2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian...

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2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea Silvestrini

Transcript of 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian...

Page 1: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

2007/80 ■

Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration

Andrea Silvestrini

Page 2: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

CORE DISCUSSION PAPER 2007/80

Testing fiscal sustainability in Poland:

a Bayesian analysis of cointegration

Andrea SILVESTRINI1

October 2007

Abstract

Fiscal sustainability is a central topic for most of the transition economies of Eastern Europe. This paper focuses on a particular country: Poland. The main purpose is to investigate, empirically, whether the post-transition fiscal policy is consistent with the intertemporal budget constraint, used as a formal theoretical framework. To test debt stabilization, the empirical analysis is made in two steps in which different inferential approaches are adopted. In the first step we perform the preliminary unit roots analysis and the selection of the cointegration rank using parametric and bootstrap procedures. In the second step we apply Bayesian inference to the estimation of the cointegrating vector and of the adjustment parameters. In this way, we experiment the usefulness of Bayesian inference in precisely assessing the magnitude of the cointegrating vector. Moreover, we show to what extent the likelihood of the data is important in revising the available prior information, relying on numerical integration techniques. Keywords: Bayesian inference, fiscal sustainability, cointegration, bootstrap.

JEL Classification: C11, C32, E62

1 CORE, Université catholique de Louvain, Belgium and Bank of Italy, Research Department, Roma, Italy.

The author is indebted to Luc Bauwens for helpful comments on an earlier draft. The author wishes to thank Pierluigi Daddi, Joanna Mlynarczyk, Gianluca Moretti, Giulio Nicoletti and David Veredas for useful discussion and suggestion. The views expressed in the paper are those of the author and do not necessarily represent those of the Bank of Italy.

This paper presents research results of the Belgian Program on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. The scientific responsibility is assumed by the author.

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Page 4: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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Page 5: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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"i = 1, · · · , p

)�QgNPOn × n

a�Qp`PN twx O v QgJMLptwv`P�rO�a�Qp¤ t a z a ywQ � y O�J � `P����Q vPv{z a�ORL7`P~�MQ¢�YO�Qgy NPORQYL+}�bcO�O�J v O�y O x `PORL���T�JrJr~p�pQp` t ~YJ v QgNPO εεεt ∼ Nn(0,Σ)

��� t `P�_c~ v{t ` t �YO�L+O��MJ t `PO x ~p�pQgN t QgJ x O�a�Qp`PN t ¤ΣQgJML t JML+O�_cO�JML+O�J&`�~p�YO�N@` t a�OY��¦�~YNPO�~p�YO�NR�

Θtwv Q

n × ma�Qp`PN t ¤ � ~YN`P�rO

mL+O�`PO�NPa t J twv ` twx �pQgN t Qgbry O v@t J

DDDt

��\B�rOm × 1

�YO x `P~YN�~ �Kx ~UO�- x�t O�J&` vDDDt

a�Q¢} x ~YJ&`|Q t J�� � ~YN t J v `|QgJ x OY� Qx ~YJ v `|QgJ&`R�rQ�y t JrORQgNB`PO�NPaEQgJML v ORQ v ~YJMQgy�L z a�a t O v �\B�rO@X I-] t J�O §&z Qp` t ~YJ;"Íd') x QgJbcO�_MQgN|Qga�O�`PO�N t � ORL t J��YO x `P~YNBO�NPNP~YN x ~YNPNPO x ` t ~YJ "¥X-Z �[¦ ) � ~YNPaöQ v

∆yyyt = ΘDDDt +

p−1∑

i=1

Πi∆yyyt−i + Πyyyt−1 + εεεt, t = 1, . . . , T − 1," �*)

Page 6: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

�#�rO�NPOΠ = −A(1) = −

(

In −p∑

i=1

Ai

) twv `P�rO#a�Qp`PN t ¤��#� twx ��L+O vPx N t bcO v `P�rOBy ~YJ � � N z J�NPO�ywQp` t ~YJ v � t _ v Qga�~YJ � `P�rO�pQgN t Qgbry O v �U�#� t y O

Πi =p∑

j=i+1

Aj , (i = 1, · · · , p− 1)QgNPO-`P�rO�a�Qp`PN twx O v Q x�x ~ z J&` t J � � ~YNB`P�rO@O � O x ` v ~ � v �r~YN{` � N z J

L+}UJMQga twx�v �� �rO�Jk`P�rOe�YO x `P~YN

yyyt

~ �nN|QgJML+~Ya �pQgN t Qgbry O vCt J "Íd') twv[t J&`PO � N|Qp`PORL�~ � ~YN|L+O�N�~YJrO,"

I(1))�QgJMLk}YO�` � ~YN v{z NPO-~YJrO~YN v O��YO�N|Qgy�y t JrORQgN x ~Ya�b t JMQp` t ~YJ v ~ � `P�rO v O��pQgN t Qgbry O v QgNPO t J&`PO � N|Qp`PORL~ � ~YN|L+O�N � O�NP~�"

I(0))����[O�QgNPO�L+ORQgy t J � � t `P�Q x ~ t J&`PO � N|Qp`PORL�X I-] v } v `PO�a��7\B�rO v O��pQgN t Qgbry O v QgNPO v Q t L�`P~!bcO x ~ t J&`PO � N|Qp`PORL���I x�x ~YN|L t J � `P~7`P�rO � N|QgJ � O�N � v]#O�_rNPO v O�J&`|Qp` t ~YJ!\B�rO�~YNPO�a�� z JML+O�N�Q v O�`@~ � Q vPv{z a�_+` t ~YJ v �c`P�rO

n−L t a�O�J v{t ~YJMQgy��YO x `P~YN

yyyt

a�Q¢}�bcO�NPO�_rNPO v O�J&`PORLQ v `P�rO v{z aö~ � Q�N|QgJML+~Ya �BQgy ��QgJMLQ v `|Qp` t ~YJMQgNP}�_rNP~ x O vPv " v O�O��Y~Y�MQgJ v O�J�� # %'%'(*)��T�J<Q x ~ t J&`PO � N|Qp`PORL7X I-] v } v `PO�a���`P�rO�N|QgJr�7~ �

Πa�Qp`PN t ¤ t J " �*) twv O §&z Qgy�`P~`P�rOkJ z a�bcO�N�~ �Bt JML+O�_cO�JML+O�J&`x ~ t J&`PO � N|Qp` t J � �YO x `P~YN v �DT � `P�rO!N|QgJr��~ �

Πtwv

rQgJML

r < n�Π�MQ v NPORL zMx ORL4N|QgJr�4QgJML�`P�rO�NPO!QgNPO

r < ny t JrORQgNPy } t JML+O�_cO�JML+O�J&` x ~Ya�b t JMQp` t ~YJ v ~ �yyyt

`P�MQp`eQgNPO v `|Qp` t ~YJMQgNP}Y��\B�rO�NPO � ~YNPOY�U`P�rO��pQgN t Qgbry O v[t Jyyyt

QgNPOI(1)

� t `P�rx ~ t J&`PO � N|Qp` t J � �YO x `P~YN v QgJML n − r

z J t `KNP~U~g` v �� �rO�J

�MQ v NPORL zMx ORL�N|QgJr��� t ` twv _c~ vPv{t bry O[`P~�L+O x ~Ya�_c~ v O t ` t J

Π = αααβββ′ �g�#�rO�NPO

αααQgJML

βββQgNPO

n×ra�Qp`PN twx O v~ � N|QgJr�

r�C\B� twv N|QgJr��NPO v `PN twx ` t ~YJa�Q¢}kbcO t J x ~YNP_c~YN|Qp`PORL t J�`P�rO�X-Z �[¦ NPO � �#N t ` t J � " �*)BQ v��

∆yyyt = ΘDDDt +

p−1∑

i=1

ΠΠΠi∆yyyt−i + αααβββ′

yyyt−1 + εεεt," � )

�#�rO�NPOβββ

yyyt−1x ~YJ v ` t ` z `PO v `P�rO�O�NPNP~YN x ~YNPNPO x ` t ~YJ!`PO�NPa�QgJML�`P�rO�O §&z Qp` t ~YJ

βββ′

yyyt = 0NPO�_rNPO v O�J&` v `P�rO�y ~YJ � � N z JO §&zrt y t brN t z a a�~+L+O�y � ~YNu`P�rO x ~Ya�_c~YJrO�J&` v ~ �

yyyt

�uT�J � O�NPO�J x O[~YJβββa�Qp`PN t ¤ twv QgJ twvPv{z O[~ � � NPORQp` O�a�_ t N twx Qgy&NPO�y O��pQgJ x OY�

βββ′

yyyt = 0O §&z Qp` t ~YJ�� t JML+O�ORL��+a�Q¢}�bcO@� t O��[ORL�Q v Q�y ~YJ � � N z JkO §&zrt y t brN t z a a�~+L+O�y�y t Jr� t J � `P�rO x ~Ya�_c~YJrO�J&` v ~ � yyyt

�¦�~YNPO�~p�YO�NR�M`P�rO�O�y O�a�O�J&` v ~ �αααQgNPO��UJr~p�#J�Q v `P�rO�QYL¢£ zMv `Pa�O�J&`�_MQgN|Qga�O�`PO�N v-t J!`P�rO�X-Z �[¦�a�~+L+O�yÍ��a�ORQ v{z N t J ��r~p� §&zrtwx �Uy }7L+O�� t Qp` t ~YJ v-� NP~Ya y ~YJ � � N z J7O §&zrt y t brN t z a � O�ORL!bMQ x � t J&`P~`P�rO v } v `PO�a���\B�rO�ywQgN � O�N-`P�rO t N�Qgb v ~Yy z `PO�pQgy z OY�+`P�rO §&zrtwx �YO�N[`P�rO�QYL¢£ zMv `Pa�O�J&`K~ � `P�rO v } v `PO�a `P~p�BQgN|L v�t ` v y ~YJ � � N z J�O §&zrt y t brN t z aö_MQp`P���

\B�rO!_MQgN|Qga�O�`PN t � Qp` t ~YJΠ = αααβββ

′ t a�_ry t O v�t L+O�J&` t � x Qp` t ~YJ twvPv{z O v ~YJβββQgJML

ααα� \B�rO v O7a�Qp`PN twx O v QgNPO7Jr~g`z J tw§&z O�y } t L+O�J&` t �MORL v{t J x O

Πx QgJ7bcO � Q x `P~YN t � ORL7Q v

(αααH−1)(Hβββ′

) = αααβββ′ � x �r~U~ v{t J � QgJU} �¥z y y N|QgJr��a�Qp`PN t ¤

H"¥~ � L t a�O�J v{t ~YJr × r

)���I x�x ~YN|L t J � `P~¡BQ z �[O�J v QgJML7V z brN|QgJr~ "$# %'%'(*)��r2 NPO v `PN twx ` t ~YJ v#� ~YN t L+O�J&` t � x Qp` t ~YJ7�MQ¢�YO`P~�bcO t a�_c~ v ORL�~YJ

αααQgJML

βββ�CI3_rN|Q x ` twx Qgy v ~Yy z ` t ~YJ t J�`PO�NPa v ~ � O x ~YJr~Ya twx-t J&`PO�NP_rNPO�`|Qp` t ~YJ twv `P~ v O�`KNPO v `PN twx ` t ~YJ v~YJ

βββ�ry ORQ¢� t J � ααα

z JrNPO v `PN twx `PORL���V t JrORQgN#NPO v `PN twx ` t ~YJ v ~YJβββx QgJbcO t J&`PNP~+L zMx ORLL+O��MJ t J �

βββ(n×r)

=

−Ir

β∗β∗β∗

((n−r)×r)

,

" 9')

�#�rO�NPOβ∗β∗β∗twv `P�rO z JrNPO v `PN twx `PORLka�Qp`PN t ¤�� T�J�#�MQp` � ~Yy y ~p� v �[O v �MQgy y v ` twx ��`P~�`P� twv L+O��MJ t ` t ~YJ�QgJML��[O v �MQgy y��YO�O�_

αααz JrNPO v `PN twx `PORL��

� ���� � ������� ���� � ��� � � � �� ���� � ��� ����4��� ��� �� � � ��� ����4���

¡BQ¢}YO v{t QgJ t J � O�NPO�J x O�~YJβββQgJML

αααtwv �rO�NPORQ � `PO�N-NPO�� t O��[ORL � ~Yy y ~p� t J � `P�rO�QgJMQgy } v{twvKx ~YJML zMx `PORL�bU}�¡BQ z �[O�J v ���3���

"$# %'%'%*)�� ]#O � QgN|L t J � `P�rO-y t �YO�y t �r~U~+L �¥z J x ` t ~YJ��Y`P�rO-_rN t ~YN�QgJML�`P�rOK_c~ v `PO�N t ~YN�L+O�J v{t `W}Y�&NPO v{z y ` v QgNPO v{z a�a�QgN t � ORL � ~YN`P�rO � O�JrO�N|Qgy x Q v O�QgJML � ~YN#`P�rO x Q v O�~ � Q v{t J � y O x ~ t J&`PO � N|Qp` t J � �YO x `P~YNR�[^K~g`PO�`P�MQp`K`P� twvKtwv �p����~ � `P�rO�_rNP~Y_c~ v Qgy v�Pò[ë�æWá è|ã æWêUé�æ � ãCïYëBà&ë�æ�ïYá å æWá à&ôRî&á å�ê �+ãPæ � ã|ã|à@ïYãPæWãPäWõ�á à&á å æWá è�âRé�äWá é��&ì ã|å�æWêUé�æ�é�äWã�äWã|å æ�äWá èPæWã�ï-æWëBì á ã�á àeæWê&ã è|ëRá àpæWã|ô�ä{é�æWá ëRàå�üUé�è|ã@é�àUï�æWê&ëRå�ãKæWêUé�æBé�äWã-à&ë�æ�� �YäWëRõ ê&ãPäWã-ëRà��é�ä{ïYå|Þ�� ãeé�å�å�î&õ�ã#æWêUé�æ�æWê&ã-ïYãPæWãPäWõ�á à&á å æWá è-âRé�äWá é��&ì ã|åBé�ì ì�ì á ã-ëRîYæWå�á ïYãKæWê&ãè|ëRá àpæWã|ô�ä{é�æWá ëRà�å�üUé�è|ã�

Page 7: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

_rNPO v O�J&`PORL t J/`P�rO¡BQ¢}YO v{t QgJ�y t `PO�N|Qp` z NPOY� v O�O � ~YN t J v `|QgJ x O �@y O t bcO�N � O�J<QgJML��pQgJ 5 t £W� "$# %'%.� )�� � O��[O��YO�"$# %'%'(*)���@y O t bcO�N � O�J�QgJML���QYQg_ "ÍdgfYf&d')��pS&`PN|Q x �MQgJ "ÍdgfYf �*)��¢X t y ywQgJ t "ÍdgfYf*9')�QgJML@`P�rO[NPO � O�NPO�J x O v `P�rO�NPO t J�� SUO�O � QgNPJrO1"ÍdgfYf'(*)� ~YN�Qe�YO�NP}�NPO x O�J&`CQg_r_ry twx Qp` t ~YJ�~ � ¡BQ¢}YO v{t QgJ�`PO x �rJ tw§&z O v t J x ~ t J&`PO � N|Qp`PORL�X I-]�a�~+L+O�y v `P~e`P�rO#O v ` t a�Qp` t ~YJ�~ � O z NP~QgNPORQ�L+O�a�QgJML � ~YNK¦ �r�\B�rO v `|QgN{` t J � _c~ t J&` twv O�¤+_rNPO vPv{t J � `P�rO�X-Z �[¦ t J " � ) t J�a�Qp`PN t ¤ � ~YNPaöQ v Q�NPO � NPO vPv{t ~YJka�~+L+O�y

Y = XΓΓΓ + Zβββααα′

+ E

=[

X Zβββ]

[

ΓΓΓ

ααα′

]

+ E

=[

X Z]

[

ΓΓΓ

ααα′

]

+ E,

"@(*)

�#�rO�NPO

Y(T×n)

=

∆y′

1���∆y

T

, E

(T×n)=

εεε′

1���εεε

T

, Z

(T×n)=

y′

0���y

T−1

.

¦�~YNPO�~p�YO�NR�XQgJML

ΓΓΓx ~YNPNPO v _c~YJML�`P~

X(T×(m+n(p−1)))

=

D′

0 ∆y′

0 . . . ∆y′

2−p

D′

1 ∆y′

1 . . . ∆y′

3−p��� ��� ���D

T−1 ∆y′

T−1 . . . ∆y′

T−p+1

, ΓΓΓ((m+n(p−1))×n)

Θ′

Π′

1���Π

p−1

." h2)

^K~g`PO�`P�MQp`βββQgJML

αααx ~Ya�_rN twv O@`P�rO�_MQgN|Qga�O�`PO�N v ~ ��t J&`PO�NPO v `R�[\B�rO�a�Qp`PN twx O v

Θ′ �

Π′

1

�. . .�Π

p−1

QgJMLΣx ~YJ&`|Q t JJ zrtwv QgJ x O@_MQgN|Qga�O�`PO�N v �

��� l����¥m����� �� � ���R Y��� �¹���������������|���������P���|���|  �¹�/�|�������Y��� ���p� � �¢�Y = WB + E,

"@&*)

! � ���P� W =[

X Zβββ] �p�� 

B =[

Γ ααα′

]

′ �"B���|� ! ���|�p�� ��¹� �·�p�G�p� βββ # � � � ���R Y��� �¹�$�&%�� �w� � �¹�����p�"�¹� �¹�·����p�������������|� ('������|� # �¹� �w�)�M�¢�|� ����� ����������*Y�,+U�����-� ��� �u� � �.*p��� ! �!�P����+�� �·�/���p�"�,+�� � �10��p� � �p�����P�&�g�P���|� �·�p� ���R Y��� ��¹�!� � �32 ���Y��� � �p���|������� ! �p��* SUO�O�IK_r_cO�JML t ¤�I�� # � ~YNB`P�rO�Qgy � O�brN|Q twx L+O�`|Q t y v �\B�rO@y t �YO�y t �r~U~+L �¥z J x ` t ~YJ54@`P�MQp` x ~YNPNPO v _c~YJML v `P~�`P�rO v `|Q x �YORL�NPO � NPO vPv{t ~YJ t J "@&*) twv

L(βββ,B,Σ | Data) ∝ |Σ|−T

2 exp

{

−1

2tr Σ−1(Y − WB)

(Y − WB)

}

,"@%*)

�#�rO�NPOData = {Y,X,Z}

t JML twx Qp`PO v `P�rO�Q¢�pQ t ywQgbry O v Qga�_ry O t J � ~YNPa�Qp` t ~YJ��I v _c~ t J&`PORL�~ z `#bU} � QgNPJrO�"ÍdgfYf'(*)��&`P�rO x �r~ twx Oe~ � Q�_rNP~Y_cO�NB_rN t ~YNBL twv `PN t b z ` t ~YJ~ � `P�rO�_MQgN|Qga�O�`PO�N v�twv Q v `PO�_~ � _MQgN|Qga�~ z J&` t a�_c~YN{`|QgJ x OY��SUO��YO�N|QgyuQgy `PO�NPJMQp` t �YO v QgNPO�Qp`@�MQgJML���SUO�OY��Qga�~YJ � ~g`P�rO�N v � �-~U~Y_=���#��� "ÍdgfYf'(*) � ~YN6 � ê&ãBå úYõ �+ëRì

L(·)ïYã|à&ë�æWã|å�æWê&ãBì á ÷¢ã|ì á ê&ëpëgï�øwî&à&èPæWá ëRà�ë�ø�ôRã|à&ãPäWá èKé�äWôRî&õ�ã|àpæ

(·)

Ø

Page 8: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

QgJ z _�LrQp`PORL v{z NP�YO�}Y� s ~Yy y ~p� t J � X t y ywQgJ t "ÍdgfYf*9')�� �[O x QgJ=L+O x ~Ya�_c~ v O�`P�rO � O�JrO�N|QgyC_rN t ~YN�L twv `PN t b z ` t ~YJG~YJ<`P�rO_MQgN|Qga�O�`PO�N v Q vφ(βββ,B,Σ, r) = φ(βββ,B,Σ | r) × φ(r),�#�rO�NPO

φ(r)twv Q�_rN t ~YN#L+O�J v{t `W} � ~YNB`P�rO x ~ t J&`PO � N|Qp` t ~YJ�N|QgJr���p�

T�J��#�MQp` � ~Yy y ~p� v �[O�Q vPv{z a�O@`P�rO x ~ t J&`PO � N|Qp` t J � N|QgJr�k`P~kbcO�Qgy NPORQYL+}k�UJr~p�#J�~YNKO v ` t a�Qp`PORL��B¦�~YNPO�~p�YO�NR�+Q v Q_rN t ~YN t J � ~YNPa�Qp` t ~YJ~YJBQgJML

�r�[O t J&`PNP~+L zMx Oe`P�rO v `|QgJMLrQgN|L�L t ��zMv O�_rN t ~YN

φ(B,Σ) = |Σ|−(n+1)

2 ."$#Rf*)

S&`PN|Q x �MQgJ<QgJML�T�JML+O�N "ÍdgfYf.� ) zMv O�Q�L t � O�NPO�J&`�Qg_r_rNP~&Q x ���uL+O��MJ t J � Q��[ORQg�Uy } t J � ~YNPa�Qp` t �YO�_rN t ~YN�L+O�J v{t `W}�~YJ ααα�I L t � O�NPO�J&`@NP~ z `PO twv Qgy v ~k`|Qg�YO�J!bU}!S&`PN|Q x �MQgJ7QgJML7�pQgJ 5 t £W�."ÍdgfYf �*)���X t y ywQgJ t "ÍdgfYf*9')KQgJML � QgNPJrO "ÍdgfYf'(*)�� � O_rNPO � O�NB`P~�NPO�a�Q t JJr~YJ � t J � ~YNPa�Qp` t �YOe~YJ�`P�rO v O�_MQgN|Qga�O�`PO�N v[v{t J x O��[O@�MQ¢�YOeJr~�_rN t ~YN t J � ~YNPa�Qp` t ~YJ~YJ�`P�rO�a��

J<`P�rO x ~YJ&`PN|QgNP}Y� �[O v O�`�QgJ t J � ~YNPa�Qp` t �YO�_rN t ~YN�~YJ<`P�rO z JrNPO v `PN twx `PORL<O�y O�a�O�J&` v ~ �βββ�YO x `P~YNR�<IKJU}<_rN t ~YNL+O�J v{t `W}�~YJ�`P�rO z JrNPO v `PN twx `PORL�O�y O�a�O�J&` v ~ �

βββx QgJ�bcO zMv ORL�� � O-L+O�Jr~g`PO t `�� t `P�

φ(βββ)� � O zMv O

βββQ v QgJkQgN � z a�O�J&`~ � `P�rOeL+O�J v{t `W} �¥z J x ` t ~YJ��&O��YO�J t ��v ~Ya�OKO�y O�a�O�J&` v QgNPO#�UJr~p�#J�� t `P�k_rNP~YbMQgb t y t `W}�~YJrO�"¥�#�rO�JrO��YO�N��[O t a�_c~ v O v ~Ya�OJr~YNPa�Qgy t � Qp` t ~YJ�NPO v `PN twx ` t ~YJ v �rQ vBt J;" 9') )���I v Q�NPO v{z y `R�+`P�rOB£W~ t J&`#_rN t ~YN t J � ~YNPa�Qp` t ~YJ twv � t �YO�JbU}�`P�rO�_rNP~+L zMx `

φ(βββ,B,Σ) ∝ φ(βββ) × |Σ|−(n+1)

2 ."$#'#0)

\B�rO�£W~ t J&`�_c~ v `PO�N t ~YN�L+O�J v{t `W}-~ �(βββ,B,Σ)

twv O §&z Qgyp`P~B`P�rOC_rNP~+L zMx `�~ � `P�rOCy t �YO�y t �r~U~+L �¥z J x ` t ~YJL(βββ,B,Σ|Data)t J "@%*)�` t a�O v `P�rOB£W~ t J&`#_rN t ~YN

φ(βββ,B,Σ)t J "$#'#0)�� t � OY�

φ(βββ,B,Σ | Data) ∝ φ(βββ) × |Σ|−(T+n+1)

2 exp

{

−1

2tr Σ−1(Y − WB)

(Y − WB)

}

."$#¢d')

V�O�` zMv L+O��MJrOE = (Y − WB)

(Y − WB)�C\B� zMv �&`P�rO[£W~ t J&`#_c~ v `PO�N t ~YNBL+O�J v{t `W}

φ(βββ,B,Σ | Data)t J;"$#¢d')a�Q¢}kbcO�O�¤+_rNPO vPv ORLQ v

φ(βββ,B,Σ | Data) ∝ φ(βββ) × |Σ|−(T+n+1)

2 exp

{

−1

2tr Σ−1(E

E)

}

.

"$#��*)

^K~g`PO�`P�MQp`R� � t �YO�J βββQgJML

B��O §&z Qp` t ~YJ."$#��*) twv _rNP~Y_c~YN{` t ~YJMQgy�`P~k`P�rO��YO�NPJrO�y�~ � QgJ t JU�YO�N{`PORL �1twv �MQgN{`@L twv `PN t �b z ` t ~YJ�� t `P��_MQgN|Qga�O�`PO�N v

E′

E"¥_c~ v{t ` t �YO�L+O��MJ t `PO0)#QgJML

T > n − 1L+O � NPO�O v ~ �u� NPO�ORL+~Ya��g� t � OY�

φ(Σ | βββ,B, Data) ∝ |Σ|−(T+n+1)

2 exp

{

−1

2tr (Σ−1E

E)

}

,"$# � )

`P�rO�NPO � ~YNPOΣ | (βββ,B, Data) ∼ IWn(E

E, T )��T�J&`PO � N|Qp` t J � "$# � )[� t `P��NPO v _cO x `#`P~

�r�[O � O�`B`P�rO t JU�YO�N v Oe~ � `P�rOt J&`PO � N|Qp` t J � x ~YJ v `|QgJ&`B~ � `P�rO t JU�YO�N{`PORL �1twv �MQgN{`KL twv `PN t b z ` t ~YJ��

φ(Σ | βββ,B, Data)dΣ =

|Σ|−(T+n+1)

2 exp

{

−1

2tr (Σ−1E

E)

}

= C(n, T ) × |E′

E|−12 T ,�#�rO�NPO

C(n, T ) = 212 Tnπ

14 n(n−1)

n∏

i=1

Γ

(

T + 1 − i

2

)

� �&ë�ä�á à&å æ{é�à&è|ãRÞ��pæ�ä{é�è{êUé�à�é�àUï�ùÍàUïYãPä =@?�������A ïYã�&Uà&ãCøwë�ä�æWê&ã�è|ëRá àpæWã|ô�ä{é�æWá ëRà�ä{é�à&÷φ(r) = (n + 1)−1

é�åCéKüYäWá ë�äCïYã|à&å�á æÍú����ã[äWãPøwãPäCæWë�óuë���é�àUï � á é�ë�= ��ÿ%$ 60A øwë�äBé@ïYã�&Uà&á æWá ëRà�é�àUï�üYäWëRü+ãPä�æWá ã|åCë�ø�æWê&á å�ïYá å æ�äWá �&îYæWá ëRà�

Page 9: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

twv Q x ~YJ v `|QgJ&`B`P�MQp`-L+~UO v Jr~g`KL+O�_cO�JML�~YJβββQgJML

B � �C\B�rO�NPO � ~YNPO-`P�rOB£W~ t J&`K_c~ v `PO�N t ~YN#L+O�J v{t `W}k~ � βββ QgJML Btwv

φ(βββ,B | Data) ∝ φ(βββ) × |E′

E|−12 T .

"$#09')

��� l����¥m������ �����p���������P���|������� � �¢�φ(βββ,B | Data) ∝ φ(βββ) × |(S + (B − B)

W′

W(B − B))|−12 T ,

"$# (*)

! � ���P� B = (W′

W)−1W′

Y�p�� 

S = (Y − WB)′

(Y − WB)

s ~YNKQ�_rNP~U~ �uv O�O�IK_r_cO�JML t ¤�I�� d+�T�J;"$# (*)�� � t �YO�J βββ

�r�[O x QgJNPO x ~ � J t � O-`P�rO��YO�NPJrO�y�~ � Q�a�Qp`PN twx �pQgN t Qp`PO�S&` z L+O�J&`-L twv `PN t b z ` t ~YJ � ~YNB

" v O�O�Q � Q t JIK_r_cO�JML t ¤�I t J�¡BQ z �[O�J v ��� ��� ��# %'%'%*)��&`P�rO�NPO � ~YNPOφ(B | βββ, Data) ∝ |(S + (B − B)

W′

W(B − B))|−12 T ,

"$#ph2)QgJML

B | βββ, Data ∼ Mt(m+n(p−1)+r)×n(B,W′

W, S, T − (m + n(p − 1)) − r)�

\B�rO@a�QgN � t JMQgy�_c~ v `PO�N t ~YN#L+O�J v{t `W}�~ � βββ twv ~Yb+`|Q t JrORL�bU} t J&`PO � N|Qp` t J � � t `P��NPO v _cO x `#`P~ BO §&z Qp` t ~YJ;"$# (*)

φ(βββ | Data) =

φ(βββ,B | Data)dB ∝ φ(βββ) ×

|(S + (B − B)′

W′

W(B − B))|−12 TdB

= φ(βββ) × CMt(B,W′

W, S) × |S|−(T−k−r)

2 |W′

W|−n

2 ,"$# &*)

�#�rO�NPO � ~YNB`P�rO�ORQ v O@~ � `P�rO�O�¤+_c~ v{t ` t ~YJ�[O@y O�`k = m + n(p − 1)

�T�J "$# &*)��

CMt(B,W′

W, S)twv `P�rO t JU�YO�N v O�~ � `P�rO t J&`PO � N|Qp` t J � x ~YJ v `|QgJ&`�~ � `P�rO!a�Qp`PN twx �pQgN t Qp`PO!S&` z L+O�J&`L twv `PN t b z ` t ~YJ�QgJML twv#t JML+O�_cO�JML+O�J&`K~YJ

βββ�

CMt(B,W′

W, S) = π12 n(k+r)

n∏

i=1

Γ(

T−k−r+1−i2

)

Γ(

T+1−i2

) .

\B�rO�NPO � ~YNPOφ(βββ | Data) ∝ φ(βββ) × |S|−

(T−k−r)2 |W

W|−n

2 ."$# %*)

��� l����¥m�� � �·�C���¹� � �7���p��� �¹�����p� ���p� �10p�.��� �·�p��p�B

�p�� Σ

�¢� �¹� ���� � # �p� �¹�����p� ���p� �10p� ��� �·�p��p�βββ

�p�� *Y�|� � �¹� �

ααα++�M�P�����Í� �·�����|  # �¹� �w�,�M�¢�|� ����� �!����� � � ! � � �p��� � � ���p� �.�¹� ��� �M�¢������� �·�p�� Y���r� �¹� �G�-�

βββ�¹� �������=���p� �|��������P���|���|  �¢�

φ(βββ | Data) ∝ φ(βββ) ×|βββ

W2βββ|l2

|βββ′

W1βββ|l1,

"Ídgf*)! � ���P�

W2 = Z′

MXZ

W1 = Z′

MY

(

IT −X(X′

MY X)−1X′

)

MY Z,"Íd #0)

l2 = T−k−r−n2 # l1 = T−k−r

2

�p�� MY = IT −Y(Y

Y)−1Y′

� ��ë�äWã|ë�â¢ãPä�ÞΓ(α) =

∫ ∞

0xα−1 exp(−x)dx, x > 0,

�pú�ïYã�&Uà&á æWá ëRà�ë�ø�æWê&ã �Bé�õ�õ�éKøwî&à&èPæWá ëRà��

Page 10: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

s ~YNKQ x ~Ya�_ry O�`PO@_rNP~U~ � ��� ~ � `P� twv NPO v{z y ` v O�O�IK_r_cO�JML t ¤�I�� �r�� OeNPO�a t JML�`P�MQp`

rtwv `P�rO@J z a�bcO�NB~ � y t JrORQgN[NPO v `PN twx ` t ~YJ v QgJML

k = m + n(p− 1)twv `P�rO@J z a�bcO�NB~ �ux ~Yy z a�J v~ � `P�rO-a�Qp`PN t ¤

X��\B�rO-O�¤+_rNPO vPv{t ~YJ�~YJ�`P�rO-N t � �&` � �MQgJML v{t L+O-~ � "Ídgf*) twv `P�rO-_rNP~+L zMx `�~ � Q�S&` z L+O�J&`[�YO�NPJrO�yr` t a�O v QN|Qp` t ~K~ � L+O�`PO�NPa t JMQgJ&` v�t JU�Y~Yy � t J � `W�[~ §&z QYL+N|Qp` twx � ~YNPa v�t J βββ

� \B�rO�NPO v{z y ` t J � L+O�J v{t `W} φ(βββ|Data)L+~UO v Jr~g`ubcO�y ~YJ �`P~@Q-�UJr~p�#J x ywQ vPv ~ � L twv `PN t b z ` t ~YJ v ��TWL+ORQgy y }Y�p�[O v �r~ z ywL�bcOBQgbry O�`P~-NPO x ~p�YO�N�`P�rO[a�QgN � t JMQgyUL+O�J v{t `W}@~ � QgJU}@O�y O�a�O�J&`~ �

βββQgJML7`P~ x Qgy x�z ywQp`PO�`P�rO�a�~Ya�O�J&` v � t J&`PO � N|Qp` t J � "Ídgf*)����KJ � ~YN{` z JMQp`PO�y }Y��L+ORQgy t J � � t `P� v O��YO�N|Qgy x ~ t J&`PO � N|Qp` t J ��YO x `P~YN v �&Jr~�QgJMQgy }&` twx QgycNPO v{z y ` v[x QgJ�bcOe~Yb+`|Q t JrORL��CSU~Ya�O@QgJMQgy }&` twx QgycNPO v{z y ` v �U�r~p�[O��YO�NR�&QgNPOeQ¢�pQ t ywQgbry O t Jk`P�rO x Q v O~ � Q v{t J � y O x ~ t J&`PO � N|Qp` t J � �YO x `P~YNR� � O�NPO � O�NR�rQ � Q t J��&`P~k¡BQ z �[O�J v ��� ��� B"$# %'%'%*)��

sut JMQgy y }Y�+�[O�L+O�N t �YO@NPO v{z y ` v[� ~YNB`P�rOB

((k+r)×n)=

[

Γ

ααα′

] a�Qp`PN t ¤�� s NP~Ya "$#ph2)��r�[O@�UJr~p�A`P�MQp`

B | βββ, Data ∼ Mt(k+r)×n(B,W′

W, S, T − k − r),�#�rO�NPOB = (W

W)−1W′

Y�S = (Y − WB)

(Y − WB)QgJML

W =[

X Zβββ] �

!KO�J x Oααα

′ Qgy v ~ � ~Yy y ~p� v Q�a�Qp`PN twx �pQgN t Qp`POkS&` z L+O�J&`�L twv `PN t b z ` t ~YJ�� ��~YJ v O §&z O�J&`Py }Y� x ~YJML t ` t ~YJMQgy y }!~YJβββ�uORQ x �O�y O�a�O�J&`B~ �

ααα′ �MQ v Q z J t �pQgN t Qp`PO@S&` z L+O�J&`#L twv `PN t b z ` t ~YJ "¹`P� twv#x QgJkbcOe_rNP~p�YORL�Qg_r_ry } t J � NPO v{z y ` v ~YJka�Qp`PN twx �pQgN t Qp`POS&` z L+O�J&`eL twv `PN t b z ` t ~YJ v )��C\B�rOe�MN v `#a�~Ya�O�J&`K~ �

Btwv

E(B | βββ, Data) = B

=

[

X′

X X′

Zβββ

βββ′

Z′

X βββ′

Z′

Zβββ

]−1 [

X′

βββ′

Z′

]

Y

=

[

(X′

X)−1 + (X′

X)−1X′

ZβββF2βββ′

Z′

X(X′

X)−1 −(X′

X)−1X′

ZβββF2

−F2βββ′

Z′X(X′

X)−1 F2

] [

X′

βββ′

Z′

]

Y,

"ÍdYd')� ~YN

T − k − r > nQgJML!�#�rO�NPO

F2 = (βββ′

Z′

Zβββ − βββ′

Z′

X(X′

X)−1X′

Zβββ)−1��V�O�` zMve� ~ x�zMv ~YJ!`P�rO�_MQgN{` t ` t ~YJx ~YNPNPO v _c~YJML t J � `P~ ααα

′ ��T `K�r~YywL v `P�MQp`B =

[

. . .

−F2Z′

X(X′

X)−1X′

+ F2Z′

]

Y

=

[

. . .

F2Z′

(

IT −X(X′

X)−1X′

)

]

Y

=

[

. . .(

Z′

Z − Z′

X(X′

X)−1X′

Z)

−1

Z′

(

IT −X(X′

X)−1X′

)

]

Y

=

[

. . .(

Z′

MX Z)

−1

Z′

MX

]

Y

=

[

. . .(

βββ′

Z′

MXZβββ)

−1

βββ′

Z′

MX

]

Y.

"Íd.�*)\B�rO v O x ~YJML�a�~Ya�O�J&`#~ �

Btwv

Var(vec(B | βββ, Data)) =1

T − k − r − n − 1S ⊗ (W

W)−1

=1

T − k − r − n − 1(Y

Y −Y′

W(W′

W)−1WY) ⊗ (W′

W)−1,

"Íd�� )Ù��|ù·æCá å�éeõ�ë�äWã#ïYãPæ{é�á ì ã�ï�ã��Yü+ëRå�á æWá ëRà�ë�ø�æWê&ãBüYäWëpë�ø�æWêUé�æCè�é�à �+ã[øwëRî&àUï�á à�ó é�î� ã|à&å�')( *�+ , = ��ÿRÿRÿ0A

Page 11: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

_rNP~p� t L+ORLk`P�MQp`T − k − r > n + 1

� � � � � ���� � � ��� ����� ��� � ��� �4��� � �8������ ��� �� ��� ��� ����� ��� ���� � � ��� �

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)KQgNPO�Q v O�`e~ � a�~YJ&`P�ry }�L+O�`PO�NPa t J twv ` twx�v ORQ v ~YJMQgyuL z a�a t O v `P~bcO v O�y O x `PORL��C\B�rO v O@ywQp`{`PO�N-QgNPOeO�¤+_cO x `PORL�`P~ x Qg_+` z NPO v ORQ v ~YJMQgy�O � O x ` v �\B�rOB~YN|L+O�N ~ � `P�rOKQ z `P~YNPO � NPO vPv{t �YOCa�~+L+O�yr�MQ v `P~@bcO x �r~ v O�J��YQ v �[O�y yÍ� \B� twv�v O�y O x ` t ~YJ twv ~ ��x N zMx�t Qgy t a�_c~YN{`|QgJ x Ov{t J x O t `�Q � O x ` v `P�rO#~ z ` x ~Ya�O#~ � `P�rO x ~ t J&`PO � N|Qp` t ~YJ�N|QgJr��`PO v `�`P�MQp` twv � ~ t J � `P~�bcO t a�_ry O�a�O�J&`PORL���\B�rOBywQ � y O�J � `P�twv L+O�`PO�NPa t JrORL�bU} � ~ x�zMv{t J � ~YJ�`P�rO�NPO v{t L z Qgy vKv `PN zMx ` z NPO�QgJML�bU} zMv{t J � ywQ � O�¤ x y zMv{t ~YJ`PO v ` v �#\B�rO�~Y_+` t a z aEywQ �y O�J � `P��~ � `P�rO�X I-] v } v `PO�a twv `W�[~�"

p = 2)��C\B�rO@a�~+L+O�y twv O v ` t a�Qp`PORL t J�y O��YO�y v bU}ka�ORQgJ v ~ � VuS��

0 \uI-¡BV�Z[S�d�I-^B5 ��I-¡ �-\ !KZ�]#Z$1\B�rO�`P~Y_�_MQgN{`�~ � \uQgbry O7d!L twv _rywQ¢} v `P�rO z JrNPO v `PN twx `PORLGX I-]öO v ` t a�Qp`PORL x ~UO�- x�t O�J&` v ��`P�rO v `|QgJMLrQgN|LGO�NPNP~YN vQgJML`P�rO�` ��v `|Qp` twv ` twx�v �B^K~g`PO�`P�MQp`K`P�rO�O v ` t a�Qp`PORLa�~+L+O�y t J x y z L+O v Q x ~YJ v `|QgJ&`#`PO�NPa��cQ�L z a�a�}�pQgN t Qgbry O � ~YN#`P�rOv O x ~YJML��MQgy � ~ � # %'%'%QgJML/Q v ORQ v ~YJMQgy�L z a�a�} � ~YN

s = 12� 5 t � O�NPO�J&`�L+O�`PO�NPa t J twv ` twx�v ORQ v ~YJMQgy�L z a�a t O v �MQ¢�YObcO�O�J�~Ya t `{`PORL7bcO x Q zMv O�`P�rO�}7QgNPO v `|Qp` twv ` twx Qgy y }�Jr~g` v{t � J t � x QgJ&`@Qp`�9 �k��\B� twvev _cO x�t � x Qp` t ~YJ7�MQ v bcO�O�J!_rNPO � O�NPNPORLv{t J x O t `eL+O�y t �YO�N v `P�rO�� t � �rO v `-QYL¢£ zMv `PORL�] ��vP§&z QgNPORL��rQ v Q�a�ORQ v{z NPO@~ � � ~U~+L+JrO vPv ~ � �r`R�cQgJML�`P�rO�y ~p�[O v `#�pQgy z O�~ �`P�rO�S x �U�BQgN �@x N t `PO�N t ~YJ���\B�rO v O v{z a�a�QgNP}kNPO � NPO vPv{t ~YJ v `|Qp` twv ` twx�v QgNPO@_rNPO v O�J&`PORL t J�`P�rO x O�J&`PN|Qgy�_MQgN{`#~ � \uQgbry O�d+�

T�Jk`P�rO@bc~g`{`P~Ya _MQgN{`B~ � \uQgbry O�d t ` twv#v �r~p�#J�`P�rO@~ z ` x ~Ya�Oe~ � `W�[~�X I-] NPO v{t L z QgycbMQ v ORL�`PO v ` v �C\B�rOeNPO v{t L z Qgyv O�N t Qgy x ~YNPNPO�ywQp` t ~YJ�V�Q � N|QgJ � O�¦ z y ` t _ry t O�N "·V�¦ )#`PO v ` twv-x Qgy x�z ywQp`PORL � ~YNeQkywQ � ~YN|L+O�N-O §&z Qgy�`P~d+�B�KJML+O�NK`P�rO�J z y y�U}U_c~g`P�rO v{twv ~ � Jr~ v O�N t Qgy x ~YNPNPO�ywQp` t ~YJ���`P�rOV�¦ v `|Qp` twv ` twx�twv Q v }Ua�_+`P~g` twx Qgy y }�L twv `PN t b z `PORLGQ v Q x � t ��vP§&z QgNPO�� t `P��/L+O � NPO�O v ~ �K� NPO�ORL+~Ya��<V�Q � N|QgJ � Ok¦ z y ` t _ry t O�N�`PO v ` v `|Qp` twv ` twx O�¤ x y z L+O v�v O�N t Qgy x ~YNPNPO�ywQp` t ~YJ t JG`P�rONPO v{t L z Qgy v Qp`9 �k��\B�rO�NPO v{t L z Qgy �&QgN §&z O � ¡[O�N|Q�Jr~YNPa�Qgy t `W}�`PO v `R� z JML+O�N-`P�rO�J z y y �U}U_c~g`P�rO v{twv " t � OY�@NPO v{t L z Qgy v QgNPO�a z y ` t �pQgN t Qp`POJr~YNPa�Qgy )�� twv Q v }Ua�_+`P~g` twx Qgy y }7L twv `PN t b z `PORL/Q v Q x � t ��vP§&z QgNPO�� t `P� ��L+O � NPO�O v ~ ��� NPO�ORL+~Ya�� � O � Q t y `P~�NPOW£WO x `@`P�rOJ z y y��U}U_c~g`P�rO v{twv ~ � Jr~YNPa�Qgy t `W}�Qp`�9 � v{t � J t � x QgJ x O�y O��YO�yÍ� sut JMQgy y }Y��\uQgbry O ��NPO�_c~YN{` v `P�rO t JU�YO�N{`PORL�NP~U~g` v ~ � `P�rOx �MQgN|Q x `PO�N twv ` twx I-]ö_c~Yy }UJr~Ya t QgyÍ�4^K~/NP~U~g`�y t O v ~ z ` v{t L+O`P�rO z J t ` x�t N x y OY���rO�J x O`P�rO�O v ` t a�Qp`PORL=X I-] v } v `PO�av Qp` twv �MO v `P�rO v `|Qp` t ~YJMQgN t `W} x ~YJML t ` t ~YJ�� �Y�

����� 9 � ��� � � ����� �$� ��� �)�$ ��6 ���� ��� � � ���)�$ ����6 ��

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L= �Cê&ãPäWã

Lá åuæWê&ã�î&å�îUé�ì+ì é�ô-ëRü+ãPä{é�æWë�ä A{Þá ã�

A(L) = I − A1L − A2L2 − · · · − ApLp

Þrá æBá å#� ã|ì ì�÷gà&ë��Cà�æWêUé�æ�æWê&ãKæWá õ�ãKå�ãPäWá ã|å[á åBå æ{é�æWá ëRàUé�ä�ú�á ø�æWê&ã-þ|ãPäWëRå[ë�ø�æWê&ãïYãPæWãPäWõ�á àUé�àpæ{é�ì+ü+ëRì úYà&ëRõ�á é�ì|A(z)|

é�äWã[ëRîYæWå�á ïYã[æWê&ã[î&à&á æ�è|á äWè|ì ãBá ã�z−1

á à|A(z)| = 0

ã�ípîUé�æWá ëRà��Ý �YüYäWã|å�å�ã�ï�ïYá �rãPäWã|àpæWì úpÞæWê&ã#ã|å æWá õ�é�æWã�ï�ý ��Ü=á åCå æ{é�æWá ëRàUé�ä�ú�á ø�é�ì ìræWê&ã[äWëpë�æWå�êUé�â¢ãBõ�ëgïYî&ì î&å�ì ã|å�å�æWêUé�à�ëRà&ã�ÔgÖ

Page 14: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

ywQ �Y� ORLL t � O�NPO�J x ORL�`PO�NPa v ~YJ`P�rO�N t � �&` � �MQgJML v{t L+O�~ � `P�rO@`PO v `-NPO � NPO vPv{t ~YJ���\�~ v O�y O x `#`P�rO�Qg_r_rNP~Y_rN t Qp`PO@J z a�bcO�N~ � ywQ �Y� ORL7�MN v `�L t � O�NPO�J x O�`PO�NPa v k`P~!QYLrL����[O�a�Qg�YO zMv O�~ � `P�rOkNPO x�z N v{t �YO�_rNP~ x ORL z NPO�_rNP~Y_c~ v ORL!bU}�^ � QgJML� O�NPNP~YJ "$# %'%*9')�� � O v `|QgN{` � NP~YaEQgJ��U}U_c~g`P�rO�` twx Qgy�a�Qp¤ t a z aöywQ � y O�J � `P��� kmax = 12

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DUMt

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Page 15: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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Page 16: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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log(L0)twv `P�rO�y ~ � � y t �YO�y t �r~U~+L�~ � `P�rO�NPO v `PN twx `PORL�a�~+L+O�y

" t J&`PO�N x O�_+` v O�`@O §&z Qgy�`P~ � O�NP~Q vev _cO x�t �MORL t J7`P�rO�J z y y��U}U_c~g`P�rO v{twv )����#�rO�NPORQ vlog(L1)

twv `P�rO�y ~ � � y t �YO�y t �r~U~+L�~ �`P�rO z JrNPO v `PN twx `PORLka�~+L+O�yÍ� �KJML+O�N[`P�rO@J z y y��U}U_c~g`P�rO v{twv �λtwv L twv `PN t b z `PORL�Q v Q x � t ��vP§&z QgNPOKN|QgJML+~Ya �pQgN t Qgbry O-� t `P�

#�L+O � NPO�O�~ �C� NPO�ORL+~Ya�� � O � Q t yu`P~�NPOW£WO x `-`P�rO�J z y y �U}U_c~g`P�rO v{twv Qp`�9 � v{t � J t � x QgJ x O�y O��YO�yÍ� � O�`P�rO�NPO � ~YNPO � Q¢�Y~ z Nv _cO x�t � x Qp` t ~YJ;#0)�� t a�_ry } t J � Jr~ � NP~p�B`P� t J`P�rO�y O��YO�y�LrQp`|Q�QgJML � O�NP~ t J&`PO�N x O�_+` t J`P�rO x ~ t J&`PO � N|Qp` t J � NPO�ywQp` t ~YJ � �g� J7`P� twv bMQ v{twv ��`P�rO�_rNPO v O�J x O�~ � Q x ~ t J&`PO � N|Qp` t J � NPO�ywQp` t ~YJ v � t _ twv `PO v `PORL!bU} zMv{t J � `P�rO�`PN|Q x O�`PO v `R��I[`@`P� twvv `|Q � OY�u`P�rO�NPO � ~YNPOY� �Y~Y�MQgJ v O�J � v ��� � � �,+�� � � *Y��� � � �R�R  ���¢���| /a�O�`P�r~+L+~Yy ~ � } twv�zMv ORL�� � O�NPO�y }�~YJ<`P� twv�v `PN|Qp`PO � }v{t J x OY��Q v O�¤+_rywQ t JrORL�bU}�¡BQ z �[O�J v ��� ��� "$# %'%'%*)��r`P�rO v O�y O x ` t ~YJ7~ � `P�rO x ~ t J&`PO � N|Qp` t ~YJ�N|QgJr��Qg_r_ry } t J � ¡BQ¢}YO v{t QgJt J � O�NPO�J x O��MQ v Jr~g`#}YO�` � t �YO�J��YO�NP} x ~YJU� t J x�t J � NPO v{z y ` v �

0 \uI-¡BV�Z 9�I-¡ �-\ !KZ�]#Z$1\B�rO~ z ` x ~Ya�O~ � `P�rO`PN|Q x O�`PO v ` twv _rNPO v O�J&`PORL t J=`P�rO`P~Y_�_MQgN{`�~ � \uQgbry O�9+��\B�rOJ z y y#�U}U_c~g`P�rO v{twv�twv Jr~x ~ t J&`PO � N|Qp` t J � �YO x `P~YN v Qga�~YJ � `P�rO@�pQgN t Qgbry O v � � O@L+~�Jr~g` t J x y z L+O@QgJ t J&`PO�N x O�_+` t J�`P�rO x ~ t J&`PO � N|Qp` t J � O §&z Qp` t ~YJ��

5-O�`PO�NPa t J twv ` twx `PNPO�JML v QgNPO�O�¤ x y z L+ORL���Q v �[O�y yÍ��\B�rO x N t ` twx Qgy �pQgy z O v QgNPO�`P�r~ v O�NPO�_c~YN{`PORL7bU}7¦7Q x � t JrJr~YJ����#��� "$# %'%'%*)��KI[`-`P�rO,9 � v{t � J t � x QgJ x O�y O��YO�yÍ�M`P� twv `PO v `eNPOW£WO x ` v `P�rO�_rNPO v O�J x O�~ � Jr~ x ~ t J&`PO � N|Qp` t ~YJ�Qga�~YJ � `P�rO v O�N t O v �t J � Q¢�Y~ z NK~ � Q v{t J � y O x ~ t J&`PO � N|Qp` t J � NPO�ywQp` t ~YJ v � t _ "

r = 1) � �¢�e\B� twvev{z �Y� O v ` v `P�MQp`@NPO��YO�J z O v QgJML�O�¤+_cO�JML t ` z NPO vv �MQgNPOe~YJrO x ~ t J&`PO � N|Qp` t J � NPO�ywQp` t ~YJ v � t _�QgJMLk`P�rO�NPO twv ~YJrO x ~Ya�a�~YJ v `P~ x �MQ v ` twx `PNPO�JML�� t � OY�

yt ∼ CI(1, 1)��\B�rO�NPOO�¤ twv ` v Q�y t JrORQgN x ~Ya�b t JMQp` t ~YJ7~ � `P�rO v O�`W�[~L t � O�NPO�J x O ��v `|Qp` t ~YJMQgNP} v O�N t O vev ~�`P�MQp`e`P�rO�NPO v{t L z Qgy v ~Yb+`|Q t JrORL � NP~Ya`P� twv y t JrORQgN x ~Ya�b t JMQp` t ~YJ�QgNPO v `|Qp` t ~YJMQgNP}Y�

\B�rOKNPO v `PN twx `PORL x ~UO�- x�t O�J&` v L+O�N t �YORL � NP~Ya `P�rO x ~ t J&`PO � N|Qp` t J � O §&z Qp` t ~YJkQgNPOKL twv _rywQ¢}YORL t J�`P�rO-bc~g`{`P~Ya _MQgN{`�~ �\uQgbry O89+��\B�rO v O x ~UO�- x�t O�J&` v QgNPO `P�rO �Y~Y�MQgJ v O�J � v a�Qp¤ t a z a3y t �YO�y t �r~U~+L,"·¦7V�)�O v ` t a�Qp`PO v ~ � `P�rOCO�¤rQ x `Py } t L+O�J&` t �MORLx ~ t J&`PO � N|Qp` t J � �YO x `P~YNR�cQ � `PO�NeJr~YNPa�Qgy t � Qp` t ~YJ��K\B�rO�O v ` t a�Qp`PORL x ~ t J&`PO � N|Qp` t J � �YO x `P~YN twv βββ = [−1 0.8638]

′ �-\B�rOQ vPv ~ x�t Qp`PORL<O §&zrt y t brN t z a O §&z Qp` t ~YJβββ

yt = 0x ~YNPNPO v _c~YJML v ��� `P~

−revt + 0.8638expt = 0�=\B�rO�Q v }Ua�_+`P~g` twxv `|QgJMLrQgN|L�O�NPNP~YNB~ � `P�rO v Qga�_ry O v `|Qp` twv ` twx@twv fr� #¢d2(&d+�

J!bMQgywQgJ x OY��NPO��YO�J z O v@v O�O�a `P~ � NP~p� v y ~p�[O�N-`P�MQgJ/O�¤+_cO�JML t ` z NPO v ��\B�rO�NP~Yb zMv `PJrO vPv ~ � `P� twv NPO v{z y `R� t J/QgJU}x Q v OY� v �r~ z ywL�bcO x QgNPO �¥z y y } x �rO x �YORL�� � OKL+~@`P� twv�t J�`P�rO v O §&z O�yÍ�&L+O�O�_cO�J t J � ~ z NCQgJMQgy } v{twv QgJML�Qg_r_ry } t J � ¡BQ¢}YO v{t QgJt J � O�NPO�J x O@`P~�`P�rO x ~ t J&`PO � N|Qp`PORLkX I-]3a�~+L+O�yÍ�

� � � � � ���� � � ��� ����� ��� ��� �4��������� ��� � � ��� ����4� �� ����� ����� � ��� ����4� ���

� � � ��� ����4� � �4��� � �8��� ���

T�JGSUO x ` t ~YJ ��[O��MQ¢�YO�L+O�`PO�NPa t JrORL7`P�rO�J z a�bcO�N@~ �Bx ~ t J&`PO � N|Qp` t J � NPO�ywQp` t ~YJ v � t _ v bU}7a�ORQgJ v ~ � `P�rO�`PN|Q x O�`PO v `R�� O��MJML7~YJrO x ~ t J&`PO � N|Qp` t J � O §&z Qp` t ~YJ�bcO�`W�[O�O�J7NPO��YO�J z O v QgJML�O�¤+_cO�JML t ` z NPO v-t J7`P�rO�O v ` t a�Qp`PORL7b t �pQgN t Qp`PO�X I-]Ù ��Ý�å æWá õ�é�æWá ëRà�äWã|å�î&ì æWå�é�äWãKé�âRé�á ì é��&ì ã#ø äWëRõDæWê&ã#é�îYæWê&ë�ä�î&ü+ëRà�äWã�ípî&ã|å æ�Ù �|ò[ë�æWã[æWêUé�æ�ÞYæWê&ãPäWãPøwë�äWãRÞ � ã#î&å�ã[æWê&ã#ì á ÷¢ã|ì á ê&ëpëgï�ä{é�æWá ë@æWã|å æCæWë@è|ëRõ�üUé�äWã[æ � ë@ê&á ãPä{é�äWè{ê&á è�é�ì ì ú�à&ã|å æWã�ï�õ�ëgïYã|ì å � ����ã-à&ë�æWã-æWêUé�æ�Þ��Cá æWê&ëRîYæBäWã|å æ�äWá èPæWá à&ô�æWê&ã-è|ëRà&å æ{é�àpæBæWãPäWõ æWë�þ|ãPäWë�á àkæWê&ãeè|ëRá àpæWã|ô�ä{é�æWá ëRàã�ípîUé�æWá ëRàMÞrá æ � ëRî&ì ïkå�î&ôRôRã|å ææWêUé�æ�á à�æWê&ã#ì ëRà&ô�ñ¥äWî&à�äWã|â¢ã|àgî&ã|å�è�é�à �+ãBü+ëRå�á æWá â¢ã#ã|â¢ã|à�á ø�ã��Yü+ã|àUïYá æWîYäWã|å�é�äWã#þ|ãPäWë%�{Ù ��ã[ü+ëRá àpæCëRîYæ�æWêUé�æ�Þ&å�á à&è|ã � ãKé�äWãBã|å æWá õ�é�æWá à&ô�é

2 × 2ý ��ÜGå úYå æWã|õ�ÞYõ�é �Yá õeî&õ@ñ·ã|á ôRã|àgâRé�ì î&ã�é�àUï�æ�ä{é�è|ãBæWã|å æWå�å�ê&ëRî&ì ïüYäWëgïYî&è|ã#á ïYã|àpæWá è�é�ìräWã|å�î&ì æWåCá ø

r = 1 �&ë�ä�æWê&á å�äWã�é�å�ëRà�æWê&ãBõ�é �Yá õeî&õ@ñ·ã|á ôRã|àgâRé�ì î&ã�æWã|å æ�å æ{é�æWá å æWá èKá åCà&ë�æ�äWã|ü+ë�ä�æWã�ï$

� ���gá à&è|ã � ã�é�äWã�ïYã�é�ì á à&ô �Cá æWê�ì ëRôRå-ë�øCïYã54+é�æWã�ïkäWã|â¢ã|àgî&ã|åKé�àUïã��Yü+ã|àUïYá æWîYäWã|å|Þ�� ã�å�ê&ëRî&ì ï��+ãPæ�æWãPä���äWá æWã−log(revt) +

βexplog(expt) = 0�ùÍà �CêUé�æCøwëRì ì ë��Cå#� ã#å�êUé�ì ìcå æWá è{÷�æWëeæWê&á åCà&ë�æ{é�æWá ëRà�

Ô �

Page 17: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

v } v `PO�a���\B�rO�NPO � ~YNPOY�Y`P�rO-N|QgJr��~ ��x ~ t J&`PO � N|Qp` t ~YJ twv O §&z QgyM`P~�~YJrO "r = 1

)�QgJMLβββtwv Q

2× 1x ~ t J&`PO � N|Qp` t J � �YO x `P~YNR�

I v Qgy NPORQYL+}/L twvPx�zMvPv ORL�� `P�rO x ~ t J&`PO � N|Qp` t J � �YO x `P~YN twv Jr~g` t L+O�J&` t �MORL z Jry O vPv �[O t J&`PNP~+L zMx O v ~Ya�OQgNPb t `PN|QgNP}Jr~YNPa�Qgy t � Qp` t ~YJ���T�J � O�JrO�N|QgyÍ��`P~/bcO t L+O�J&` t �MORL�� r2NPO v `PN twx ` t ~YJ v �MQ¢�YO�`P~/bcO t a�_c~ v ORL=~YJ

αααQgJML

βββa�Qp`PN twx O v �S t J x O t JG~ z N x Q v O

r = 1��`P�rO�NPO twv Q v{t J � y OkJr~YNPa�Qgy t � Qp` t ~YJ � ~YN�`P�rO x ~ t J&`PO � N|Qp` t J � �YO x `P~YNR��`P�MQp` x ~YNPNPO v _c~YJML v`P~

βββ = [−1 βexp]′ ��Jr~YNPa�Qgy t ��t J � ~YJ�NPO��YO�J z O v ��¦�~YNPO�~p�YO�NR� ααα′ twv Q

1 × 2�YO x `P~YNe~ � QYL¢£ zMv `Pa�O�J&`�_MQgN|Qga�O�`PO�N v �

5 t � O�NPO�J&`Py } � NP~Yaβββ�rQ v Qgy NPORQYL+}�O�¤+_rywQ t JrORL�� t ` twv y O � ` z JrNPO v `PN twx `PORL��

T�J��#�MQp` � ~Yy y ~p� v �&�[O � ~ x�zMv ~YJk`P�rO x Q v Oe~ � Q v{t J � y O x ~ t J&`PO � N|Qp` t J � �YO x `P~YNR� \B�rO v O�` z _ twv `P�rO@X I-]Aa�~+L+O�y t J"Íd')K`P�MQp`R� t �[x ~ t J&`PO � N|Qp` t ~YJ��r~YywL v ��a�Q¢}��#N t `{`PO�J t J!O�NPNP~YN x ~YNPNPO x ` t ~YJ � ~YNPa Q v-t J " � )���\B� twv ywQp`{`PO�N x ~YNPNPO v _c~YJML v`P~�`P�rO@NPO � NPO vPv{t ~YJ�a�~+L+O�y t J "@(*)��C\B� twv NPO�_rNPO v O�J&`|Qp` t ~YJ twv JrO�ORL+ORL�`P~�_cO�N � ~YNPa `P�rO@¡BQ¢}YO v{t QgJ�O�a�_ t N twx Qgy�QgJMQgy } v{twv �^K~g`POC`P�MQp`R�p�rO�J x O � ~YN{`P���¢�[O�Q vPv{z a�O � ~YN�`P�rO��YO x `P~YN v

DDD0�DDD1�. . .�DDDT−1

t J " h2)�`P�rO v Qga�O�L+O�`PO�NPa t J twv ` twx�v `PN zMx ` z NPONPO�_c~YN{`PORL t J "Íd2(*)��S&` twx � t J � `P~�`P�rO�NPO � NPO vPv{t ~YJ!a�~+L+O�y t J "@&*)�QgJML zMv{t J � `P�rOk_rN t ~YN t J "$#'#0)�� ¡BQ z �[O�J v ������� ."$# %'%'%*)e_rNP~p� t L+Ov ~Ya�O zMv O �¥z y[NPO v{z y ` v�x ~YJ x O�NPJ t J � `P�rOQgJMQgy }&` twx Qgy � ~YNPa ~ � `P�rO_c~ v `PO�N t ~YN�L+O�J v{t `W}/~ � `P�rO x ~ t J&`PO � N|Qp` t J � �YO x `P~YN

βββ��T�J<S z b v O x ` t ~YJ;9+� #��[O�NPO�� t O��(_MQgN{`@~ � `P�rO v O twvPv{z O v ��T�J/S z b v O x ` t ~YJ 9+� d��[O�L+ORQgy � t `P��O�y twx�t `|Qp` t ~YJ!~ � _rN t ~YN_MQgN|Qga�O�`PO�N v �Y�#� t y O t J�S z b v O x ` t ~YJ 9+� ���[OKbrN t O��M}�O�¤+_rywQ t J�`P�rOea�O�`P�r~+L+~Yy ~ � } zMv ORL�`P~ x Qgy x�z ywQp`PO-a�~Ya�O�J&` v ~ � `P�rOa�QgN � t JMQgy�_c~ v `PO�N t ~YN#L twv `PN t b z ` t ~YJ � ~YN βββ

� T�J7S z b v O x ` t ~YJ�9+� ���[O�_rNPO v O�J&`KQgJMLL twvPx�zMvPv O v ` t a�Qp` t ~YJNPO v{z y ` v �

�)� 9 � �6 �� ����� � �$�������) ������� ����� � � ��� OkQgNPO�L+ORQgy t J � � t `P�/Q v{t J � y O x ~ t J&`PO � N|Qp` t J � �YO x `P~YNR��IKJU}�_rN t ~YN�L+O�J v{t `W}7~YJ7`P�rO z JrNPO v `PN twx `PORL7O�y O�a�O�J&`�~ � βββ

�φ(βexp)

� x QgJ!bcO zMv ORL���Qp`@y ORQ v ` t J�_rN t J x�t _ry OY��T � �[O�Q vPv{z a�O�Q v _rN t ~YN@L+O�J v{t `W}�O t `P�rO�N�Q��cQp`φ(βexp) ∝ 1

~YN@QS&` z L+O�J&`[L twv `PN t b z ` t ~YJ��Y`P�rO#_c~ v `PO�N t ~YNCL+O�J v{t `W} twvCt J&`PO � N|Qgbry O#QgJML�bcO�y ~YJ � v `P~�Q@�UJr~p�#J x ywQ vPv �g`P�rO#_c~Yy } � `[L+O�J v{t ` t O v~YJrOY��I v O�¤+_rywQ t JrORL�bU} 5-N�� � O�"$# %UhYh2)��gJr~�QgJMQgy }&` twx Qgy+O�¤+_rNPO vPv{t ~YJ v O�¤ twv ` � ~YNCa�~Ya�O�J&` v ~ � `P�rO v O-L+O�J v{t ` t O v �!KO�J x OY��[O-�MQ¢�YO#`P~�NPO v ~YN{`C`P~�J z a�O�N twx Qgy t J&`PO � N|Qp` t ~YJ�� S t J x O-�[O x ~Ya�_ z `PO-_c~ v `PO�N t ~YN�L twv `PN t b z ` t ~YJ v Qg_r_ry } t J � v{t a z ywQp` t ~YJa�O�`P�r~+L v ��`P�rO�NPO twv Jr~NPORQ v ~YJ�`P~�NPO v `PN twx `e~ z N�QgJMQgy } v{twv `P~7S&` z L+O�J&`�~YN#�cQp`@_rN t ~YN v � `P�rO�N@_rN t ~YN�L twv `PN t b z ` t ~YJ va�Q¢}kbcO zMv ORL��MQ v �[O�y yÍ�V�O�` zMv L+O�Jr~g`PO�� t `P�

s`P�rO@J z a�bcO�N#~ � y t JrORQgNBNPO v `PN twx ` t ~YJ v[t a�_c~ v ORL~YJ

βββt J�QYLrL t ` t ~YJ�`P~�`P�rO�Jr~YNPa�Qgy t � Qp` t ~YJx ~YJML t ` t ~YJ��

stwv �UJr~p�#J�Q v `P�rOK~YN|L+O�N�~ � ~p�YO�N � t L+O�J&` t � x Qp` t ~YJ�� � OKNPO�_c~YN{`C� t `P�r~ z `�_rNP~p� t J � `P�rO � ~Yy y ~p� t J � NPO v{z y ` v �`P�MQp`-QgNPO���~YNP~Yy ywQgNP}�%r� � t J�¡BQ z �[O�J v ��� ��� B"$# %'%'%*) � �p�

#Y�eT � `P�rO#_rN t ~YN�~YJ�`P�rO z JrNPO v `PN twx `PORL�O�y O�a�O�J&` v ~ �βββtwv L t ��zMv OY� t � OY�

φ(βexp) ∝ 1�p`P�rOK_c~ v `PO�N t ~YN

φ(βexp|Data)twv Q # � #�_c~Yy } � ` t J&`PO � N|Qgbry O�L+O�J v{t `W}Y��T � ��QYLrL t ` t ~YJMQgy y }Y��`P�rO�~YN|L+O�Ne~ � ~p�YO�N � t L+O�J&` t � x Qp` t ~YJ twv J z y y>"s = 0

)��`P� twv L+O�J v{t `W}�L+~UO v Jr~g`K_c~ vPv O vPv �MJ t `PO�a�~Ya�O�J&` v �d+�eT � �r~YJ`P�rO x ~YJ&`PN|QgNP}Y�

φ(βexp)twv S&` z L+O�J&`K� t `P�

ν0L+O � NPO�O v ~ �u� NPO�ORL+~Ya��U`P�rO�_c~ v `PO�N t ~YN φ(βexp|Data)

twv Qd � #e_c~Yy } � ` t J&`PO � N|Qgbry O�L+O�J v{t `W}k� t `P��MJ t `PO�a�~Ya�O�J&` v ~ � ~YN|L+O�N n + ν0 − 1�

S t J x O@�[O t a�_c~ v O@Jr~�QYLrL t ` t ~YJMQgy�NPO v `PN twx ` t ~YJ v ~YJ�`P�rOβββ�YO x `P~YNR�

s = 0t J`P�rO x Q v O z JML+O�N v ` z L+}Y�C\B�rO�NPO � ~YNPOY�� t `P��Q@L t ��zMv OB_rN t ~YNR�¢`P�rO4# � #[_c~Yy } � `CL+O�J v{t `W}��MQ v Jr~e�MJ t `PO#a�~Ya�O�J&` v � � � \�~�Q¢�Y~ t L@`P� twv L+N|Q¢�#bMQ x ���¢y O�` zMv Q vPv{z a�OQ�S&` z L+O�J&`#_rN t ~YNBL+O�J v{t `W}�~YJ

βexp

� t `P�ν0L+O � NPO�O v ~ ��� NPO�ORL+~Ya���\B�rOe_c~ v `PO�N t ~YNBL+O�J v{t `W} t J;"Ídgf*) x ~YNPNPO v _c~YJML v `P~Qkd � #e_c~Yy } � `B`P�MQp` x QgJbcO v _cO x�t �MORL�Q v

φ(βexp | Data) ∝ [1 + m0(βexp − β0)2]−

12 (ν0+1) ×

|β2expW2|

l2

|β2expW1|l1

,"Íd2&*)

�#�rO�NPOβ0�m0�ν0QgNPO�_rN t ~YN�_MQgN|Qga�O�`PO�N v QgJML

[1 + m0(βexp − β0)2]−

12 (ν0+1) twv `P�rO x ~YNPNPO v _c~YJML t J � S&` z L+O�J&`�YO�NPJrO�yÍ�

�&1 �&ë�ä�æWê&ã#üYäWëpë�ø � ãBäWãPøwãPä�æWëeæWê&á å��+ëpëR÷�� 2 ![ë�� ã|â¢ãPä�Þ&ïYã�&Uà&á à&ô@é-æ�äWî&à&è�é�æWã�ï+4+é�æ�üYäWá ë�ä�Þ%&Uà&á æWã#õ�ëRõ�ã|àpæWå ïYë@ã��Yá å æ�

ÔgØ

Page 18: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

sut N v `R���[O � ~ x�zMv ~YJ!`P�rO�N|Qp` t ~~ � L+O�`PO�NPa t JMQgJ&` v ~YJ7`P�rO�N t � �&` � �MQgJML v{t L+O�~ � "Íd2&*)���]#O�a t JML � NP~Ya "Íd #0)#`P�MQp`W2 = Z

MXZQgJML

W1 = Z′

MY (IT −Y(Y′

MY Y)−1Y′

)MY Z��\B�rO v O

2×2a�Qp`PN twx O v a�Q¢}@bcOBNPO�_rNPO v O�J&`PORL

Q vW2 =

[

W(1,1)2 W

(1,2)2

W(2,1)2 W

(2,2)2

] QgJMLW1 =

[

W(1,1)1 W

(1,2)1

W(2,1)1 W

(2,2)1

] �Y�#�rO�NPOB� t `P�W

(i,j)k

�[O t JML twx Qp`PO[`P�rO(i, j)

� `P�O�J&`PNP}k~ � `P�rO

k� `P�

Wa�Qp`PN t ¤��

I v{t L+O � NP~Ya x ~YJ v `|QgJ&`#`PO�NPa v � t ` x QgJ�bcO v �r~p�#J`P�MQp`K`P�rO�QgJMQgy }&` twx Qgy � ~YNPaö~ � `P�rO�_c~ v `PO�N t ~YN#a�Q¢}�bcO �¥z N{`P�rO�NO�¤+_rNPO vPv ORLQ v

φ(βexp | Data) ∝ [1 + m0(βexp − β0)2]−

12 (ν0+1) ×

(s2 + m2(βexp − β2)2)l2

(s1 + m1(βexp − β1)2)l1,

"Íd2%*)�#�rO�NPO

m2 = W(2,2)2

�m1 = W

(2,2)1

�β2 = m−1

2 W(1,2)2

�β1 = m−1

1 W(1,2)1

�s2 = W

(1,1)2 − β2

2m2QgJML

s1 =

W(1,1)1 −β2

1m1��^K~g`POB`P�MQp`C_c~Yy } � `CL twv `PN t b z ` t ~YJ v QgNPO x �MQgN|Q x `PO�N t � ORL�bU}�`P�rO#_rNP~Y_cO�N{`W}@`P�MQp`�`P�rO t N�L+O�J v{t `W}��YO�NPJrO�y vQgNPO[_rNP~+L zMx ` v �p~YNu_rNP~+L zMx ` v ~ � N|Qp` t ~ v �¢~ � S&` z L+O�J&`CL+O�J v{t `W}��YO�NPJrO�y v �¢O�¤rQ x `Py }�Q vut J "Íd2%*)�� � O[NPO � O�Nu`P~ 5-N�� � O "$# %UhYh2)� ~YNKQ�NPO�� t O��A~ � _c~Yy } � `KL+O�J v{t ` t O v#t J`P�rO � N|Qga�O��[~YNP��~ � ¡BQ¢}YO v{t QgJkNPO � NPO vPv{t ~YJ�QgJMQgy } v{twv �

�Cx ~ z N v OY�r~g`P�rO�N-_rN t ~YNKL+O�J v{t ` t O vex QgJ�bcO t J&`PNP~+L zMx ORL t J "Íd2&*)�� s ~YN t J v `|QgJ x OY�cQ v{zrt `|Qgbry O x �r~ twx O v O�O�a v Qgy v ~`P~�bcOφ(βexp | Data) ∝ [βι−1

exp (1 − βexp)κ−1] ×

(s2 + m2(βexp − β2)2)l2

(s1 + m1(βexp − β1)2)l1,

" �Yf*)�#�rO�NPO

ι > 0QgJML

κ > 0QgNPO#_rN t ~YNC_MQgN|Qga�O�`PO�N v QgJML

[βι−1exp (1−βexp)

κ−1]twv `P�rO x ~YNPNPO v _c~YJML t J � �YO�NPJrO�yr~ � Q�¡[O�`|QL+O�J v{t `W}Y� t � OY�

βexp ∼ Beta(ι, κ)� t `P�

0 < βexp < 1�-^K~g`POY���r~p�[O��YO�NR�+`P�MQp`e� t `P�!Qk¡[O�`|Q�_rN t ~YNK`P�rO�_MQgN|Qga�O�`PO�NL+~Ya�Q t J twv[x ~YJ v `PN|Q t JrORL�~YJ�`P�rO z J t ` v{t a�_ry O�¤�� !K~p�[O��YO�NR�p`P�rO@¡[O�`|Q�L+O�J v{t `W}�~YJ "·fr� #0) x QgJkbcOeORQ v{t y }�`PN|QgJ vW� ~YNPa�ORL`P~kQ�¡[O�`|Q�L+O�J v{t `W}�~YJ�QgJU}��MJ t `PO t J&`PO�NP�pQgy�bU}�Q�y t JrORQgN x �MQgJ � Oe~ � �pQgN t Qgbry OY�

�)�� ��� � ��� ���� � ��� � ����� � � � ��� � � �� ���

V�O�` zMv Jr~p�(` z NPJ!`P~�`P�rO v _cO x�t � x Qp` t ~YJ!~ �[v O�J v{t bry O�_rN t ~YN�L twv `PN t b z ` t ~YJ v QgJML7_MQgN|Qga�O�`PO�N v ���#�r~ v O�O�y twx�t `|Qp` t ~YJ twv_cO�NP�MQg_ v `P�rO�y O vPv ~YbU� t ~ zMv `|Q v � t J t a�_ry O�a�O�J&` t J � ¡BQ¢}YO v{t QgJ t J � O�NPO�J x OY��T�J "Íd2%*)��+`P�rO�S&` z L+O�J&`e_rN t ~YN#a�ORQgJ�a�Q¢}NPORQ v ~YJMQgbry }kbcOβ0 = 1

�B\B� twvKtwv-v O�`-~YJ�`P�rO � NP~ z JML v `P�MQp`R��Qp`-y ORQ v ` t J�`P�rO�y ~YJ � � N z J��r�[O�O�¤+_cO x `-NPO��YO�J z O v QgJMLO�¤+_cO�JML t ` z NPO v `P~ � NP~p�1Qp`�`P�rO v Qga�O-N|Qp`POeQgJML�� vPx Qgyc_c~Yy twx }�`P~�bcO v{zMv `|Q t JMQgbry OY�uT � `P�rO@_rN t ~YN[L+O�J v{t `W} twv[v _cO x�t �MORLbU}�a�ORQgJ v ~ � Q!S&` z L+O�J&`�L+O�J v{t `W} x O�J&`PNPORL<Qp`�~YJrOY�u�#�rO�NPO twv `P�rO�_c~ v `PO�N t ~YN�L+O�J v{t `W}φ(βexp | Data)

y ~ x Qp`PORL��\B� twv#twv `P�rO�� t JML~ ��§&z O v ` t ~YJ�[Oe`PNP}�`P~kQgJ v �[O�NB�rO�NPO t J��\�~!O�y twx�t `

m0�C�[O x QgJ v ~Yy �YO � ~YN�`P�rO z Jr�UJr~p�#JG_MQgN|Qga�O�`PO�N�`P�rOO §&z Qp` t ~YJ

m0 = 1ν0−2Var(βexp)

−1� t � �[O�MQ¢�YOK_rN t ~YN t J � ~YNPa�Qp` t ~YJ�~YJ

Var(βexp)��V t `{`Py O t JML twx Qp` t ~YJ twv � t �YO�J�bU}�`P�rOeLrQp`|Q x ~YJ x O�NPJ t J � `P�rO-_rN t ~YN��pQgN t QgJ x OY�

JrO�_c~ vPv{t b t y t `W} twv `P~ v ~Yy �YOe`P� twv O §&z Qp` t ~YJ � ~YN-L t � O�NPO�J&`-�pQgy z O v ~ �Var(βexp)

�BT�J`P� twv �BQ¢}��[O x QgJ�Q vPv O vPv �r~p�`P�rO�_rN t ~YN#L twv _cO�N v{t ~YJ t J�� z O�J x O v `P�rO@_c~ v `PO�N t ~YN#L+O�J v{t `W}QgJML t ` v _MQgN|Qga�O�`PO�N v � IKy `PO�NPJMQp` t �YO�y }Y�+�[O x QgJ � ~Yy y ~p�A`P�rOa�O�`P�r~+LGL+O vPx N t bcORL�bU}�¡BQ z �[O�J v ������� "$# %'%'%*)���_�� #Rf �r� � ~YN�`P�rOkO�y twx�t `|Qp` t ~YJ<~ � _rN t ~YN�_MQgN|Qga�O�`PO�N v ~ � a�QgN � t JMQgyS&` z L+O�J&`eL+O�J v{t ` t O v �I x�x ~YN|L t J � `P~�`P� twv ywQp`{`PO�N�Qg_r_rNP~&Q x �����[O v O�` s0 = 1

�ν0 = 12

�β0 = q0.025+q0.975

2 = 0.5+1.52 = 1

�m0 =

1ν0

(

2t0.95

q0.975−q0.025

)2

= 1.0589�g�#�rO�NPO

t0.95 = 1.7823twv `P�rO %*9@_cO�N x O�J&` t y O-~ � `P�rO v `|QgJMLrQgN|L t � ORLkS&` z L+O�J&`#L+O�J v{t `W}

� t `P� #¢d�L+O � NPO�O v ~ ��� NPO�ORL+~Ya�����~YJ v O §&z O�J&`Py }Y�Var(βexp) = 0.0944

�u^K~g`POK`P�MQp`[�[O-�MQ¢�YO x �r~ v O�J � ~YNβexp

`P�rO1%*9_cO�N x O�J&`-� t � �rO v `K_rN t ~YNKL+O�J v{t `W} t J&`PO�NP�pQgy Pr[q0.025 < βexp < q0.975] = Pr[0.5 < βexp < 1.5] = 0.95�[I(�[~YN|L~ ��x Q z ` t ~YJ�Qp`C`P� twv _c~ t J&`R� T�JS z b v O x ` t ~YJ 9+� �M�Y�[OeQgNPO � ~ t J � `P~��Í��++�����p���#`P�rO@S&` z L+O�J&` "·QgJMLk¡[O�`|Q*)�_rN t ~YN v `P~ " � #Y�

�*)���\B�rO�a�~Ya�O�J&` vβ0 = 1

QgJMLVar(βexp) = 0.0944

QgNPO@Jr~g`#O�¤rQ x ` � ~YN��Í��++�����p���| S&` z L+O�J&`�"·QgJML�¡[O�`|Q*)[_rN t ~YN v �!K~p�[O��YO�NR�&`P�rO v O��pQgy z O v QgNPO x y ~ v Oe`P~�`P�rO@O�¤rQ x `K~YJrO vBv{t J x O@`P�rO@`PN z J x Qp` t ~YJ twv � v a�Qgy y � " v O�O sut � z NPO v ��QgJML 9')��

0 s T � �-]#Z �kI-¡ �-\ !KZ�]#Z$1Ô �

Page 19: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

sut � z NPO0��L twv _rywQ¢} v `P�rO�S&` z L+O�J&`(β0 = 1, s0 = 1, m0 = 1.0589, ν0 = 12)

_rN t ~YNR��`P�rO�a�QgN � t JMQgy y t �YO�y t �r~U~+LQgJML x ~YNPNPO v _c~YJML t J � _c~ v `PO�N t ~YNKL+O�J v{t `W}�~ � `P�rO v Qga�_ry ORL�LrQp`|Qr�[\B�rO v O�QgNPO�~ z NKNPO � O�NPO�J x O@�pQgy z O v �B\B�rO�_c~ v `PO�N t ~YNL+O�J v{t `W} twv�v y t � �&`Py }�N t � �&` ��v �YO��[ORL��gNPO��MO x ` t J � `P�rO t J�� z O�J x O-~ � `P�rO-y t �YO�y t �r~U~+L�~ � `P�rOeLrQp`|Qr� \uQ t y v QgNPOKJr~g` v ~�y ~YJ �v{t J x OY� t J!`P�rO x Q v O z JML+O�N v ` z L+}Y��`P�rO�_c~ v `PO�N t ~YN@L+O�J v{t `W} twv@t J&`PO � N|Qgbry O�QgJML!_c~ vPv O vPv O v �MJ t `PO�a�~Ya�O�J&` v ��I v �[Oa�Q¢}�O�¤+_cO x `R�&� t `P��`P� twv�x �r~ twx O#~ � `P�rOβ0_rN t ~YNC_MQgN|Qga�O�`PO�NR�p`P�rOK_rN t ~YN t J � ~YNPa�Qp` t ~YJ�~YJ

βexp

a�~p�YO v `P�rOK_c~ v `PO�N t ~YNO�¤+_cO x `PORL4�pQgy z O`P~p�BQgN|L v `P�rO�_rN t ~YN�O�¤+_cO x `PORL��pQgy z OY� ��y ORQgNPy }GO�Jr~ z � ���[a�~YNPO�~p�YO�NR��`P�rO�_c~ v `PO�N t ~YN�L+O�J v{t `W} twvy ~ x Qp`PORLbcO�`W�[O�O�J�`P�rO�y t �YO�y t �r~U~+LQgJML�`P�rO�_rN t ~YN#L+O�J v{t `W}Y�CIKy y�`P� twvBtwv � twv{t bry OY� �·��� +!�R� +�� �¹� t J sut � z NPO��M�srz N{`P�rO�NPa�~YNPOY��`P�rO v a�Qgy y O�Ne`P�rO

m0_MQgN|Qga�O�`PO�NR��`P�rO�b t �Y� O�N twv `P�rO�_rN t ~YN@�pQgN t QgJ x O

Var(βexp)��\B�rO�b t �Y� O�Ntwv

Var(βexp)�[`P�rO�y O vPv `P�rO�_rN t ~YN twvkx ~YJ x O�J&`PN|Qp`PORL�QgNP~ z JML t ` v a�ORQgJ��A\B� zMv �#Qgy v ~!`P�rO7_c~ v `PO�N t ~YN�L+O�J v{t `W} twvt J�� z O�J x ORLbU}k`P�rO x �r~ twx Oe~ �

m0_rN t ~YNB_MQgN|Qga�O�`PO�NR�

s ~ x�zMv{t J � ~YJ�`P�rO�¡[O�`|Q�_rN t ~YNKL+O�J v{t `W} t J " �Yf*)��r�[O t J t ` t Qgy y } v{z �Y� O v `KQ�¡[O�`|Q (2, 2)v _cO x�t � x Qp` t ~YJ;"

ι = κ = 2)��

t � OY�/Q v }Ua�a�O�`PN twx L twv `PN t b z ` t ~YJ�QgNP~ z JML�`P�rO�a�ORQgJ (

E(βexp) = ιι+κ

= 0.5) �;!K~p�[O��YO�NR� Q7_rN t ~YN�L twv `PN t b z ` t ~YJ

~YJ "·fr� #0) twv O�¤ x O vPv{t �YO�y }kNPO v `PN twx ` t �YO�Q v �U}U_c~g`P�rO v{twv �#\B�rO�NPO � ~YNPOY�r�[O�Qg_r_ry }Q�y t JrORQgN x �MQgJ � O�~ � �pQgN t Qgbry O@`P~ βexp

�� O�JrO�N|Qp` t J � Q�JrO�� �pQgN t Qgbry O βexp = c × βexp + d

��`P~�bcOkL+O��MJrORL t J<QywQgN � O�N t J&`PO�NP�pQgy� s ~YN t J v `|QgJ x OY��`|Qg� t J �c = 3.5

QgJMLd = −1

�+�[O��MQ¢�YOe`P�MQp`E(βexp) = c × E(βexp) + d = 0.75

�CIKy `PO�NPJMQp` t �YO�y }Y� t �c = 4

QgJMLd = −1

�E

(

βexp

)

= c × E(βexp) + d = 1�C\B� twv ywQ v ` x �r~ twx O � ~YN

cQgJML

dx ~YJ v `|QgJ&` v[twv#t a�_ry O�a�O�J&`PORL�rO�NPORQ � `PO�NR�

��~Ya t J � `P~�`P�rO�_rN t ~YN#�pQgN t QgJ x OY�Var(βexp) = ικ

(ι+κ+1)(ι+κ)2 = 0.05��T �

c = 3.5QgJML

d = −1�Var(βexp) =

c2 × Var(βexp) = 0.6125�CbcO t J � `P�rO��pQgN t QgJ x O�Q §&z QYL+N|Qp` twx ~Y_cO�N|Qp`P~YNR�AS t a t ywQgNPy }Y� t � c = 4

QgJMLd = −1

�Var(βexp) = c2 ×Var(βexp) = 0.8

� \B�rOey t �YO�y t �r~U~+L�L+~Ya t JMQp`PO v `P�rO@¡[O�`|Q�_rN t ~YN[a zMx �ka�~YNPOK`P�MQgJk`P�rO�S&` z L+O�J&`_rN t ~YNR� v{t J x O�`P�rO�¡[O�`|Q twv a�~YNPO�L twv _cO�N v ORL�`P�MQgJ7`P�rO�S&` z L+O�J&`R� � O�Jr~g`PO�`P�MQp`e`P�rO�¡[O�`|Q(2, 2)

_rN t ~YN-�pQgN t QgJ x O twvQgy a�~ v `BJ t JrO-` t a�O v b t �Y� O�NC`P�MQgJ�S&` z L+O�J&` (1, 1, 1.0589, 12)_rN t ~YN��pQgN t QgJ x O "·fr� f'%.� � )��u\B�rO@¡[O�`|Q

(2, 2)L+O�J v{t `W} twv`P~U~�L twv _cO�N v ORL�QgJML��Y`P�rO�NPO � ~YNPOY�Y`P�rO-_c~ v `PO�N t ~YNCNPO v{z y ` v QgNPOKJr~g` �¥z y y } x ~Ya�_MQgN|Qgbry O#� t `P��`P�r~ v OK~Yb+`|Q t JrORLkQ vPv{z a t J �Q�S&` z L+O�J&`e_rN t ~YNR� s ~YN#`P� twv NPORQ v ~YJ��r�[O x �MQgJ � O@`P�rO�_MQgN|Qga�O�`PO�N v ~ � `P�rO�¡[O�`|Q�_rN t ~YN#`P~ka�Qg�YO t ` v _rN t ~YNK�pQgN t QgJ x Ox y ~ v O�N�`P~�`P�MQp`k~ � `P�rO7S&` z L+O�J&`R� � O��YO�O�_=`P�rO�¡[O�`|Q/L+O��MJrORL=~YJ=`P�rO " � #Y�/�*) t J&`PO�NP�pQgyKQgJML=�[O t J x NPORQ v O�`P�rO`W�[~�_MQgN|Qga�O�`PO�N v

ιQgJML

κ`P~7L+O x NPORQ v O�`P�rOk_rN t ~YN@�pQgN t QgJ x OY�k¦�~YNPO�~p�YO�NR�c`P~�O�J v{z NPO�Q v }Ua�a�O�`PN twx�v �MQg_cO�~ � `P�rOL twv `PN t b z ` t ~YJ��r�[O@NPO §&zrt NPO

ι = κ�

¡[} zMv{t J � `P�rOeO�¤+_ry twx�t ` � ~YNPa z ywQ � ~YN�`P�rO-�pQgN t QgJ x OK~ � Q�¡[O�`|Q�_rN t ~YN�QgJML�`P�rO � Q x `�`P�MQp` ιtwv O §&z QgyM`P~

κ�&�[OK�MJML`P�MQp`

Var(βexp) = c2 × Var(βexp) = 16 ×

(

ικ

(ι + κ + 1)(ι + κ)2

)

= 0.0944 ⇒ ι = κ = 20.6864.

��~YJ v O §&z O�J&`Py }Y�+�[O x �r~U~ v O�Q�¡[O�`|Q(ι = 20, κ = 20)

�U`PN z J x Qp`PORL~YJ;" � #Y� �*)��rQ v `P�rO@�MJMQgy�_rN t ~YN v _cO x�t � x Qp` t ~YJ��0 s T � �-]#Z 9�I-¡ �-\ !KZ�]#Z$1

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Page 20: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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Page 21: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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βexp) = 0.2818�\B� twv a�Qg�YO vKv O�J v OY�M~ �Cx ~ z N v OY��T � `P�rO�_rN t ~YN twv y O vPv-x ~YJ x O�J&`PN|Qp`PORL;" t � OY�

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� O�Qgy v ~�_rNPO v O�J&`�~g`P�rO�N�L twv `PN t b z ` t ~YJMQgyC�U}U_c~g`P�rO v O v�x ~YJ x O�NPJ t J � `P�rO�_rN t ~YN�L+O�J v{t `W}Y��� v{t J � Q�¡[O�`|Q (20, 20)" x Q v O�d')KL+O��MJrORL t J." � #Y� �*)��c`P�rO�_c~ v `PO�N t ~YNKa�ORQgJ twv fr� %&d.�&d�QgJML�`P�rO v `|QgJMLrQgN|L�L+O�� t Qp` t ~YJ t J x NPORQ v O v `P~fr� d.�'%&d+�s ~YNeQ�`PN z J x Qp`PORL �cQp`e_rN t ~YN�" x Q v O �*)��M`P�rO�O v ` t a�Qp`PORL�_c~ v `PO�N t ~YNKa�ORQgJ twv O §&z Qgy�`P~�fr� &'% �'(r�#\B�rO x ~YNPNPO v _c~YJML t J �v `|QgJMLrQgN|L�L+O�� t Qp` t ~YJ twv fr� �*%/# &r��I v O�¤+_cO x `PORL���� t `P�<Q �cQp`�_rN t ~YNR��_c~ v `PO�N t ~YN@O v ` t a�Qp` t ~YJ twv y O vPv Q x�x�z N|Qp`PO�`P�MQgJ�#�rO�JGQ vPv{z a t J � t J � ~YNPa�Qp` t �YO�_rN t ~YN v " x Q v O v #�QgJML<d')���^K~g` twx O�`P�MQp`�`P�rO�_c~ v `PO�N t ~YN�L+O�J v{t `W}!� t `P�<Q`PN z J x Qp`PORL�cQp`e_rN t ~YN x ~YNPNPO v _c~YJML v `P~�`P�rO�a�QgN � t JMQgy�y t �YO�y t �r~U~+L �¥z J x ` t ~YJ t J sut � z NPO��k~YN t J sut � z NPO 9�"·LrQ v �rORL�y t JrO0)��K\B� twvL+O�J v{t `W}��MQ v _c~ v{t ` t �YO v �YO��#JrO vPv QgJML � Qp`B`|Q t y v �

��y ORQgNPy }�O�Jr~ z � ����`P�rO_c~ v `PO�N t ~YN�a�ORQgJE(βexp)

twv y ~ x Qp`PORLG_rNPO�`{`W} � QgN�Q¢�BQ¢} � NP~Ya #Y�C�#�MQp`PO��YO�N�`P�rO�_rN t ~YNL+O�J v{t `W}L+O��MJrORL " x Q v O v #Y� d+� �*)�� v{t J x O@`P�rO�a�QgN � t JMQgy�y t �YO�y t �r~U~+L~ � `P�rO x ~ t J&`PO � N|Qp` t J � �YO x `P~YN#�MQ v a�~YNPO@a�Q vPv ~YJ`P�rO�y O � `�~ � #kQgJML7`P�rO v _cO x�t �MORL�_rN t ~YN v QgNPO §&zrtwx �Uy }!L+~Ya t JMQp`PORL!bU}�`P�rOkLrQp`|Qr� J�bMQgywQgJ x OY��`P�rO�O v ` t a�Qp`PO v ~ �`P�rO x ~ t J&`PO � N|Qp` t J � x ~UO�- x�t O�J&` � ~YN�NPO��YO�J z O � O�¤+_cO�JML t ` z NPO��pQgN t Qgbry O v �pQgNP}!bcO�`W�[O�O�J=fr� &'% �'(�QgJML<fr� %&d2%'(r��\B�rO v OO v ` t a�Qp`PO vev �r~ z ywL�bcO�NPO � QgN|L+ORL�Q v Qka�ORQ v{z NPO�~ � βexp

t J�`P�rO�y ~YJ � � N z J � O §&z Qp` t ~YJ log(revt) = βexplog(expt)��#�rO�NPO�`P�rO v O��pQgN t Qgbry O v �MQ¢�YO�bcO�O�J!O�¤+_rNPO vPv ORL t J!NPORQgy�`PO�NPa v L t � t L t J � bU}�!KT ��� ��T�J&`PO�NP_rNPO�` t J � `P�rO v O�O v ` t a�Qp`PO vt JG`PO�NPa v ~ � O�ywQ v ` twx�t ` t O v �u`P� twv�t JML twx Qp`PO v `P�MQp`R� t J<`P�rO�y ~YJ � � N z J�� Q #RfYf�_cO�N x O�J&`�N twv O t JGb z L � O�`�O�¤+_cO�JML t ` z NPO v� t y y�Qg_r_rNP~¢¤ t a�Qp`PO�y }�`PN|QgJ v ywQp`PO t J&`P~�Q %Yf�_cO�N x O�J&` � NP~p�B`P��~ � NPO��YO�J z O v � IKJMLk`P�MQp`B� ~Yy twv �k� vPx Qgyc_c~Yy twx }kL+~UO v Jr~g`v Qp` twvW� }7`P�rO t J&`PO�N{`PO�a�_c~YN|Qgy b z L � O�` x ~YJ v `PN|Q t J&`�" t J v `PNP~YJ � � ~YNPa )���\B� zMv ��~YJ!`P�rO�bMQ v{twv ~ � `P�rOkQ¢�pQ t ywQgbry O�LrQp`|Qr��[O��MQ¢�YO�O�� t L+O�J x Ok`P�MQp`�O�¤+_cO�JML t ` z NPO v " t J x y zMv{t �YOk~ �Kt J&`PO�NPO v `�_MQ¢}Ua�O�J&` v )�QgNPO � NP~p� t J � � Q v `PO�N�`P�MQgJGNPO��YO�J z O v �

�KJML+~ z b+`PORL+y }Y�Y`P� twv�x ~YJ v ` t ` z `PO v Q@�BQgNPJ t J � v{t � JMQgy+`P~�`P�rOK� ~Yy twv � � ~p�YO�NPJra�O�J&`R��\B�rO v OKNPO v{z y ` vCx Q v `�L+~ z b+`�~YJ�`P�rO_c~ vPv{t b t y t `W}�`P�MQp` � ~p�YO�NPJra�O�J&`CNPO��YO�J z O v QgNPO x Qg_MQgbry O-~ � a�O�O�` t J ��� ~p�YO�NPJra�O�J&`CO�¤+_cO�JML t ` z NPO v � t J�O §&zrt y t brN t z a " t J`P�rO�y ~YJ � � N z J6)��\B�rO�bc~g`{`P~Ya _MQgN{`�~ � \uQgbry O�(/_rNPO v O�J&` v `P�rO�¡BQ¢}YO v{t QgJ=_c~ v `PO�N t ~YNka�~+L+O�~ � `P�rO z JrNPO v `PN twx `PORL4O�y O�a�O�J&`�~ �`P�rO x ~ t J&`PO � N|Qp` t J � �YO x `P~YN�"

βexp

)���Q vPv{z a t J � QgJ z J&`PN z J x Qp`PORL �cQp`e_rN t ~YNR�KT�J�`P� twvex Q v O�`P�rO�a�QgN � t JMQgy�_c~ v `PO�N t ~YNL+O�J v{t `W} twv�t J&`PO � N|Qgbry O#b z `#L+~UO v Jr~g`[_c~ vPv O vPv �MJ t `PO-a�~Ya�O�J&` v ��\B�rO-_c~ v `PO�N t ~YNCa�~+L+O twv[x Qgy x�z ywQp`PORL�bU}�a�Qp¤ t a z ay t �YO�y t �r~U~+L�� t � OY��bU}ka�Qp¤ t a t ��t J � � t `P��NPO v _cO x `B`P~ βexp

`P�rO�a�QgN � t JMQgy�y t �YO�y t �r~U~+L �¥z J x ` t ~YJ

L(βexp | Data) ∝(s2 + m2(βexp − β2)

2)l2

(s1 + m1(βexp − β1)2)l1,

" �/#0)

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Page 22: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

�#�rO�NPOY�KQ v O�¤+_rywQ t JrORL t J "Íd #0)��m2 = W

(2,2)2

�m1 = W

(2,2)1

�β2 = m−1

2 W(1,2)2

�β1 = m−1

1 W(1,2)1

�s2 =

W(1,1)2 −β2

2m2QgJML

s1 = W(1,1)1 −β2

1m1� \B�rO#a�Qp¤ t a z a y t �YO�y t �r~U~+L�O v ` t a�Qp`POKQgJML�`P�rOKQ vPv ~ x�t Qp`PORL�Q v }Ua�_+`P~g` twxv `|QgJMLrQgN|L�O�NPNP~YN��MQ¢�YOCbcO�O�J�~Yb+`|Q t JrORL zMv{t J � `P�rO[¦7I�\#V�I-¡ �¥z J x ` t ~YJ � ~YNu~Y_+` t a t � Qp` t ~YJ�¦7Qp¤ *By t ��� a�� �p�u\B�rO�a�~+L+O~ � `P�rO-a�QgN � t JMQgyry t �YO�y t �r~U~+L �¥z J x ` t ~YJ " t � OY�u`P�rO-_c~ v `PO�N t ~YN�L+O�J v{t `W} z JML+O�NBQ��cQp`�_rN t ~YN ) twv�v � t � `PORL�`P~�`P�rO-y O � `[� t `P�NPO v _cO x `e`P~k`P�rO�_c~ v `PO�N t ~YN-a�ORQgJ z JML+O�N@Q�S&` z L+O�J&`@_rN t ~YN "·fr� %&d2%'(*)#QgJML z JML+O�N@Q�¡[O�`|Q�_rN t ~YN "·fr� %&d.�&d')��BT�JML+O�ORL��Q v NPO�_c~YN{`PORL t J=\uQgbry O (r�u`P�rOa�~+L+O�~ � `P�rO�a�QgN � t JMQgy�y t �YO�y t �r~U~+L �¥z J x ` t ~YJ twv fr� &YfYf'(�QgJML/`P�rO x ~YNPNPO v _c~YJML t J �Q v }Ua�_+`P~g` twx@v `|QgJMLrQgN|L�O�NPNP~YN twv fr� d2&'& �r�

JrO x ~Ya�a�O�J&`KQgbc~ z `#`P�rO�¡BQ¢}YO v{t QgJ�_c~ v `PO�N t ~YNBa�~+L+O�"·fr� &YfYf'(*)�QgJML�`P�rO �Y~Y�MQgJ v O�J � v ¦7V/O v ` t a�Qp`PO�"·fr� &'( �'&*)NPO�_c~YN{`PORL t J\uQgbry O49+� \B�rO@L t � O�NPO�J x O-bcO�`W�[O�O�Jk`P�rO�a twv Jr~g`[JrO � y t � t bry OeQgJML�Q � O��A�[~YN|L v ~ �ux ywQgN t � x Qp` t ~YJkQgNPOeQp`~YN|L+O�NR�[I v O�¤+_rywQ t JrORL�bU}¡BQ z �[O�J v QgJML�V z brN|QgJr~�"$# %'%'(*)��r_�� #'#Y�+`P�rO �Y~Y�MQgJ v O�J � v ¦7V<O v ` t a�Qp`PO x ~YNPNPO v _c~YJML v `P~`P�rO�a�Qp¤ t a z aE~ � `P�rO�|�p���|���M�Í���p���| �y t �YO�y t �r~U~+L �¥z J x ` t ~YJ��c�#� t y O@`P�rO�_c~ v `PO�N t ~YN#a�~+L+O z JML+O�NeQ0�cQp`K_rN t ~YN twv `P�rOa�Qp¤ t a z a ~ � `P�rO ���p� �.�¹� ���uy t �YO�y t �r~U~+L �¥z J x ` t ~YJ���T�J_MQgN{` twx�z ywQgNR� t ` twv _c~ vPv{t bry O-`P~ v �r~p�1`P�MQp`B`P�rO x ~YJ x O�J&`PN|Qp`PORLy t �YO�y t �r~U~+L �¥z J x ` t ~YJ twv

L(βexp | Data) ∝(s2 + m2(βexp − β2)

2)l1

(s1 + m1(βexp − β1)2)l1,

" �&d')t � OY� t `eL t � O�N v[� NP~Ya `P�rO�a�QgN � t JMQgy�y t �YO�y t �r~U~+L �¥z J x ` t ~YJ t J " �/#0)�bU}k`P�rO�O�¤+_c~YJrO�J&` t J`P�rO�J z a�O�N|Qp`P~YN4"

l1t J v `PORQYL~ �

l2)��

0 s T � �-]#Z (�I-¡ �-\ !KZ�]#Z$1\B�rO�L+O�J v{t ` t O v#t J " �/#0)BQgJML;" �&d')BQgNPO�L twv _rywQ¢}YORL t J sut � z NPO�(r�[^K~g`PO@`P�MQp`-bc~g`P��QgNPO�# � #@_c~Yy } � `R��T ` twv O�� t L+O�J&``P�MQp`@`P�rO x ~YJ x O�J&`PN|Qp`PORL�y t �YO�y t �r~U~+L �¥z J x ` t ~YJ twv@v � t � `PORL7`P~k`P�rO�N t � �&`e� t `P�!NPO v _cO x `e`P~k`P�rO�a�QgN � t JMQgy�y t �YO�y t �r~U~+L�¥z J x ` t ~YJ��B¦�~YNPO�~p�YO�NR�Y`P�rO x ~YJ x O�J&`PN|Qp`PORL�y t �YO�y t �r~U~+L �¥z J x ` t ~YJ��MQ vB� Qp`{`PO�N#`|Q t y v QgJML��MQ v O�¤+_cO x `PORL�� t ` v a�Qp¤ t a z a

"·fr� &'( �'&*) twv y ~ x Qp`PORL�~YJ�`P�rO�N t � �&`#~ � `P�rO�a�Qp¤ t a z aö~ � `P�rO�a�QgN � t JMQgy�y t �YO�y t �r~U~+L �¥z J x ` t ~YJ "·fr� &YfYf'(*)��sut JMQgy y }Y�+y O�` zMv ` z NPJ�`P~�`P�rO�L+}UJMQga twx�v ~ � v �r~YN{` � `PO�NPaöQYL¢£ zMv `Pa�O�J&`B`P~p�BQgN|L v `P�rO�y ~YJ � � N z J�O §&zrt y t brN t z a��T�J!\uQgbry O�h��[O�_rNPO v O�J&`-_c~ v `PO�N t ~YN-NPO v{z y ` v#� ~YNK`P�rO�QYL¢£ zMv `Pa�O�J&` x ~UO�- x�t O�J&` v " t � OY�

αrev

QgJMLαexp

)��-T�J7~g`P�rO�N�[~YN|L v � � ~ x�zMv{t J � ~YJ1`P�rO�X-Z �[¦ O §&z Qp` t ~YJ t J+" � )��-�[O�Qg_r_ry }A¡BQ¢}YO v{t QgJ t J � O�NPO�J x O7`P~=`P�rO�O v ` t a�Qp` t ~YJA~ �αααx ~UO�- x�t O�J&` v#t J�`P�rO�y t JrORQgNKQYL¢£ zMv `Pa�O�J&`K_rNP~ x O vPv

αααβββ′

yyyt−1�+�#� twx � x QgJbcO v _cO x�t �MORL�Q v

[

αrev

αexp

]

[

−1 βexp

]

[

revt−1

expt−1

]

=

[

αrev

αexp

]

[−revt−1 + βexpexpt−1] ." � �*)

T�J." � �*)��αrev

QgJMLαexp

QgNPO�Q�a�ORQ v{z NPO�~ � `P�rO v _cO�ORL�~ � QYL¢£ zMv `Pa�O�J&`-~ �revt

QgJMLexpt

�cNPO v _cO x ` t �YO�y }Y�+`P~p�BQgN|L v`P�rO t NKy ~YJ � � N z J�O §&zrt y t brN t Qr��\B�rO�� t � �rO�NB`P�rO t N#�pQgy z O v �U`P�rO §&zrtwx �YO�NB`P�rO�QYL¢£ zMv `Pa�O�J&`K_rNP~ x O vPv �0 \uI-¡BV�Z h�I-¡ �-\ !KZ�]#Z$1

\B�rO�O v ` t a�Qp` t ~YJ twv _cO�N � ~YNPa�ORL<bcORQgN t J � t J=a t JML<`P�rO�L twv `PN t b z ` t ~YJMQgyBNPO v{z y ` t J "$#ph2)�� t � OY�B | βββ, Data ∼

Mt(m+n(p−1)+r)×n(B,W′

W, S, T − (m+n(p−1))−r)��Z�¤+_rNPO vPv{t ~YJ v � ~YN

E(α′

α′

α′

| βββ, Data)QgJML

Var(vec(α′

α′

α′

|βββ, Data))

QgNPO � t �YO�J t J "Íd.�*)�QgJML "Íd�� )�� \B�rO�NPO � ~YNPOY�#Qp`ORQ x �4_c~ t J&`�`P�rO!_c~ v `PO�N t ~YN�L+O�J v{t `W} � ~YNβββ�YO x `P~YN twvO��pQgy z Qp`PORL�� t ` twv _c~ vPv{t bry OB`P~ x Qgy x�z ywQp`PO#_c~ v `PO�N t ~YN x ~YJML t ` t ~YJMQgyrNPO v{z y ` v � ~YN

ααα�CS z b v O §&z O�J&`Py }Y�YbU}�a�QgN � t JMQgy t � Qp` t ~YJ���[O x QgJ�Qg_r_rNP~¢¤ t a�Qp`PO

E(α′

α′

α′

|Data)� t `P�

N∑

i=1

E(α′

α′

α′

| βββi, Data)

N,

1PÙWùÍõ�ü&ì ã|õ�ã|àpæWã�ï �pú ��á è{êUé�ã|ìcÜ�ëpè{÷gá à&ôRãPä�é�àUï�é�âRé�á ì é��&ì ãKé�æ�êpæ�æWü � 232������� ê&ã|è� î&à&á ì è{ê�2�õ�é�æWì é��+è|ëgïYã|å2Ö�Ô

Page 23: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

�#�rO�NPOβββi

twv `P�rOBa�QgN � t JMQgyU_c~ v `PO�N t ~YN�L+O�J v{t `W}�~ � βββ O��pQgy z Qp`PORL�Qp` `P�rO t � `P��LrQp`|Qe_c~ t J&`>"i = 1, 2, . . . , N

)�� � ~Yy y ~p� t J �`P�rO�_c~ v `PO�N t ~YNBO v ` t a�Qp` t ~YJ�a�O�`P�r~+L v ~ z `Py t JrORL t J7S z b v O x ` t ~YJ�9+� �r�\B�rO�O v ` t a�Qp`PORL�_c~ v `PO�N t ~YNKa�ORQgJ�~ �

αrev

twvE(αrev) = 0.3899

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twvE(αexp) = −0.0561

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�Ué�å�ã�ï�ëRà�üUé�à&ã|ìMï&é�æ{é�$��á õ�ã|ë%� C�$��0�<á ì è|ë���û � 6= ��ÿ��Rÿ0A � ê&ã#å�î&å æ{é�á àUé��&á ì á æÍú�ë�ø�ôRë�â¢ãPäWà&õ�ã|àpæ�ïYã�&Uè|á æWå ��ùÍõ�ü&ì á è�é�æWá ëRà&åCë�ø�æWê&ã#üYäWã|å�ã|àpæ�ñ·âRé�ì î&ã �+ë�ä�äWë��Cá à&ôè|ëRà&å æ�ä{é�á àpæ�E� ��! ��� *�+ �#" $ ��� ' ;�� )8� ' � � ( � * ��� 9�* �/d � � \�ÞF%F- �?�ÿ��Pñ 6���-�

Ö �

Page 27: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

� ����� È������. � � ­&¬�¬���¬��� �©M¯��[Ë·Ñ������� ê&ã#ì á à&ã�é�ä[õ�ëgïYã|ìrá à = -0Auõ�é|ú �+ã ��äWá æ�æWã|à�é�å

∆y1(n×1)

= ΘΘΘD0 + Π1∆y0 + . . . + Πp−1∆y2−p + αααβββ′

y0 + εεε1

∆y2(n×1)

= ΘΘΘD1 + Π1∆y1 + . . . + Πp−1∆y2−p+1 + αααβββ′

y1 + εεε2

∆yT−1(n×1)

= ΘΘΘDT−2 + Π1∆yT−2 + . . . + Πp−1∆yT−p + αααβββ′

yT−2 + εεεT−1

∆yT(n×1)

= ΘΘΘDT−1 + Π1∆yT−1 + . . . + Πp−1∆yT−p+1 + αααβββ′

yT−1 + εεεT .= 6���A

ò[ë�æWã�æWêUé�æ-ã�é�è{ê�ã�ípîUé�æWá ëRà�è|ë�ä�äWã|å�ü+ëRàUïYåKæWëké�æWá õ�ã@ë��&å�ãPäWâRé�æWá ëRà��Ý�é�è{ê7ï&é�æ{é�ü+ëRá àpæeá å-é(n × 1)

â¢ã|èPæWë�ä�Þ�å�á à&è|ã � ã�é�äWãïYã�é�ì á à&ô �Cá æWê�éeõeî&ì æWá âRé�äWá é�æWãBå úYå æWã|õ ��ãBõ�é|ú�øwîYä�æWê&ãPä�ã��YüUé�àUï = 6���A�é�å

∆y1,1 = c1,1 + Π(1,1)1 ∆y0,1 + . . . + Π

(1,n)1 ∆y0,n + . . . + Π

(1,1)p−1 ∆y2−p,1 + . . . + Π

(1,n)p−1 ∆y2−p,n

+αααβββ′

y0,1 + ε1,1

. . .

∆y1,n = c1,n + Π(n,1)1 ∆y0,1 + . . . + Π

(n,n)1 ∆y0,n + . . . + Π

(n,1)p−1 ∆y2−p,1 + . . . + Π

(n,n)p−1 ∆y2−p,n

+αααβββ′

y0,n + ε1,n

∆yT,1 = cT,1 + Π(1,1)1 ∆yT−1,1 + . . . + Π

(1,n)1 ∆yT−1,n + . . . + Π

(1,1)p−1 ∆yT−p+1,1 + . . . + Π

(1,n)p−1 ∆yT−p+1,n

+αααβββ′

yT−1,1 + εT,1

. . .

∆yT,n = cT,n + Π(n,1)1 ∆yT−1,1 + . . . + Π

(n,n)1 ∆yT−1,n + . . . + Π

(n,1)p−1 ∆yT−p+1,1 + . . . + Π

(n,n)p−1 ∆yT−p+1,n

+αααβββ′

yT−1,n + εT,n,= 60C A

�Cê&ãPäWã�ã�é�è{ê@äWë���à&ë���è|ë�ä�äWã|å�ü+ëRàUïYå�æWëKé(1×1)

ã|ì ã|õ�ã|àpæ�ë�øræWê&ãTâ¢ã|èPæWë�äWå�á à�= 6���A �<á æWê

Π(i,j)k

� ã�á àUïYá è�é�æWã�æWê&ã(i, j)

ñ¥æWêã|àpæ�ä�ú�ë�ø�æWê&ãkñ¥æWê

Πõ�é�æ�äWá �$ � ê&ãKïYãPæWãPäWõ�á à&á å æWá èBè|ëRõ�ü+ëRà&ã|àpæWå�á à = 60C A�é�äWãKã�ípîUé�ì+æWë

c1,1 = Θ1,1D0,1 + . . . + Θ1,mD0,m

. . .

c1,n = Θn,1D0,1 + . . . + Θn,mD0,m

cT,1 = Θ1,1DT−1,1 + . . . + Θ1,mDT−1,m

. . .

cT,n = Θn,1DT−1,1 + . . . + Θn,mDT−1,m.

Ö+Ø

Page 28: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

. �� � ­&¬�¬���¬��� �©M¯��[Ë·Ñ������ûBã�&Uà&á à&ô

E = (Y − WB)′

(Y − WB)Þ�� ã#è�é�à�ã��YüYäWã|å�å

E′

Eá à = ��C ACé�å�é@ípîUéRïgä{é�æWá è[øwë�äWõ á à

B

E′

E = Y′

Y − Y′

WB − B′

W′

Y + B′

W′

WB

= S + (B − B)′

W′

W(B − B),= 6�-0A

�Cê&ãPäWãB = (W

W)−1W′

Yá å�æWê&ã#ì ã�é�å æ�åWípîUé�äWã|åCã|å æWá õ�é�æWë�ä[é�àUï

S = (Y − WB)′

(Y − WB).

ò[ë�æWã[æWêUé�æCå�î �&å æWá æWîYæWá à&ôB = (W

W)−1W′

Yá à�æWê&ãBã��YüYäWã|å�å�á ëRà�øwë�ä

Sá à = 6�-0A � ã#êUé�â¢ã

S = (Y − W(W′W)−1

W′Y)′(Y − W(W′

W)−1W

′Y)

= (Y′ − Y′W(W′

W)−1W

′)(Y − W(W′W)−1

W′Y)

= Y′Y − Y

′W(W′

W)−1W

′Y − Y

′W(W′

W)−1W

′Y + Y

′W(W′

W)−1W

′W(W′

W)−1W

′Y

= Y′Y − 2Y′

W(W′W)−1

W′Y + Y

′W(W′

W)−1W

′Y

= Y′Y − Y

′W(W′

W)−1W

′Y.

= 6%$ A

��ë�äWã|ë�â¢ãPä�ÞYå�î �&å æWá æWîYæWá à&ôB = (W

W)−1W′

Yá à�æWê&ã�ã��YüYäWã|å�å�á ëRà

(B− B)′

W′

W(B− B)á à = 6�-0Aué�àUï�ïYã|â¢ã|ì ëRü&á à&ôYÞ� ãôRãPæ

(B − (W′W)−1

W′Y)

W′

W(B − (W′W)−1

W′Y)

= (B′

W′

W − Y′

W(W′W)−1

W′

W)(B − (W′W)−1

W′Y)

= (B′

W′

W − Y′

W)(B − (W′W)−1

W′Y)

= B′

W′

WB − Y′

WB − B′

W′

W(W′

W)−1W

Y + Y′

W(W′

W)−1W

Y

= B′

W′

WB − Y′

WB − B′

W′

Y + Y′

W(W′

W)−1W

Y.= 6��0A

��á àUé�ì ì úpÞUå�î&õ�õ�á à&ô�= 6%$ ACé�àUï = 6��0A{Þ%� ã#ôRãPæY

′Y − Y

′W(W′

W)−1W

′Y + B

W′

WB − Y′

WB − B′

W′

Y + Y′

W(W′

W)−1W

Y

= Y′Y − Y

WB − B′

W′

Y + B′

W′

WB,é�àUï�æWêgî&å � ã#êUé�â¢ãBüYäWë�â¢ã�ï�æWêUé�æE

E = S + (B − B)′

W′

W(B − B)Þ&é�å�á à = 6�-0A

. ��� � ­&¬�¬���¬��� �©M¯��[Ë·Ñ���� ���ãKé�ì äWã�éRïgú�üYäWë�â¢ã�ï�æWêUé�æ

S = (Y − WB)′

(Y − WB)

= Y′

Y −Y′

WB − B′

W′

Y + B′

W′

WB.

�gî �&å æWá æWîYæWá à&ôeá àSê&ãPäWãKé��+ë�â¢ãBæWê&ãBã��YüYäWã|å�å�á ëRà

B = (W′

W)−1W′

YÞ%� ã#ôRãPæ

S = Y′

Y − Y′

W(W′

W)−1W

Yé�àUïrÞYæ{é�÷gá à&ôeæWê&ã#ïYãPæWãPäWõ�á àUé�àpæ�ëRà �+ë�æWê�å�á ïYã|å|Þ|S| = |Y

Y − Y′

W(W′

W)−1W

Y|.

Ö �

Page 29: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

ùÍà�ôRã|à&ãPä{é�ì¹ÞUá æ�ê&ëRì ïYå�æWêUé�æ�æWê&ãKïYãPæWãPäWõ�á àUé�àpæ�ë�ø�æWê&ã#üUé�ä�æWá æWá ëRà&ã�ï�õ�é�æ�äWá � [

W′

W W′

Y

Y′

W Y′

Y

] á å∣

[

W′

W W′

Y

Y′

W Y′

Y

]∣

= |W′

W||Y′

Y − Y′

W(W′

W)−1W

Y|

⇒ |W′

W|−1

[

W′

W W′

Y

Y′

W Y′

Y

]∣

= |Y′

Y − Y′

W(W′

W)−1W

Y|

⇒ |W′

W|−1|Y′

Y||W′

W − W′

Y(Y′

Y)−1Y

W| = |Y′

Y −Y′

W(W′

W)−1W

Y|

⇒ |W′

W|−1|Y′

Y||W′

(IT −Y(Y′

Y)−1Y

)W| = |Y′

Y −Y′

W(W′

W)−1W

Y|

⇒ |W′

W|−1|Y′

Y||W′

MY W| = |Y′

Y − Y′

W(W′

W)−1W

Y|,

�Cê&ãPäWãMY = IT − Y(Y

Y)−1Y′�pú�ïYã�&Uà&á æWá ëRà�ë�ø�æWê&ã#é�à&à&á ê&á ì é�æWë�ä�

� ê&ãPäWãPøwë�äWã�� ã#è|ëRà&è|ì îUïYã[æWêUé�æ|S| = |Y

Y − Y′

W(W′

W)−1W

Y| = |W′

W|−1|W′

MY W||Y′

Y|.= 6Rÿ0A

��ãBà&ë��=à&ã|ã�ï�æWë�ïYãPäWá â¢ã[æWê&ãBã��YüYäWã|å�å�á ëRà&åCøwë�ä|W

W|−1é�àUï

|W′

MY W| � ê&ãPäWãPøwë�äWãRÞ

|W′

W|á å

|W′

W| =

[

X′

X X′

Zβββ

(Zβββ)′

X (Zβββ)′

Zβββ

]∣

=

[

X′

X X′

Z

Z′

X Z′

Z

]∣

= |X′

X||Z′

Z − Z′

X(X′

X)−1X

Z| = |X′

X||Z′

(IT − X(X′

X)−1X

)Z| = |X′

X||Z′

MXZ|, = � �0A

�+ã|á à&ôMX = (IT − X(X

X)−1X′

)Þ �pú�ïYã�&Uà&á æWá ëRà� ��ë�äWã|ë�â¢ãPä�ÞUè|ëRõ�á à&ô-æWë

|W′

MY W|Þ%� ã#êUé�â¢ã

|W′

MY W| =

[

X′

MY X X′

MY Zβββ

(Zβββ)′

MY X (Zβββ)′

MY Zβββ

]∣

= |X′

MY X||Z′

MY Z − Z′

MY X(X′

MY X)−1X

MY Z|.= �%��A

ùÍà�æWê&ãBõ�é�äWôRá àUé�ìMü+ëRå æWãPäWá ë�äBïYã|à&å�á æÍú�øwë�äβββá à = ��ÿ0A{Þ&æWêUé�æ�� ã#ì á å æ[ê&ãPäWã �+ã|ì ë��=øwë�äCæWê&ãBåWé�÷¢ã#ë�ø�æWê&ã[äWã�éRïYá à&ôYÞ

φ(βββ | Data) ∝ φ(βββ) × |S|−(T−k−r)

2 |W′

W|−n

2

� ã#è�é�à�ü&ì î&ô@á à�ã�ípîUé�æWá ëRà&å>= 6Rÿ0A�é�àUï = � �0A ��ãBè�é�à�ë��Yæ{é�á à

φ(βββ | Data) ∝ φ(βββ) ×(

|W′

W|−1|W′

MY W||Y′

Y|)−

(T−k−r)2

(

|X′

X||Z′

MXZ|)− n

2= ��? A

��á àUé�ì ì úpÞ � ã#è�é�à�å�î �&å æWá æWîYæWãBøwë�äB= �%��A á à = ��? A{ÞYúYá ã|ì ïYá à&ô

φ(βββ | Data) ∝ φ(βββ) ×(

|W′

W|−1|X′

MY X||Z′

MY Z − Z′

MY X(X′

MY X)−1X

MY Z||Y′

Y|)−

(T−k−r)2

×(

|X′

X||Z′

MXZ|)− n

2

= φ(βββ) ×(

|W′

W|−1|X′

MY X||Z′

MY (IT − X(X′

MY X)−1X

)MY Z||Y′

Y|)−

(T−k−r)2

×(

|X′

X||Z′

MXZ|)− n

2.

��ã#å�ãPæW1 = Z

MY (IT − X(X′

MY X)−1X′

)MY Zé�àUï

W2 = Z′

MXZ ��ãBôRãPæ

φ(βββ | Data) ∝ φ(βββ) ×(

|W′

W|−1|X′

MY X||W1||Y′

Y|)−

(T−k−r)2

×(

|X′

X||W2|)− n

2.

= �%60A

Ö �

Page 30: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

Ü�ã|õ�á àUïYá à&ô-ø äWëRõ = � �0AuæWêUé�æ|W

W| = |X′

X||Z′

MXZ|Þ&ã�ípîUé�æWá ëRà = �%60A �+ã|è|ëRõ�ã|å

φ(βββ | Data) ∝ φ(βββ) ×(

|X′

X|−1|Z′

MXZ|−1|X′

MY X||W1||Y′

Y|)−

(T−k−r)2

×(

|X′

X||W2|)− n

2

= φ(βββ) ×|X

X|(T−k−r−n)

2 |W2|(T−k−r−n)

2

|X′

MY X|(T−k−r)

2 |Y′

Y|(T−k−r)

2 |W1|(T−k−r)

2

= φ(βββ) × k ×|W2|

(T−k−r−n)2

|W1|(T−k−r)

2

,= ����A

�Cê&ãPäWãk = |X

X|(T−k−r−n)

2

|X′MY X|

(T−k−r)2 |Y

′Y|

(T−k−r)2

á å�éCæWãPäWõ�æWêUé�æ�ïYëpã|å�à&ë�æ�ïYã|ü+ã|àUïKëRàβββ �gî �&å æWá æWîYæWá à&ô�á à = ����AMæWê&ã ã��YüYäWã|å�å�á ëRà&å

øwë�äW2

é�àUïW1

Þ%� ã#ôRãPæ

φ(βββ | Data) ∝ φ(βββ) × k ×|Z

MXZ|(T−k−r−n)

2

|Z′

MY(IT − X(X′

MYX)−1X′)MYZ|

(T−k−r)2

= φ(βββ) × k ×|βββ

Z′

MXZβββ|(T−k−r−n)

2

|βββ′

Z′

MY(IT − X(X′

MYX)−1X′)MYZβββ|

(T−k−r)2

= φ(βββ) × k ×|βββ

Z′

MXZβββ|l2

|βββ′

Z′

MY(IT − X(X′

MYX)−1X′)MYZβββ|l1

,

l2 = T−k−r−n

2

é�àUïl1 = T−k−r

2

��åCî&å�îUé�ì¹ÞMX = IT − X(X

X)−1X′

Ö �

Page 31: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

�[²&Â�¿¥´!Ô+°���´pµ�¼¢»�¶¹Å�Á�¶��r´�µ�Á�²&Á�¶¥µ�Á�¶¥¼pµ

����� ��� ��� ��������������������� ���"! �# �ρ1 ρ2 ρ12 ρ24$&%(' $&%(' $&%(' $&%(')Q1 * )

Q2 * )Q12 * )

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Page 38: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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Page 39: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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Page 40: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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Page 41: 2007/80 · CORE DISCUSSION PAPER 2007/80 Testing fiscal sustainability in Poland: a Bayesian analysis of cointegration Andrea SILVESTRINI1 October 2007 Abstract Fiscal sustainability

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