Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL...

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Département des Sciences Économiques de l'Université catholique de Louvain Testing Fiscal Sustainability in Poland : A Bayesian Analysis of Cointegration A. Silvestrini Discussion Paper 2007-40

Transcript of Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL...

Page 1: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

Département des Sciences Économiquesde l'Université catholique de Louvain

Testing Fiscal Sustainability in Poland :A Bayesian Analysis of Cointegration

A. Silvestrini

Discussion Paper 2007-40

Page 2: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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!-Q z � "$# %'%/#0)-`PO v ` v�� ~YN z J t `�NP~U~g` v QgJML x ~ t J&`PO � N|Qp` t ~YJ/~ � NPORQgy[L+O�b+`�QgJML/NPORQgy�_rN t a�QgNP} v{z NP_ry zMv �pQgN t Qgbry O v �x Qgy x�z ywQp` t J � `P�rOK�MJ t `PO v Qga�_ry O x N t ` twx Qgyc�pQgy z O v � t Q�¦�~YJ&`PO4�[QgNPy ~�a�O�`P�r~+L v �87 zrt J&`P~ v "$# %'%*9') t J&`PNP~+L zMx O v � v `PNP~YJ � �QgJML:� �[ORQg��� x ~YJML t ` t ~YJ v�� ~YN�� vPx Qgy v{zMv `|Q t JMQgb t y t `W}G`P~<�r~YywL��3\B�rO v `PNP~YJ � ~YJrO twv O §&zrt �pQgy O�J&`�`P~ x ~ t J&`PO � N|Qp` t ~YJbcO�`W�[O�O�J!NPO��YO�J z O v QgJML�O�¤+_cO�JML t ` z NPO v Q v QkJrO x O vPv QgNP}�QgJML v{z - x�t O�J&` x ~YJML t ` t ~YJ � ~YN v{zMv `|Q t JMQgb t y t `W}Y�@\B�rO��[ORQg�~YJrOeNPO §&zrt NPO v "·Q v JrO x O vPv QgNP}�QgJML v{z - x�t O�J&` x ~YJML t ` t ~YJ6)�`P�rO�L+O�b+`B_rNP~ x O vPv `P~ � NP~p� v y ~p�[O�NC`P�MQgJk`P�rO � NP~p�B`P��N|Qp`PO~ � `P�rO�a�ORQgJ t J&`PO�NPO v `#N|Qp`POY��\B�rO v O x ~YJML t ` t ~YJ v QgNPO@Qgy v ~�L twvPx�zMvPv ORL�bU}�¦7QgN{` t J;"ÍdgfYfYf*)��Ù��gã|ãRÞYøwë�ä�á à&å æ{é�à&è|ãRÞUð2< î&à&ô¢ípâgá å æ[é�àUï �Yé�äWôRã|àpæ>=@?�������ACé�àUï�óuî&á æWãPäB=@?����0C A

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5-O v _ t `PO�`P�rO�ywQgN � O�J z a�bcO�Ne~ � O�a�_ t N twx Qgy QgJML�`P�rO�~YNPO�` twx Qgyu_MQg_cO�N v-� ~ x�zMv{t J � ~YJ7� vPx Qgy v{zMv `|Q t JMQgb t y t `W} t J7`P�rO�eS!QgJML t J � O v `PO�NPJ!Z z NP~Y_cORQgJ x ~ z J&`PN t O v�� � v{t a t ywQgN t JU�YO v ` t � Qp` t ~YJ v ~YJ�L+O��YO�y ~Y_ t J � QgJML�`PN|QgJ v{t ` t ~YJ!O x ~YJr~Ya t O v�MQ¢�YO@bcO�O�J�bU} � QgN#y O vPv J z a�O�NP~ zMv �CIKa�~YJ � `P�rO v OY����Qgy }Y~YJ x�z "ÍdgfYf*9')�O�¤rQga t JrO v#v{zMv `|Q t JMQgb t y t `W}k~ � � vPx Qgy v `|QgJ x O vt J<�M�YOkL+O��YO�y ~Y_ t J � x ~ z J&`PN t O v "·¦�O�¤ twx ~M��SU~ z `P�GI � N twx Qr��`P�rO��C� t y t _r_ t JrO v �CSU~ z `P���-~YNPORQQgJML<\ z NP�YO�}/)���¡ zrt `PO�N"$# %'%'(*)BQgJML�¡ z L t JMQ�QgJML�pQgJ �1t £WJUbcO�N � O�J "$# %'%Uh2) x ~YJ v ` t ` z `PO�NPO�_rNPO v O�J&`|Qp` t �YOeO�¤rQga�_ry O v ~ ��v ` z L t O v-x ~YJ x O�NPJ t J �� vPx Qgy v ~Yy �YO�J x } t J/Z�Q v `PO�NPJ�Z z NP~Y_cOY�k¦�~YNPO�NPO x O�J&`Py }Y� � NPO�O�J1������� "ÍdgfYf/#0)KO��pQgy z Qp`PO�L+O�� x�t ` v{zMv `|Q t JMQgb t y t `W} t J� ~YywQgJMLbcO�`W�[O�O�J # %'%/#@QgJML # %'%'&r�

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QgJML�bU}v ~Yy � t J � O §&z Qp` t ~YJ "$#0) � ~YNP�BQgN|L+y }Y�+IKN � �U}UNP~ z QgJML�V zrt J&`PO�y � O�`B`P~�O §&z Qp` t ~YJ "Íd')��U`P�MQp`#�[O@y twv `#�rO�NPOebcO�y ~p� � ~YN[`P�rOv Qg�YO@~ � x ywQgN t `W}

GBt = Et

∞∑

j=0

(1 + R)−(j+1)(revt+j − expt+j)

+ limj→∞

Et

[

(1 + R)−(j+1)GBt+j+1

]

,

�#�rO�NPOGBt

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revt

QgJMLexpt

QgNPO�NPORQgy#NPO��YO�J z O v QgJML=NPORQgyO�¤+_cO�JML t ` z NPO v Qp`` t a�Ot�

Rtwv `P�rO/NPORQgy t J&`PO�NPO v `�N|Qp`PO "¥�#�r~ v O x ~YJML t ` t ~YJMQgyeO�¤+_cO x `|Qp` t ~YJ twv Q vPv{z a�ORL4`P~�bcOx ~YJ v `|QgJ&` )CQgJML

Et

twv `P�rOeO�¤+_cO x `PORL��pQgy z O-~Y_cO�N|Qp`P~YN x ~YJML t ` t ~YJMQgyM~YJ�`P�rO t J � ~YNPa�Qp` t ~YJ v O�`BQ¢�pQ t ywQgbry OKQp`t��\�~�N z y O~ z `� #�p����������������� t � OY��`P~7Q¢�Y~ t L 3�\B�rO�_c~ vPv{t b t y t `W}!~ � `P�rO � ~p�YO�NPJra�O�J&`@�MJMQgJ x�t J � ~YywL/L+O�b+`�`P�MQp`�a�Qp` z NPO v bU}twvPv{zrt J � JrO��(L+O�b+`�3 "@!-Qg�U� t ~�QgJML�] zMv ���# %'%/#Y�r_�� ���/#0)�� t `ea zMv `-�r~YywL�`P�MQp`-`P�rO�y t a t `e~ � `P�rO�O�¤+_cO x `PORL��pQgy z O�~ �`P�rO�L twvPx ~ z J&`PORL �¥z ` z NPO v `P~ x �k~ � NPORQgy�L+O�b+` � ~UO v `P~ � O�NP~M�

limj→∞

Et

[

(1 + R)−(j+1)GBt+j+1

]

= 0."$#0)

T�J!`P� twv �BQ¢}`P�rO t J&`PO�N{`PO�a�_c~YN|Qgy b z L � O�` x ~YJ v `PN|Q t J&` twv@v Qp` twv �MORL!QgJML7`P�rO v{zMv `|Q t JMQgb t y t `W}��U}U_c~g`P�rO v{twv �r~YywL v ��T�J~ z N[�[~YNP���[O t JU�YO v ` t � Qp`POK�#�rO�`P�rO�N[`P�rO-� ~Yy twv �LrQp`|Q�QgNPO x ~YJ v{twv `PO�J&`��-� t `P� "$#0)C~p�YO�N�`P�rO v Qga�_ry O-_cO�N t ~+L���\�~�`P� twvQ t a���Qka�O�`P�r~+L+~Yy ~ � } v{z �Y� O v `PORL�bU}`P�rO t J�� z O�J&` t Qgy y t `PO�N|Qp` z NPO twv `P~�`PO v ` � ~YN x ~ t J&`PO � N|Qp` t ~YJ�bcO�`W�[O�O�J7NPO��YO�J z O vQgJML�O�¤+_cO�JML t ` z NPO v QgJMLk`P~�O v ` t a�Qp`POe`P�rO@_MQgN|Qga�O�`PO�Nβexp

t J�`P�rO@Jr~YNPa�Qgy t � ORL x ~ t J&`PO � N|Qp` t J � �YO x `P~YN [−1 βexp]′ �

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Page 5: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

T�JGQ7J z ` v �rO�y yÍ�u`P�rO�NPO � ~YNPOY� �#�MQp`��[OL+~ twv `P~�`PO v ` #0) x ~ t J&`PO � N|Qp` t ~YJ�bcO�`W�[O�O�JGNPO��YO�J z O v QgJML<O�¤+_cO�JML t ` z NPO v" t J x y zMv{t �YO�~ �Ct J&`PO�NPO v ` )KQgJML`P~�O v ` t a�Qp`PO�d')[`P�rO�_c~ v `PO�N t ~YN-L twv `PN t b z ` t ~YJ7~ � _MQgN|Qga�O�`PO�N

βexp

t J�`P�rO�Jr~YNPa�Qgy t � ORLx ~ t J&`PO � N|Qp` t J � �YO x `P~YN [−1 βexp]

′ �u\�~-bcO � t J�� t `P���¢`P�rO[_rNPO�y t a t JMQgNP} z J t ` NP~U~g` v QgJMQgy } v{twv QgJML�`P�rO v O�y O x ` t ~YJ�~ � `P�rOx ~ t J&`PO � N|Qp` t ~YJ�N|QgJr��QgNPO x QgNPN t ORL~ z `e� t `P� t J!Q�Jr~YJ � ¡BQ¢}YO v{t QgJ t J � O�NPO�J&` t Qgy v O�`{` t J � �KT�J7Q v O x ~YJML v `PO�_���¡BQ¢}YO v{t QgJt J � O�NPO�J x O twv Qg_r_ry t ORL!`P~7L+O�N t �YO�`P�rO�_c~ v `PO�N t ~YN�L twv `PN t b z ` t ~YJ/~ � `P�rO x ~ t J&`PO � N|Qp` t J � �YO x `P~YN�QgJML!`P~ x Qgy x�z ywQp`PO t ` v�MN v `K`W�[~�a�~Ya�O�J&` v � �1t `P�!Q v{t J � y O x ~ t J&`PO � N|Qp` t J � NPO�ywQp` t ~YJ��r�[O�a�Qg�YO zMv O�~ �Cv ~Ya�O��UJr~p�#J�NPO v{z y ` v-x ~YJ x O�NPJ t J �`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~YNR�u�#� twx �<bcO�y ~YJ � v `P~7`P�rO�_c~Yy } � `�L+O�J v{t ` t O v�x ywQ vPv � � OQgy v ~v �r~p�D`P~�#�MQp`@O�¤U`PO�J&`e`P�rO�y t �YO�y t �r~U~+L�~ � `P�rO�LrQp`|Q twvet a�_c~YN{`|QgJ&` t J!NPO�� twv{t J � `P�rO�Q¢�pQ t ywQgbry O�_rN t ~YN t J � ~YNPa�Qp` t ~YJ��\�~kL+~�`P� twv �r�[O@NPO�y }k~YJJ z a�O�N twx Qgy t J&`PO � N|Qp` t ~YJk`PO x �rJ tw§&z O v bMQ v ORL�~YJ�L+O�`PO�NPa t J twv ` twx a�O�`P�r~+L v �\B�rOCa�Q t J x ~YJ&`PN t b z ` t ~YJ�~ � `P�rOC_MQg_cO�N twv `P�rO�O�a�_ t N twx Qgyg_rNP~Ybry O�aA`P~KbcO[QgJMQgy } � ORL@QgJML-`P�rO�� t JML@~ � O x ~YJr~Ya�O�`PN twxv `PN|Qp`PO � }7O�a�_ry ~p}YORL�� J<Q`P�rO�~YNPO�` twx Qgy � NP~ z JML���`P�rOka�O�`P�r~+L+~Yy ~ � t O v QYL+~Y_+`PORL/QgNPO�Qgy y�`|Qg�YO�J � NP~Ya `P�rOkNPO x O�J&`y t `PO�N|Qp` z NPOY� 5 z a�a�}��pQgN t Qgbry O v QgNPO t J&`PNP~+L zMx ORL t J4`P�rO!L+O�`PO�NPa t J twv ` twx _MQgN{`k~ � `P�rO�a�~+L+O�yK`P~GL+ORQgyK� t `P�4`P�rOv ORQ v ~YJMQgy�JMQp` z NPO�~ � `P�rOkLrQp`|Qr��\B�rO z J t `�NP~U~g`�`PO v ` t J � twv�x QgNPN t ORL!~ z `�bc~g`P� t JGQ�_MQgN|Qga�O�`PN twx bc~g`P� t JGQ�Jr~YJ �_MQgN|Qga�O�`PN twx-� N|Qga�O��[~YNP���C\B�rO�NPO v{z y ` v L+O�N t �YORL�bU} zMv{t J � `P�rO�_MQgN|Qga�O�`PN twx Qg_r_rNP~&Q x �QgNPO x QgNPO �¥z y y } x ~Ya�_MQgNPORL�`P~`P�r~ v O�~ � Qgy `PO�NPJMQp` t �YO�bc~U~g` v `PN|Qg_�_rNP~ x ORL z NPO v �[T�J�`P� twv �BQ¢}Y�r�[O�bcO�y t O��YO�`P�MQp`eQ�a�~YNPO�Qg_r_rNP~Y_rN t Qp`POe`PO v `eL+O x�twv{t ~YJa�Q¢}�bcO�`|Qg�YO�J���\B�rO�¡BQ¢}YO v{t QgJ7QgJMQgy } v{twv ~ �[x ~ t J&`PO � N|Qp` t ~YJ��c� t `P�/Q�`PN|QgJ v _MQgNPO�J&` x �r~ twx O�~ � `P�rO�_rN t ~YN@QgJML7� t `P�ORQ v } � `P~ � t a�_ry O�a�O�J&`B_c~ v `PO�N t ~YN x Qgy x�z ywQp` t ~YJ v �UO�JMQgbry O v[zMv `P~ x ~YJ x y z L+O-`PO�J&`|Qp` t �YO�y }k�#�rO�`P�rO�NKQgJMLk`P~��#�MQp`BO�¤U`PO�J&`� vPx Qgy v{zMv `|Q t JMQgb t y t `W} x QgJbcO�Q x � t O��YORL t J�`P�rO�y ~YJ � � N z J��^K~g` twx O�`P�MQp`eQgy y�`P�rO�_MQg_cO�N v-v ~ � QgN x�t `PORL x ~Ya�_rN twv O�Jr~YJ � ¡BQ¢}YO v{t QgJ�QgJMQgy } v O v ~ �Cx ~ t J&`PO � N|Qp` t ~YJ�� !K~p�[O��YO�NR�rQ v_c~ t J&`PORL�~ z `-bU}X t y ywQgJ t "ÍdgfYf*9')��M_�� �&d&hU��3��C� t y ~ v ~Y_r� twx Qgy twvPv{z O v Q v{t L+OY�cQk¡BQ¢}YO v{t QgJ�Qg_r_rNP~&Q x � twv QYL+�pQgJ&`|Q � O�~ zMv 3v{t J x O 3�T `�_rNP~+L zMx O v �#�r~Yy O�_rNP~YbMQgb t y t `W}/L twv `PN t b z ` t ~YJ v�� ~YN�ORQ x � z Jr�UJr~p�#JG_MQgN|Qga�O�`PO�N�`P�MQp`�QgNPO��pQgy t L � ~YN�QgJU}v Qga�_ry O v{t � O 3p�u\B� twv twv `P�rO � Q t J�~ � `P�rO#¡BQ¢}YO v{t QgJ�Qg_r_rNP~&Q x ��� t `P��NPO v _cO x `u`P~e`P�rO � NPO §&z O�J&` twv `R�u\�~e~ z N �UJr~p�#y ORL � OY�� vPx Qgy v{zMv `|Q t JMQgb t y t `W} twv QgJ twvPv{z O@`P�MQp`e�MQ v JrO��YO�NKbcO�O�J�`|Q x �Uy ORL� t `P� t J�`P�rO�¡BQ¢}YO v{t QgJ � N|Qga�O��[~YNP���[\B� twv QgN{` twx y Otwv QgJ�Qp`{`PO�a�_+`#`P~��My y�`P� twv � Qg_��\B�rO!_MQg_cO�N twv ~YN � QgJ t � ORL1Q vk� ~Yy y ~p� v � T�JDSUO x ` t ~YJ3d<�[O t J&`PNP~+L zMx O v ~Ya�O7Jr~g`|Qp` t ~YJ QgJML §&zrtwx �Uy }�NPO�� t O��x ~ t J&`PO � N|Qp`PORL/X I-] v } v `PO�a v ��T�J�SUO x ` t ~YJ �7�[O�~ z `Py t JrOQ!¡BQ¢}YO v{t QgJGQgJMQgy } v{twv ~ �-x ~ t J&`PO � N|Qp`PORL/X I-] a�~+L+O�y v �SUO x ` t ~YJ � x ~YJ&`|Q t J v Q�_rNPO�y t a t JMQgNP}�O�a�_ t N twx Qgy�QgJMQgy } v{twv�x ~YJ x O�NPJ t J � LrQp`|Q twvPv{z O v � z JrNPO v `PN twx `PORL�X I-] O v ` t a�Qp` t ~YJ��z J t `BNP~U~g`�`PO v ` v QgJML v O�y O x ` t ~YJ�~ � `P�rO x ~ t J&`PO � N|Qp` t ~YJ�N|QgJr���uT�J�SUO x ` t ~YJ 9@�[O-Qg_r_ry }�¡BQ¢}YO v{t QgJ t J � O�NPO�J x OK`P~�Q vPv O vPv`P�rO v{zMv `|Q t JMQgb t y t `W}�~ � � vPx Qgy�_c~Yy twx } t JG� ~YywQgJML��!SUO x ` t ~YJ.(r���MJMQgy y }Y��L+N|Q¢� v ~ z N x ~YJ x y zMv{t ~YJ v ��\B�rO�_rNP~U~ �·v QgNPO

� Qp`P�rO�NPORL t J`P�rO�IK_r_cO�JML t ¤��

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

\B� twv SUO x ` t ~YJ!brN t O��M}�NPO�� t O�� v-z JrNPO v `PN twx `PORL�X I-] a�~+L+O�y v QgJML x ~ t J&`PO � N|Qp` t J � X I-] v } v `PO�a v ��\�~ v `|QgN{`R��y O�` zMvt J&`PNP~+L zMx O@`P�rO � O�JrO�N|Qgy�X I-]3~ � ~YN|L+O�N pQ v

yyyt = ΘDDDt +

p∑

i=1

Aiyyyt−i + εεεt,"Íd')

�#�rO�NPO{yyyt}

T

t=1

twv Qn−L t a�O�J v{t ~YJMQgyB�YO x `P~YN�~ � �pQgN t Qgbry O v �

Ai

"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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

�#�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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

_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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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

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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,

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DUMt

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t �ttwv O §&z Qgy�`P~ � z y }

# %'%'%r�cI z � zMv ` # %'%'%r��SUO�_+`PO�a�bcO�N # %'%'%r� x `P~YbcO�N�# %'%'%r�c^K~p�YO�a�bcO�N�# %'%'%r��5-O x O�a�bcO�N # %'%'%r� �&QgJ z QgNP}�dgfYfYf�QgJMLDUMt = 0

O�y v O��#�rO�NPOY�s NP~Ya Q x QgNPO �¥z yey ~U~Y���#a�~YNPO�~p�YO�NR�B�[O x QgJ v O�O7`P�MQp`O�¤+_cO�JML t ` z NPO vk� QgNO�¤ x O�ORL1NPO��YO�J z O v �#� t `P�A�YO�NP} � O��O�¤ x O�_+` t ~YJ v �B\B� twv � Qg_ twv a t J t a z a Qp`#`P�rO�bcO � t JrJ t J � ~ � `P�rO v Qga�_ry O�_cO�N t ~+L�QgJML t J x NPORQ v O v#v `PORQYL t y }Q � `PO�N#`P�rOO�JML~ � }YORQgNKdgfYfYfr�uT ` v O�O�a v#v y t � �&`Py }�`P~�NPORL zMx O@~YJry }�Qp`#`P�rO�O�JML~ � dgfYf'(r�\B� twv `PO�JML+O�J x } twv Qgy v ~ x ~YJ+�MNPa�ORL�bU} sut � z NPO�d+���#� twx �<_rNPO v O�J&` v `P�rO x ~YJ v ~Yy t LrQp`PORL � NP~ vPv L+O�b+` "¥bMQ v ORL�~YJZ[S+I # %'%*9�L+O��MJ t ` t ~YJ6)���Q v Q�_cO�N x O�J&`|Q � O�~ � � 5-�(Qp`�a�QgNP�YO�`@_rN twx O v ��~p�YO�N-`P�rO�_cO�N t ~+L.# %'%'( � dgfYf'(r��^K~g`PO�`P�MQp``P�rOL twv _rywQ¢}YORL/LrQp`|Q�_cO�N{`|Q t J/`P~7`P�rO � O�JrO�N|Qgy � ~p�YO�NPJra�O�J&` v O x `P~YNR�u�#� twx � x ~Ya�_rN twv O v `P�rO x O�J&`PN|QgyCy O��YO�yÍ��\B�rOLrQp`|Q v ~ z N x O twv `P�rO�Z �[¡1� v������ ,)" "¥Z �[¡-*KI-¦�Z>#0)��@\B�rO�NPO twv Q�L+~p�#JU�BQgN|L�`PNPO�JML z _�`P~�dgfYfYfr� � ~Yy y ~p�[ORL�bU}QgJ z _U�BQgN|Lk`PNPO�JML�� t JML twx Qp` t J � Q x y ORQgNKL+O�`PO�N t ~YN|Qp` t ~YJ�~ � b z L � O�`-L+O�� x�t `R��I[`#`P�rO�O�JML~ � dgfYf'(r�&`P�rO�L+O�b+`-N|Qp` t ~ twvQg_r_rNP~&Q x � t J � 9gf�� ~ � � 5-� �

0 s T � �-]#ZAd�I-¡ �-\ !KZ�]#Z$1��~YJ x O�NPJ v ~p�YO�NKQ�a�~ z J&` t J � L+O�b+`R�c`P�rO�NPO � ~YNPOY�My O � t ` t a�Qp`PO�y }N|Q twv O@`P�rO twvPv{z O�~ � � vPx Qgy v{zMv `|Q t JMQgb t y t `W}Y�#T�J�`P� twv� N|Qga�O��[~YNP���&`P�rO�Q t a ~ � `P�rO�O�a�_ t N twx Qgy�QgJMQgy } v{twv[twv `P~�Q vPv O vPv �#�rO�`P�rO�N#`P�rO@� ~Yy twv � � ~p�YO�NPJra�O�J&` x QgJ x ~YJ&` t J z Oe`P~~Y_cO�N|Qp`PO z JML+O�N t ` v#x�z NPNPO�J&`[� vPx Qgy�_c~Yy twx } t JML+O��MJ t `PO�y }Y� zMv{t J � x ywQ vPv{twx Qgy x ~ t J&`PO � N|Qp` t ~YJkQgJMQgy } v{twv `P~�L+O�`PO�NPa t JrOe`P�rOx ~ t J&`PO � N|Qp` t ~YJ�N|QgJr��QgJML¡BQ¢}YO v{t QgJ t J � O�NPO�J x Oe`P~kL+N|Q¢�1`P�rO�_c~ v `PO�N t ~YN#L twv `PN t b z ` t ~YJ�~ � `P�rO z JrNPO v `PN twx `PORL�O�y O�a�O�J&`~ � `P�rO x ~ t J&`PO � N|Qp` t J � �YO x `P~YNR�

���� .4H ���6 ���� �� ���� �

I v Q@�C�|��� v `PO�_��+�[O@O v ` t a�Qp`POe`P�rO z JrNPO v `PN twx `PORLkX I-]3a�~+L+O�y t J;"Íd')�~YJ�NPO��YO�J z O v QgJML�O�¤+_cO�JML t ` z NPO v ��\B�rO@X I-]v } v `PO�a t J twv b t �pQgN t Qp`PO v{t J x O

yt =

(

revt

expt

) �u�#�rO�NPOkQ v Qgy NPORQYL+}7O�¤+_rywQ t JrORLrevt

NPO�_rNPO v O�J&` v `P~g`|QgyCNPO��YO�J z O vQgJML

expt

`P~g`|Qgy�O�¤+_cO�JML t ` z NPO v Qp`#` t a�Ot�C\B�rO@O v ` t a�Qp`PORL v } v `PO�a twv

[

revt

expt

]

=

[

DETrev

DETexp

]

+ A1

[

revt−1

expt−1

]

+ . . . + Ap

[

revt−p

expt−p

]

+

[

εrevt

εexpt

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,

"Íd'9')

Ô+Ô

Page 13: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

� t `P�`P�rO�L+O�`PO�NPa t J twv ` twx _MQgN{`KL+O��MJrORL�Q v

DETrev = arev + drevDUMt +

12∑

s=2

κrevs δs,t

DETexp = aexp + dexpDUMt +12∑

s=2

κexps δs,t, t = 1, . . . , T,

"Íd2(*)�#�rO�NPO

DUMt

twv Q�L z a�a�}��pQgN t Qgbry O-`P�MQp`KNPO�a�~p�YO v `P�rO t J�� z O�J x O�~ � `P�rO�QgJr~Ya�Qgy ~ zMv ` t a�O v O�N t O v bcO��MQ¢� t ~ z N t J`P�rO v O x ~YJML��MQgy � ~ � # %'%'%�QgJMLδs,t

"s = 2, . . . , 12

)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�

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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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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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′ � I v Q x ~YJ v O §&z O�J x OY�p`P�rO v `PNP~YJ � v{zMv `|Q t JMQgb t y t `W}x ~YJML t ` t ~YJ�NPO §&zrt NPO v `P~�`PO v `K`P�rO�J z y y��U}U_c~g`P�rO v{twv `P�MQp`K`P�rO v O�`W�[~ v O�N t O v QgNPO x ~ t J&`PO � N|Qp`PORLQgJML`P�MQp`#`P�rO x ~ t J&`PO �� N|Qp` t J � �YO x `P~YN twv [−1 1]′ ��\B� twv�twv L+~YJrO-N z JrJ t J � `P�rO x ~ t J&`PO � N|Qp` t ~YJ�`PO v `4" �Y~Y�MQgJ v O�J��6# %'%'(*)��Y�#�r~ v OK_ z NP_c~ v O twv`P~L+O�`PO�NPa t JrO��#�rO�`P�rO�NeQ � NP~ z _�~ � Jr~YJ ��v `|Qp` t ~YJMQgNP} v O�N t O v QgNPO x ~ t J&`PO � N|Qp`PORL��KI v Qgy NPORQYL+}L+O�`|Q t y ORL���QgJ t a�_MQ x `a�Qp`PN t ¤

Π = −A(1)t J " �*)-~ � NPORL zMx ORL7N|QgJr��� t � OY� v a�Qgy y O�N-`P�MQgJ t ` v L t a�O�J v{t ~YJ�� twv Q � ORQp` z NPO�~ �[x ~ t J&`PO � N|Qp`PORLX I-] v } v `PO�a v �#T�J!Q�J z ` v �rO�y yÍ� �Y~Y�MQgJ v O�J � v a�O�`P�r~+L+~Yy ~ � }�NPO §&zrt NPO v `P~�O v ` t a�Qp`PO�`P�rO

Πa�Qp`PN t ¤�QgJML`P~�`PO v `R��Qp``P�rO v Qga�Oe` t a�OY�r�#�rO�`P�rO�N#`P�rO�NPO v `PN twx ` t ~YJ v[t a�_ry t ORLbU}�`P�rO�NPORL zMx ORLN|QgJr��~ �

Πx QgJbcO�NPOW£WO x `PORL~YNBJr~g`R�

� N|Qg_r� twx Qgy�O�� t L+O�J x O v O�O�a v `P~ v{z �Y� O v `#`P�MQp`-NPO��YO�J z O v QgJML�O�¤+_cO�JML t ` z NPO v y O��YO�yuLrQp`|Q��MQ¢�YO�Q�Jr~YJ � � O�NP~ x ~YJ �v `|QgJ&`#QgJML�Jr~�L+O�`PO�NPa t J twv ` twx `PNPO�JML v � � �rO�`P�rO�N[`P� twv�� ORQp` z NPOeQg_r_ry t O v `P~�QgJkO��YO�J&` z Qgy x ~ t J&`PO � N|Qp` t J � NPO�ywQp` t ~YJ v � t _bcO�`W�[O�O�J<`P�rO�a twv Jr~g`�ORQ v }!`P~�L twvPx O�NPJGbU} � N|Qg_r� twx Qgy t J v _cO x ` t ~YJ��CQgy `P�r~ z � �=Q7L+O�`PO�NPa t J twv ` twx `PNPO�JMLGL+~UO v Jr~g`v O�O�aE`P~�bcO�_rNPO v O�J&`R� J�`P�rO x ~YJ&`PN|QgNP}Y�+`P�rO�NPO�QgNPO�L+~ z b+` v NPO � QgN|L t J � Q�Jr~YJ � � O�NP~ t J&`PO�N x O�_+` t J v{t L+O�`P�rO x ~ t J&`PO �� N|Qp` t ~YJ v _MQ x OY�K\B� twv-twv �[~YN{`P��_c~ t J&` t J � ~ z ` v{t J x OY� t J � O�JrO�N|QgyÍ�r`P�rO�Q v }Ua�_+`P~g` twx L twv `PN t b z ` t ~YJ�~ � `P�rO�N|QgJr�k`PO v `v `|Qp` twv ` twxe� ~YN x ~ t J&`PO � N|Qp` t ~YJ twv Jr~g` v `|QgJMLrQgN|L�QgJML v `PN twx `Py }�L+O�_cO�JML v ~YJ�`P�rO�Q vPv{z a�_+` t ~YJ v a�QYL+O@� t `P��NPO v _cO x `B`P~`P�rO�L+O�`PO�NPa t J twv ` twx@x ~Ya�_c~YJrO�J&` v ~ � `P�rO�X-Z �[¦ a�~+L+O�y t J " �*)�� Jry }k�#�rO�J`P�rO�a�~+L+O�y v _cO x�t � x Qp` t ~YJ twv L+O��MJrORL��t ` twv _c~ vPv{t bry Oe`P~kL+O�`PO�NPa t JrOe`P�rO�N|QgJr�k~ � `P�rO�y ~YJ � � N z J�a z y ` t _ry t O�NΠa�Qp`PN t ¤��

\�~ v O�y O x `Qga�~YJ � _c~ vPv{t bry O7a�~+L+O�y v �#�[O!Qg_r_ry }�`P�rO7y ~ � � y t �YO�y t �r~U~+L�N|Qp` t ~/`PO v `R� t � OY� Q/`PO v ` t JA�#� twx �4`P�rOa�Qp¤ t a z a ~ � `P�rO@y t �YO�y t �r~U~+L �¥z J x ` t ~YJ~ � Q�NPO v `PN twx `PORLka�~+L+O�y twv#x ~Ya�_MQgNPORL�`P~�`P�rO@a�Qp¤ t a z a ~ � `P�rO@y t �YO�y t �r~U~+LÙ � � ê&ã#ïYã|â¢ã|ì ëRü+ã�ï � � � ð �[ó�è|ëgïYã#á å�é�âRé�á ì é��&ì ã#ø äWëRõ(æWê&ã#é�îYæWê&ë�äCî&ü+ëRà�äWã�ípî&ã|å æ�

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Page 16: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

�¥z J x ` t ~YJ<� t `P��`P�rO x ~YJ v `PN|Q t J&` v NPO�ywQp¤+ORL���T�J/_MQgN{` twx�z ywQgNR���[O�O v ` t a�Qp`PO�`W�[~�L t � O�NPO�J&`�X-Z �[¦ v _cO x�t � x Qp` t ~YJ v � � �#0)kQ x�x ~YN|L t J � `P~<`P�rO7�MN v `�X-Z �[¦ a�~+L+O�yÍ�B`P�rO!y O��YO�yeLrQp`|Q<L+~GJr~g`�_c~ vPv O vPv Q<L+O�`PO�NPa t J twv ` twx `PNPO�JML1QgJML�`P�rOx ~ t J&`PO � N|Qp` t J � O §&z Qp` t ~YJ<L+~UO v Jr~g`��MQ¢�YO�QgJ t J&`PO�N x O�_+`��Cd')eQ x�x ~YN|L t J � `P~�`P�rO v O x ~YJML�X-Z �[¦ a�~+L+O�yÍ��`P�rO�y O��YO�yLrQp`|Q�L+~�Jr~g`#_c~ vPv O vPv Q�L+O�`PO�NPa t J twv ` twx `PNPO�JMLb z `B`P�rO x ~ t J&`PO � N|Qp` t J � O §&z Qp` t ~YJ�MQ v QgJ t J&`PO�N x O�_+`R�C\B�rO v OY� t J~ z N� t O�����QgNPOk`P�rOa�~ v `�_rywQ zMv{t bry O�Qgy `PO�NPJMQp` t �YO v ��\B�MQp` twv � `P�rOa�~+L+O�y t J #0)�L t � O�N v�� NP~Ya `P�rOa�~+L+O�y t J4d')�~YJry }bU}/`P�rOQYLrL t ` t ~YJ=~ � ~YJrO�_MQgN|Qga�O�`PO�NR�u`P�rO x ~YJ v `|QgJ&`�`PO�NPa t JG`P�rO x ~ t J&`PO � N|Qp` t ~YJ v _MQ x O�� � � � O�`P�rO�J x Qgy x�z ywQp`PO`P�rO�`PO v ` v `|Qp` twv ` twx

λ = −2(log(L0) − log(L1))���#�rO�NPO

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�

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Page 17: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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

sut � z NPO 9-L twv _rywQ¢} v Qe¡[O�`|Q (ι = 20, κ = 20)_rN t ~YNR�¢`P�rOBa�QgN � t JMQgy&y t �YO�y t �r~U~+L�QgJML�`P�rO x ~YNPNPO v _c~YJML t J � _c~ v `PO�N t ~YNL+O�J v{t `W}Y��\B�rOe¡[O�`|Q�_rN t ~YN twv[x O�J&`PNPORL�Qp`1#Y�&O�¤rQ x `Py }�Q v `P�rO�S&` z L+O�J&`[_rN t ~YN t J sut � z NPO3�M��\B�rOe¡[O�`|Q�_rN t ~YN��pQgN t QgJ x Otwv

Var(βexp) = 0.0976� \B�rO�_rN t ~YN#L+~Ya�Q t J twv " � #Y���*)��

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

¦�~Ya�O�J&` v ~ � `P�rO�_c~ v `PO�N t ~YN@L+O�J v{t ` t O v@t J "Íd2%*)eQgJML " �Yf*)eQgNPO�Jr~g`��UJr~p�#J�QgJMQgy }&` twx Qgy y }Y��^ z a�O�N twx Qgy t J&`PO � N|Qp` t ~YJ�MQ v `P~7bcO zMv ORL�`P~�~Yb+`|Q t J�`P�rO�a ��� �T�J/_rN t J x�t _ry OY�u~YJrO��BQ¢}�`P~ x ~Ya�_ z `PO�_c~ v `PO�N t ~YN�NPO v{z y ` v ~YJ/_MQgN|Qga�O�`PO�N v@twvbU}k_c~ v `PO�N t ~YN v{t a z ywQp` t ~YJa�O�`P�r~+L v �C¡BQ z �[O�J v QgJML � t ~g`�"$# %'%'&*) v{z �Y� O v `[`P~�Qg_r_ry }k¦7QgNP�Y~p� x �MQ t J¦�~YJ&`PO��[QgNPy ~a�O�`P�r~+L v " v _cO x�t � x Qgy y }Y� � t brb v�v Qga�_ry t J � QgJML � N t LrL+} � � t brb v�v Qga�_ry t J � `PO x �rJ tw§&z O v )�`P~�Q x ~ t J&`PO � N|Qp`PORL/X I-]� � ó é�î� ã|à&å�é�àUï�Ü�á è{êUé�ä{ï = ��ÿ��0C A�üYäWëRü+ëRå�ãKé�à�é�ì ôRë�äWá æWê&õ(æWë@å�á õeî&ì é�æWãBü+ëRì úgñ¥æ�ïYã|à&å�á æWá ã|å

Ô �

Page 20: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

v } v `PO�a ~ � Q �Y� NPO � Qp`PO t a�_c~YN{` v QgJML t a�_c~YN{`�_rN twx O v�� ~YN�¡[O�y � t z a��!IKJr~g`P�rO�N�_c~ vPv{t b t y t `W} twv `P~ zMv O t a�_c~YN{`|QgJ x Ov Qga�_ry t J � "¥OY� � �k¡BQ z �[O�J v QgJML/V z brN|QgJr~M� # %'%'(*)���\B� twv�v Qga�_ry t J � vPx �rO�a�O�NPO §&zrt NPO v `P~ x �r~U~ v O�QgJ t a�_c~YN{`|QgJ x O�¥z J x ` t ~YJ�`P�MQp` x QgJ@bcO v{t a z ywQp`PORL@L t NPO x `Py }@Q v Q � ~U~+L@Qg_r_rNP~¢¤ t a�Qp` t ~YJ-`P~B`P�rOC_c~Yy } � `�_c~ v `PO�N t ~YN�L+O�J v{t `W}K`P~KO��pQgy z Qp`POY�\B� twvKx �r~ twx OY� � O�JrO�N|Qgy y } v _cORQg� t J � � twv Jr~g`KQgJ�ORQ v }�`|Q v ��`P~kQ x�x ~Ya�_ry twv ���IKJr~g`P�rO�N��BQ¢}-`P~ x Qgy x�z ywQp`PO�_c~ v `PO�N t ~YN�NPO v{z y ` v ~YJ�_MQgN|Qga�O�`PO�N v ��O v _cO x�t Qgy y }@�#�rO�J�`P�rOBL t a�O�J v{t ~YJ�~ � `P�rO t J&`PO � N|Q �` t ~YJ twv Jr~g`BywQgN � OY� twv `P~�a�Qg�YO zMv Oe~ � L+O�`PO�NPa t J twv ` twx-t J&`PO � N|Qp` t ~YJ�� \�~Ya�_MQ "$# %Uh��*)�_c~ t J&` v ~ z `[`P�MQp`#L+O�`PO�NPa t J twv ` twxt J&`PO � N|Qp` t ~YJ twv �YO�NP}�O�- x�t O�J&` � ~YN x ~Ya�_ z ` t J � `P�rO t J&`PO � N|QgyÍ��`P�rO�a�~Ya�O�J&` v QgJML7`P�rO�a�QgN � t JMQgy L+O�J v{t ` t O v ~ � a � #QgJMLka � f�_c~Yy } � `#L+O�J v{t ` t O v[t J

k�pQgN t Qgbry O v "¥�#�MQp`PO��YO�NC`P�rO@�pQgy z O-~ �

mQgJML

k)���¦�~YNPO�~p�YO�NR�g`P� twv a�O�`P�r~+L twv[zMv O �¥z yQgJML�ORQ v }�`P~ t a�_ry O�a�O�J&` � ~YN x ~Ya�_ z ` t J � QgJ t J&`PO � N|Qgy+~ � L t a�O�J v{t ~YJ�~YJrOY�&Q v t J�`P�rO x Q v O z JML+O�N v ` z L+}Y��\B�rO�NPO � ~YNPOY�t J/`P�rO v O §&z O�yÍ�u�[OkQgNPO � ~ t J � `P~ zMv OkL+O�`PO�NPa t J twv ` twx�t J&`PO � N|Qp` t ~YJ!`P~7L+O�N t �YO�`P�rOk_c~ v `PO�N t ~YN�L twv `PN t b z ` t ~YJ<QgJML�`P~O��pQgy z Qp`PO@`P�rOe�MN v `#`W�[~�_c~ v `PO�N t ~YNBa�~Ya�O�J&` v �

s ~ x�zMv{t J � ~YJ x ~Ya�_ z `|Qp` t ~YJMQgy twvPv{z O v � _c~ v `PO�N t ~YN�NPO v{z y ` v�zMv{t J � L+O�`PO�NPa t J twv ` twxkt J&`PO � N|Qp` t ~YJ<QgNPOk~Yb+`|Q t JrORL/bU}t a�_ry O�a�O�J&` t J � `P�rO�O�¤U`PO�JML+ORL=S t a�_ v ~YJ � v a�O�`P�r~+L��7T ` x ~YJ v{twv ` v�t J t J&`PO�NP_c~YywQp` t J � `P�rO t J&`PO � N|QgJML�bU}/Q � O�JrO�N twx_c~Yy }UJr~Ya t Qgy-`P�MQp`�a�Qp` x �rO v `P�rO t J&`PO � N|QgJMLAQp` v ~Ya�O v O�y O x `PORL1_c~ t J&` v � V�O�`φ(βexp)

bcO!`P�rO �¥z J x ` t ~YJ `P~=bcOQg_r_rNP~¢¤ t a�Qp`PORL�QgJMLβexp

Q vPx QgywQgNR� � O�Q vPv{z a�O�`P�MQp`e`P�rO t J&`PO � N|Qp` t ~YJ�N|QgJ � O twv (0, 1)� s ~YNeL t � O�NPO�J&`eN|QgJ � O v�� 4¢�t ` twv Qgy �BQ¢} v _c~ vPv{t bry O-`P~�Qg_r_ry }kQ�~YJrO � `P~ � ~YJrOK`PN|QgJ vW� ~YNPa�Qp` t ~YJk~ � �pQgN t Qgbry OK`P~ x Qgy x�z ywQp`POK`P�rO t J&`PO � N|Qgyc~p�YO�N (0, 1)

�\B�rO �¥z J x ` t ~YJ twv Qg_r_rNP~¢¤ t a�Qp`PORL�`P�rNP~ z � � 2n + 1O §&z Qgy y } v _MQ x ORL�_c~ t J&` v �MQ x�x ~YN|L t J � `P~�`P�rO�NPO�ywQp` t ~YJ

∫ 1

0

φ(βexp)dβexp u1

6n

[

φ(β(0)exp) + 4φ(β(1)

exp) + 2φ(β(2)exp) + . . . + 2φ(β(2n−2)

exp ) + 4φ(β(2n−1)exp ) + φ(β(2n)

exp )]

,

�#�rO�NPO@~YJ�`P�rO�N t � �&` � �MQgJML v{t L+O��[Oe�MJML�`P�rO t J&`PO�NP_c~YywQp` t J � _c~Yy }UJr~Ya t Qgy�Qp` 2n + 1LrQp`|Q�_c~ t J&` v � bU� t ~ zMv y }Y�&`P�rOywQgN � O�N[`P�rO�J z a�bcO�N#~ � _c~ t J&` v �U`P�rO�bcO�`{`PO�N#`P�rO §&z Qgy t `W}k~ � `P�rO�Qg_r_rNP~¢¤ t a�Qp` t ~YJ � �¢�

^K~g`PO#`P�MQp`R�+Q v O�¤+_rywQ t JrORL t J�¡BQ z �[O�J v QgJMLkV z brN|QgJr~ "$# %'%'(*)��&Qp`�`P�rO v Qga�OK` t a�O-J z a�O�N twx QgyM_c~ v `PO�N t ~YNCNPO v{z y ` v~YJβexp

QgNPO x Qgy x�z ywQp`PORL�� t ` twv _c~ vPv{t bry OK`P~�~Yb+`|Q t JkO v ` t a�Qp`PO v ~ �ααα�YO x `P~YNR� s NP~Ya "$#ph2)�� t JML+O�ORL��U�[OeQgy NPORQYL+}��UJr~p�

`P�MQp`B | βββ, Data ∼ Mt(m+n(p−1)+r)×n(B,W

W, S, T −(m+n(p−1))−r)� t � OY�

B"·QgJML��rO�J x O

α′

α′

α′ ) � ~Yy y ~p� v Q

a�Qp`PN twx �pQgN t Qp`POeS&` z L+O�J&`#L twv `PN t b z ` t ~YJ�� x ~YJML t ` t ~YJMQgy y }�~YJβββ��¦�~YNPO�~p�YO�NR�gORQ x �kO�y O�a�O�J&`[~ �

α′

α′

α′ � ~Yy y ~p� v Q z J t �pQgN t Qp`POS&` z L+O�J&`eL twv `PN t b z ` t ~YJ��

\B�rO-�MN v `B`W�[~ x ~YJML t ` t ~YJMQgy�a�~Ya�O�J&` v ~ �ααα�YO x `P~YN4" � t �YO�J βββ

)[QgNPOeNPO�_c~YN{`PORL t J;"Íd.�*)[QgJML "Íd�� )���I[`#ORQ x ��_c~ t J&``P�rOB_c~ v `PO�N t ~YN

φ(βββ|Data)twv O��pQgy z Qp`PORL��p`P�rO[_c~ v `PO�N t ~YN

φ(α′

α′

α′

| βββ, Data)twv Qgy v ~eO��pQgy z Qp`PORL���T�J�`P� twv �BQ¢}Y�p_c~ v `PO�N t ~YN

NPO v{z y ` v ~YJα

α′

α′ a�Q¢}kbcO�~Yb+`|Q t JrORL v{t a�_ry }�a�QgN � t JMQgy t ��t J � `P�rO x ~YJML t ` t ~YJMQgy�NPO v{z y ` v ~YJ α

α′

α′ � t �YO�J βββ

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

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(a, b)Þ(−∞, b)

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∫ 1

0

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exp) + 2φ(β(2)exp) + . . . + 2φ(β(2n−2)

exp ) + 2φ(β(2n−1)exp ) + φ(β(2n)

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.

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Page 21: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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x ~Ya�_rN twv ORL t J!`P�rOka�Qp`PN t ¤Xt J " h2)x ~YJ&`|Q t J�Q x ~YJ v `|QgJ&`B`PO�NPa��MQ�L+O�`PO�NPa t J twv ` twx�v ORQ v ~YJMQgy�L z a�a�}QgJMLQ�L z a�a�}��pQgN t Qgbry O-`P~�NPORL zMx Oe`P�rO t J�� z O�J x O~ � `P�rO�QgJr~Ya�Qgy ~ zMv ` t a�O v O�N t O v bcO��MQ¢� t ~ z N t J�`P�rO v O x ~YJML��MQgy � ~ � # %'%'%r�UQ v#v �r~p�#J t J "Íd2(*)��

0 \uI-¡BV�Z (�I-¡ �-\ !KZ�]#Z$1I v \uQgbry O�(�NPO�_c~YN{` v ��� t `P�/Q�S&` z L+O�J&`

(1, 1, 1.0589, 12)_rN t ~YN�" x Q v O #0)���`P�rO�O v ` t a�Qp`PORL!_c~ v `PO�N t ~YN@a�ORQgJ�~ �

βexp

twvE(βexp) = 0.9296

��\B�rO x ~YNPNPO v _c~YJML t J � v `|QgJMLrQgN|L!L+O�� t Qp` t ~YJ twv�v � L�� " βexp) = 0.2285� � Oka�Q¢}7� twv �`P~<�pQgNP}<`P�rO!S&` z L+O�J&`�_rN t ~YN�_MQgN|Qga�O�`PO�N v "

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m0)�`P~GQ vPv O vPv �r~p� `P�rO�} t J�� z O�J x O�`P�rO�O v ` t a�Qp`PO v ~ � `P�rO_c~ v `PO�N t ~YN-a�~Ya�O�J&` v � �Y�@IKy `P�r~ z � �7Jr~g` v �r~p�#J t J!\uQgbry O (r� x Qgy t brN|Qp` t J � Q (1, 1, 1.0589, 6)

S&` z L+O�J&`�L+O�J v{t `W}Y���[O� O�`

E(βexp) = 0.9288QgJML v � L�� "

βexp) = 0.2345�1!KO�J x O�`P�rO t J�� z O�J x O v O�O�a v `P~�bcO�y t a t `PORL���Qp`@y ORQ v ` t J7`P� twv_MQgN{` twx�z ywQgN x Q v OY� �1t `P�kQ

(1, 1, 0.5, 12)S&` z L+O�J&`�_rN t ~YNR� x ~YJU�YO�N v O�y }Y�

E(βexp) = 0.9104QgJML v � L�� "

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�

m0twv-v a�Qgy y O�N )��r`P�rO�_c~ v `PO�N t ~YNKbcO x ~Ya�O vy O vPvKx ~YJ x O�J&`PN|Qp`PORL��+Q v �[O�y y�

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

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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 �

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L(βexp | Data) ∝(s2 + m2(βexp − β2)

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Page 22: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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Page 23: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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-/% � $�$ C�ñ&$�$�ÿ�� �%6(� � åWé�ä{éRï&é�÷gá å� �=@?�������A �óuëpë�æWå æ�ä{é�ü�æWã|å æWå#øwë�äeé�à�é�îYæWë�äWã|ô�äWã|å�å�á â¢ã�î&à&á æ#äWëpë�æ-á à�æWê&ã@üYäWã|å�ã|à&è|ã@ë�ø � ã�é�÷gì ú�ïYã|ü+ã|àUïYã|àpæãPä�äWë�äWå E� ��! ��� *�+ �#"D? � �"' , '�� � '��5�0� *�+ ; � � �WÞ�%'% �C�$�$�ñ C�ÿ��%� �����@Z[î&á àpæWëRå�Ú# Ý�8= ��ÿRÿ0C A ��gî&å æ{é�á àUé��&á ì á æÍú7ë�ø[æWê&ã�ïYã�&Uè|á æ�üYäWëpè|ã|å�å �Cá æWê!å æ�äWî&èPæWîYä{é�ìCå�ê&á ø æWå �� ��! ��� *�+ �#" 9 ! � � � '���� * ���

����������� � � , ( * ( � � ( � ���WÞE-/. � � �Rÿ�ñ �%��$� ��C���Ü�ã|á à&êUé�ä�æ�Ú# �� ÞCÜ�ëRôRë �:� �� Þ �Yé�âRé�å æ{é�à&ë �� � �=@?�����60A ûBã �Yæ�á àpæWëRì ãPä{é�à&è|ãRÞCòBó Ý�Ü ��ë�äW÷gá à&ô � é�ü+ãPä�ò[ë% ÿRÿ����gÞ

��î&ôRî&å æ�� �%-(���Yé�á ï��� Ý� ÞrûBá è{÷¢ãPú�û � = ��ÿ�����A � ã|å æWá à&ô�øwë�ä#î&à&á æ[äWëpë�æWåBá à�é�îYæWë�äWã|ô�äWã|å�å�á â¢ã@õ�ë�âgá à&ô�é�â¢ãPä{é�ôRãeõ�ëgïYã|ì å[ë�ø î&à&÷gà&ë��Càë�ä{ïYãPä� 9�� ���"')(3� � d * Þ���-UÞ C�ÿRÿ�ñ -��%$� ��$����pæ�ä{é�è{êUé�à!Ü � �=@?�����60A �ýué�ì á ï!ó é|úpã|å�á é�à�ã|å æWá õ�é�æWá ëRà!ë�ø[æWê&ã�è|ëRá àpæWã|ô�ä{é�æWá à&ôãPä�äWë�ä�è|ë�ä�äWã|èPæWá ëRà�õ�ëgïYã|ì 0� ��! ��� *�+ �#"

9 ! � � � '���� * ��������������� � � , ( * ( � � ( � ���WÞF%F- � ���0C�ñ ��ÿ0C� � �(���pæ�ä{é�è{êUé�à�Ü � ÞpâRé�à@ûBá < ÷ ! � =@?�����60A gó é|úpã|å�á é�à�õ�ëgïYã|ìYå�ã|ì ã|èPæWá ëRà �Cá æWê�é�à�î&à&á àYøwë�äWõ�é�æWá â¢ã�üYäWá ë�ä��H*I#"J��� � 9 ! + + ')( � �

�#"+����������� � ��� * ��� , ( * ( � � ( � ���WÞFb'Y � ��-�6�ñ �%$ -�� �¢ÿ(���pæ�ä{é�è{êUé�àkÜ � ÞrùÍàUïYãPäBó# =@?�������A có é|úpã|å�á é�àé�àUé�ì úYå�á åBë�ø�æWê&ãKãPä�äWë�ä#è|ë�ä�äWã|èPæWá ëRà�õ�ëgïYã|ì 8� ��! ��� *�+ �#"X�����������"')(3� � ���WÞ

-/%'. � 6��%$�ñ 60? C

Ö �

Page 26: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

� C��(� � ëRõ�üUé0! �= ��ÿ%$ 60A � ê&ã�á æWãPä{é�æWá â¢ã �gá õ�ü&å�ëRàõ�ãPæWê&ëgï�ë�øCàgî&õ�ãPäWá è�é�ì�á àpæWã|ô�ä{é�æWá ëRà��Ú�Û[Ü[Ý=ûBá å�è|î&å�å�á ëRà � é�ü+ãPä�$ 6�6�-gÞß[à&á â¢ãPäWå�á æWç-Ú é�æWê&ëRì á ípî&ãKïYã#ðMëRî&âRé�á àMÞUóuã|ì ôRá î&õ � C��� � äWã|êUé�àGó# Þ���é�ì å�ê=Ú# Ý� = ��ÿ����0A �ÚuëRõ�õ�ëRà�æ�äWã|àUïYå|ÞCá àpæWãPä�æWã|õ�ü+ë�ä{é�ì �&îUïYôRãPæ �Ué�ì é�à&è|ãRÞBé�àUï!äWã|â¢ã|àgî&ã�å�õ�ëpë�æWê&á à&ô%

� ��! ��� *�+ �#"+����������� � �XA ; � * � � ��� * ���G)*����(3� � + Þ�-/% � ��? C�ñ �����%� C ?�� � äWã|êUé�à!ó# Þ ��é�ì å�ê�Ú# Ý��= ��ÿRÿ���A � ã|å æWá à&ôká àpæWãPä�æWã|õ�ü+ë�ä{é�ì �&îUïYôRãPæeè|ëRà&å æ�ä{é�á àpæWå � � ê&ã|ë�ä�ú�é�àUï�é�ü&ü&ì á è�é�æWá ëRà&åeæWë�ß��øwã�ïYãPä{é�ì �&îUïYôRãPæ�é�àUï�è|îYä�äWã|àpæ�é�è|è|ëRî&àpæ[ïYã�&Uè|á æWå �� ��! ��� *�+ �#" $ ��� ' ;�� )8� ' � � ( � * ��� 9�* �/d � � \�Þ�%'. �?���-�ñ ? ? 6�� C 6(��ýué�à&ê&ë�äWã �+ã|ã|÷ � Þ@ýué�à4Ü�ëRõ�ü&îYú � 4= ��ÿRÿ0C A �gëRì â¢ã|à&èPú4é�àUï4å�î&å æ{é�á àUé��&á ì á æÍú1ë�ø�&Uå�è�é�ì�ü+ëRì á è|á ã|å�á à4æWê&ã!Ý�ß [A '

����������� � � (¹Þ�- UF. � ��C�$�ñ ��$ 6�� C ����ý[á ì ì é�à&á ��6=@?����0C A +ó é|úpã|å�á é�à�äWãPøwãPäWã|à&è|ãKé�àUé�ì úYå�á å�ë�ø�è|ëRá àpæWã|ô�ä{é�æWá ëRà�������������"')(3� � ��?�<�' ��� ; ÞF%F- � 60? -�ñ 60C�$� C C��0��é�äWà&ã�� �=@?�����-0A Uó é|úpã|å�á é�à�á àYøwãPäWã|à&è|ã#á à�è|ëRá àpæWã|ô�ä{é�æWã�ï�ý ��ÜGõ�ëgïYã|ì å �Cá æWê�é�ü&ü&ì á è�é�æWá ëRà&å�æWë-æWê&ã#ïYã|õ�é�àUï@øwë�ä[Ý�îYäWëé�äWã�é�� 6�+ÝuÚ ó ��ë�äW÷gá à&ô � é�ü+ãPä�ò[ë% -Rÿ0?� C -(�0��ã|å æWãPäWì î&àUï��% Þ � äWëRê&ì �� =@?����%$ A � ã|å æWá à&ô�øwë�ä�&Uå�è�é�ì�å�î&å æ{é�á àUé��&á ì á æÍú��Cá æWê&á àkæWê&ãeÝ�îYäWëRü+ã�é�àß[à&á ëRà �[ò[ã��1ã|âgá ïYã|à&è|ã

�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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

� ����� 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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

. �� � ­&¬�¬���¬��� �©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: Testing Fiscal Sustainability in Poland : A Bayesian Analysis ...s ~YN@aQgJU} t JML zMv `PN Qgy ORL x ~ z J&`PN O v ~YJ O NPJ ~p YO N-`P rO ORQ v{t b y `W}~ Q_cO NPaQgJrO J&` L+O x

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′

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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′

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[

X′

MY X X′

MY Zβββ

(Zβββ)′

MY X (Zβββ)′

MY Zβββ

]∣

= |X′

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MY Z − Z′

MY X(X′

MY X)−1X

MY Z|.= �%��A

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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) ×(

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φ(βββ | Data) ∝ φ(βββ) ×(

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MY X(X′

MY X)−1X

MY Z||Y′

Y|)−

(T−k−r)2

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2

= φ(βββ) ×(

|W′

W|−1|X′

MY X||Z′

MY (IT − X(X′

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)MY Z||Y′

Y|)−

(T−k−r)2

×(

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X||Z′

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2.

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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

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= �%60A

Ö �

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X|(T−k−r−n)

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|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

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MYX)−1X′)MYZ|

(T−k−r)2

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2

|βββ′

Z′

MY(IT − X(X′

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MXZβββ|l2

|βββ′

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,

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2

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Ö �

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Département des Sciences Économiquesde l'Université catholique de Louvain

Institut de Recherches Économiques et Sociales

Place Montesquieu, 31348 Louvain-la-Neuve, Belgique

ISSN 1379-244X D/2007/3082/040