hni al hange on the produ tivity of the ith voivodship

3

As far as the e ien y of the produ tion pro ess is on erned we an dierentiate

betweente hni ale ien y that results from theproperuse ofthe produ tionte hnology

giveninputsandallo ativee ien ywhi hisa onsequen eoftheproperallo ationofinputs

(see [Osiewalski 2001℄). The subje t matter of the sto hasti frontier model is te hni al

e ien y.

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by the geometri mean ofx

it andx

i;t+1

.Hen e weobtain the te hni al hange

TC

i;t+1

given by

TC

i;t+1

=exp

1

2 (x

i;t+1 +x

it )

0

(

t+1

t )

(3.14)

and the hange in inputsIC

i;t+1

IC

i;t+1

=exp

1

2 (x

i;t+1 x

it )

0

(

t+1 +

t )

: (3.15)

The entire produ tivity hange of the ithvoivodship an be al ulated as

PC

i;t+1

=TC

i;t+1

IC

i;t+1

: (3.16)

Thethird growth omponentisthee ien y hange EC

i;t+1

that an bewrit-

ten as

EC

i;t+1

=exp [(u

i;t u

i;t+1 )℄=

i;t+1

i;t

: (3.17)

Cumulated te hni al, input and e ien y hanges for all periods onsidered

are given by

CTC

i

= T 1

Y

t=1 TC

i;t+1

; (3.18)

CIC

i

= T 1

Y

t=1 IC

i;t+1

; (3.19)

and

CEC

i

= T 1

Y

t=1 EC

i;t+1

=exp (u

i;1 u

i;T

): (3.20)

The average hange in all periods are al ulated as a geometri mean of all

annual hanges. Thus, for te hni al hangewe an write

ATC

i

=(CTC

i )

1

T 1

; (3.21)

and for e ien y and input hangewe have

AEC

i

= (CEC

i )

1

T 1

; (3.22)

AIC

i

= (CIC

i )

1

T 1

(3.23)

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respe tively. The expe ted (predi ted) average annualgrowth an be derived

fromequation 3.12

AGC

i

=ATC

i

AIC

i

AEC

i

; (3.24)

whereas the average annualprodu tivity be omes

APC

i

=ATC

i

AEC

i

: (3.25)

In order to fa ilitate interpretation, the nal results are given in per entage

points:ATG=100(ATC

i

1)forte hni al hange,AEG=100(AEC

i 1)

for e ien y hange, et .

4 Bayesian model

To investigate the produ tion growth in Polish voivodships in 2000 - 2004

from models 2.8 and 2.11 we use a Bayesian framework. We have N = 16

voivodships and T = 5 periods (annual data from years 2000 - 2004). The

observation ve tor is anoutput ve tor y with the exogenous variables matrix

X given (dened asX =fX A

;X B

g fora A or B version of the model):

y =(y;X):

Ve torof parameters that are the obje t of inferen eis given by

=

0

2

u

;

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where is dened (depending onthe modelversion) as 4

A

or B

. Given the

above mentioned notationwe dene the likelihoodfun tion as 5

L(yj;X)=f TN

N

(yjX u; 2

TN I

TN

): (4.26)

The prior density for the ve tor of produ tion fun tion parameters p() is

an(improper) trun ated uniform distribution,whi htakes the value p()=1

whentheregularity onditionsgivenin2.3aresatisedandp()=0otherwise.

For the posterior density to be well-dened our prior densities of parameters

2

and must beproper (informative), otherwise the posteriordensity is not

- omplete 6

(see[Fernandez,Osiewalski,Steel1997℄).Inthemodelweuse the

following prior densitiesfor 2

and

p(

2

)=(

2

) n

0

=2 1

exp a

0

2

; (4.27)

and

p(

1

)=f

G (

1

j

01

;

02

) (4.28)

with predened hyperparameters a

0 , n

0 ,

01

and

02

. Koop et al. [1999℄ take

a

0

> 0,

01

>0,

02

> 0 and n

0

0. For n

0

= 0 the prior is improper yet it

impliesaproperposteriordistribution (seeFernandez etal.[1997℄).Following

Koop et al. [1999℄ we assume a

0

= 10 6

. The resulting prior density is lose

to the non-informative prior p() / 1

but it assigns small weights on big

realizations of 2

. The hyperparameter

01

= 1 was hosen for the prior

4

Introdu tionofsu hnotationshouldnot ompli atetheentirereasoningsin einalmost

allfun tionsdis ussedX and appearastheprodu tX,thedimensionofwhi hisequal

(NT 1)forbothmodels(A andB).In aseof ajointdistribution ofallparametersora

distributionoftheparameter weshouldrememberthatformodelAthesedimensionsare

equal(NT+TJ+2)andTJ respe tively,whereasforB theyare(NT+2J+2)and2J.

5

FordenitionsseeAppendixA.1

6

Ameasuredenedonaspa eX is- ompleteithere existsa(innite)sequen eof

subsetsX

1 ,X

2

,:::summinguptoX,that(X

i

)isniteforalli.

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density of to be at.

02

= log(

) through

ree ts prior beliefs about

the median e ien y. Koop et al. [1999℄ perform a sensitivity analysis of the

model with respe t to dierent values of

. The results show that '(...) the

model displays an impressive degree of robustness a ross these very dierent

priors'.Furtheronin the analysis we assume

=0:75.

The prior density for u is assumed to be the Gamma distribution (see

Se tion2)

T

Y

t=1 N

Y

i=1 f

G (u

it j1;

1

): (4.29)

The jointprior density is aprodu tof priordistributions ofallparameters

p(; 2

; 1

;u)=p()p(

2

)p(

1

)p(u): (4.30)

Having dened all priorswe an writethe posteriordensity as:

p(jy;X)/f TN

N

(yjX u; 2

I

TN

)p()p(

2

)p(

1

) T

Y

t=1 N

Y

i=1 f

G (u

it j1;

1

):

(4.31)

4.1 Gibbs sampler

In order to approximate our posterior density we use Gibbs sampler, the

Markov Chain Monte Carlo algorithm (for details see e.g. [Tierney 1994℄,

[Bernardo, Smith 1994℄, [Casella, Robert 1999℄, [Roberts, Rosenthal 2004℄).

First we spe ify onditional densities of all parameters. The derivations here-

afterare basedon the hintsgiven by Koopet al.[1999℄.

To obtain the onditional density for 7

p(ju; 2

; 1

;y;X) note that

^

=(X 0

X) 1

X 0

(y+u)

and

(y+u) 0

X

^

=

^

0

X 0

X

^

:

7

FormodelAitis30-andformodelB 12-dimensionaldensity.

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Then we al ulate

p(ju; 2

; 1

;y;X)/exp

"

(y X+u) 0

(y X+u)

2

#

p()

= exp

"

(y+u) 0

(y+u) 2(y+u) 0

X+ 0

X 0

X

2

#

p()

= e h

(y+u) 0

(y+u) 2(y+u) 0

X

^

+2

^

0

X 0

X

^

2(y+u) 0

X+ 0

X 0

X

2

i

p()

= exp

"

(y+u X

^

) 0

(y+u X

^

)+(

^

) 0

X 0

X(

^

)

2

#

p()

/ exp

"

(

^

) 0

X 0

X(

^

)

2

#

p() (4.32)

sin e(y+u X

^

) 0

(y+u X

^

)isindependentof (itisapartofanormalising

onstant).Obviously4.32isanormaldistributionwithmean

^

and ovarian e

matrix 2

(X 0

X) 1

.The next onditional density isp(

2

j;u; 1

;y;X).De-

noting as an(NT 1)ve tor of ones we al ulate

p(

2

j;u; 1

;y;X)/(

2

) TN =2

e h

(y X+u) 0

(y X+u)

2

i

2

e 10

6

2

= (

2

) TN

2 1

exp

1

2 [10

6

+(y X+u) 0

(y X+u)℄

2

:(4.33)

Thedistributiongivenin4.33isaGammadistributionwithashapeparameter

TN

2

and s aleparameter 10 6

+(y X+u) 0

(y X+u) (see the denition

A.3 inAppendix A.1). Forour ine ien y ve tor u the onditional density is

trun ated normal.

p(uj; 2

; 1

;y;X)/

/ exp

"

(y X+u) 0

(y X+u)

2

#

T

Y

t=1 N

Y

i=1

exp ( u

it

1

)I(u

it

0)

= exp

"

(y X+u) 0

(y X+u)

2

#

exp ( u 0

1

)I(u

it

0)

= exp

"

(y X+u) 0

(y X+u)+2

2

u 0

1

2

#

I(u

it

0)

/ e

(y X+u) 0

(y X+u)+2

2

u

0

2

(X y)

0

+(

2

)

2

I(u

it

0):