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Bank Ked 47(5 6 365-394 Do ake pe pove e aua o naon oean n Poand? A daeaed appoa Łukaz Lena* Anezka Lezzńka-Pazena # Subed: Ma 6. Aeped: Auu 6. Aba T pape nveae o-e oea o Po ea-on-ea (-o- naon un uen ake daa and a daeaed on-on-on (-o- onue pe nde (CP. We popoe a ode baed on a e o uvaae eponena oon ode (ESM n o and a pe nonnea n ode. To end e oa -o- CP daeaed o COCOP (4-d oponen ( an appo. 5% onbuon n e oa CP and e eann pa o e CP. To pove oea aua (n paua n noan o ea COCOP e ue e avaabe uen ake daa on ee a ood and peo pe. We nveae and e e oean aua o e ode ake daa aan benak ode (ou ake pe n a peudo ea-e aeok. Ou ndn ue a o o o e -o- oponen e ode ake pe oupeo e ondeed benak ode a ue COCOP daa e on. Keod: naon oean uvaae eponena oon ode n ode noan uen ake pe JEL: E3 E37 C53 * Naodo Bank Pok Eono nue; Cao Unve o Eono Depaen o Maea; e-a: ukaz.ena@nbp.p. # Naodo Bank Pok Eono nue; Unve o Lodz Depaen o Eonoe; e-a: anezka.ezznka-pazena@nbp.p.

366 Ł. Lena A. Lezzńka-Pazena. noduon naon oean an be appoaed n o aenave a: udeen and ode oean poedue. Foea aua de ao ozon epe oea uua oupeo ode n o ozon a e nopoae addona noaon ndependen o e ode. T epa a ea douened n a nube o ude (ee MNee 99; Sande Rzan. On e oe and a onen ode appoa ao a bee and oe oa epeenaon o ode and oea unean. Te queon a ondeed n pape o o eend ode oea o a e nude noaon avaabe o epe e keepn e pue ode appoa? n ou oean eee e ue uen ake pe and peudo epe knoede a an be anaed o a e ee e.. announeen b e ee ene and a pe euao ae knon po o naon announeen. Hene e a o pape a a an paa one. We bud and app a oean ode an advanae n noan povde a dbuon oea a ao pobab-baed ana. To aeve e oa o pape e popoe a pe ode bu o e daeaed CP un e au noaon e avaabe o e ake.e. uen ene and a pe o oueod pe o e ood ake and peo aon. Foean eono e ee uua baed on ode a daabae o e ae equen. Hoeve opae un uen noaon epea o o equen eono ee u a quae DP o on naon. douened a uen ake daa ondeed n a ne-ode aeok pove oea aua bo o DP (ee annone Ren Sa 8 and naon (ee Moduno 3. Un uen daa pobe ed equen ode (e.. MDAS ee e Sana-Caa Vakanov 6 o e dna ao ode (Banbua Moduno. n pape e onde uen ake pe o e equen an e on CP. We ano eek daa o a o-aed peudo on daa bae b a pe eu ao. Fna e popoed ode baed on a e o ndependen ode apped o e daeaed CP and uen ake pe e ae equene. We ue e uvaae eponena oon ode (ESM a e oean oo n ou eee due o p and eave advanae ove oe pe e ee ode. Noe a e ESM peo e n oean opeon (ee Makdak Hbon. Tee e nueou eponena oon eod onenn a o un eve end and eaona oponen (ee Hndan e a. 8. oud be epazed a ESM ae an ued o nonaona e ee and eeoe a o oden no ve popua n ena bank n e o-e naon oean. Teeoe e onep o aona and oeaab a been de eaned n e ondeed ode. n a pe and naua a e eneaze e ondon o aona and oeaab peened b Hndan e a. (8 o a uvaae ae. Hene e aeok peened n pape enabe u o ea e popua aona aupon n e o e naon oean and a oe enea dna ode aoed n paua o e eaona paen. To o e CP oponen (a and ee pe ae odeed un a pe n ode a ao u o oban a ed pe o pedve dbuon due o e dee-onnuou ounepa o e daa. Te oune o e pape a oo. Seon onan e daa peenaon. Te oon pa ove e ba ode eup: eponena oon eod e n ode e ode

Do ake pe pove e aua 367 ondeed o -o- CP and e na ode o -o- CP. Te a pa douen e oean peoane o e popoed ode and opae e benak.. Daa Te daeaon o e oa -o- CP ued n e oean eee ued b e avaab o ake pe (Subeon.. Noe a e ake pe ae eoded deen equene: eek o on. Te eek daa ae anoed e ue o a eu eod o oban daa e ae on equen n Subeon.... CP daeaon We onde -o- CP daa a COCOP (aaon o ndvdua onupon b pupoe 4-d eve. We ee oe COCOP o e eevan noaon abou uen ake pe avaabe (ee Tabe. Spea e onde e oon ub-aeae o CP ( an appo. 5% oa onbuon n e oa -o- CP. Te ee ae aken o e ood oup: bead and eea; ea; k eee and e. Te ne ue and uban o peona anpo and e o a ae: ee and a. Cuen ake pe ued n e oean eee o e aoeenoned nde ae: eek daa pe aeed o peo aon (oue: p://.e-peo.p/ and auua ake (oue: p://.no.ov.p/rnk-one and on daa (deon o e ee and a pe euao ade b e Ene Reuao Oe; oue: p://.ue.ov.p. Takn no aoun e ane a and andad devaon o e ondeed COCOP e ue a and ee pe ae e o poan dve o oa naon (ee Tabe. Te pe nde o e eann pa o e CP bake a an appo. 75% onbuon n e oa -o- CP. n ode o e a ea-e daabae a doe no nude epe knoede o oeon ea uen ake pe a oeed up o a e da ae e CP announeen (b Cena Saa Oe o Poand: p://a.ov.p/. Hoeve e u-o dae a ea on doe no eeed e 7 da o ea on. Fo bead and eea; ea; k eee and e e uen ake pe ae eek daa appea ea Tuda ( e eepon o non-okn da n Poand. Te uen ake pe onenn ue and uban o peona anpo ae ao eek and appea ea Wedneda. Ea CP oponen avaabe o Janua 999 bu e avaab o ake pe deen o deen COCOP and peened n Tabe... Daa anoaon Te eek ee ued n e oean eee a been anoed o a on equen n a pe eu a. n e ep o ea eek ake pe ( ubp e dene a peudo on pe { : Z} p :

368 Ł. Lena A. Lezzńka-Pazena p e aveae pe o e a o avaabe eek beoe e u-o dae ee ubp Z ee o e nube o e on (ee uave Fue 4. Te avaab o eek daa deened b e e a e peudo on pe auaed (e u-o dae appoae a e bennn o e eond a o e on. n e eond ep e peudo { p on : Z} o o e ake pe a auaed va: ee p p C p p C p / p e aveae (obaned va nea appoaon pe o e pevou on. / p p T nea appoaon a ade b appn e equaon: p Σ p nube o da n a on J ee J a e o a oneuve eek o daa ae avaabe n on and W obevaon equaon a nube o da n { pe ven eek and on. { p : Z} F { p : Z} { p ae equaon : Z} : } Z p n p { p a : Z} e peudo on daa avaabe po o e CP daa announeen no pubaon p a (ee Fue. p [ ] an C p / C p p / C p / C p p / p p p p C p p / p an 3. Mode eup p p p p p p ( p p 3.. Muvaae eponena oon ode ba p p Σ p p p W Σ J p We onde a Σ nube o da n a on p J uvaae J nube o da n p a eponena on oon J nube ode o n da a ae n F pae a on epeenaon (ee Hndan p J nube o da n a on Σ p e a. 8 o e o: J nube o da n a on W obevaon equaon W obevaon equaon W obevaon equaon F W obevaon ae equaon equaon ( F W ae equaon obevaon F equaon ae equaon F ae equaon ( ee: F ae equaon e noe a zeo ean veo and an ovaane a Σ [ ] an [ W ] an oun [ o obevaon ] equaon [ ] an an an F ae equaon ( oun o unobeved ae ( an ( e nube o unobeved ae [ ] an an ( an a ea ( oodnae. an ( n ou ondeaon e aue a: W W F F W W W F F F W F ( ( ( ( ( ( : N Σ [ ] o obevaon equaon W obevaon W W obevaon equaon F ae equaon obevaon equaon equaon W

p p p p { p : Z} p Σ p p J nube Σ o da n a on p J nube Do ake o da p pe n a pove on e aua 369 C p / p W obevaon equaon W obevaon equaon and o a o veo a quae a a oun veo. F ae equaon F ae equaon p p n u a ae ea oodnae o oun eaed b a nea eaon va a W on o a e o unobeved ae. n oe od (. Teeoe ode an be dvded no deen ae pae ode (one p o ea oodnae o veo ndependen eo e bu [ onepoaneou ] an Σ p [ ] oeaon an beeen J nube eo o da n deen n a on ode. Noe a e uvaae ae pae ode u a pope a ondeed n e Baean aeok an ( an ( n Cobeán-Vae Beúdez Vee (. Te ena dea onenn aona and W obevaon equaon oeaab n e uvaae ae an be ound n Pa A o e Append. enea n e eponena oon F appoa e ae ae equaon veo onan unobeved oponen a oepond o e eve end and eaona oponen (ee e aaon n W F Hndan e a. W 8 p.. Sne n e -o- o F o e Po CP nde and n o -o- oponen o e CP e end paen no ea obeved e onde eponena oon [ ] an ode eve and eaona oponen on. Te oon eape o a bae o enea ode ouaon n e ne eon. an ( Eape. Conde ( a uvaae ( eponena daped eve oon ode o e o: W obevaon equaon W obevaon F obevaon equaon ae equaon equaon ( W W F F ae equaon ae equaon F ( ee W W F F W obevaon equaon F ae equaon Σ ( : N Σ [ ] o : N( Σ [ ] o W F ξ T knd ξ o ode ξ n a unvaae ae ondeed n Hndan e a. (8 eon 3.5.. aed a daped eve ode. Te ena γ dea onenn ode an be ound n Pa B o e Append. γ γ Eape. n e ne eape e onde e ode a eaona paen. Fo p e : N( Σ [ ] o ake ne e eneazaon o > naua. Le u onde a unvaae daped eve ode addve eaona o e ee ξ o e o: ξ ξ [ [ ] ] ξ ξ (3 γ γ γ γ ξ [ ]

Ł. Lena A. Lezzńka-Pazena 37 equvaen o: (4 [ ] ξ γ T ode a no ondeed n dea n Hndan e a. (8. Te ena dea onenn ode an be ound n Pa C o e Append. 3.. Eponena oon ode epanao vaabe Le epeen e unvaae -o- pe nde and e epeen e noaon (epanao o nevenon vaabe o e ake a e ue n e ode o epan e dna o. n ou oean eee a an e e obevaon o e ee avaabe beoe e avaab o. n e ep e u aue a e ae neeed n e noan pedve dbuon o a e ondona on (. We ue e oon unvaae nea nnovaon ode eeo (ee Hndan e a. 8 p. 38: K Eu u V u [ ] R F W K Eu u V u / a a ( N : ( N :. ξ ξ σ σ γ Σ γ ψ ψ ψ (5 ee a equene o..d. ando vaabe equaon ae obevaon equaon F W W F ( N : [ ] o [ ] ( K Eu u V u [ ] R F W K Eu u V u / a a ( N : ( N :. ξ ξ σ σ γ Σ γ ψ ψ ψ R ae e paaee and u a veo o unobeved ae. e ae neeed n e pedve dbuon o e ne ozon ( > e onde e oon on ode o ( equaon ae obevaon equaon ( F W W F ( N : [ ] o [ ] ( K Eu u V u [ ] R F W K Eu u V u / a a ξ ξ γ Σ γ ψ ψ ψ (6

Do ake pe pove e aua 37 ee aued a ando equene and ae..d. o noa dbuon and ae on ndependen. Te above ode (6 no apped o ea ondeed COCOP. Teeoe n e ne eon a uppeena ode nodued. 3.3. Nonnea n ode epanao vaabe No a e ondeed vaabe and ake pe ave a onnuou ounepa. To o e (ee and a pe ae euaed ean a e -o- pe o onan n a e ubequen on. T a ue e ue o a ode ed pe dbuon. We popoe a nonnea n ode o e o: / a a (7 ee epeen e ondeed COCOP and epeen e R k vaued veo o avaabe ake pe (ee dea n Tabe ( N : ( N :. σ σ Te eo e a a ed pe dbuon a onenaed a pon a R k pobab p and noa dbued ( k N / a a σ σ γ Σ ψ ψ ψ pobab p. Teeoe e veo dened e veo o ake pe a a a ed pe dbuon ounepa. Te and ae e unknon ea vaued paaee and e unknon veo o ea vaued paaee. n uue ondeaon e a pon a R k and epeen a onan pe (. Noe a en a a onnuou ounepa. Denoe [ ] ( K Eu u V u [ ] R F W K Eu u V u / a a ( N : ( N :. ( k N / a a ξ σ σ γ Σ ψ ψ ψ We aue a e equene ( and e equene ae..d. equene. T ode ea aona o. 3.4. Mode peaon o daeaed on CP and eaon poedue n eon e popoe a e o ode o ea ondeed daeaed -o- CP. We popoe ode o even vaabe: COCOP and e eann pa o e CP. A enoned beoe e end oponen no naua o pe o nde. A eaonabe pope n e pedve dbuon o o o pe dna e pope o eunn o e onan ean vaue o eaona uuaon aound a onan ean vaue. n oe od ee no eaon o uppon e eene o end paen n pe dna. Teeoe e end paen

37 Ł. Lena A. Lezzńka-Pazena no ondeed. Wa oe on e eann pa o e CP uppo e eene o a eaona paen. Teeoe e do no onde eaona paen n oden COCOP and eaed ake pe. Te e o e ondeed COCOP and e eaed ake pe a dvded no o an oup. n e oup e pu vaabe an be nepeed a eazaon o e onnuou dbuon. n oup e onde e ou COCOPS (and e oepondn ake pe o Tabe. We denoe ee vaabe a ( 3 4. We aue a ea ( oo o e enea uvaae ode (6 ode peaon ( o u [ ] a veo o unobeved ae [ ( eve oponen ean vaue and [ V ] E K. ean a: W F (8 ee aued a ando equene and ae..d. o noa dbuon and ae on ndependen. Te ae W F and ae ven b ode (. Fo e ou COCOP o Tabe e onde e peaon o Tabe. n Pee aaon e end and e eaona oponen pa a ena oe. n ou eee o an daa e o Tabe e ue ode no end and eaona oponen. We ake no ondeaon eve oponen o e ondeed ee and e oepondn epanao vaabe. n eaon o Pee ona aaon and ae eenon and odaon (ee aedne 985; Hndan e a. e above ode M M6 an be aed n Tabe 3. Fo ea COCOP o oup e opaon a benak ode ou ake pe ondeed. A e benak ode e ue e ARMA (p q ode nu AC ee p q 5. n e eond oup e onde a and ee ode peaon (7. Fo ea ake pe o oup a nonzeo pobab a a e pon aued. Te vaue oepond o no ane o e pe. T uaon uua o e ondeed vaabe beaue o adnave euaon. Fo a and e on ake pe onenn a ( 5 5 e onde ode (7 a ed dbuon o.e. onenaed a pon pobab p and dbued unde a noa N( σ dbuon pobab p. Fo ee and on ake pe onenn ee.e. PE ene pe Tauon ene pe and Enea ene pe ( 6 6 e onde a ed dbuon o.e. onenaed pobab p e a pon ( and dbued unde a noa N3( µ e Σ e dbuon pobab p e. A a benak ode o ee o vaabe e ao onde e ARMA (p q ode e nu AC ( p q 5.

Do ake pe pove e aua 373 Fna e ( 7 be R 7 vaued e ee (a daeaed oa -o- CP oe oodnae onan: oneuve COCOP o Tabe ( 6 n e ae ode a n Tabe and e 7 be -o- CP o e oe COCOP a on ( 7 eneae e oa -o- CP poe { : Z}. Fo 7 e onde o peaon o e ode (3: non onan eaona ee (γ and ed eaona (γ (ee Tabe 4. A a enoned n Hndan e a. (8 p. 65: Soe ee paua n e aea o aoeono poe que abe eaona paen. n pape e o nd evdene o aeen n appaon o oa -o- CP. n e eaon poedue e ue e MLE (ee Hndan e a. 8 eon 5. oon pae n appaon o ee e na vaue (oodnae o e eed veo b a eu eod (ee Hndan e a. 8 eon 5.. We ue a eu eod n e ae o ea eve oponen (n daped and non-daped veon and e ake. Noe a n ea eaed ode a eve oponen e paaee a eaed. Teeoe (A edue o E(. n e ae o na eaona oponen e ake ee - - and _ a ean o daa e }. { 3.5. Fna ode o -o- CP Unde e above noaon e oa -o- CP an be epeened a a nea obnaon o e veo : ( 7 (9 7 7 e van e ean onan dun ea ea. ( 7 Te CP announeen n Febua o ea ea.e. a daa o Janua doe no nude COCOP (4-d ub-aeae and e van e ( 7. Te (4-d COCOP o ea Janua ae knon one addona on dea n opaon oe on (announed n Ma. Ou oean eee peudo ea e. Teeoe e a obevaon o Janua e ue ode (3 o e oa -o- CP. Sun up e na ode o : Z popoed a oo: ode (3 o oa CP ode (9 o a obevaon o e CP o Janua o oe on ( ( We aue a e eo e o ode (3 (a obevaon o naon o Janua ndependen o e eo e o ode (9 (a obevaon no o Janua a e. Noe a u a ode a ( enea no aona. n pae e e van e ( 7 ae knon up un e end o an ven ea. Teeoe o auae e pedve dbuon o one ozon an o e end o e uen ea e aue a e e ae onan (equa o e a knon vaue. Te e ae announed b e Cena Saa Oe o Poand ea Ma.

374 Ł. Lena A. Lezzńka-Pazena Mode ( eneae e pedve dbuon o -o- CP. B Y e denoe e -o- oa CP a an be evauaed un e a eve oa -o- CP va e eaon: Y Π ( Te appoaon (baed on n( o oe o zeo: ( Y ( pobe bu no ued n ou epeen due o ben a pobe oue o eo. Wen e pedve dbuon o e oa -o- CP avaabe e pedve dbuon o Y appoaed e ue o a pe Mone Cao uaon. Σ 4. Epa eu 4.. Condeed eaue o aua e douened n e eaue a a bee n e ape doe no neea povde oe auae oea. Teeoe apa o oodne-o- ea e onde e peoane o ou- -o-ape oea. n e ondeed ape peod e a peod.e. Janua Deebe 4 (e oean ubape o e qua o oea a eaned. We aue a onan en o e ape (a 9-ea on ndo e end o e ape n n peod. A a oodne-o- ea e ue e popua AC. On e ba o pon oea e onde e o popua eaue n e ae o eponena oon ode e RMSE (ee Hndan e a. auaed o oea ozon:. Te RMSE deene beeen e ESM and benak ode ee ao evauaed e ue o e HLN-DM e a (Have Leboune Nebod 997. Fna o ae e qua o pobab oea e ue o popua on ue: a oa oe and a onnuou anked pobab oe (CRPS (ee nen Rae 7 o dea. Te neave oenaon o CPRS: CRPS* -CRPS enea ued n pae. Tee eaue ave been apped o ude e qua o noan ( unvaae pedve dbuon. We peen ee eaue n oe dea o e onvenene o e eade n Pa E o e Append. We ue e oon eana ue: e e e aveae oa oe e bee e ode and e oe e aveae CRPS* e bee e ode. Noe a n e ae o e ondeed oea aua eaue (RMSE aveae oa oe CRPS* eoea ve e ae eu o -o- and -o- CP dun ode opaon. T a naua onequene o eaon (. Moeove ode M and M o -o- daa ve e ae oa oe (o and e ae CRPS*. T ea oo o e ode uue (ee Tabe. Te ae uaon onen ode M3 and M4 a e a ode M5 and M6. Te RMSE eaue ued n ode opaon o M M6 aan e benak ARMA ode. Te oa oe and CRPS* ae ued o ek e een o noan n M M6 and S S ode. Te AC ued n benak eeon.

Do ake pe pove e aua 375 4.. Foea aua o daeaed -o- CP Te pobe ondeed n ubeon onen e ueune o ode ake pe. n ou opaon e ue e RMSE and HLN-DM e a. n e ae o bead and eea e ode ake pe M M6 (ee Tabe 6 do no oupeo e benak ode. Te ae onuon an be ouaed n e ae o ea (ee Tabe 7 and k eee and e (ee Tabe 8. n e ae o ue and uban o peona anpo ode M M6 ave a nan oe RMSE an e benak ode ean a n ae ode ake pe oupeo e benak ode (ee Tabe 9. Fo e ee and a e onuon ae e ae ode ake pe ea oupeo e benak ode (ee Tabe. n onuon on n e ae o ee o e ondeed COCOP.e. ue and uban o peona anpo; ee and a e popoed ode ake pe ea oupeo e benak ode. Te ne pobe a e onde e be ode n e M M6 e (oea aua and n e ape. F e onde ue baed on e aveae oa oe and e aveae CRPS*. Ne e onde e AC. Te aveae oa oe (ee Tabe A uppo ode M M n e ae o ea and ue and ode M3 M4 n e ae o bead and eea and k eee and e. Te CRPS* uppo ode M M o o ae (bead and eea; ea; ue and uban o peona anpo. On n e ae o k eee and e e aveae CRPS* uppo M5 M6 ode bu e deene aveae CRPS* n e ae o M M ode ea o be nebe. e ake no ondeaon e AC n ubape (Tabe 4 en n e ae o ue and uban o peona anpo ode M (55% o e ubape and M6 (45% ae peeabe. n e ae o ea ode M3 (% M5 (7% and M6 (7% ae peeabe. n e ae o bead and eea e o peeabe ode ae M (98% and M (%. Fna n e ae o k eee and e ode M a e oe AC o ea ondeed 9-ea on ndo. Sun up e AC doe no uppo e aona o e ondeed ode o -o- daa. n e ne eon e onde ue n oe dea and e o e dadvanae o un nonaona ode o -o- daa e e -o- pedve dbuon uae auaed. Fo e eann pa o e daeaed -o- CP e oea o ode S ( ed eaona ave e oe RMSE a (ee Tabe A. Hoeve opan e aveae oa oe and e aveae CRPS* (o e onuon ae no e ae. Un eana ue: n e ae o e aveae oa oe e S ode oe peeabe and n e ae o e aveae CRPS* e nvee ue (ee Tabe A. 4.3. Foea aua o oa -o- CP n eon e onde e pobe o oon e be e o ode o e oponen o e daeaed -o- CP o ave a ood qua o -o- CP oea. F o a noe a n e ae o e CP e pedve dbuon o -o- CP u oe peeabe an -o- oea. Moeove e poedue o ode eeon o -o- CP oud no be baed on on e oodne- -o- and oean aua bu ao on e obeve o oea. Fo eape oon knon a o ena bank pon o ve pon oea o -o- CP oud nvove a eun o a onan vaue a a on onneon naon aen. T evon o onan vaue

376 Ł. Lena A. Lezzńka-Pazena an be aeved o eape en ode aona popee ae ued. Teeoe n e o e naon oean ena bank ue an aona ode. Te pedve dbuon o e oa -o- CP an be obaned e ue o o aenave a: de b oden -o- CP o nde b oden -o- CP. e pedve dbuon o -o- CP avaabe en un e eaon beeen -o- and -o- CP (ee oua ( e pedve dbuon o -o- CP an be obaned. Unounae e anoaon o -o- pedve dbuon o -o- pedve dbuon o ozon > a deae ue (ee oua (. To o e dadvanae o annuazaon poedue auoa ode eeon eon o -o- CP e onde an ea uave eape n Pa D o e Append. Le N be a equene o N obevaon o CP -o- pe nde and e N be a equene o N obevaon o ake pe (ee Tabe. Ten on e ba o ode ( e pedve dbuon o ake e o: ( N N N L(( L(( N N N N N N ( ( N N N N oepond o Janua N oepond o oe on T done due o e ed noaon on naon n Janua epoed b e aa oe n Febua ea ea (noaon on daeaed COCOP no avaabe. n pae o oban pedve dbuon e ue pu-n enque (e epae e veo o paaee e MLE. Ten e -o- dbuon auaed b annuazn e -o- pedve dbuon e ue o (. We auae e -o- pedve dbuon baed on ode ( un a 9-ea on eaon ndo ( e eepon o ue ee a oe daa ape a avaabe. On e ba o e above eu e ue ode M (o e ou COCOP o Tabe ode (7 o a and ee and ode S. Ten e annuaze e -o- pa o oban o 49 o 6 ea e ou o ape -o- oea o e CP (dependn on e oea ozon. Fo ea oea ozon e opae e RMSE o e na ode baed on ake pe.e. ode ( ou benak ode ( do no ake eoue o uen ake daa. Te oup o benak on o auoeeve ode eaona onan: AR( AR(6 and AR( apped o e oa -o- CP. Te oea o -o- CP ae en annuazed o -o- CP oea b appn (. Te a benak ode e ARMA(p q ode e na AC (ee p q apped de o -o- CP (ee Tabe 5. Te eu ea o a e ode ake pe a e be peoane n noan. Te ARMA ode apped de o -o- daa a e be RMSE n noan. Teeoe n e o e naon oean ode baed on -o- daa ae oe peeabe. Sne e ake pe a e ue ae avaabe on one on aead o one ozon e ode ake pe oe pedve poe. 5. Conuon n e ae e popoe a ode baed on a uvaae eponena oon ode a a oo o oean Po naon. Te COCOP o an uen ake noaon avaabe ae odeed epaae (oee e oepondn ake daa e o uabe peaon. Toe COCOP ae: bead and eea; ea; k eee and e; ee; a; and ue and uban o peona anpo. Te eann pa o e CP bake odeed a ue o a unvaae ESM.

Do ake pe pove e aua 377 n e na ae e pedve dbuon o CP oponen ae aeaed and annuazed. Te eu o nan and obu poveen ove pe benak n oe ozon. Hoeve n one ozon e oea o e popoed ESM ode eae o oupeo e benak. Reeene Bańbua M. Moduno M. ( Mau keood eaon o ao ode on daa e aba paen o n daa ECB Wokn Pape 89. Cobeán-Vae A. Beúdez J. Vee E. ( Foean oeaed e ee eponena oon ode nenaona Jouna o Foean 7( 5 65. aedne E. (985 Eponena oon: e ae o e a Jouna o Foean 4( 8. e E. Sana-Caa O. Vakanov R. (6 Pedn voa: en e o ou o eun daa aped a deen equene Jouna o Eonoe 3( 59 95. annone D. Ren L. Sa D. (8 Noan: e ea-e noaona onen o aoeono daa Jouna o Monea Eono 55(4 665 676. nen T. Rae A. (7 S pope on ue pedon and eaon Jouna o e Aean Saa Aoaon (477 359 378. Have D. Leboune S. Nebod P. (997 Ten e equa o pedon ean quaed eo nenaona Jouna o Foean 3( 8 9. Hndan R. Koee A. Od J. Snde R. (8 Foean eponena oon Spne. Hndan R. Koee A. Snde R. oe S. ( A ae pae aeok o auoa oean un eponena oon eod nenaona Jouna o Foean 8(3 439 454. Kan R. (8 Fo oen o u o oen o podu Jouna o Muvaae Ana 99(3 54 554. Laon R. (998 Ho oud addve Ho-Wne eae be oeed? nenaona Jouna o Foean 4(3 393 43. Makdak S. Hbon M. ( Te M3-opeon: eu onuon and paon nenaona Jouna o Foean 6(4 45 476. MNee S. (99 Te oe o uden n aoeono oean aua nenaona Jouna o Foean 6(3 87 99. Moduno M. (3 No-an naon un equen daa nenaona Jouna o Foean 9(4 664 675. Sande N. Rzan L. ( Judena aduen o aa oea n: J.S. Aon (ed. Pnpe o oean Spne. See A. (985 Copun e vaane o e oea eo o e Ho-Wne eaona ode Jouna o Foean 4( 35 43. Aknoedeen We oud ke o ank Pae Maa o e ep n onun ea e daa e. Te ve epeed ee ae oe oe o e auo and do no epeen e ve o Naodo Bank Pok.

378 Ł. Lena A. Lezzńka-Pazena Append Pa A n Hndan e a. (8 e onep o aona and oeaab o unvaae ( eponena ode a eaned. Pevou e onep o oeaab a eaned n eaon o eponena oon ode n See (985 and Laon (998. Baed on ee denon e eneaze e Σ denon o aona and oeaab o e uvaae ae. Sne e E Σ C ee E W F and C C W F [ ( k ](o > a daona a eeen ( k k k k k e eneaze n naua a e onep o aona o e uvaae ae. We a a e ode ( ( aona e en Σ ( < k k o an k { } and E ne. Te pedon Te E( W an be evauaed a Σ A B ee A W D B W D [ b( ] and k D F W d d ee d. d Sne B a daona a eeen [ b( k k ] k dk k e eneaze e onep o oeaab o e unvaae o e uvaae ae n e oon naua a. We a a e ode ( oeaabe Σ b( k k < o an Σ k { } and A ne. Te enea ondon unde a unvaae ode oeaabe ae ouaed n eoe. n Hndan e a. (8. Noe a n pape e do no onde e o-aed ab o e ode. We e oueve o aona and oeaab ne e ab pope oe eve an oeaab.e. ea abe ode oeaabe bu e nvee no neea ue. Pa B Sne ( k k k k and b ( k k k k( k k k e ondeed ode aona o a e ave < (a oepondn ondon a obaned n e unvaae ae.e. and oeaabe o a e ave <. Te oeaab ondon a no eaned n Hndan e a. (8 o ode. Te aona and oeaab eon o ea ae no ep. e ake (o en e oban e uvaae eponena oon ode eve oponen a no aona. n ae e ondeed ode oeaabe < o a. On e oe and e an ake

Do ake pe pove e aua 379 a an eape.5 and 4 o en e ondeed ode aona bu no oeaabe. Noe 4 a o e aona ondon aed ( < o a en b naua eneazaon o e eu obaned n eon 3.5. n Hndan e a. (8 e e a: E( (A and ov( k k k k[ ( ] k k k k σ k σ (A k k ee and σ k ov ( k (eeen o e ovaane a Σ. oud be epazed a e above uvaae ode an be equvaen en a a equene o ARMA( ode (one ode o one oodnae o oeaed eo e beeen ee ode: ( L ( L ( (A3 ee : N (. Ten e ondon < o a eaed o e aona o ode and ondon < p eaed o e nveb o ode (ee eaed pobe n Hndan e a. p. 69. Pa C Sne γ { } (ee {A} e ndao unon o e even A and ean a e dvbe b e ondeed ode no aona. Sne de( D λ ( λ [ λ λ ( λ ( γ ] ee D γ γ γ λ λ ( λ( γ e ondeed ode oeaabe oo o λ λ ( λ( γ e nde e un e (ee eoe. n Hndan e a. 8. Bu pobe oo du o be oved γ

38 Ł. Lena A. Lezzńka-Pazena anaa. Sun up e ondeed ode n eape no aona and oeaabe 3 oo o λ λ ( λ( γ e nde e un e. T ode an be epeened a a edued o o e eaona ARMA ode va equaon.7 n Hndan e a. ( p. 7. Ae auaon e e: ( L ( L [ L ( γ L ( γ L( ] (A4 Fo pepeve ode no aona and nvebe oo o [ L ( γ L ( γ L( ] e nde e un e. Noe a ode an ARMA ( p q( P Q ode unde e addona eon o ( ( ( γ. We ue e above ode on n e oean eee ou an pea nepeaon o e eaona oponen eeoe no neea o e eaona oponen o be noazed (ee Hndan e a. 8 ape 8 o oe dea. e aue a γ (onan eaona ee e above ode edue o e ARMA( ode eaona ean unon. Pa D n eape e ue e R pakae e auoa eon baed on e noaon eon. We onde 5 deen ode o e oa -o- CP (ee Hndan e a. p.. Foon e ue o e auoa ode oe eon and e appaon o e ae 9-ea on ndo a n e pevou eon e ea o e be ode podued ode (N A o e o: γ (A5 T ode ea nonaona. Te pon oea (ondona epeaon vaue o ozon equa o e a auazed oepondn eaona oeen aound e a auazed eve.e. E ( F. Teeoe e annuazed pon oea o -o- CP an be on baed b e a auazed eve vaue. T ee ea vbe n e oon Fue 3a. Te pon oea e a obevaon dan o 3 Ju 3 de o e oe ne June 3 bou an unepeed ok n adnave pe (eed b e deeuaon o e ae dpoa ake and e enun ke n abae oeon ee. To be oe obu o ee ok e an onde a daped eve ode (DL n o: γ (A6 non aona bu e pon oea (epeaon vaue o ozon equa o e oepondn a auazed eaona oeen aound e (unde e aupon < e equene oape o.

Do ake pe pove e aua 38 n an noa anne E ( F o ae. Fue 3b o e eu ee ode (A6 a apped o -o- CP and ne oua ( a ued. Te pobe a e ae n o onde no o o epan e π deene n pon oea n Fue 3a and 3b. T pobe oo du o be oved anaa ( e onde pon oea.e e ondona epeaon vaue baed on ode (A5 and (A6 and oua (. Teeoe e onde a o uave uaon eape ee e o o e pobe nuene o paaee o ee ode o e -o- pon oea. Eape 3. n ou uaon eape e aue a. eaed n a eu a o e.5 naon ae o e Po -o- CP. e aue a eo e auan en un pepoon o Kan (8 e -o- oen an be evauaed. Bu no ea o epan e apo popee o ee epeaon vaue o e -o- daa unde ode (A5 o (A6. Teeoe e ue a Mone Cao uaon. We auae e ean and auoovaane a o e pedve dbuon and en un ape o dbuon e auae e pa o -o- dbuon. Fna e auae e MC appoaon o e ean veo o -o- dbuon a a uua ean vaue. n e ae o ode (A5 e ue deen vaue o eve and deen vaue o paaee. n e ae o ode (A6 e ue deen and deen paaee (.5. n bo ae e ue σ.8 and γ.5 and e ae a auazed eaona oeen: { } { 3n(π /: }. Fo ea deen e o paaee e a avaabe vaue o -o- daa (.e. { } ae uaed o e ode: d ee d..d. o N( /. Te eu o ou uaon ae peened n Fue 4. T uaon ea o a n ode (A5 e -o- pon oea ae on baed b e a auazed eve o e -o- vaabe. We ue ou deen a auazed eve: -.5 (ee S S4; -.5 (ee S5 S8;.5 (ee S9 S;.5 (ee S3 S6. Noe a e pon oea do no abze aound e ae vaue o deen eve. Te e e a auazed e e e pon oea. T uaon ea no obeved o ode (A6 ee o deen a auazed and paaee e pon oea ee o abze nea e ae onan vaue. T onan vaue oepond n a eu a o.5. Fo pon o ve ode (A6 eneae oe abe pon oea o -o- CP. Noe a ean eveon n e ae o -o- CP an be nepeed b ena bank a an epeaon o eun o ae naon. Teeoe epeed a -o- CP pon oea onenae nea ae naon. Hene e oud onenae ou aenon n paua on eponena ode u popee. Sun up e noa eon (baed on e obeve o oea a oud be nopoaed n pape a ode even o e ean vaue popee o -o- daa oud be oe peeabe an oe.

38 Ł. Lena A. Lezzńka-Pazena Pa E We auae: Nov4 ( ˆ RMSE Σ (A7 6 De9 ee ˆ a pon oea (ondona epeaon vaue o ozon { } ondona on F and ˆ a eazaon. On e ba o ode M e aveae oa oe auaed va: Nov4 aveae oa oe o( ( M F 6 Σ (A8 De9 ee ( M F a pobab den unon (ondona on F o e pedve dbuon o on e ba o ode M and a eazaon a e. On e ba o ode M a an end o e ape e: Deebe 9 Janua Novebe 4 e CRPS ae auaed va e oon nea: CRPS( F M F M F [ ( F { }] d (A9 ee F ( M F a uuave dbuon unon (ondona on F o e pedve dbuon o obaned on e ba o ode M an obevaon a e and {A} an ndao unon o e even A. e pedve dbuon a auan ean and vaane σ en e above nea pe o: σ CRPS( F M F σψ ( (A π σ σ Σ ee ψ(. and (. denoe e pobab den unon and e uuave dbuon unon o a andad auan vaabe epeve. Baed on ode M e aveae CRPS* auaed va: Nov4 aveae CRPS* CRPS* ( F M F 6 Σ (A De9

Do ake pe pove e aua 383 Tabe COCOP ondeed ake pe Coponen (4-d COCOP Bead and eea (.. Mea (.. Mk eee and e (..4 Fue and uban o peona anpo (7.. Aveae e n CP bake n 4 (% Rane o e daa n peenae pon (ane a / andad devaon n e ape L o ake pe o ea COCOP Make pe avaabe o Make daa equen Make daa pubaon edue* 4. 4.4 /.64 Fou pe 5 June 3 eek Tuda 6.5 7.7 /.3 3.3 4.9 /.7 5.5 5.8 /.46 a (4.5..7 7. /.6 Ee (4.5. 4.6. /.45 Pok d auae ken bee Medu e paeued k Edak eee RON 98 ue Dee LP PN a pe PE ene pe Tauon ene pe Enea ene pe June 4 eek Tuda June 4 eek Tuda Feb 7 eek Tuda Jan Jan on on appoved n. 4 da beoe an pe ane appoved n. 4 da beoe an pe ane * Fo on n a ea e daa on CP oponen avaabe a 5 da a. Te daa o Janua avaabe a 45 da a due o e COCOP e eauaon.

Ł. Lena A. Lezzńka-Pazena 384 Tabe Mode peaon o e ou ondeed COCOP (e ubp 3 4 a been dopped Labe Reon eaed o Reon eaed o Mode peaon M none none F W M none o ea F W M3 none F W M4 M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea F W M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea F W M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea F W M5 M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea F W M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea F W M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea F W M6 M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea F W M none none F W M none o ea F W M3 none F W M4 o ea F W M5 none F W M6 o ea F W B benak ode ARMA (p q ode nu AC ee p q 5

Do ake pe pove e aua 385 Tabe 3 Caaon o eponena oon ode epanao vaabe eve oponen Leve oponen o e ee Leve oponen o epanao vaabe A (addve A d (addve daped N none N A N A d A addve A A A A d A d addve daped A d A A d A d Tabe 4 Mode ondeed o e 7 eann pa o oa -o- CP Labe Reon Mode peaon Paaee S non onan eaona ee 7 γ 9 9 va eu eod e (ee Seon. γ eaed paaee S ed eaona 7 γ va eu eod γ (ee Seon.. 9 eaed paaee

386 Ł. Lena A. Lezzńka-Pazena Tabe 5 RMSE o ozon Mode ESM baed on ake pe Benak ode: annuazed AR( eaona due annuazed AR(6 eaona due annuazed AR( eaona due ARMA(p q o -o- CP n 6 n 59 3 n 58 4 n 57 5 n 56 6 n 55 7 n 54 8 n 53 9 n 5 n 5 n 5 n 49.4.38.56.7.9.6..36.5.65.78.94.3***.47***.65**.83..6.3.47.63.77.9.4.3***.47***.66**.84**.*.7.3.48.65*.78.9.4.3***.47**.64.8.96.9..34.49.6.7.84.38***.57***.75*.93.9.4.36.47.59.68.76.86 Noe: ozon n nube o oneuve oea eo. Te HLN-DM e (Have Leboune Nebod 997: e oea aua o a ode baed on ake pe nan bee an a ven benak oea a: *** % eve ** 5% eve *% eve. Tabe 6 RMSE o aenave oean ode o e bead and eea CP oponen 3 4 5 6 7 8 9 M.494.594.6.63.67.63.63.634.64.458.49.43 M.494.63.66.6.63.65.646.654.665.487.453.457 M3.54.576.69.65.68.64.68.63.638.458.49.433 M4.54.63.6.65.66.67.64.656.668.493.458.46 M5.549.645.687.69.699.75.73.73.79.533.489.49 M6.549.675.689.689.694.75.74.738.746.55.499.5 B.56.598.64.68.65.634.64.647.653.466.433.436 Noe: Te HLN-DM e (Have Leboune Nebod 997: e oea aua o ode M nan bee an e benak oea a: *** % eve ** 5% eve *% eve.

Do ake pe pove e aua 387 Tabe 7 RMSE o aenave oean ode o e ea CP oponen 3 4 5 6 7 8 9 M.6.77.73.734.738.746*.73.74.747.755.74.78 M.6.75.769.735.76.757*.7.76.78.84.77.87 M3.646.77.733.734.739.746.73.74.748.756.74.78 M4.646.78.769.743.766.769.695.763.73.87.773.799 M5.686.75.77.7.757.85.795.778.79.85.85.775 M6.686.7.76.77.795.839.79.85.789.885.845.865 B.79.7.747.75.758.767.75.76.766.777.768.763 Noe: Te HLN-DM e (Have Leboune Nebod 997: e oea aua o ode M nan bee an e benak oea a: *** % eve ** 5% eve * % eve. Tabe 8 RMSE o aenave oean ode o e k eee and e CP oponen 3 4 5 6 7 8 9 M.54.7.78.7.73.76.73.74.79.74.73.738 M.54.757.743.74.737.73.73.7.73.78.737.746 M3.577.638.668.678.689.75.76.7.76.73.735.738 M4.577.75.73.75.7.7.79.73.73.739.748.754 M5.55.65.659.673.685.74.73.78.73.74.747.748 M6.55.66.68.689.699.79.75.77.73.738.749.753 B.65.84.788.749.74.74.77.7.76.73.74.75 Noe: Te HLN-DM e (Have Leboune Nebod 997: e oea aua o ode M nan bee an e benak oea a: *** % eve ** 5% eve * % eve.

388 Ł. Lena A. Lezzńka-Pazena Tabe 9 RMSE o aenave oean ode o e ue and uban o peona anpo CP oponen 3 4 5 6 7 8 9 M.77***.875.868.85.857.859.88.895.88.894.9.95 M.77***.93.84.93.84.879.93.84.83.836.99. M3.779***.86.868.85.858.86.88.897.883.894.9.95 M4.779***.95.863.9.863.855.877.873.86.85.99.8 M5.8***.89.97.98.97.99.949.958.943.959.975.3 M6.8***.976.886.96.94.93.983.973.98.939.5.65 B.879.958.965.95.96.897.8.8.84.83.87.97 Noe: Te HLN-DM e (Have Leboune Nebod 997: e oea aua o ode M nan bee an e benak oea a: *** % eve ** 5% eve * % eve. Tabe RMSE o aenave oean ode o e ee CP oponen 3 4 5 6 7 8 9 M.69**.84.47**.58*.67*.77.8.93.33.34.36.338 B.8.77.5.6.7.8.8.9.33.34.3.337 Noe: Te HLN-DM e (Have Leboune Nebod 997: e oea aua o a ode M nan bee an e benak oea a: *** % eve ** 5% eve * % eve. Tabe RMSE o aenave oean ode o e a CP oponen 3 4 5 6 7 8 9 M.949*.488.58.5.534.54**.536.549.565.574.563*.58 B.48.494.57.5.53.545.539.55.565.576.567.58 Noe: Te HLN-DM e (Have Leboune Nebod 997: e oea aua o a ode M nan bee an e benak oea a: *** % eve ** 5% eve * % eve.

Do ake pe pove e aua 389 Tabe RMSE o aenave oean ode o e eann pa o e CP 3 4 5 6 7 8 9 S.55.48.54.59.64.65.6.65.67.67.69.73 S.44.4.4.46.48.5.5.5.55.54.56.59 Tabe 3 Aveae oa oe and CRPS* o e oean ode o e eann pa o e CP ( Aveae oa oe Aveae CRPS* S -.65.4 S -.4.37 Tabe 4 Te peenae o nzed vaue o AC o e ondeed ode Fue Mea Bead and eea Mk eee and e M6.45.7.. M5..7.. M4.... M3.... M.55..98. M.... Tabe 5 Aveae oa oe and CRPS* o e oean ode o e CP oponen ( CP oponen Aveae oa oe Aveae CRPS* M M M3 M4 M5 M6 M M M3 M4 M5 M6 Bead and eea -.85 -.789 -.99.4.75.67 Mea -.95 -.6 -.5.35.367.389 Mk eee and e -.3 -.949 -.44.53.9.5 Fue and uban o peona anpo -.89 -.79 -.7.39.439.458

39 Ł. Lena A. Lezzńka-Pazena Fue Coponen o -o- CP nde (COCOP ake pe dna 5 5 5 95 9 85 8 A. Bead and eea B. Mea 8 7 6 5 5 4 3 95 9 99 98 85 3 4 5 6 7 8 9 3 4 34 5 6 7 8 9 3 4 8 3 98 CP: bead and eea (RHS Fou pe 5 (LHS CP: ea Pok on (LHS D auae (LHS Cken (LHS Bee (LHS 4 3 9 8 7 6 5 4 C. Mk eee and e D. Fue 5 4 5 6 7 8 9 3 4 5 5 95 9 5 5 95 9 85 8 7 8 9 3 4 5 5 95 9 CP: k eee and e (RHS Paeued k (RHS Medu e (LHS Edak eee CP: ue (RHS Dee (LHS Pb98 (LHS LP (LHS 9 8 7 E. a F. Ee 5 6 9 345678934 7 97 5 5 95 9 85 8 345678934 5 3 9 7 5 3 99 97 CP: a (RHS a eudn dbuon (LHS CP: ee (RHS Tauon ene pe (LHS Enea ene pe (LHS PE ene pe (LHS

Do ake pe pove e aua 39 Fue Pon oea ( 4 o deen paaee 5 A. Mode (A5 {-.5 -.5.5.5} {...3.4} 4 3 6 8 4 S S S3 S4 S5 S6 S7 S8 S9 S S S S3 S4 S5 S6 B. Mode (A6 {-.6 -...6} {-.6 -...6}.5 5 4 3 6 8 4 R R R3 R4 R5 R6 R7 R8 R9 R R R R3 R4 R5 R6

39 Ł. Lena A. Lezzńka-Pazena Fue 3 Pon oea ( un 9-ea on ndo o oa -o- CP baed on ode o -o- CP daa and eaon ( A. Mode (A5 NA 6 5 4 3-3 4 5 B. Mode DL 5 4 3-3 4 5 Noe: o Janua o Deebe 5. Bak ne: -o- CP. a ne: pon oea (epeaon vaue.

Do ake pe pove e aua 393 Fue 4 Daa avaab edue CP announeen CP announeen o pevou on u-o dae o pevou on u-o dae on ( - oepondn p - on oepondn p eek eek eek eek eek eek eek eek eek o a announen o eek daa beoe u-o dae o a announen o eek daa beoe u-o dae e eek

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