F esoli, Diego
A icle
Spanish GDP sho - e m poin and densi y o ecas ing
using a mixed- equency dynamic ac o model
SERIEs - Jou nal o he Spanish Economic Associa ion
P o ided in Coope a ion wi h:
Spanish Economic Associa ion
Sugges ed Ci a ion: F esoli, Diego (2024) : Spanish GDP sho - e m poin and densi y o ecas ing
using a mixed- equency dynamic ac o model, SERIEs - Jou nal o he Spanish Economic
Associa ion, ISSN 1869-4195, Sp inge , Heidelbe g, Vol. 15, Iss. 2, pp. 145-177,
h ps://doi.o g/10.1007/s13209-024-00297-3
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SERIEs (2024) 15:145–177
h ps://doi.o g/10.1007/s13209-024-00297-3
ORIGINAL ARTICLE
Spanish GDP sho - e m poin and densi y o ecas ing
using a mixed- equency dynamic ac o model
Diego F esoli1
Recei ed: 23 Sep embe 2022 / Accep ed: 26 Feb ua y 2024 / Published online: 10 Ap il 2024
© The Au ho (s) 2024
Abs ac
We ha e assessed he e ec o da a eleases when cons uc ing sho - e m poin and
densi y o ecas s o he Spanish g oss domes ic p oduc g ow h. Fo his pu pose, we
conside ed a eal- o ecas ing exe cise in which we de ined se e al pseudo-da a in-
ages ha had a mix u e o mon hly and qua e ly equencies and we e unbalanced
owa ds he end o he sample. We implemen ed a mixed- equency dynamic ac o
model o deal wi h da a ea u es and o p oduce g oss domes ic p oduc o ecas s.
We e alua ed he p edic i e con en o da a eleases om poin and densi y o ecas
pe spec i es, he la e aspec o he analysis being p e iously unexplo ed in he li -
e a u e p oducing Spanish g oss domes ic p oduc sho - e m o ecas s. We obse ed
signi ican imp o emen s in poin o ecas s as in o ma ion is eleased h oughou he
qua e , con i ming exis ing esul s. Addi ionally, ou indings indica ed subs an ial
enhancemen s in he accu acy o densi y o ecas s as new da a eleases ma e ialized.
Keywo ds Sho - e m g oss domes ic p oduc (GDP) poin o ecas ·Densi y
o ecas ·Mixed- equency dynamic ac o model
JEL Classi ica ion C32 ·C530 ·E39 ·E370
1 In oduc ion
One o he well-known ac s abou g oss domes ic p oduc (GDP) is ha i s alues a e
eleased wi h a delay ha exceeds 3 weeks o he i s ad ance es ima e o he p e ious
qua e . Fo his eason, many economic agen s, cen al banks, go e nmen s a is ical
o ices, and o he inancial ins i u ions spend much o hei ime and esou ces con-
The eplica ion ma e ial o he s udy is a ailable a h ps://zenodo.o g/ eco ds/10691459.
BDiego F esoli
[email p o ec ed]
1Facul ad de Ciencias Económicas y Emp esa iales, Dp o. Anáisis Económico: Economía
Cuan i a i a, Uni e sidad Au onóma de Mad id, Mad id, Spain
123
146 SERIEs (2024) 15:145–177
s uc ing GDP sho - e m o ecas s. Fo his pu pose, hey use a a ie y o mon hly
indica o s, mo e equen ly and mo e imely a ailable han he a ge qua e ly GDP.
The combina ion o he qua e ly GDP and mon hly indica o s used as p edic o s gen-
e a es a mixed- equency na u e o da a. Fu he mo e, as a iables a e asynch onous,
wi h hei la es da amadea ailable a di e en poin sin imeandwi hdi e en delays,
a mix o a ailable and missing obse a ions comes up owa ds he end o he sample,
a pa e n known as unbalancedness o agged edge da a s uc u e. Hence, cons uc ing
GDP sho - e m o ecas s in ol es coping wi h missing obse a ions as an essen ial
pa o da a p ocessing.
In Spain’s case, moni o ing and acking he sho - e m e olu ion o GDP is o en
suppo ed by he dynamic ac o model (DFM) and i s s a e-space ep esen a ion.
These a e, o ins ance, he Spain-STING (sho - e m indica o o g ow h) and MID-
P ed (in eg a ed model o p edic ion), which a e used by ins i u ions like he Bank
o Spain and he Independen Au ho i y o Fiscal Responsibili y, espec i ely, o he
FASE model ( ac o analysis o he Spanish economy) which is o iginally designed o
o e see and o ecas GDP wi hin he Minis y o Economy; see Camacho and Qui os
(2011), Cue as and Quilis (2012), Cue as e al. (2017), and Pa eja e al. (2020). Mo e-
o e , he MICA-BBVA (Model o economic and inancial Indica o s used o moni o
he Cu en Ac i i y by Banco Bilbao Vizcaya A gen a ia) model has also been p o-
posed o ob ain GDP p ojec ions h ough eal and se e al inancial indica o s; see
Camacho and Doménech (2012). An essen ial pa o he ask o all hese eal- ime
o ecas ing applica ions is he cons uc ion o Spanish GDP sho - e m poin es ima es
and e alua ing hei accu acy. A common inding is he exis ence o accu acy gains
as new da a become a ailable du ing he qua e . Ne e heless, in an en i onmen o
inc easing economic unce ain y, use s o GDP o ecas s demand poin p ojec ions o
i s sho - e m e olu ion and, mo e assiduously, he comple e u u e cha ac e iza ion
p o ided by i s sho - e m densi y o ecas s; see, o ins ance, Mazzi e al. (2014),
Aas ei e al. (2014,2018), Bäu le e al. (2021), Mi chell e al. (2022), and Çakmakl i
and Demi can (2022). Consequen ly, i is c ucial o assess he in o ma ion con en o
da a a each poin in ime om he pe spec i e o he GDP sho - e m densi y o ecas s.
In he p esen pape , we ha e aimed o add ess he in o ma ion when cons uc ing
Spanish GDP sho - e m o ecas s, using he s anda d e alua ion app oach h ough
poin o ecas s and, mo e ele an ly and inno a i ely, densi y o ecas s. Fo his pu -
pose, we ha e combined hi y- wo mon hly indica o s om he Spanish economy,
including eal-ac i i y, inancial and su ey a iables, and he qua e ly employmen
and GDP g ow h a es wi hin a mixed- equency model-based o ecas ing me hodol-
ogy. The da a- ich amewo k has enabled us o co e mos o he indica o s p e iously
conside ed in he e e enced models o Spain. As we did no ha e access o published
his o ical eco ds o he a iables, we ha e assumed ha he publica ion calenda on a
ecen da e, i.e. beginning o Oc obe 2023, has been s able o e he en i e pe iod and
used his o de ine pa e ns o he missing obse a ions owa ds he end o he sample.1
Based on he da a elease imings, we i s conside ed h ee empo al blocks, he ea ly,
he middle, and he la e, which we u he disagg ega ed in o sub-blocks acco ding o
1We canno add ess he e ec o he da a e isions on GDP o ecas pe o mance h ough pseudo-da a
in ages gene a ed in his way, lea ing i open o u u e esea ch.
123
SERIEs (2024) 15:145–177 147
hei economic con en . Finally, he unbalancedness pa e ns o he empo al blocks
and economic sub-blocks ha e been eplica ed in all qua e s o he1995–2023 pe iod,
p oducing pseudo- eal- ime da a in ages ha we ha e used o add ess he in o ma ion
in low; see, o ins ance, Giannone e al. (2008) o USA, Kuzin e al. (2011) o he
Eu o a ea, and Cue as e al. (2017) o Spain.
Apa om some speci ic nuances, he essence o he model we ha e used is he
same as hose e e enced o Spain. We ha e cons uc ed GDP o ecas s using ac o s
ha summa ize he o e all s a e o he business cycle. The ac o ex ac ion is based
on p incipal componen s (PC) and DFM; see Ba´nbu a and Modugno (2014). The
ad an age o implemen ing a DFM app oach is ha i enables a la ge panel o a iables
o be modelled in a pa simonious way. E en mo e impo an ly, he DFM and i s s a e-
space ep esen a ion deal wi h he mixed- equency and unbalanced na u e o he da a.
Fo his pu pose, we an he Kalman il e o ill any missing obse a ions h oughou ,
and a he end o he sample, and o p oduce a poin o ecas and condi ional a iance.
We ha e used he la e wi h a no mal assump ion o p o ide a condi ional dis ibu ion
o he u u e GDP gi en he da a a ailabili y, o densi y o ecas .
We ha e pe o med a ecu si e es ima ion scheme inco po a ing he in o ma ion
low and ob ained poin o ecas s, as mos pape s on sho - e m Spanish GDP e olu-
ion ha e. Fu he mo e, we ha e in es iga ed he pe o mance o sho - e m densi y
o ecas s ob ained by mixed- equency DFM, an aspec almos always o e looked by
he li e a u e. Ou indings con i med he exis ence o aluable in o ma ion eleased
du ing he qua e om poin o ecas s pe spec i e. Mo eo e , we obse ed subs an ial
enhancemen s in he accu acy o Spanish GDP sho - e m densi y o ecas s ob ained
h ough mixed- equency DFM as he qua e p og esses, and new in o ma ion is
inco po a ed in o he da a.
I is wo h s essing ha he li e a u e conce ning Spanish GDP sho - e m o ecas -
ing has ocused solely on cons uc ing and e alua ing sho - e m poin o ecas s. The
p esen pape adds he pe spec i eo ane alua ion h ough heGDPsho - e mdensi y
o ecas s ha can be gene a ed h ough DFM and i s s a e-space ep esen a ion. By
doing his, we ha e no sough an analysis o he ela i e me i s o he di e en models
o he phenomenon o Spanish GDP sho - e m o ecas ing. The eal- ime o ecas ing
exe cise and he e alua ion o poin and densi y o ecas s o di e en da a in ages
may as well be employed in he me hodological amewo ks o he e e enced pape s.
We ha e o ganized he emainde o he pape as ollows. The second sec ion
desc ibes he da a. The hi d explains how we a i icially gene a ed he pseudo-da a
in ages and desc ibes he no a ion ela ed o hem. The ou h sec ion p esen s he
mixed- equency DFM, i s s a e-space ep esen a ion, and how he Kalman il e p o-
duces poin and densi y o ecas s unde da a i egula i ies. Then, i discusses he
o ecas design and how we e alua ed poin and densi y o ecas s. The i h sec ion
p esen s and discusses he main indings. We ha e concluded in he six h sec ion.
2 Da a desc ip ion
Table 1shows he basic se o indica o s, consis ing o hi y- wo mon hly a iables
and he qua e ly employmen and GDP g ow h a es. The a ge a iable is he qua -
123
148 SERIEs (2024) 15:145–177
e ly GDP o which we ha e conside ed i s eal (chained olume index) e sion.
The indica o s used as p edic o s o GDP ep esen a b oad spec um o he business
cycle ac i i ies. Acco dingly, we ha e included ha d indica o s ela ed o eal eco-
nomic ac i i y. These ep esen he labou ma ke (such as egis e ed unemploymen
and social secu i y con ibu o s), domes ic economic ac i i y ( he manu ac u e indus-
ial p oduc ion index, elec ic powe consump ion, appa en consump ion o cemen ,
e c.), and he o eign sec o ( eal expo and impo ). In addi ion, we ha e conside ed
he inancial sec o h ough he in e es a e sp ead and he c edi o companies and
households; see Camacho and Doménech (2012) who ound ha inancial indica o s
a e specially ele an in p oducing a GDP o ecas du ing pe iods o ecession. Finally,
we ha e also conside ed some o he so indica o s ha po en ially cap u e agen s’
sen imen and con idence abou he sho - e m u u e o he economy. These indica-
o s co e opinion polls, i.e. su ey-based indica o s such as he pu chasing manage s’
index, economic sen imen and indus y p oduc ion pe spec i e indica o s.
In selec ing indica o s, we ha e been guided by he ollowing c i e ia. Fi s ly,
al hough i was no a equi emen , we a emp ed, as much as possible, o ha e a bal-
anced pa e n o obse a ions a he beginning o he sample pe iod.2Thus, all mon hly
indica o s ha e obse a ions spanning om 1995, excep o he egis e ed unemploy-
men , la ge companies’ sales a iables, whose igu es began in Janua y 1996, he
pu chasing manage s’ index, o e nigh s, u no e index se ice, and u no e index
indus y, whose i s obse a ions we e made in Augus 1998, Janua y 1999, 2000
and 2002, espec i ely. Secondly, as a as possible, we wan ed o ha e a ich da a
amewo k co e ing mos o he indica o s p e iously conside ed in he li e a u e ha
has deal wi h Spanish GDP sho - e m o ecas ing. Consequen ly, we ha e conside ed
indica o s p e iously used in he la ge-scale app oach by Cue as and Quilis (2012),
o he small-scale DFMs o he Spanish economy by Camacho and Qui os (2011),
Cue as e al. (2017) and Pa eja e al. (2020).3
We conside ed he seasonally adjus ed a iables o iden i y a sys ema ic measu e
unde lying he pu e economic luc ua ions o he GDP g ow h a e.4Mo eo e , he
ype o DFM implemen ed in his pape equi es s a iona i y in o de o iden i y and
es ima e ac o s, which ma e ialize h ough app op ia e ans o ma ions ha emo e
secula ends. We ha e ob ained s a iona y luc ua ions o mos indica o s by aking
he i s di e ences o hei le el o loga i hm. The excep ions a e he sp ead a e, and
wo so indica o s ob ained om su eys, he pu chasing manage s’ index and he
indus ial p oduc ion pe spec i e indica o , o which we ha e conside ed hei le els,
2In he o ecas ing exe cise we ha e ca ied ou , he model pa ame e s a e es ima ed ecu si ely, s a ing
wi h a gi en sample pe iod. The pu pose o ha ing a balanced panel is o a oid a iables con ibu ing
signi ican ly di e en obse a ions o he es ima ion o he model pa ame e s.
3Ne e heless, he e a e some excep ions wi h espec o Cue as and Quilis (2012), weha e no conside ed:
(i) he a ailabili y o consume and capi al goods, whose i s obse a ion s a s in Janua y 2005, and
he e o e, we ha e le hem aside o a oid ha ing oo many missing a he beginning o he sample; (ii)
indus ial o de book index and eal g oss wage da a, o which da a is no longe eely accessible. On he
o he hand, we could no ep oduce MICA da a because some o he inancial a iables i included a e no
eely a ailable.
4We collec ed seasonally adjus ed a iables i hey we e al eady a ailable in his o ma . When hey we e
no , we ob ained seasonally adjus ed se ies using TRAMO-SEATS. In Table 1, he a iables wi h codes
ending in a capi al le e D a e al eady adjus ed, while we ha e co ec ed hose wi hou i .
123
SERIEs (2024) 15:145–177 149
as well as he egis e ed unemploymen and he c edi o companies and households,
o which we ha e induced s a iona i y by aking a second di e ence o i s loga i hm.
Finally, he qua e ly employmen and GDP exhibi end beha iou s ha led us o
conside hei i s di e ence o hei log ans o ma ions as an app oxima ion o hei
qua e ly g ow h a e.5
In he p esen pape , we ha e analysed wo da a amewo ks in a eal- ime GDP
o ecas ing exe cise in he ollowing sec ions. We ha e conside ed a da a- ich en i-
onmen , including all a iables in Table 1. This comp ehensi e da ase has enabled
us o analyse he in o ma ion con en o e he qua e o a b oad se o indica o s
encompassing hose p e iously used in he li e a u e cons uc ing Spanish GDP now-
cas s. I is he p ima y da a amewo k o which we ha e conduc ed mos o he
sho - e m o ecas e alua ion analysis. On he o he hand, o compa ison, we ha e
also conside ed a small-scale da a amewo k, ep oducing he indica o s and da a
ans o ma ion o he MIDP ed; see Cue as e al. (2017).6The a iables wi h an ‘X’
in Table 1co espond o hose in he MIDP ed da a amewo k.
3 Pseudo- eal- ime da a in ages
Ideally, we would like o wo k wi h da a in ages cons uc ed wi h eco ds o a iables
a he poin s in ime when hey we e eleased, also including any e isions o p e i-
ously published alues ha may ha e been made. Ne e heless, we canno wo k wi h
pu e eal- ime in ages since we canno access his o ical eco ds o mon hly indica-
o s, employmen and GDP. Ins ead, we ha e elied on pseudo- eal- ime in ages ha
we e gene a ed a i icially o esemble he publica ion pa e n o a iables. The da a
we e collec ed a he beginning o Oc obe 2023, jus a e he i s ad ance o he
hi d qua e o 2023, oge he wi h he publica ion da es o all indica o s. The six h
column in Table 1p o ides he publica ion calenda ob ained in Oc obe 2023, i.e. he
app oxima e numbe o days a e he beginning o a mon h.
The c i ical assump ion behind he pseudo- eal- ime in ages is ha he publica ion
calenda has always been he same o all mon hs o he sample pe iod 1995–2023, o ,
i i has changed, he iming basically emained unal e ed. This assump ion has se ed
as he pu pose o in es iga ing he consequences ha a a he new publica ion schedule
has o GDP sho - e m o ecas ing, assuming ha such a calenda will no unde go
u he al e a ions. We ha e no sough a his o ical assessmen o he in o ma ion
con en o pas da a in ages, bu a he an analysis o wha is implied by a ecen
publica ion schedule, keeping in mind o ecas s ha may be made wi h a simila
schedule in he ime o come. O cou se, using pseudo- eal- ime da a in ages did no
enable us o ackle he e ec o da a e isions on o ecas pe o mance. We will lea e
his aspec o u u e esea ch.
5The choice o da a ans o ma ion applied o each indica o is based on an analysis o i s end beha iou
and augmen ed Dickey–Fulle (ADF) es o he le el (o loga i hms o i s le el) and i s i s di e ences.
Fo mo e de ailed in o ma ion on ADF es esul s and da a ans o ma ions, see Table 2in Appendix B.
6Al hough we ha e ocused on a iables included in he MIDP ed o compa ison, hose in he Spain-
STING da a amewo k o he app oach by Pa eja e al. (2020) may also be conside ed.
123
150 SERIEs (2024) 15:145–177
Table 1 Lis and in o ma ion o mon hly indica o s and GDP: n, a iable name, label, sou ce code, uni o measu emen , publica ion calenda (app oxima e numbe o days
a e he beginning o a mon h), composi ion o empo al blocks (i.e. ea ly, middle, and la e), composi ion o economic sub-blocks, (i.e. PMI, TRAIN, FUEL, e c.), and
publica ion delay, in mon hs, o which he las da a elease e e s
n Va iable Label Code Uni Pub. calenda B oad block Economic sub-block Pub. delay
1 Pu chasing Manage s Index,
Se ice
xPMI S 688124 Poin s +1 days EARLY PMI 1 mon h
2 En y o Tou is s (F on u ) TOUR 247100 Thousands o people +2 days EARLY TOUR 2 mon hs
3 T ain T a ic. Goods. To al. TRAIN 242200 Million Passenge s/Km +2 days EARLY TRAIN 2 mon hs
4 Gasoline Consump ion Au o GAS 256100 Thousands Tm +2 days EARLY FUEL 2 mon hs
5 Diesel Consump ion DIESEL 256200 Thousands Tm +2 days EARLY FUEL 2 mon hs
6 Elec ic Powe Consump ion
(mainland)
xENERGY 255100 Millions KWh +2 days EARLY ENERGY 1 mon h
7 Regis e ed unemploymen .
To al.
UNEM 170000N.D Thousands +2 days EARLY LABOUR 1 mon h
8 Regis e ed Con ac s. CONTRACTS 173100 Numbe +2 days EARLY LABOUR 1 mon h
9 Social Secu i y A ilia es xSSAFI 190000 Thousands +2 days EARLY LABOUR 1 mon h
10 Indus ial P oduc ion Index. xIPI 229000CC.D Index (2015=100) +5 days EARLY IPI 2 mon hs
11 Indus ial P oduc ion Index.
Manu ac u e
IPIM 229200CC.D Index (2015=100) +5 days EARLY IPI 2 mon hs
12 C edi o Companies and
Households
xCREDIT 807104 Millions e+ 5 days EARLY FIN 2 mon hs
13 Sp ead: Di . be ween in e es
a e 1 yea and 3 mon hs (*)
SPREAD D_DTES00B7
-D_DTES00U7
% + 5 days EARLY FIN 1 mon h
14 Sea T a ic. Goods. To al SEA 244200 Thousands Tm +6 days EARLY SEA 2 mon hs
15 Appa en Cemen Consump ion xCEMENT 236000 Thousands Tm + 14 days MIDDLE CEMENT 1 mon h
16 Ai T a ic. Passenge s. To al.. AEREO 245100.D Thousands + 14 days MIDDLE AEREO 1 mon h
17 Ca Regis a ion CAR 271200 Numbe + 15 days MIDDLE VEHICLE REG 1 mon h
123
SERIEs (2024) 15:145–177 151
Table 1 con inued
n Va iable Label Code Uni Pub. calenda B oad block Economic sub-block Pub. delay
18 T uck Regis a ions TRUCK 282100 Numbe + 15 days MIDDLE VEHICLE REG 1 mon h
19 La ge Companies Sales xSALES 25D100 Index (2013=100) + 15 days MIDDLE SALES 2 mon hs
20 La ge Companies Sales.
Compensa ion o Employees
xSALES C 25M500 Index (2013=100) + 15 days MIDDLE SALES 2 mon hs
21 Cons uc ion Index. To al IND CONST 235500 Index (2015=100) +20 days MIDDLE CONSTRUCT 2 mon h
22 Tu no e Index Indus y TURNOVER I 223100.D Index (2015=100) +21 days MIDDLE TURNOVER 2 mon hs
23 Tu no e Index Se ice TURNOVER S 249000.D Index (2015=100) +21 days MIDDLE TURNOVER 2 mon hs
24 Expo s (cons an p ices) EXPORTS 502R00 Thousands e+ 22 days MIDDLE FOREIGN 2 mon hs
25 Impo s (cons an p ices) x IMPORTS 506R00 Thousands e+ 22 days MIDDLE FOREIGN 2 mon hs
26 Numbe o O e nigh S ays.
Residen s Ab oad.
OVERNIGHTS 241421 Thousands People + 23 days MIDDLE OVERNIGHTS 1 mon h
27 Employed Labou Fo ce Su ey xEMPLOYMENT 120000
/1h20000
Thousands People + 26 days LATE EMPLOYMENT 3 mon hs
28 G oss Domes ic P oduc . Index
o chained olumes.
xGDP 940000D Index (2015=100) + 29 days LATE GDP 3 mon hs
29 Economic Sen imen Indica o S ESI EI_BSSI_M_R2 Ne % + 29 days LATE SURVEY 0 mon hs
30 Indus y P oduc ion Pe spec i e S INDUPP EI_BSIN_M_R Ne % + 29 days LATE SURVEY 0 mon hs
31 Cons uc ion o Residen ial
Building Pe mi s (Num.)
BUILDING 230111 Thousands e+ 30 days LATE BUILDING 2 mon hs
32 Real es a e mo gages. U ban
eal es a e (Value.)
MORTGAGE 258020 Thousands e+ 30 days LATE BUILDING 2 mon hs
33 Re ail ade index la ge business RETAIL-LS 272801.D Index (2015=100) + 30 days LATE RETAIL 1 mon h
34 Re ail T ade Index (wi hou gas
s a ion)
REATIL 272701G.D Index (2015=100) + 30 days LATE RETAIL 1 mon h
Xma ks he indica o s conside ed by he MIDP ed model. (*) Mon hly in e es a e (Spanish T easu y bonds, sec. ma ke ) is cons uc ed as a e ages o daily obse a ions. We p o ide he da a
sou ces in Appendix A.
123
152 SERIEs (2024) 15:145–177
Using he assumed publica ion calenda , we g ouped he a iables acco ding o he
imeo hemon hinwhich hei new igu eswe epublished,whichledus odis inguish
h ee empo al blocks o publica ion ha included indica o s whose alues a e pub-
lishedea ly(E) in hemon h,i.e. app oxima elyin he i s hi do i ,in he middle(M),
i.e. in he second hi d, and la e (L) in he mon h, i.e. in he las hi d. The se en h col-
umn in Table 1shows he a iables ha ha e made up each empo al block. Finally, he
las column o Table 1shows he delay in mon hs, o which he las da a elease e e s.
We obse ed ha he indica o s’ published alues e e o di e en pe iods. In blocks
‘E’ and ‘M’, da a eleases e e ei he o 1-lag o 2-lag mon hs o egis e ed unem-
ploymen and indus ial p oduc ion index, o example. On he o he hand, in block ‘L’,
he alues eleased also e e o he cu en mon h, as is he case o so su ey-based
economic sen imen and indus ial p oduc ion pe spec i e indica o s. As a esul , each
block is cha ac e ized by a ce ain pa e n o a ailable and missing obse a ions.
The publica ion delay o he empo al blocks a e eplica ed in he i s mon h
(0q/1m), second (0q/2m)and hi d (0q/3m)o he cu en qua e , gene a ing a
ce ain pa e n o a ailable and missing obse a ions. In addi ion, we conside ed he
las mon h o he p eceding qua e (−1q/3m), which illus a es he e olu ion o he
in o ma ion be o e he cu en qua e began, i.e. i gi es a ime pe spec i e, and helps
o seize he alue o he in o ma ion eleased in subsequen mon hs, and he blocks
‘E’ and ‘M’ o he ollowing qua e ’s i s mon h (+1q/1m) ha illus a e he da a
a ailabili y jus be o e he elease o he GDP i s ad ance.
No e ha each mon h has h ee empo al blocks o da a eleases, wi h he excep ion
o he las mon h, which has only wo. As a esul , we had υ=1,2, ..., 14 in ages,
i.e. h ee o he i s ou mon hs, and wo o he las mon h.
The main issue o he empo al blocks is ha hey end up including a iables wi h
e y di e se de ini ions. To o e come his issue, we ha e u he di ided hem acco d-
ing o he economic con en and empo a ily o de ed hem, espec ing hei publica ion
da es as much as possible.7The e o e, we spli he in age ‘E’ in o six sub-blocks: pu -
chasing manage s’ index (PMI), consump ion o elec ic powe (ENERGY), labou
ma ke (LABOUR), en y o ou is (TOUR), ain a ic (TRAIN), consump ion
o gasoline and diesel (FUEL), sp ead and c edi (FIN), indus ial p oduc ion (IPI),
and sea a ic (SEA); ‘M’ in cemen consump ion (CEMENT), ai a ic (AEREO),
ca and uck egis a ion (VEHICLE R), la ge companies sales and compensa ion
(SALES), cons uc ion p oduc ion (CONSTRUCT), u no e se ice and indus y
indexes (TURNOVER),and he o eignsec o (FOREIGN); inally, ‘L’includes nigh s
s ays (OVERNIGHTS), employmen , GDP, he su ey-based indica o as economic
sen imen and indus ial pe spec i e (SURVEY), ac i i y ela ed o he e ail ade
(RETAIL), and building pe mi s and mo gage (BUILDING). In any case, he esul s
we e obus o changes in he o de chosen o he sub-blocks. The eigh column in
Table 1shows he esul ing economic sub-blocks wi hin each empo al block.
G ouping he indica o s acco ding o hei economic con en enabled us o in es-
iga e wi hin he empo al blocks. We used economic sub-blocks in mon hs wi hin
qua e s, ollowing he same easoning explained abo e o he empo al blocks. The e
7In he case o any ambigui y due o simila publica ion da es, we o de ed he economic sub-blocks aking
in o accoun he pe iod o he eleased alue, pu ing i s hose da a eleases ha showed he la ges ime
delay.
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SERIEs (2024) 15:145–177 159
ALS =1
W
W
w=1
log ˆ
φυ˜
Y(w)
2,T∗+h∗,(12)
whe e ˆ
φυ(·)deno es he no mal densi y o ecas o Y(w)
2,T∗+h∗. Indeed, i is la ge
when he densi y o ecas s assign a high p obabili y o he obse ed GDP g ow h a es
and, consequen ly, can be used o ank hem; see Bao e al. (2007). No e ha in Eqs.
(11) and 12, he poin and densi y o ecas depend on he es ima ed model and he
in o ma ion se . As a esul , MSFE and ALS e lec he unce ain y associa ed wi h
pa ame e es ima ion and model ins abili y, and he in o ma ion low.
Fu he mo e, we ha e p o ided s a is ical es s o assess he p edic i e con en
o new da a eleases. The c ucial poin is whe he he in o ma ion in low imp o es
he o ecas pe o mance. Rele an new in o ma ion, when included in an expanding
in o ma ion se , should lead o a o ecas wi h lowe MSPE. On he con a y, in o ma-
ion lacking a p edic i e alue should al e nei he he poin o ecas no he esul ing
MSPE. We implemen ed a Mince –Za nowi z ype o eg ession o es he p edic i e
con en o new in o ma ion, as ollows:
e(w)
T∗+h∗|υ=β0+β1ˆ
˜
Y(w)
2,T∗+h∗|υ+1+uT∗+h∗,w=1, ..., W,(13)
whe e ˆ
˜
Y(w)
2T∗+h∗|υ+1=ˆ
˜
Y(w)
2T∗+h∗|υ+1−ˆ
˜
Y(w)
2,T∗+h∗|υ, is he o ecas e ision; see
Mince and Za nowi z (1969), and Ellio and Timme mann (2016), pp. 355–358. The
no ele an new in o ma ion hypo hesis can be exp essed in e ms o he slope coe -
icien β1, which mus be ze o. Rejec ing he null hypo hesis means ha he o ecas
e o ob ained using he in o ma ion in υis co ela ed wi h he new in o ma ion
in υ+1 ha is comp ised in he o ecas e ision. The e o e, we would conclude
ha he in low o in o ma ion is ele an o he GDP o ecas , and including i in
he in o ma ion se can lead o a MSPE educ ion. We implemen ed a simila a gu-
men o es whe he new in o ma ion signi ican ly changes ALS. The eg ession used
is o he same ype as (13) bu i has as dependen a iable he scaled o ecas e o
e(w)
T∗+h∗|υ/V(˜
Y2,T∗+h∗|υ). Once mo e, ejec ing he null o a slope coe icien equal
o ze o implies ha he new in o ma ion imp o es he expec ed log-sco ed. 13
Finally, we assessed he densi y o ecas in e ms o i scalib a ion.Awell-calib a ed
densi y o ecas easonably es ima es he unknown condi ional dis ibu ion o he
u u e a iable (Gnei ing e al. (2007) and Mi chell and Wallis (2011)). We used
he p obabili y in eg al ans o m numbe s (PITs), he in e se o ecas cumula i e
dis ibu ion e alua ed a he ealized obse a ions, and he in e se PITs (INTs). I
he densi y o ecas shows con o mi y wi h he ue and unknown one, i.e. i is well-
calib a ed, he PITs a e i.i.d. uni o m o e he in e al (0,1) and, hus, he INTs a e
i.i.d. s anda d no mal; see Diebold e al. (1998) and Be kowi z (2001). We conside ed
he likelihood a io es de ised by Be kowi z (2001), which is based on he INTs and
exploi s hei h ee cha ac e is ics unde he null hypo hesis o co ec calib a ion: ze o
13 See Appendix D o a o mal a gumen o why ejec ing he null hypo hesis o β1=0 implies a MSFE
(ALS) imp o emen .
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160 SERIEs (2024) 15:145–177
mean, uni a iance, and absence o se ial co ela ion. Be kowi z’s es s a is ics is
B=−2L(0,1,0)−L(ˆc,ˆσ2,ˆρ)∼χ2
3,
whe e L(c,σ2,ρ) is he exac log-likelihood when we assume ha INT
=c+
ρINT
−1+e ,e
i.i.d.
∼N(0,σ2), and ha s deno e es ima ed pa ame e s. Acco d-
ingly, L(0,1,0)is he log-likelihood unc ion ob ained unde he assump ion ha
INTs a e i.i.d. no mal ze o mean and uni a iance; see Be kowi z (2001). An inspec-
ion o he cons an , a iance and au o eg essi e coe icien es ima es can e eal he
easons behind he quali y o calib a ion, i.e. causes ha lead away om he null
hypo hesis o con o mi y be ween he es ima ed densi y o ecas and he ue and
unknown one.
The asymp o ic chi-squa ed dis ibu ion o Be kowi z’s s a is ics is ob ained assum-
ing ha he sample size is la ge enough so ha he es ima ion e o s o he cons an ,
a iance, and au o eg essi e coe icien anish. Ne e heless, in ou case, he sample
size (i.e. numbe o he INTs) was small, so his assump ion was di icul o main-
ain. Also, es ima ion e o s unde lies he INTs as hey a e based on a MFDFM wi h
pa ame e s ixed a some es ima es. Thus, pa ame e unce ain y makes he asymp-
o ic dis ibu ion o he Be kowi z’s s a is ic e y conse a i e. Fo his eason, we
also cons uc ed i s boo s ap dis ibu ion o ackle pa ame e and es ima ion e o s
and ob ained i s c i ical alues o in e es ; see Hall and Wilson (1991) and K eiss and
F anke (1992).
5 Empi ical esul s
To implemen he MFDFM, we i s need o se he numbe o common ac o s ( ) and
he lag o de o hei dynamics (p). He e, we ha e ollowed he e e enced li e a u e
p oducing Spanish GDP sho - e m o ecas s and chosen =1 and p=2 o p esen
he esul s o he eal- ime o ecas ing exe cise; see Camacho and Qui os (2011),
Cue as and Quilis (2012), Cue as e al. (2017), and Pa eja e al. (2020).14
To cha ac e ize he ac o , i.e. o gi e i an economic in e p e a ion, we conside ed
as an illus a ion he es ima ion esul s when we used he comple e da a se co e ing
he pe iod 1995Q2–2023Q2. The i s ac o explained 57% o he o al a iabili y and
desc ibed he business cycle, as sugges ed by Fig. 1, wi h la ge posi i e weigh s o
social secu i y a ilia es (SSAFI), indus ial p oduc ion pe spec i e (S INDUPP), and
pu chasingmanage s’index(PMI),andla genega i eweigh o hesp ead a e(INT),
and medium-size posi i e weigh s o la ge companies’ sales (SALES), u no e index
14 An anonymous e e ee has commen ed on ha he chance o some indica o s ha e unde gone big
s uc u al changes is high, e en mo e so conside ing he pe iod s a ing in 2020. Unde big s uc u al b eaks,
inc easing he numbe o ac o s may be necessa y o cons uc mo e accu a e o ecas s; see B ei ung and
Eickmeie (2011) and Chen e al. (2014), who poin ou ha big s uc u al b eaks in he loading o he
indica o s may inc ease he dimension o he ac o space. Al hough mo e ac o s esul in sligh ly di e en
quan i a i e esul s, hey yield he same conclusions as when a DFM wi h one ac o and a lag o de o wo
is used. The e o e, o he sake o compa ison and simplici y, we ha e se =1andp=2. The esul s o
al e na i e alues o and pa e a ailable upon eques .
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SERIEs (2024) 15:145–177 161
Fig. 1 Es ima ed loadings ob ained using a MFDFM wi h =1andp=2 and no modelling o he
idiosync a ic e o s applied o he la ge da ase . The es ima ion pe iod is 1995Q2–2023Q2
Fig. 2 Qua e ly Spanish GDP g ow h a e and es ima ed common componen ob ained using a ob ained
using a MFDFM wi h =1andp=2 and no modelling o he idiosync a ic e o s applied o he la ge
da ase . The es ima ion pe iod 1995Q2–2023Q2
se ice (TURNOVER S), indus ial p oduc ion indexes (IPI and IPIM), e ail ade
index (RETAIL), and consump ion o cemen (CEMENT). Fu he mo e, Fig. 2shows
he es ima ed e-cen ed and scaled common componen
¯
˜
Y2+σ˜
Y2׈
Y2×(ˆ
F +2ˆ
F −1+3ˆ
F −2+2ˆ
F −3+ˆ
F −4)
acks qui e well he GDP g ow h a e and i is e ec i e in cap u ing in ad ance
he mo emen o he la e . Bo h a e highly co ela ed con empo aneously, wi h an
es ima ed co ela ion coe icien o 0.95. They a e also dynamically co ela ed, wi h
es ima ed c oss-co ela ion coe icien s be ween GDP and he common componen ’s
one- and wo-qua e lags o 0.92 and 0.86, espec i ely.
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162 SERIEs (2024) 15:145–177
5.1 Poin and densi y o ecas pe o mance
Figu e 3plo s he oo MSPE (RMSPE) o e he empo al blocks and hei economic
disagg ega ion ob ained o he la ge da a amewo k wi h he MFDFM wi hou mod-
elling he dynamics o he idiosync a ic e o s. Fo compa ison, we ha e also included
he poin accu acy when we es ima ed and o ecas ed he GDP g ow h a e using MID-
P ed app oach. Fu he mo e, we also epo ed he AR(1) o ecas as a benchma k. The
i s panel o Fig. 3shows ha publica ion eleases h oughou he qua e a e essen ial
as hey educe he RMSPE. Howe e , he a es a which RMSPE dec eases a e no con-
s an o all eleases. We obse e ha he s eepes declines in poin o ecas accu acy
a ained by he MFDFM occu a in ages −1q/3m, and 0q/1m, hen dampen be o e
disappea ing owa ds mid +0q/3m. The i s panel o Fig. 3also sugges s ha when
he pu pose is o p oduce Spanish GDP sho - e m es ima es, he ich da a en i onmen
and he MIDP ed app oaches p o ide somewha simila pe o mance o all da a in-
ages. Ne e heless, hese app oaches sys ema ically ou pe o m he AR(1), a aining
be e accu acy han he benchma k o all publica ion eleases. In e es ing o no e
is ha MDDFM a e compa a i ely mo e a ac i e han he AR(1) o da a in ages
be o e he elease o he i s ad ance o GDP han o he subsequen eleases.
The i s panelo Fig.3also epo s heobse ed es s a is ics o henullhypo hesis
o no in o ma ional gain be ween consecu i e in ages. We ha e epo ed hese es
s a is ics o he la ge da a ame app oach. In he i s panel in Fig. 3, we obse ed ha
hose in ages wi h s eep d ops in MSPE a e associa ed wi h posi i e and signi ican
obse eds a is ics,meaninga ejec ion o henullo noin o ma ionalgain.Incon as ,
con ainedd opsinMSPEbe weenconsecu i e in agesshows a is ics ha a esmall in
magni ude and non-signi ican , i.e. wi h no s ong e idence agains he null. The e o e,
we co obo a ed ha e e ing o he p onounced educ ions in MSPE and he p esence
o signi ican in o ma ional gains a e equi alen .
We can analyse which ype o se ies con ibu es he mos o he imp o emen o
o ecas pe o mance using he second panel in Fig. 3, which epo s he RMSPE, and
he obse ed es s a is ic o he economic sub-blocks. The i s aspec o highligh
is ha he o e all beha iou o he poin o ecas accu acy. Mo eo e , we obse ed
ha MFDFM app oaches a e be e han AR(1) o all in ages, al hough much mo e
a ac i e be o e he elease o he la es GDP da a in he cu en qua e ’s i s mon h.
Rema kably, he closes app oach be ween he pe o mances o he MFDFM me hods
and he AR(1) models is p oduced when he ad anced o he p e ious qua e ’s GDP
is eleased.
The labou ma ke (LABOUR) da a has a p onounced impac on MSPE. The
LABOUR sub-block includes he egis e ed unemploymen , social secu i y a ilia es,
and egis e ed con ac s, indica o s ha a e published as soon as he mon h begins,
wi h he la es alues e e ing mainly o he p e ious mon h. Also impo an in e ms
o RMSPE educ ion is SURVEY, composed o su ey-based indica o s, which a e
eleased a he end o he mon h and a e he mos imely since hey a e he i s o ha e
in o ma ion abou i . The pu chasing manage s’ index se ice (PMI S) also p o ides
p onounced d ops in RMSPE. Beyond LABOUR, SURVEY, and PMI S, we obse ed
a subs an ial educ ion in he RMSPE when he la es GDP da a is eleased owa ds he
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SERIEs (2024) 15:145–177 163
end o he qua e ’s i s mon h. Ano he sub-g oup ha p o ides addi ional poin o e-
cas accu acy, al hough less p onounced, is IPI, which gi es posi i e and s a is ically
signi ican MSPE educ ions in he i s wo mon hs.
I is wo h no ing ha he exis ing li e a u e on Spanish GDP sho - e m o ecas s
has consis en ly obse ed imp o emen s in he accu acy o poin o ecas s as new da a
ela ed o GDP and i s p edic o s becomes a ailable h oughou he qua e . Fu he -
mo e, ou indings ha e emphasized he signi icance o LABOUR, SURVEY, IPI, and
PMI indica o s, which ha e al eady been ecognized as ele an o p oducing Spanish
GDP sho - e m o ecas s, while also p omp ing ques ions abou he con ibu ions o
e ail sales, o e nigh s ays, ene gy consump ion, and la ge companies’ sales.
We ha e u he e alua ed he p edic i e con en o he in o ma ion low o GDP
sho - e m des iny o ecas s. The panels in Fig. 4 epo he ALS o he Spanish GDP
g ow h a e densi y o ecas s o he MFDFM app oaches and he AR(1) model o e
he empo al blocksand hei economicdisagg ega ion.Theyalsoinclude he obse ed
es s a is ics o henullhypo hesiso no ele an in o ma ion o hedensi y o ecas s,
ob ained o la ge da a ame. Once mo e, a ejec ion o he null hypo hesis poin s o
a po en ial imp o emen in he densi y o ecas ha we could achie e by ecompu ing
he poin o ecas and, consequen ly, he densi y, leading o an imp o emen in log
sco e.
The i s panel in Fig. 4shows ha he ALS inc eases as he new da a becomes
a ailable, i.e. Spanish GDP sho - e m densi y o ecas s imp o e wi h he in o ma ion
in low. Mo e speci ically, he ALS a ies only sligh ly in −1q/3m, hen changes mos
sha ply du ing he 0q/1mand 0q/2m, s abilizing i s g ow h om 0q/3monwa d.
Rema kably, mos o he signi ican empo al blocks in he i s panel in Fig. 4a e also
ele an in he i s one in Fig. 3when we ocus on poin o ecas accu acy. The second
panel o Fig. 4shows posi i e changes in densi y o ecas accu acy du ing 0q/1m ha
mode a e a e i , con i ming he esul s o he i s panel. The mos ele an economic
in ages in e ms o log sco e imp o emen a e he LABOUR, SURVEY, GDP, and
IPI, which some imes p oduce posi i e mode a e, and signi ican changes in he log
sco e, in ha o de .
Fu he mo e, Fig. 4sugges s ha he la ge da ase and he MIDP ed app oaches
p o ide simila ALS, pe haps sligh ly be e o he o me han he la e om mid
0q/1monwa ds. The compa ison o he ALS o MFDFM me hods and he AR(1)
model also e ealed, on one side, ha he o me p o ides, on a e age, much be e
densi y o ecas s and, on he o he , ha he di e ence be ween he wo is mo e p o-
nounced un il he elease o he i s GDP ad ance o he p e ious qua e . Mo eo e ,
his poin in he cu en qua e p oduces he closes app oxima ion be ween he ALS
o bo h models.
An explana ion is ha be o e he elease o he la es GDP da a, he mo e imely
mon hly indica o s signal he e olu ion o he business cycle, eplacing he mos ecen
and una ailable GDP in o ma ion. Acco dingly, mon hly p edic o s p oduce subs an-
ial ela i e gains wi h espec o an AR model in which only GDP in o ma ion is
exploi ed. The la es GDP igu e b ings in o ma ion on business cycle condi ions ha
we e summa ized, un il ha ime in he mon hly indica o s and p oduces an app oxi-
ma ion o he pe o mances o he MFDFM and he AR model.
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164 SERIEs (2024) 15:145–177
Fig. 3 RMSPE o e empo al blocks ( i s ) and economic sub-g oups (second) be ween −1q/3mand
+1q/1m o qua e ly GDP g ow h a e ob ained using a: (i) MFDFM wi h =1andp=2 and no
modellingo heidiosync a ic e o s applied o hela geda ase (con inuous-do edblue line), (ii) a MFDFM
wi h =1andp=2 and AR(1) idiosync a ic e o s applied o he small-da a se , i.e. MIDP ed model
(con inuous blue line), and, (iii) uni a ia e AR(1) model (dashed blue line). The e ical ba s show he
obse ed es s a is ics ob ained usinga MFDFM wi h =1andp=2 andno modelling o he idiosync a ic
e o s applied o he la ge da ase o he null hypo hesis o no in o ma ion gain be ween he da a in age
on he x-axis and he one immedia ely p eceding i , and he e ical a ea ou side he dash ho izon al lines
ep esen a 5% Rejec ion Region (colo igu e online)
123
SERIEs (2024) 15:145–177 165
Fig. 4 ALS o e empo al blocks ( i s ) and economic sub-g oups (second) be ween −1q/3mand +1q/1m
o qua e ly GDP g ow h a e ob ained using a: (i) MFDFM wi h =1andp=2 and no modelling o
he idiosync a ic e o s applied o he la ge da ase (con inuous-do ed blue line), (ii) a MFDFM wi h =1
and p=2 and AR(1) idiosync a ic e o s applied o he small-da a se , i.e. MIDP ed model (con inuous
blue line), and (iii) uni a ia e AR(1) model (dashed blue line). The e ical ba s show he obse ed es
s a is ics ob ained using a MFDFM wi h =1andp=2 and no modelling o he idiosync a ic e o s
applied o he la ge da ase o he null hypo hesis o no in o ma ion gain be ween he da a in age on he
x-axis and he one immedia ely p eceding i , and he e ical a ea ou side he dash ho izon al lines ep esen
a 5% Rejec ion Region (colo igu e online)
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166 SERIEs (2024) 15:145–177
In summa y, he e a e poin o ecas pe o mance measu es e ealed he impo ance
o he in o ma ion low Spanish GDP g ow h a e nowcas s. Likewise, wi h some
nuances, hey poin ou ha he mos p onounced imp o emen a e a ained owa ds
he end o he cu en qua e ’s i s mon h. This momen in he qua e eme ges
as he pe iod o sepa a ion be ween da a in ages ha p oduces, i s ly, he g ea es
imp o emen in o ecas pe o mances and, secondly, he la ges di e ence be ween
he MFDFM app oaches and he AR(1) model. An explana ion is ha be o e he
elease o he la es GDP da a, he imely mon hly indica o s signal he e olu ion
o he business cycle, p o iding a summa y o he una ailable p e ious qua e ’s
GDP in o ma ion. Acco dingly, ele an mon hly p edic o s p oduce a subs an ial
imp o emen in o ecas accu acymeasu esandgene a e he mos e iden ela i egain
wi h espec o an AR model in which only he GDP in o ma ion has been exploi ed.
Finally, hose indica o s wi h a high in o ma ion con en a e hose in he LABOUR,
SURVEY, PMI S, and IPI.
5.2 Densi y o ecas calib a ion
We now u n o he calib a ion o densi y o ecas s. Figu e 5shows Be kowi z’s
obse ed es s a is ics ob ained o densi y o ecas s o e economic da a in ages
and i s 5% asymp o ic and boo s ap c i ical alues. An obse ed es s a is ic la ge
han he c i ical alue a a gi en signi icance le el implies ha we canno ejec he
null hypo hesis o good calib a ion o he densi y o ecas . Bu beyond his igid in e -
p e a ion, we implemen he c i ical alues as a e e ence o globally di e en ia e da a
in ages ha gene a e an inadequa e densi y o ecas calib a ion om hose in which
he e is no e idence agains a good one. In Fig. 5, we obse e less and less e idence
agains good calib a ion as mo e and mo e da a becomes a ailable in he i s h ee
mon hs. In o he wo ds, he GDP sho - e m densi y o ecas s s eadily imp o e and
app oach he egion o no e idence agains good calib a ion wi h he low o in o ma-
ion. I is also ema kable ha he s eepes changes in he obse ed Boccu h oughou
0q/1m, gene a ing an appa en clus e ing be ween he in ages o which we clea ly
ejec he null o good calib a ion hypo hesis and hose o which he e is no s ong
e idence agains i .
Be kowi z’s es signals a iola ion o he hypo hesis o good calib a ion, bu i does
no p o ide easons o i . The la e need o be ound in he cha ac e is ics ha INTs
should ha e and may lack. The eby, we included he second panel in Fig. 5which plo s
he es ima ed alues ˆc,ˆσ2, and ˆρunde lying he es . Fi s ly, he es ima ed cons an is
always nega i e, showing li le app oxima ion owa ds i s null alue o ze o wi h da a
eleases. A nega i e cons an indica es a densi y o ecas gi ing high p obabili ies
o la ge alues o he u u e ou come, i.e. i has a la ge mean, signalling a posi i e
bias o poin o ecas s. Secondly, he es ima ed a iance dec eases wi h da a eleases,
especially since he beginning o he cu en qua e , indica ing ha he eason o he
close p oximi y be ween he densi y o ecas s and he unknown ones a chi ed wi h
da a eleases is a be e es ima ion o he condi ional a iance. Fu he mo e, he ac
ha he es ima ed au o-co ela ion o INTs diminishes o e da a in ages indica es
123
SERIEs (2024) 15:145–177 167
Fig. 5 Fi s , Be kowi z’s obse ed es s a is ics o he null o good densi y o ecas calib a ion o e da a
in ages be ween −1q/3mand +1q/1mob ained using a MFDFM wi h =1andp=2 and no modelling
o he idiosync a ic e o s applied o he la ge da ase applied. The e ical a eas abo e he ho izon al
con inuous lines and he dash line ep esen he 5% asymp o ic and boo s ap ejec ion egions. Second,
es ima es o cons an (ˆc), a iance ˆσ2and au o eg essi e coe icien (ˆρ) o each economic sub-block
ha he mixed- equency DFM used o cons uc GDP o ecas s be e cap u es he
GDP dynamics, as mo e in o ma ion becomes a ailable. 15
15 I is ema kable ha obse ed Be kowi z es s a is ics a e a he la ge o da a in ages be ween −1q/3m
and la e 0q/1m, jus be o e he elease o las qua e ’s GDP alue. A eason is ha be o e ha poin in
ime, densi y o ecas s a e wo-pe iod ahead conce ning he las obse a ion a ailable o GDP. Fo mo e
han one-pe iod ahead o ecas s, he PITs and INTs may exhibi se ial co ela ion, and he la ge obse ed
123
168 SERIEs (2024) 15:145–177
O e all, he es ima ed alues o ˆc,ˆσ2and ˆρindica e ha he less e idence agains
he null o con o mi y be ween he densi y o ecas and he unknown achie ed wi h
da a eleases is mainly due o a educ ion in he dispe sion and he au o-co ela ion o
he INTs. In o he wo ds, calib a ion imp o es wi h da a eleases because he densi y
o ecas s be e cap u e he condi ional a iance o he unknown ones.
As an illus a ion, Figs. 6and 7show he GDP g ow h a es densi y o ecas s and
i s ealized alue o se e al qua e s. The i s no able imp ession om hese igu es
is ha GDP sho - e m densi y o ecas s e ol e wi h he da a eleases h oughou he
qua e , as he in o ma ion low a ec he condi ional mean and a iance es ima es
unde lying he condi ional no mal densi ies. The cen e o he la e , i.e. he poin
o ecas , mo es as i ying o ca ch he ealized GDP g ow h a e, and he dispe sion
na ows wi h new da a eleases. Fo example, Fig. 6displays he si ua ion du ing he
g ea economic ecession o 2008/2009. The ou h qua e o 2008 and he i s qua e
o 2009 show a con inuous adjus men owa ds he le in sho - e m densi y o ecas s
as he in o ma ion low p oceeds, bu no enough because he ealized GDP g ow h
a es a e well loca ed on he le ail o he densi ies. In he second and hi d qua e s
o 2009, he si ua ion e e ses, and he low o in o ma ion pushes densi ies om he
le o GDP. Mo eo e , he educ ion in dispe sion wi h da a eleases is appa en .
Changes in GDP densi y o ecas s wi h he in o ma ion low a e also e iden in
panels o Fig. 7, which desc ibe he qua e s in 2019, al hough compa a i ely less
p onounced han hose seen in Fig. 6. A sha p con as in hese igu es shows us ha
he esponses o he condi ional means and a iances a e much mo e p onounced in
pe iods o economic u bulence han in pe iods o ela i e calm when he eleases o
he la es da a e eal mo e ab up changes in he e olu ion o he economic indica o s.
Finally, i is no ewo hy ha ealized GDP g ow h a es a e gene ally below he
cen e o i s sho - e m densi y o ecas s in all bu he second and hi d qua e s o
2009. Hence, he p obabili y o obse ing a alue less han he obse ed GDP o mos
densi ies is smalle han 0.5, i.e. PITs a e smalle han 0.5, implying nega i e INTs.
This exempli ies wha happens on a e age o da a in ages o all qua e s and is he
eason behind he nega i e es ima es o cons an s in models o INTs.
6 Conclusion
We ha e in es iga ed he p edic i e con en o da a eleases o sho - e m Span-
ish GDP o ecas ing. We used in o ma ion om a wide g oup o mon hly a iables
ha we e hen implemen ed as p edic o s o he Spanish GDP g ow h a e wi hin a
MFDFM. We based ou analysis on a ecu si e es ima ion scheme, using pseudo eal-
ime da a in ages ha eplica ed he publica ion calenda o he mon hly indica o and
he GDP. Then, we e alua ed om he mo e common pe spec i e o poin o ecas s
and he mo e no el one o densi y o ecas s.
As ime passes and mon hly indica o s and GDP a e eleased he e a e no able
imp o emen s in sho - e m GDP o ecas s. Howe e , a signi ican po ion o he
Bs a is ics be o e he elease o he p e ious qua e ’s GDP alue can be o igina ed in he ac ha we ha e
eques ed absence o au o-co ela ion.
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SERIEs (2024) 15:145–177 175
:EF(YT∗+h∗(ωυ+1), YT∗+h∗)−EF(YT∗+h∗(ωυ), YT∗+h∗)=0.(15)
Taking expec a ions on bo h sides o equa ion (14), we ob ain ha (15) implies ha
EF(YT∗+h∗(ωυ), YT∗+h∗)(
YT∗+h∗(ωυ+1)−YT∗+h∗(ωυ))=0.(16)
The las can be iewed as an o hogonali y condi ion. Unde he hypo hesis o no
change in he expec ed loss, F(·)(i.e. in wo ds o G ange , “ he gene alized o ecas
e o ”) is unco ela ed o he change in poin o ecas s.
The o hogonali y condi ion (16) pu o wa d Mince -Za nowi z ype o eg ession
o he ollowing o m:
FYT∗+h∗|υ,YT∗+h∗=β0+β1YT∗+h∗| +1−YT∗+h∗|υ+uT∗+h∗,
and es ing whe he he slope coe icien β1is equal o ze o. The ejec ion o null
hypo hesis means ha he addi ional ele an in o ma ion in υ+1imp o es he poin
o ecas and, consequen ly, helps achie e a lowe expec ed loss.
Anin ui i eexplana ionisas ollows.The o ecas e o eT∗+h∗|υis heunexpec ed
componen YT∗+h∗wi h espec o he in o ma ion se υ,comp ised in he poin
o ecas . On he o he hand, YT∗+h∗|υ+1is he pa o he new o ecas o hogonal
o he p e ious one, hus co esponding o he change in poin o ecas s ha can be
a ibu ed o he new in o ma ion in +1. I he o ecas e o eT∗+h∗|υ ela es o
YT∗+h∗|υ+1, hen i means ha υ+1con ains new ele an in o ma ion ha can
lead o a MSPE imp o emen .
The inal o m o he Mince -Za nowi z eg ession, i.e. i s dependen a iable,
depends on he loss unc ion. Conside i s he quad a ic loss unc ion
Q(YT∗+h∗|υ+1,YT∗+h∗|j+1)=(YT∗+h∗−YT∗+h∗|j)2,
in which case we ha e ha
Q(YT∗+h∗|υ+1,YT∗+h∗|υ+1)=−2(YT∗+h∗−YT∗+h∗|υ).
On he o he hand, in he case he loss is he loga i hmic sco e,
Sφυ(YT∗+h∗), YT∗+h∗, whe e φυ(·)is o ecas densi y unde υ. As be o e, we
assume ha new in o ma ion a i es and he densi y o ecas is ecompu ed, gi ing
ise o φυ+1(·). I is s aigh o wa d o ob ain
Sφυ(YT∗+h∗), YT∗+h∗=1
φυ
δφυ
δYT∗+h∗|υ
,
which educes, in he case o a No mal o ecas densi y, o
YT∗+h∗−YT∗+h∗|υ
V(YT∗+h∗|υ).
123
176 SERIEs (2024) 15:145–177
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