Globalisation and density forecasts of Euro-area inflation from Phillips Curve models : 2016 edition.
Date: 2016-09-28
Institutions: Eurostat
Subjects: business activity, econometrics, economic analysis, economic fluctuation, economic forecasting, economic model, euro area, globalisation, inflation, statistical method

Globalisation and density 
forecasts of Euro-area 
inflation from  
Phillips Curve models

GIAN LUIGI MAZZI AND JAMES MITCHELL 

2016 edition

S T A T I S T I C A L 
W O R K I N G   P A P E R S

STATISTICAL WORKING PAPERSMain title 2015 editionGlobalisation and density  
forecasts of Euro-area  
inflation from  
Phillips Curve models 
GIAN LUIGI MAZZI AND JAMES MITCHELL 

2016 edition 

 
 
 
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Table of contents 

Abstract ............................................................................................................................... 4 

1 

2 

Introduction ................................................................................................................ 5 

Globalisation, output gaps and inflation density forecasting ................................. 6 

2.1 

2.2 

2.3 

Introduction ................................................................................................................................... 6 

In search of an open-economy Phillips curve ............................................................................ 7 

Indicators of globalisation ........................................................................................................... 8 

2.3.1 
2.3.2 
2.3.3 

KOF indicator 
Foreign output: and the global output gap 
Import and oil prices 

8 
10 
11 

2.4 

The VAR model: common trends in the Euro area .................................................................. 11 

2.4.1 
2.4.2 
2.4.3 
2.4.4 
2.4.5 

Cointegration 
The multivariate BN decomposition 
Bivariate Phillips Curve models 
Combination forecasts 
Occam’s Window: excluding bad models and grand ensembles 

14 
14 
16 
17 
19 

2.5  Density forecast evaluation ....................................................................................................... 20 

2.5.1 
2.5.2 

The continuous ranked probability score (CRPS) 
Tail scores 

21 
21 

2.6 

Empirical Results ........................................................................................................................ 22 

2.6.1 
2.6.2 

Estimating the Euro area cycle 
Forecasting inflation in a globalised world 

22 
26 

2.7  AR and Output Gap Ensembles ................................................................................................. 28 

2.7.1 
2.7.2 
2.7.3 
2.7.4 

Forecasting the GDP deflator 
Controlling for changes in import prices and oil prices 
Grand ensemble 
Forecasting the import price deflator 

28 
35 
39 
41 

2.8  Conclusion................................................................................................................................... 45 

References ........................................................................................................................ 46 

3 

Appendix: Real-time data vintages ......................................................................... 50 

3.1  Real-time GDP data ..................................................................................................................... 50 

3.1.1 

Creation of real-time GDP data triangles 

50 

 Globalisation and density forecasts 

3 

Abstract 

Abstract 

In  this  paper  we  test  whether  globalisation  has  affected  the  relationship  between  the  output  gap  and 
inflation  in  the  euro  area.  Then  we  investigate  how  the  traditional  Phillips  Curve  relating  the  inflation  and 
the  domestic  output  gap  can  be  enhanced  incorporating  globalisation  effects.  We  analyse  the  effect  of 
different  specification  of  the  Phillips  Curve  in  an  out-of-sample  real-time  simulation  exercise,  focusing  on 
the inflation density forecast.  

Keywords: globalisation, input output gap, business cycles 

Acknowledgements: 

Paper first presented at the 31st CIRET conference in Vienna (September 2012). The paper has benefitted 
from the outcomes of the multi-annual PEEIs project financed by Eurostat.  

Authors: 

Gian Luigi Mazzi (1) and James Mitchell (2) 

(1) Gian Luigi  Mazzi, Eurostat, Luxembourg; email: gianluigi.mazzi@ec.europa.eu. 
(2 )James Mitchell University of Warwick,; email: James.Mitchell@wbs.ac.uk. 

 Globalisation and density forecasts 

4 

Introduction 

1 

1  Introduction  

We first consider how inference about the euro area business cycle, specifically the output gap, is affected 
by  globalisation.  Examination  of  traditional indicators  of globalisation,  such  as  the  KOF  indicator which  is 
based on the share of trade in GDP, indicates that globalisation has increased dramatically in the last three 
decades.  We  consider  contrasting  means  of  accommodating  any  globalisation  effects  -  namely  of 
accommodating  increased  cross-sectional  dependence  across  countries  -  when  estimating  and  using 
business cycles. In particular, we focus on the alleged effect globalisation is said to have on the dynamics 
of Euro area inflation and its relationship with traditional (domestic) measures of the output gap. 

Traditionally,  inflation  is  modelled  by  macroeconomists  with  respect  to  the  (domestic)  output  gap,  often 
estimated using a univariate filter. But this relationship is well-known to be unstable and furthermore many 
believe it has weakened over time. Globalisation is often cited as one explanation for this. If a multivariate 
filter  is  estimated,  it  tends  to  be  one  which  in  addition  to  domestic  GDP  data  considers  other  domestic 
variables,  such  as  inflation  and  say  unemployment.  But  increasingly,  globalisation  has  led  to  interest  in 
examining whether domestic inflation also has foreign determinants. 

Accordingly  in  this  paper  we  test  whether  globalisation  has  affected  the  relationship  between  the  output 
gap and inflation in the Euro area. Has it led to the breakdown of the traditional Phillips curve relationship 
and  the  emergence  of  an  open-economy  (global)  Phillips  Curve  where  global  output  gaps,  or  other 
indicators of globalisation, affect domestic inflation? As well as constructing a global output gap estimate, 
and considering the informational content of US output gaps when forecasting Euro area inflation, we also 
consider whether import and oil prices have had an effect. Importantly we test for globalisation effects not 
just by looking at the central tendency of inflation, but its whole density. 

As  well  as  testing  for  globalisation  effects  by  augmenting  the  traditional  Phillips  Curve  with  these 
globalisation  indicators,  and  testing  their  significance  in  out-of-sample  simulations,  we  also  consider  re-
estimating  the  Euro  area  cycle  itself  but  allowing  for  cross-country  dependence.  This  is  achieved  by  de 
trending output using a multivariate estimator which allows for cross-country linkages. Specifically, we use 
a possibly co-integrating VAR model, in the GDP of each Euro area country. We also allow oil prices, US 
GDP  or  global  output  to  enter  the  VAR  as  exogenous  I(1)  regressors.  We  look  at  the  output  gap  in  the 
European VAR as a function of the rank of the VAR - i.e. as a function of the number of exactly identified 
co-integrating vectors, the number of common trends. 

We  then  propose  a  density  forecast  combination  approach  to  forecast  Euro  area  inflation  using  (a)  Euro 
area  output  gaps;  (b)  Global  output  gaps  and  (c)  import/oil  prices.  Thereby  we  integrate  out  uncertainty 
about both the preferred regressor when forecasting inflation and the form of its relationship with inflation. 

Our  analysis  shows  that  domestic  inflation  over  the  last  decade  has  clearly  been  influenced  by  foreign 
economic conditions, working principally through import and oil prices rather than global output gaps. But 
the  domestic  output  gap  has  played  a  role  since  late  2006.  Consideration  of  import  and  oil  prices,  in 
conjunction  with  domestic  and  global  output  gap  data,  does  help  deliver better  fitting  density forecasts of 
inflation than autoregressive models. Therefore it is important to allow for the process determining inflation 
to  vary  over  time,  as  documented  in  other  studies;  but  there  is  not  clear  evidence  that  it  is  important  to 
consider global output gaps when forecasting Euro area inflation. This contrasts the widely cited findings of 
Borio & Filardo (2007). 

 Globalisation and density forecasts 

5 

 
 
 
 
 
 
 
 
2  Globalisation,  output  gaps  and 

inflation 

Globalisation, output gaps and inflation density forecasting 

2 

density forecasting 

2.1  Introduction 

Traditionally,  inflation  is  modelled  by  macroeconomists  with  respect  to  the  (domestic)  output  gap,  often 
estimated using a univariate filter. But this relationship is well-known to be unstable and furthermore many 
believe it has weakened over time. Globalisation is often cited as one explanation for this. If a multivariate 
filter  is  estimated,  it  tends  to  be  one  which  in  addition  to  domestic  GDP  data  considers  other  domestic 
variables,  such  as  inflation  and  say  unemployment.  But  increasingly,  globalisation  has  led  to  interest  in 
examining  whether  domestic  inflation  also  has foreign  determinants.  This  has  led  Borio  &  Filardo (2007), 
Calza  (2008)  and  others  to  add  global  output  gaps  to  the  inflation  equation  in  an  attempt  to  find  a  more 
stable  relationship  between  the  output  gap  and  inflation.  However,  focus  has  remained  on  testing  for  a 
relationship  under  quadratic  loss,  usually  based  on  tests  of  in-sample  predictability  based  on  OLS 
estimation.  Studies  do  not  seem  to  test  for  out-of-sample  predictability  by  looking  at  RMSE  ratios. 
Moreover,  no  attempt  has  been  made  to  model  the  domestic  output  gap  explicitly  accommodating  the 
cross-country dependencies implied by  globalisation. In this paper we therefore look at density forecasts, 
and  also  in  addition  to forecasting  inflation  using  domestic  and  global  output  gaps,  based  on  considering 
domestic  and  foreign  GDP  data  separately,  consider  estimating  the  domestic  output  gap  using  a 
multivariate  model.  This  model  allows  domestic  GDP  to  depend  on  foreign  GDP  data,  and  indeed  on 
additional explanatory variables. We concentrate on the use of the Beveridge-Nelson decomposition as a 
means of identifying and estimating the output gap. The BN offers a unified means of extracting the cycle in 
univariate and multivariate models, and involves conditioning on different information sets when taking the 
long-run forecast of output which is used to define the trend component to GDP. 

The motivation for considering this multivariate estimator of the output gap is based on the belief that these 
foreign variables may help explain domestic GDP growth. If they do, and output growth, to some extent, is 
forecastable,  then  there  is  a  temporary  component,  i.e.  there  is  an  “output  gap”,  which  in  due  course 
reverts  to  its  unconditional  mean  (of  zero).  The  output  gap  is  larger  the  greater  the  degree  of  long-run 
forecastability; only when output growth is unforecastable does output follow a random walk and there is no 
output gap. When output growth, to some extent, is forecastable there is a temporary component, i.e. there 
is an output gap, which in due course reverts to its mean. When the output gap is positive (output is above 
trend)  we  are  requiring  future  growth  at  a  below  trend  rate.  Analogously,  if  the  output  gap  is  negative 
recovery  requires  future  growth  at  an  above  trend  rate.  As  Nelson  (2008)  says:  “predictability  is  the 
essence  of  transitory  variation”.  So  the  finding,  in  some  studies,  that  global  output  gaps  do  matter  may 
simply  be  a  result  of  not  estimating  the  domestic  output  gap  allowing  for  the  fact  that  domestic  output 
depends  on  foreign  output.  That  is,  globalisation  may  mean  that  global  output  gaps  affect  domestic 
inflation; but it might mean that the domestic output gap does influence domestic inflation when one allows 
for linkages between domestic and foreign output. By considering both global and multivariate output gap 
estimates we attempt to distinguish these two effects. 

Empirically  we  focus  on  testing  whether  Euro  area  inflation,  as  an  aggregate,  has  been  affected  by  the 
globalisation process. Previously Calza (2008) studied the Euro area, as did Borio & Filardo (2007) in their 
panel  study.  Our  paper  tests  out-of-sample  predictability  using  real-time  data.  In  contrast  the  extant 
literature has considered in-sample tests for predictability using final-vintage data, despite the fact that we 
know the output gap in particular is subject to considerable measurement error in real-time due to both data 
revisions and the one-sided nature of the detrending filter when applied at the end of the sample. We also 
extend  previous  work  by  looking  not  just  at  the  central  forecast  for  inflation  −  but  its  uncertainty.  This  is 
important because it may well be that the output gap, whether measured domestically or globally, matters 
more for inflation uncertainty than the central estimate of inflation. Indeed globalisation is also often claimed 
to  have  reduced  macroeconomic  uncertainty.  That  is,  globalisation  may  matter  more  for  second  moment 
rather  than  first  moment  inflationary  dynamics.  Importantly  our  methodology  lets  us  test  whether  foreign 

 Globalisation and density forecasts 

6 

 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

factors  are  affecting  inflation  to  an  increasing  extent  over  time.  We  ascertain  whether  the  dynamics  of 
inflation have changed over time. 

We distinguish between forecasts of the GDP deflator and of the import price deflator. Globalisation might 
be  thought  to  mean  that  the  prices  of  domestically  produced  and  imported  goods  are  determined 
differently,  with  the  traditional  Phillips  curve  model  only  useful  in  explaining  the  evolution  of  the  prices  of 
domestically produced goods. The prices of imported goods are instead determined on world markets. 

Our  econometric  methodology  explicitly  accommodates  the  fact  that  there  is  uncertainty  about  the 
determinants  of  domestic  inflation.  We  adopt  a  density  forecast  combination  approach  which  involves 
combining inflation density forecasts which exploit domestic, multivariate and global output gaps. While we 
do not seek to identify structurally the channels through which global factors may affect domestic inflation, 
from  a  practical  perspective  we  seek  to  identify  whether  consideration  of  global  variables  helps  deliver 
improved density forecasts in an out-of-sample context. 

We find that global output gaps play little, if any, role in the determination of  Euro area. inflation, certainly 
independent of any role it might play in the determination of the  Euro area output gap. But import and oil 
price do matter, though to varying degrees over time. Importantly, we find that consideration of  Euro area 
gaps does help deliver improved density forecasts for inflation relative to benchmark AR models. 

2.2  In search of an open-economy Phillips curve 

Romer  (1993)  documents  that  there is  a  negative  correlation  between  openness  and  inflation;  he  argues 
that as the degree of openness of an economy increases, the cost of inflationary surprises rises, thereby 
reduces the incentive to inflate. Borio & Filardo (2007) suggest that in an integrated global economy where 
many goods can be produced and consumed anywhere in the world, it is the global output gap, defined as 
the  deviation  of  world  demand  from  world  potential  supply,  and  measured  by  aggregating  output  gaps 
across  different  economies,  that  should  be  considered  when  assessing  global  influences  on  domestic 
inflation.  Similarly  Lucas  Papademos,  Vice  President  of  the  European  Central  Bank  has  said:  “One 
conclusion that has some empirical support is that domestic inflation is no longer determined predominantly 
by domestic demand and supply constraints, but seems to depend more on the degree of global economic 
slack”  (2006,  page  6).  In  contrast,  Ball  (2006)  writes  “there  is  little  reason  to  think  that  globalization  has 
changed  the  structure of  the  Phillips curve  or the  long-run  level  of inflation” (Page  15).  Ihrig  et  al.  (2010) 
also find little evidence that globalization has increased the role of international factors and decreased the 
role of domestic factors in the inflation process in industrial economies. 

Measures of the domestic output gap should already contain some information about global influences on 
domestic  costs  and  prices  in  the  sense  that  net  exports  are  part    of  GDP.  So  strong  global  demand  for 
goods/services produced in the domestic country should affect the domestic output gap. But a global output 
gap  measure  might  contain  additional  information  in  two  ways.  First,  a  positive  global  output  gap  could 
contain information about future import price inflation; e.g. a positive global output gap could signal rising 
cost pressure in these economies, which could be transmitted to the EA via higher import prices. Secondly, 
the  global  output  gap  may  affect  the  labour  market,  and  potential  supply  in  the  domestic  economy.  For 
instance,  spare  capacity  overseas  could  imply  a  low  cost  of  replacing  domestic  workers  with  foreign 
workers and therefore be associated with weak bargaining power of domestic workers. These effects would 
make domestic labour supply and wages respond to the cyclical position of the world economy. They also 
imply that, with migrant labour flowing into and out of the domestic labour force, domestic potential output 
and the domestic output gap are less well defined. 

As the introduction notes, there have been a number of attempts to isolate globalisation’s effects on U.S. 
inflation. An early study was Tootell (1998), who asks whether globalisation could account for the missing 
inflation of the late 1990s in the US. Using a standard Phillips curve approach, he finds little evidence that 
globalisation—specifically,  measures  of  foreign  slack—helps  determine  U.S.  inflation.  Previously  Calza 
(2008)  and  Borio  &  Filardo  (2007)  have  studied  the  Euro  area,  with  Calza  (2008)  again  finding  little 
evidence that global gaps matter. 

 Globalisation and density forecasts 

7 

 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

2.3  Indicators of globalisation 

However measured it is clear that globalisation has increased in recent decades. The production of many 
goods and services has become increasingly internationalised and the level of trade in goods and services 
has increased. The most commonly used measure of openness, which we might equate with globalisation, 
is the sum of imports and exports divided by gross domestic product; this is picked up by the KOF indicator, 
discussed further below. 

Another aspect  of  globalisation  is  the  growth in  commodity  prices  estimated  to  have resulted from  strong 
output growth in developing economies, and a decline in the average rate of non-commodity import price 
inflation  that  is  estimated  to  have  resulted  from  higher  levels  of  trade  with  developing  countries.  More 
generally  these  price  variables  pick  up  supply-side  shocks.  In  the  analysis  below  we  accordingly  also 
consider measures of both oil and commodity prices. 

2.3.1  KOF INDICATOR 

To  measure  the  degree  of  economic  globalisation  KOF  measure  actual  flows:  trade,  foreign  direct 
investment  and  portfolio  investment  (all  in  percent  of  GDP).  Income  payments  to  foreign  nationals  and 
capital employed (in percent of GDP) are included to proxy for the extent a country employs foreign people 
and capital in its production processes. 

They also measure restrictions on trade and capital using hidden import barriers, mean tariff rates, taxes on 
international trade (as a share of current revenue) and an index of capital controls. Given a certain level of 
trade, a country with higher revenues from trade taxes is less globalised. Openness is measured based on 
the IMF’s Annual Report on Exchange Arrangements and Exchange Restrictions and includes 13 different 
types  of  capital  controls.  The  index  is  constructed  by  subtracting  the  number  of  restriction  from  13  and 
multiplying the result by 10. 

Figure 1 plots the KOF indicators of (economic) globalisation for the nine European countries; these data 
are  interpolated  to  the  quarterly  frequency,  from  the  available  annual  estimates,  by  fitting  a  flexible  UC 
model  to  each  series  separately.  Across  the  9  countries  we  clearly  see  that  globalisation  has  increased, 
and  appears  to  exhibit  a  stochastic  trend.  Interestingly,  since  about  2000  globalisation  appears  to  have 
slowed according to this estimate. 

 Globalisation and density forecasts 

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Globalisation, output gaps and inflation density forecasting 

2 

Figure 1: The KOF indicator of globalisation 

Since this indicator is a compiled one, in our econometric work we seek to pick up the increased evidence 
for  globalisation  and  international  cross-sectional  dependence  by  including  foreign  output,  the  so-called 
global output gap, in our models. Globalisation is believed to have increased cross-sectional dependence; 
therefore  models  which  allows  for  this  dependence  internationally,  perhaps  by  including  a  global  output 
gap, are one means of implementing globalisation when modelling. 

 Globalisation and density forecasts 

9 

 AUT−Level 95 BEL−Level 80 DE−Level 80 90 70 60 85 60 1980 80 ES−Level 2000 1980 2000 1980 2000 FI−Level GR−Level 80 70 60 60 60 1980 2000 1980 2000 1980 2000 70 IT−Level 90 NET−Level 80 FR−Level 80 70 50 60 70 1980 2000 1980 2000 1980 2000  
 
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

2.3.2  FOREIGN OUTPUT: AND THE GLOBAL OUTPUT GAP 

Following  Calza  (2008)  we  measure  foreign  (as  far  as  the  Euro  area  is  concerned)  output  based  on  the 
global  VAR  model  of  Dees  et  al.  (2007)  Foreign  output  is  obtained  by  aggregating  data  on  25  foreign 
countries  (Argentina,  Australia,  Brazil,  Canada,  Chile,  China,  India,  Indonesia,  Japan,  Malaysia,  Mexico, 
New  Zealand,  Norway,  Peru,  Philippines,  Saudi  Arabia,  Singapore,  South  Africa,  South  Korea,  Sweden, 
Switzerland,  Thailand,  Turkey,  UK  and  US)  using  trade-weights  derived  from  bilateral  trade  volumes. 
These weights can be seen as a measure of economic distance. 

Figure  2  provides  estimates  of  the  global  (foreign)  output  gap.  It  can  be  seen  that  apart  from  a  sharp 
downturn in 2008, the global output gap has experienced little persistence and small amplitude. 

Figure 2: The global output gap: BN estimates of the output gap for y∗ using AR(8) and ARIMA(1,1) 
specifications 

It  is  interesting  to  compare  the  global  output  gap  estimates  with  those  for  the  EA9,  computed  using  a 
univariate  BN  decomposition  via  both  the  direct  and  indirect  approaches.  In  both  cases  estimates  using 
AR(8) and ARMA(1,1) specifications are provided. Comparison of the top and middle panels shows that the 
European and global cycles appear quite similar, in the sense that both are dominated by the recession of 
2007-9.  But  it  is  also  seen,  and  this  is  clearer  in  the  bottom  panel  of  Figure  2  which  excludes  the 
recessionary  period,  that  from  1980-2006  the  global  cycle  and  European  cycles  do  exhibit  differences, 
especially in the early 1990s and again in the 2000s when the global output gap is more positive than the 
European one. 

 Globalisation and density forecasts 

10 

 AR(8) ARMA(1,1) 0.01 0.00 −0.01 −0.02 −0.03 −0.04 1980 1985 1990 1995 2000 2005 2010  
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

2.3.3 

IMPORT AND OIL PRICES 

Globalisation is commonly claimed to be reflected in asset price inflation, particularly growth in commodity 
prices, such as oil. Accordingly, another possible means of accommodating globalisation in business cycle 
analysis is to incorporate import and oil price variables into the model. Below we describe how this can be 
done. 

2.4  The VAR model: common trends in the Euro area 

∞  
Consider the following VAR model of order p, [VAR(p)] in the m-vector process {𝑧𝑡}𝑡=1

𝚽(𝐿)(𝒛𝑡 −  𝝁 −  𝜸𝑡) = 𝒖𝑡, 

(1) 

for t = 1,  2,  ...,  T  ,  where  Φ(L) is  a  (m×m)  matrix  polynomial  such  that 𝚽(𝐿) =   𝑰𝑚 − ∑ 𝚽𝑗𝑳𝑗

𝑝
𝑗=1

, Lj  zt  =  zt−j,  

µ and γ are m-vectors of intercepts and trend coefficients and ut is a serially uncorrelated m-vector of errors 

with a zero mean and a constant positive definite variance-covariance matrix, Σ = (σij). 

 Globalisation and density forecasts 

11 

 
 
 
 
 
 
 
 
 
 
 
 
Figure 3: The European and the global output gaps: BN estimates of the output gap using AR(8) 
and ARIMA(1,1) specifications 

Globalisation, output gaps and inflation density forecasting 

2 

From (1) the more conventional representation of the VAR model is derived 

zt = a0  + a1t + Φ1zt−1  + ... + Φpzt−p + ut, 

where the intercept and trend coefficients, a0 and a1, are given by 

a0 

a1 

= 

= 

−Πµ + (Γ + Π)γ, 

−Πγ , 

(2) 

(3) 

(4) 

such that 𝜞 =   −𝚷 + ∑ 𝑖𝚽𝒊
important when there is cointegration. 

𝑝
𝑖=1

 and Π = −Φ(1). a0  and a1 are restricted when Π is rank deficient. This proves 

 Globalisation and density forecasts 

12 

 0.000   −0.025 European cycle    Direct:_ARMA(1,1) Direct:_AR(8) Indirect:_AR(8)  Indirect:_ARMA(1,1) 1980 1985 1990 1995 2000 2005 2010 0.000   −0.025 Global cycle    Global AR(8) Global (1,1) 1980 1985 1990 1995 2000 2005 2010 0.01  0.00  −0.01 Global: ARMA (1,1) European: ARMA (1,1) 1980 1985 1990 1995 2000 2005  
 
 
 
 
 
 
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

Two  alternative  but  equivalent  representations  of  the  VAR  model  that  prove  useful  are  the  Vector  Error 
Correction Model [VECM] and the Vector Moving Average [VMA] representation: 

1. 

The VECM representation of {zt}t=1

∞ , is obtained by rewriting (2) as3 

𝛥 𝑧𝑡 = 𝑎0 + 𝑎1𝑡 + 𝚷𝑧𝑡−1 + ∑

𝑝−1
𝑖=1

𝚪𝑖

𝛥 𝒛𝑡−𝑖 + 𝒖𝑡, or 

𝛥 𝑧𝑡 = 𝑎0 + 𝚷∗𝑧𝑡−1

∗ + ∑

𝑝−1
𝑖=1

𝚪𝑖

𝛥 𝒛𝑡−𝑖 + 𝒖𝑡 

(5) 

(6) 

𝑝
 for i = 1, ..., p−1, Π* = Π(−γ, Im) and z*
where 𝜞𝑖 = − ∑
𝑗=𝑖+1
mapping between the parameters of the VAR model and that of the VECM formulation: 

t-1=(t,z'

𝚽𝑖

t-1)'. The following relations define the 

Φ1  = 

Im − Π + Γ1 , 

Φi 

=  Γi − Γi−1, i = 2, 3, ..., p − 1 , 

Φp  =  −Γp−1  . 

2. 

The infinite VMA representation of {zt}t=1

∞  is defined as: 

∆𝐳𝑡  =  𝐂(𝐿) (𝐚0  +  𝐚1𝑡  +  𝐮𝑡)   =  𝐛0  +  𝐛1𝑡  +  𝐂(𝐿)𝐮𝑡 

where 

𝐛0   =  𝐂(1)𝐚0  +   𝐂∗(1)𝐚1;  𝐛1   =  𝐂(1)𝐚1 

The matrix lag polynomial 𝐂(𝐿) is given by4 

𝐂(𝐿) = 𝑰𝑚 + ∑ 𝑪𝑖𝐿𝑖

∞
𝑖=1

= 𝑪(1) + (1 − 𝐿)𝑪∗(𝐿), 𝑪∗(𝐿) =   ∑ 𝑪𝑖

∞
𝑖=0

∗𝐿𝑖

𝐂(1) = ∑ 𝑪𝑖,

∞
𝑖=1

∗
𝑪∗(1) =   ∑ 𝑪𝑖

∞
𝑖=0

. 

The matrices {Ci} derive from the recursion 

𝑝
𝑪𝑖 = ∑ 𝑪𝑖−𝑗
𝑗=1

𝚽𝑗, i > 0, 𝐶0 = 𝐼𝑚 and 𝐶1 = 0 for i < 0 

Similarly for the matrices {C*

i} 

∗ = 𝑪𝑖−1 + 𝑪𝑖−1
𝑪𝑖

∗

, i > 0, 𝑪0

∗ = 𝑪0 − 𝑪(1). 

(7) 

(8) 

(9) 

(10) 

(11) 

(12) 

(13) 

(14) 

(15) 

3 See, for example, Hamilton (1994) [p.549] for this alternative representation of a VAR(p) model. 
4 See Johansen [(1995a), p.36] or Pesaran et al. (1998). Equation (12) is analogous to a multivariate Beveridge 
and  Nelson  decomposition  of  zt  into  its  permanent  and  transitory  components.  See  Section  2.4.2  for  further 
discussion in the multivariate case. 

 Globalisation and density forecasts 

13 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                           
It follows from (2) and (10) that C(L)Φ(L) = Φ(L)C(L) = (1 − L)Im implying 

Globalisation, output gaps and inflation density forecasting 

2 

ΠC(1) 

C(1)a1 

= 

= 

C(1)Π =  0 , 

0. 

Therefore b1 = 0 and (10) is given by 

∆zt  

=  

b0 + C(L)ut   

2.4.1  COINTEGRATION 

The cointegration rank hypothesis is defined by5 

H0  :  Rank [Π] = r,  r <  m. 

Under H0 of (19), Π maybe expressed as 

Π =  αβ' 

(16) 

(17) 

(18) 

(19) 

(20) 

where α and β are (m × r) matrices of full column rank.6 Let α┴ and β┴ denote (m × m − r) matrices defined 
as  the  bases  for  the  orthogonal  complements  of  α  and  β,  such  that  α'α┴  =  0  and  β'β┴  =  0.  Furthermore, 
C(1) is a singular matrix which can be written as 

C(1) = β┴ (α'

┴ Γ β┴)-1 α'

┴ 

The cointegration rank hypothesis, (19), can therefore be equivalently expressed as 

H'

0  : Rank [C(1)] = m - r,  0 < r <  m. 

(21) 

(22) 

2.4.2  THE MULTIVARIATE BN DECOMPOSITION 

Substituting (12) into the VMA representation, (18), and solving for the level, zt, yields the common trends 
representation or the multivariate BN decomposition7 

zt  =  z0 + b0t + C(1)su

t + C∗(L)(ut − u0), 

zt  =  µ + γt + C(1)su

t + C∗(L)ut, 

(23) 

(24) 

5 See Engle & Granger (1987). 
6 Note  that  rank(Π) = rank(Π*). 
7 See King et al. (1987) and Stock & Watson (1988) for seminal contributions deriving common stochastic trend 
representations. 

 Globalisation and density forecasts 

14 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                           
Globalisation, output gaps and inflation density forecasting 

2 

where {zt} is initialised by z0 = µ + C∗(L)u0, su
b0 since C(1)Π = 0, C*(1)Π = Im and C(1)Γ− C*(1)Π = Im. 

t is the partial sum defined by, 𝑠𝑡

𝑡
𝑢 = ∑ 𝒖𝑗
𝑗=1

, t = 1, 2, ... and γ = 

Substituting (21) in (24) yields 

zt 

=  µ + γt + ψα'

┴su

t + C*(L)ut, 

(25) 

┴su

t are (m × r) independent "“common stochastic trends” and ψ=  β┴(α'

┴ Γ β┴)-1 is a factor loading 
where α'
matrix of order (m × m − r) measuring the long run response on zt of innovations to the common stochastic 
trends.  The  (m−r)  independent  common  stochastic  trends  are  supplemented  by  r  linearly  independent 
deterministic  trends,  see  (2)  and  (4).  Crucially  there  are  fewer  common  stochastic  trends  than  variables, 
the difference being the number of co-integrating relations, r. 

t = ψα'

C(1)su
stochastic trends are permanent since 

┴su

t represents the permanent or long run component of zt: the shocks that drive the common 

ψα'

┴su

t = ψ τt 

(26) 

where  τt  is  is  a  (m  −  r)  vector  of  random  walks  given  by  τt  =  τt-1  +  u∗
┴  ut  denoting  (m−r) 
permanent shocks. The common stochastic trends are consequently also interpreted as a characterisation 
of  the  “long  run”  in  cointegrating  VAR  models.  Note  that  in  (25)  the  permanent  shocks,  u*
t,  and  the 
transitory shocks, ut, are colinear. But (25) is only one possible permanent/transitory decomposition of the 
shocks; see King et al. 1991), pp. 40-2] for an alternative decomposition based on a Jordan decomposition 
of C(1). The lack of uniqueness of any particular decomposition is only resolved by imposing restrictions on 
(ψ, α'

┴) motivated by a priori theory. 

t,  with  u∗

t  =  α'

Importantly as shown say by Garratt et al. (2006), BN trends are the limiting trends of any decomposition 
method. 

 Globalisation and density forecasts 

15 

 
 
 
 
 
 
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

2.4.3  BIVARIATE PHILLIPS CURVE MODELS 

Having extracted the (many) cyclical estimates they are related to inflation using a bivariate VAR model in 
the  cycle  and  inflation.  Following  van  Norden  &  Orphanides  (2005),  we  consider  linear  Phillips  curve 
forecasting models of the form: 

𝜋𝑡+ℎ = 𝛼1

𝑗 + ∑
𝑃
𝑝=0

𝑗
𝛽1,𝑝

𝑃
𝜋𝑡−𝑝 + ∑
𝑝=0

𝑗
𝛾1,𝑝

𝑗 + 𝜀1,𝑡+ℎ
𝑦𝑡−𝑝

𝑗

(27) 

𝑗, with j = 1… , J; P + 1 
where inflation is denoted 𝜋𝑡 and the various output gap measures are denoted 𝑦𝑡
denotes  the  maximum  number  of  lags  in  inflation  and  the  output  gap  measures,  and  h  is  the  forecast 
horizon. 

We augment this specification with the corresponding output gap equation: 

𝑗 = 𝛼2
𝑦𝑡+ℎ

𝑗 + ∑
𝑃
𝑝=0

𝑗
𝛽2,𝑝

𝑃
𝜋𝑡−𝑝 + ∑
𝑝=0

𝑗
𝛾2,𝑝

𝑗 + 𝜀2,𝑡+ℎ
𝑦𝑡−𝑝

𝑗

(28) 

to  create  a  bivariate  VAR  system.  For  simplicity,  we  assume  that  the  lag  structure  is  identical  in  the  two 
equation VAR system. 

We emphasise that although each VAR component uses a particular output gap measure, our aim is not to 
find the “best” single measure of the output gap. Rather, we wish to build ensemble forecast densities for 
inflation, conditional on a number of candidate measures. 

The  two-equation  recursive  structure  described  by  these  equations  is  common  to  many  more  detailed 
models of inflation determination. For example, Rudebusch and Svensson (2002) and Laubach & Williams 
(2003) start with similar bivariate structures, and add additional explanatory relationships and restrictions; 
Garratt, Koop, Mise & Vahey (2008) consider bigger VARs to assess whether money causes inflation and 
output.  In  principle,  additional  equations  and  the  imposition  of  identifying  conditions  pose  no  conceptual 
problems for our methodology, although the computational burden would increase. We prefer to restrict our 
attention  to  a  two-equation  model  space  which,  as  Sims  (2008)  notes,  lies  at  the  heart  of  many 
explanations of inflation determination. 

If it is believed that inflation is determined by a New Keynesian Phillips Curve such that 

𝜋𝑡 = 𝛽𝐸(𝜋𝑡+1⃓ Ω 𝑡)𝛾𝑗𝑦𝑡

𝑗  

then if it is assumed that the output gap follows an AR(1) process 

𝑗 = 𝛾2
𝑦𝑡

𝑗𝑦𝑡−1

𝑗  
𝑗 + 𝜀2,𝑡

(29) 

(30) 

Then 

 Globalisation and density forecasts 

16 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

𝜋𝑡 = 𝛾 ∑

∞
𝑘=0

𝛽𝑘

𝐸(𝑦𝑡+𝑘

𝑗 ⃓ Ω 𝑡) 

𝜋𝑡 =

𝛾

𝑗 
𝑗 𝑦𝑡

1−𝛽𝛾2

(31) 

(32) 

implying inflation and the output gap follow identical univariate process, subject to a scalar. So the 1-step 

ahead  forecast  of  inflation  would  be 

,  showing  that  the  VAR model  is  perfectly  consistent  with 

𝛾

𝑗 𝛾2

𝑗

𝑗𝑦𝑡−1

1−𝛽𝛾2
forward as well as backward looking Phillips Curves. It offers a reduced-form. 

Armed  with  our  many  VAR  specifications,  it  is  straightforward  to  produce  forecast  densities  for  both 
variables of interest. Given non-informative priors, the predictive densities for both inflation and the output 
gap, for a given VAR specification, (32), are multivariate Student-t; see Zellner (1971), pp. 233-236 and, for 
a  more  recent  application,  Garratt,  Koop,  Mise  &  Vahey  (2008).  Appendix  2  provides  details  of  how  the 
predictive densities are constructed from our VARs. 

These forecasts are then combined as discussed below. 

2.4.4  COMBINATION FORECASTS 

In statistics, Bayesian Model Averaging (BMA) offers a conceptually elegant means of dealing with ‘model 
uncertainty’.  BMA  forecasts  condition  not  on  a  single  ‘best’  model  but  take  a  weighted  average  over  a 
range  of  candidate  models;  see  Hoeting  et  al.  (1999).  This  follows  from  appreciation  of  the  fact  that, 
although one model may be ‘better’ than the others, we may not select it with probability one.  We  may  not 
be  sure  that  it  is    the  best  forecast.  Therefore,  if  we  considered  this  single  forecast  alone,  we  would  be 
overstating its precision. 

Similarly in practical macroeconomic forecasting exercises, whether within a Bayesian context or not, it is a 
stylised fact that combination forecasts are hard to beat. The estimated parameters of a single forecasting 
model  are  commonly  found  to  exhibit  instabilities  and  these  can  be  difficult  to  identify  in  real-time.  In  the 
presence  of  these  so-called  ‘uncertain  instabilities’  it  can  be  helpful  to  combine  the  evidence  from  many 
models. For example, Clark & McCracken (2009) examine the scope for taking linear combinations of point 
forecasts  in  real  time,  motivated  by  the  desire  to  circumvent  the  uncertain  instabilities  in  any  particular 
specification.  In  a  series  of  influential  papers,  Stock  &  Watson  (2004)  have  documented  the  robust 
performance  of  point  forecast  combinations  using  various  types  of  models  for  numerous  economic  and 
financial variables. Selecting a single model has little appeal under ‘uncertain instabilities’ when the single 
best model suffers from instability. This might happen either if the ‘true’ model is not within the model space 
considered  by  the  modeller,  or  if  the  model  selection  process  performs  poorly  on  short  macroeconomic 
samples.  We  may  better  approximate  the  truth,  and  account  for  the  uncertainty  in  model  selection,  by 
combining forecasts. 

While methods for combining point forecasts are well established and much exploited, less direct attention 
has  been  given  in  econometrics  to  the  combination  of  density  forecasts.8 Although,  as Wallis  (2005)  has 
noted,  density  forecasts,  in  fact,  have  been  combined  in  the  ASA-NBER  Survey  of  Professional 
Forecasters  since  1969.  The  forecasters’  densities  are  combined  by  taking  a  linear  average,  a  so-called 
“linear  opinion  pool”,  as  in  BMA.  Mitchell  &  Hall  (2005)  and  Hall  &  Mitchell  (2007)  used  this  combination 
rule to combine and then analyse density forecasts from the Bank of England and the National Institute of 

8 Outside the econometrics literature, the benefits of producing density forecasts by combining information across 
different  models  have  been  recognised  for  some  time.  For  reviews  see  Genest  &  Zidek  (1986)  and  Clemen  & 
Winkler (1999). 

 Globalisation and density forecasts 

17 

 
 
 
 
 
 
 
 
 
 
                                                           
Globalisation, output gaps and inflation density forecasting 

2 

Economic and Social Research. They considered how, in practice, the densities in the combination might 
be  weighted,  considering  alternatives  to  equal  weights. Jore et  al.  (2010)  examine  linear combinations  of 
densities from VAR models, and Bache et al. (2009) take a linear combination of VAR and DSGE densities. 
Alternatives include consideration of logarithmic pooling rules; see Kascha & Ravazzolo (2010) and Wallis 
(2010).  Using  several  examples,  and  theoretical  analysis,  Geweke  &  Amisano  (2008)  demonstrate  the 
scope  for  pooled  forecast  densities  to  produce  superior  predictions,  even  if  the  set  of  components  to  be 
combined  excludes  the  ‘true’  model.  Geweke  (2010),  Chapter  5,  stresses  the  efficacy  of  density  forecast 
combination in the case where the ‘true’ model is absent from the model space—sometimes referred to as 
an ‘incomplete model space’. 

In this report we follow in the spirit of this forecasting literature and use the linear opinion pool to combine 
density nowcasts. 

We  formalise  density  combination  in  a  way  that  extends  the  commonly-adopted  convex  mix  of  point 
forecasts  by  utilising  the  linear  opinion  pool  approach;  see  Timmermann  (2006),  (p.177),  and  the 
references described  therein. 

Given  i  =  1…  ,  Nj  component  models,  the  combination  densities  for  GDP  growth  are  given  by  the  linear 
opinion pool: 

𝑝(𝛥𝑦𝜏) = ∑ 𝜔𝑖,𝜏,𝑗

𝑁𝑗
𝑖=1

𝑔(𝛥𝑦𝑡⃓ Ω𝜏

𝑗), 𝜏 = 𝜏, … , 𝜏, 

(33) 

𝑗) are  the  nowcast  forecast  densities  from  component 
where  Nj  (j  =  1,  ...,  5)  where  Nj+1  >  Nj  ; 𝑔(𝛥𝑦𝑡⃓ Ω𝜏
model i, i = 1… , Nj of 𝛥𝑦𝜏 conditional on the information set Ωj. These densities, as we discuss below, are 
obtained  having  estimated  (33).  The  non-negative  weights,  𝜔𝑖,𝜏,𝑗 ,  in  this  finite  mixture  sum  to  unity. 9 
Furthermore, the weights may change with each recursion in the evaluation period 𝜏 = 𝜏, … , 𝜏. 

𝑗), allowing for small sample issues, 
The predictive densities for 𝛥𝑦𝜏 (with non-informative priors), 𝑔(𝛥𝑦𝑡⃓ Ω𝜏
are  Student-t;  see  Zellner  (1971),  pp.  233-236  and,  for  a  more  recent  application,  Garratt,  Koop,  Mise  & 
Vahey  (2008).  Since  each  component  model,  (29),  considered  produces  a  forecast  density  that  is  t,  the 
combined  density  defined  by  equation  (33)  will  be  a  mixture  —accommodating  skewness  and  kurtosis. 
That is, the combination delivers a more flexible distribution than each of the individual densities from which 
it was derived. As Nj increases, the combined density becomes more and more flexible, with the potential to 
approximate non-linear specifications. 

We construct the weights 𝜔𝑖,𝜏,𝑗, in three ways. 

First, we consider equal weights (EW). The EW strategy attaches equal (prior) weight to each model with 
no  updating  of  the  weights  through  the  recursive  analysis: 𝜔𝑖,𝜏,𝑗 = 𝜔𝑖,𝑗 = 1/𝑁𝑗.  We  present  results  for  the 
EW  strategy  without  (prior)  truncation  of  the  set  of  models  to  be  included.  The  EW  strategy  is  often 
recommended  when  combining  point  forecasts,  although  its  effectiveness  for  density  forecasts  has  been 
questioned (see Jore et al. (2010) and Garratt et al. (2010)). 

9 The  restriction  that  each  weight  is positive could  be  relaxed;  for  discussion  see  Genest &  Zidek  (1986).  Note 
that in (33) the only unknown parameters to be estimated are the 𝜔𝑖,𝜏,𝑗. The N component densities are taken as 
given.  Somewhat  confusingly,  in  “mixture  models”  these  weights  are interpreted  on  the  basis  of  a  latent  binary 
random  variable,  which  is  often  assumed  to  have  a  Markov  structure;  see  Geweke  &  Amisano  (2008)  and 
Mitchell  &  Wallis  (2010).  But  in  these  models  the  parameters  of  the  component  models  are  often  estimated 
simultaneously  with 𝜔𝑖,𝜏,𝑗 .  In  so-called  BMA  for  ensemble  forecasting  models  (see  Raftery  et  al.  (1995)),  the 
component  densities  g(.)  are  centered  on  the  point  forecasts  from  the  competing  component  models,  but  the 
variance  of  the  component  density forecasts  is  assumed common across  N and  estimated  simultaneously  with 
𝜔𝑖,𝜏,𝑗. 

 Globalisation and density forecasts 

18 

 
 
 
 
 
 
 
 
                                                           
Globalisation, output gaps and inflation density forecasting 

2 

Secondly,  we  construct  the  weights 𝜔𝑖,𝜏,𝑗 based  on  the  fit  of  the  individual  model  forecast  densities:  the 
Recursive  Weight  (RW)  strategy.  Following  Jore  et  al.  (2010)  and  Garratt  et  al.  (2009),  we  use  the 
logarithmic  score  to  measure  density  fit  for  each  model  through  the  evaluation  period.  The  logarithmic 
scoring  rule  is  intuitively  appealing  as  it  gives  a  high  score  to  a  density  forecast  that  assigns  a  high 
probability  to  the  realised  value.10 Specifically,  the  recursive  weights  for  the  nowcast  densities  take  the 
form: 

𝜔𝑖,𝜏,𝑗 =

exp[∑

𝜏−1
𝜏−8

∑

𝑁
𝑖=1

exp[∑

𝑗
ln 𝑔(𝛥𝑦𝑡⃓ Ω𝜏
𝜏−1
𝜏−8

𝑗
ln 𝑔(𝛥𝑦𝑡⃓ Ω𝜏

)

]

 ,   𝜏 = 𝜏, … , 𝜏, 
]

)

(34) 

where  the  𝜏 − 8 to 𝜏 comprises  the  two-year  training  period,  since  we  employ  quarterly  data,  used  to 
initialise  the  weights.  Computation  of  these  weights  is  feasible  even  for  large  Nj.  Given  the  uncertain 
instabilities problem, the recursive weights should be expected to vary across 𝜏. 

From a Bayesian perspective, density combination based on recursive logarithmic score weights has many 
similarities with an approximate predictive likelihood approach (see Raftery & Zheng (2003), and Eklund & 
Karlsson  (2007)).  Given  our  definition  of  density  fit,  the  model  densities  are  combined  using  Bayes’  rule 
with  equal  (prior)  weight  on  each  model—which  a  Bayesian  would  term  non-informative  priors.  (Koop 
(2003)  (chapter  11)  and  Geweke  &  Whiteman  (2006)  provide  recent  general  discussions  of  Bayesian 
model averaging methods.) Andersson & Karlsson (2007) propose Bayesian predictive likelihood methods 
for  forecast  combination  with  Bayesian  VARs  but  do  not  consider  forecast  density  evaluation.  Hall  & 
Mitchell  (2007)  and  Geweke  &  Amisano  (2008)  consider  iterative  algorithms  to  select  weights  that 
maximize the logarithmic score, suitable for small Nj. We note that instead of looking at fit over the entire 
density the component models, with a larger out-of-sample window than available in our application, could 
be  scored  according  to  their  ability  to  forecast  tail  events.  Nevertheless,  there  are  important  differences 
with  (predictive)  BMA  as  Geweke  (2010)  explains.  For  example,  since  we  assume  that  our  component 
models represent an incomplete model space, the conventional Bayesian interpretation of the weights as 
reflecting  the  posterior  probabilities  of  the  components  is  inappropriate.  Accordingly,  we  do  not  consider 
model  selection  using  the  combination  weights;  nor  do  we  consider  strategies  averaging  a  selection  of 
component models. We are interested in the performance of the density from the linear opinion pool. 

Thirdly, as discussed below, we consider combining the  models using their continuous ranked probability 
score (CRPS) - instead of their log score. The CRPS is believed to be a more robust measure of density fit. 

2.4.5  OCCAM’S WINDOW: EXCLUDING BAD MODELS AND GRAND ENSEMBLES 

There is always a question about how one should choose the set of models over which one combines. Our 
strategy  is  to  select  a  model  space,  namely  VAR  models  linking  inflation  and  the  output  gap  that  is 
commonly  used. We  then  consider a  range  of  output  gap  estimators  distinguished by  the  information  set 
which they use to define the long-run output forecast used to define the BN cycle. We start by employing 
an uninformative prior on all models and use the data (Bayes rule) to update the weight on each model as 
evidence accumulates. 

There  may  be  empirical  benefits,  however,  to  excluding  some  bad  models  prior  to  taking  the  ensemble. 
Occam’s  window  is  one  option;  it  is  an  empirically  driven  strategy  to  improve  forecast  performance  by 
excluding  bad  models  from  the  set  of  model.  Madigan  &  Raftery  (1994)  propose  an  algorithm  which  is 
based on selecting a small set of models by using posterior sampling and treats all the worst fitting models 
outside the subset as having zero posterior probability. We have not explored this strategy to-date. 

10 The logarithmic score of the density forecast, ln 𝑔(𝛥𝑦𝑡⃓ Ω𝜏
𝑔(. ⃓ Ω𝜏

𝑗), evaluated at the outturn 𝛥𝑦𝑡. 

𝑗 ), is the logarithm of the probability density function 

 Globalisation and density forecasts 

19 

 
 
 
 
 
 
 
 
                                                           
We  do  however  experiment  with  taking  combinations  of  combinations;  i.e.  we  take  the  forecast  density 
combinations at a second step combine. This amounts to employing a different prior. 

Globalisation, output gaps and inflation density forecasting 

2 

2.5  Density forecast evaluation 

In  constructing  the  combined  densities  using  the  linear  opinion  pool,  we  evaluate  the  density  forecasts 
using  the  logarithmic  score  at  each  recursion.  We  emphasise  that  in  deriving  the  weights  based  on  this 
measure  of  density  fit,  the  component  models  are  repeatedly  evaluated  using  real-time  data.  These 
weights provide an indication of whether the support for the component models is similar, or not, based on 
the  score  of  the  individual  densities.  A  finding  of  similar  weights  across  component  models  would  be 
consistent with the equal-weight strategy. 

A  common  approach  to  forecast  density  evaluation  provides  statistics  suitable  for  tests  of  (absolute) 
forecast  accuracy,  relative  to  the  “true”  but  unobserved  density.  A  popular  method,  following  Rosenblatt 
(1952), Dawid (1984) and Diebold et al. (1998), evaluates using the probability integral transforms (pits) of 
the realisation of the variable with respect to the forecast densities. A density forecast can be considered 
optimal  (regardless  of  the  user’s  loss  function)  if  the  model  for  the  density  is  correctly  conditionally 

calibrated. We gauge calibration by examining whether the pits 𝑧𝜏, where 𝑧𝜏 = ∫
independently  and  identically  distributed  (see  Diebold  et  al.  (1998)).  In  practice,  therefore,  density 
evaluation with the pits requires application of tests for goodness-of-fit and independence at the end of the 
evaluation period.11 Mitchell & Wallis (2010) refer to this two component condition as “complete calibration”. 

, are uniform and 

𝑝(𝑢)𝑑𝑢

𝛥𝑦𝑡
−∞

The  goodness-of-fit  tests  employed  include  the  Likelihood  Ratio  (LR)  test  proposed  by  Berkowitz  (2001). 
We  use  a  three  degrees-of-freedom  variant  with  a  test  for  independence,  where  under  the  alternative 
𝑧𝜏 follows an AR(1) process. We also follow Berkowitz (2001) and report a censored LR test which focuses 
on  the  10%  top  and  bottom  tails.  This  is  designed  to  detect  forecast  failure  in  the  tails  of  the  forecast 
density. We also consider the Anderson-Darling (AD) test for uniformity, a modification of the Kolmogorov-
Smirnov  test,  intended  to  give  more  weight  to  the  tails  (and  advocated  by  Noceti  et  al.  (2003)).  We  also 
follow  Wallis  (2003)  and  employ  a  Pearson  chi-squared  test  which  divides  the  range  of  the 𝑧𝜏 into  eight 
equiprobable classes and tests whether the resulting histogram is uniform. 

Turning  to  the  test  for  independence  of  the  pits,  we  use  a  Ljung-Box  (LB)  test,  based  on  autocorrelation 
coefficients  up  to  four.  To  investigate  possible  higher  order  dependence  we  also  undertook  tests  in  the 
second and third powers of the pits.in the histogram. Turning to the test for independence of the  pits, we 
use a Ljung-Box (LB) test, based on (up to) fourth-order autocorrelation. 

We  also  investigate  relative  predictive  accuracy  by  considering  a  Kullback-Leibler  information  criterion 
(KLIC)-based test, utilizing the expected difference in the log scores of candidate densities; see Bao et al. 
(2007), Mitchell & Hall (2005) and Amisano & Giacomini (2007). Suppose there are two forecast densities, 
𝑝(𝜋𝜏⃓ 𝐼1,𝜏), and 𝑝(𝜋𝜏⃓ 𝐼2,𝜏), so that the KLIC differential between them is the expected difference in their log  
′⃓ 𝐼2,𝜏) .  The  null  hypothesis  of  equal  forecast  performance  is  𝐻0 =
scores:  𝑑𝜏 = ln 𝑝(𝜋𝜏
𝐸(𝑑𝜏) = 0.  A  test  can  then  be  constructed  since  the  mean  of 𝑑𝜏  over  the  evaluation  period, 𝑑̅
𝜏 ,  under 
appropriate assumptions, has the limiting distribution: √𝑇𝑑𝜏 → 𝑁(0, Ω), where Ω is a consistent estimator of 
the asymptotic variance of 𝑑𝜏.12 

′⃓ 𝐼1,𝜏) − ln 𝑝(𝜋𝜏

Mitchell  & Wallis  (2010)  discuss  the  value  of  information-based  methods  for evaluating  forecast  densities 
that look well-calibrated from the perspective of the pits. 

11  Given  the  large  number  of  component  densities  under  consideration,  we  do  not  allow  for  estimation 
(parameter)  uncertainty  when  evaluating  the  pits.  Corradi  &  Swanson  (2006)  review  pits  tests  computationally 
feasible for small N. 
12 When  evaluating  the  forecast  densities  we  abstract  from  the  method  used  to  produce  them.  Amisano  and 
Giacomini (2007) and Giacomini and White (2006) discuss more generally the limiting distribution of related test 
statistics. 

 Globalisation and density forecasts 

20 

 
 
 
 
 
                                                           
Globalisation, output gaps and inflation density forecasting 

2 

2.5.1  THE CONTINUOUS RANKED PROBABILITY SCORE (CRPS) 

Scoring rules can be used to assess the quality of weather forecasts by assigning a numerical score based 
on  the  forecast  and  the  value  or  event  that  materializes.  A  proper  scoring  rule  maximizes  the  expected 
reward (or minimizes the expected penalty) for forecasting one’s true beliefs, thereby discouraging hedging 
or cheating. For more details see Gneiting & Raftery (2007). 

Although  the  logarithmic  score  is  trivial  to  calculate  and  is  also  proper,  it  allocates  large  penalties  to  low 
probability  events,  making  it  more  sensitive  to  outliers  than  CRPS.  If  F  is  the  cumulative  distribution 
function of the forecast distribution and x verifies, the CRPS is defined as 

𝐶𝑅𝑃𝑆 (𝐹, 𝑥) = ∫ [𝐹(𝑦) − 1{𝑦 ≥ 𝑥}]2

∞
−∞

𝑑𝑦 

(35) 

where 1{𝑦 ≥ 𝑥} denotes a step function along the real line that attains the value 1 if {𝑦 ≥ 𝑥} and the value 0 
otherwise. The continuous ranked probability score can be written equivalently as 

𝐶𝑅𝑃𝑆 (𝐹, 𝑥) = 𝐸{|𝑋 − 𝑥|} − 0.5𝐸{|𝑋 − 𝑋∗|} 

(36) 

where  X  and  X*  are  independent  copies  of  a  linear  random  variable  with  distribution  function  F,  and  E{·} 
denotes the expectation operator. The CRPS is proper and is expressed in the same unit as the observed 
variable.  The  CRPS  generalizes  the  absolute  error,  and  therefore  provides  a  direct  way  of  comparing 
various deterministic and probabilistic forecasts using a single metric. 

The  linear  CRPS  is  defined  in  terms  of  the  forecast  cumulative  distribution  function  F  and  the  verifying 
observation  x  of  a  linear,  real-valued  variable  as  in  (35).  Traditionally,  this  score  is  interpreted  as  the 
integral of the Brier score for binary probability forecasts at all real-valued thresholds y.  The alternative, yet 
equivalent, representation given by (36) makes the interpretation of CRPS lucid.  The first term on the right-
hand side of (36) is the expected value of the absolute error, and the second term is a correction factor that 
measures  the  sharpness  of  the  probabilistic  forecast  F  and  renders  the  score  proper.  The  linear  CRPS 
generalizes the absolute error, to which it reduces if F is a deterministic forecast. 

An  analytic  expression  for  the  integral  in  (35)  when  the  forecast  distribution  function  F  is  Gaussian  with 
mean µ and variance 𝜎2, using (36), is given as: 

𝐶𝑅𝑃𝑆 (𝑁[𝜇, 𝜎2], 𝑥) = 𝜎  (

𝑥−𝜇

𝜎

[2Φ (

𝑥−𝜇

𝜎

) − 1] + 2𝜙 (

𝑥−𝜇

𝜎

) −

1

√𝜋

) 

where 𝜙 and Φ represent  the  standard  Gaussian  probability  density  and  cumulative  distribution  functions, 
respectively. 

2.5.2  TAIL SCORES 

We use a new scoring rule proposed by Diks et al. (2011) based on conditional and censored likelihood for 
assessing  the  predictive  accuracy  of  competing  density  forecasts  over  a  specific  region  of  interest, 
specifically the left tail (deflationary events). 

 Globalisation and density forecasts 

21 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

2.6  Empirical Results 

Before turning to forecasting inflation we consider estimation of the cycle - the output gap. 

2.6.1  ESTIMATING THE EURO AREA CYCLE 

Reflecting the fact that Euro area GDP involves the aggregation of GDP data for the member countries, we 
consider estimating the Euro area cycle using three different approaches: 

1.  The “direct” approach consists of decomposing the raw data of the aggregate itself. 

2.  The  “indirect”  approach  consists  of  decomposing  the  raw  data  corresponding  to  the  sub-
components  (national  GDP)  and  then  aggregating  the  county-series  using  their  current  quarter 
shares of aggregate GDP as weights. 

3.  The 

“multivariate”  or  simultaneous  approach 

involves  decomposing 

the  disaggregates 

simultaneously. 

In  general,  these  approaches  will  deliver  different  aggregate  cycles.  Only  for  so-called  uniform 
decomposition filters does the order of decomposition and aggregation not matter. 

But when the filters differ, as they will when an optimal signal extraction method is used like an unobserved 
components  model,  the  order  is  crucial.  For  further  discussion  see  Ghysels  (1997)  and  Planas  & 
Campolongo  (2001).  The  multivariate  approach  has  certain  optimality  properties,  although  since  it  is 
computationally  demanding  and  requires  hard  choices  to  be  made  about  the  appropriate  information  set 
this approach is rarely considered; see Geweke (1978). We however do consider it using the VAR model 
mentioned above. 

Since we wish to use real-time GDP data, availability at the national level limits us to the nine EA countries 
(Austria,  Belgium,  Germany,  Spain,  Finland,  France,  Greece,  Italy  and  The  Netherlands),  for  which  real-
time GDP data are available on a consistent basis. The data are available in Euros; therefore the Euro area 
aggregate  (in  levels)  can  be  obtained  through  simple  aggregation  of  the  nine  countries  data.  See  the 
Appendix for details on the real-time data from EuroIND. 

The data are then backcast, when necessary, to 1980q1.13 In total we have 40 out-of-sample data vintages 
containing  successive  first  releases  of  GDP,  starting  in  May  2001  and  ending  in  November  2010,  over 
which to evaluate the quality of our estimates. The first vintage, contains quarterly GDP data from 1980q1 
until 2000q4. The last vintage contains GDP data from 1980q1 to 2010q3. 

13 German data backdated with data available from the Bundesbank (http://www.bundesbank.de/statistik/statistik 
zeitreihen.en.php?lang=en&open=&func=list&tr=www s311 lr bip). 

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22 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                           
Globalisation, output gaps and inflation density forecasting 

2 

indirect  and  multivariate  cyclical  estimates  are  all  produced  via 

the  Beveridge-Nelson 
Direct, 
decomposition.  For  the  direct  and  indirect  approaches  this  involves  recursively  estimating  univariate 
models  in  GDP  growth.  Given  the  lag  order  of  the  underlying  ARMA  model  in  GDP  growth  is  known  to 
affect inference we consider two variants both of which have been used in the literature. First we consider 
AR(8)  models  as  in  Garratt,  Lee,  Mise  &  Shields  (2008).  Secondly,  we  consider  ARMA(1,1)  models  as 
suggested by Robertson & Wright (2009). The multivariate estimates are the multivariate BN cycles from a 
9-variable  VAR  (of  lag  order  2)  in  the  9  European  countries’  GDP  with  as  an  exogenous  I(1)  regressor 
either: (i) global output, as measured by Dees et al. (2007)14; (ii) US output as measured in real-time by the 
Federal Reserve Bank of Philadelphia; (iii) oil prices.  

If this additional series has significant explanatory power for European GDP, then the Euro area output gap 
should be larger than otherwise expected. 

14 These data are not available in real-time; and we therefore consider only the most recent vintage of data. 

 Globalisation and density forecasts 

23 

 
 
 
 
 
 
 
                                                           
Globalisation, output gaps and inflation density forecasting 

2 

To  provide  an  indication  of  what  the  cyclical  estimates  for  Euro  area  GDP  look  like  from  these  three 
approaches,  Figure  plots  direct,  indirect  and  multivariate  estimates  using  the  final  vintage  data.  The 
multivariate estimates are those from the VAR with global GDP as an exogenous I(1) regressor. 

Figure 4: Direct, Indirect and Multivariate based estimates of the Euro area (9 countries) Beveridge-
Nelson cycle using final vintage data 

Figure 4 shows that while the direct and indirect approaches give a similar picture of the European cycle, 
with correlation coefficients at 0.83 or higher, the multivariate approach can paint quite a different picture. 
But  this  does  depend  on  the  number  of  cointegrating  vectors  imposed  in  the  VAR.  Given  inspection  of 
information  criteria  clearly  indicates  uncertainty  about  the  number  of  cointegrating  vectors,  and  since  this 
choice  does  effect  the  look  of  the  Euro  area  cycle,  in  the  analysis  below  we  accommodate  uncertainty 
about the cointegrating rank. 

 Globalisation and density forecasts 

24 

Multivariate 0.05 0.00 −0.05 1980 1985 1990 1995 2000 2005 2010 0.000 −0.025    Direct: ARMA(1,1) Direct: AR(8) 1980 1985 1990 1995 2000 2005 2010 0.01 0.00 −0.01 −0.02 1980 1985 1990 1995 2000 2005 2010    r=0 r=1 r=2 r=3 r=4 r=5 r=6 r=7   r=8 r=9 Indirect: AR(8) Indirect: ARMA(1,1)  
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

Table 1 shows quite clearly that the correlation between the VAR-based cycle and the direct and indirect 
estimates declines as the number of cointegrating vectors rises above 3 or 4. That is, imposing more and 
more  commonality  on  the  VAR  reduces  the  similarity  with  the  direct  and  indirect  cycles.  But  there  is 
evidence that imposing a degree of commonality does increase correlation against the direct and indirect 
estimates, since a cointegrating rank of about 3 trends to deliver increased correlation relative to the case 
of assuming less or no cointegration. 

Table 1: Correlation between EA-9 cycles, using final vintage data 1980q1-2010q3 

r=0 

r=1 

r=2 

r=3 

r=4 

r=5 

r=6 

r=7 

r=8 

r=9 

Direct: 

Direct: 

ARMA(1,1)  AR(8) 

r=0 

r=1 

r=2 

r=3 

r=4 

r=5 

r=6 

r=7 

r=8 

r=9 

1 

0.9 

0.9 

0.8 

0.4 

0.7 

1 

0.9 

0.8 

0.3 

0.7 

1 

0.9 

1 

0.4  0.52 

1 

0.6  0.51  0.34 

1 

0.5  0.5 

0.3  0.34  0.45  0.8 

0.4 

0.3  0.23 

0.4  0.8 

0.5 

0.2 

0.3 

0.2 

0.3 

Direct: 

0.7  0.6 

0.7 

0.7 

0.3  0.4 

Direct: 

0.6 

0.6 

0.6  0.58 

0.3  0.4 

Indirect:  0.8 

0.7 

0.6  0.61  0.39  0.5 

0 

-0 

-0.2  0.4  0.6 

0.6 

1 

0.1  0.13 

-0  0.2 

1 

1 

1 

0.3 

0.3 

0.3 

0.5 

0.3  0.5 

0.2  0.1 

1 

-0 

0.3  0.1  0.05 

0.4  0.2  0.19 

Indirect:  0.8  0.7 

0.8 

0.8  0.36  0.6  0.4 

0.3  0.2  0.05 

1 

0.94 

0.83 

0.95 

1 

0.88 

0.87 

But it is important to apply the decomposition methods in real-time. We find that there is a general tendency 
for both the direct and indirect methods to overstate the size of the (negative) output gap in the recession. 
The  direct  and  indirect  approaches,  in  real-time,  gave  very  contrasting  impressions  of  the  depth  of  the 
negative  output  gap  during  the  recession.  The indirect  approach  suggested  an  output  gap  of  around  4%; 
but  the  direct  approach  indicated  about  10%. Typically  they  indicated  that  output  was  5% or  more below 
trend in real-time; when the final vintage estimates suggest a more moderate deterioration of around 2%. 
This is because with the advantage of hindsight much of the dramatic fall in European GDP is attributed to 
a  trend  movement,  rather  than  a  cyclical  movement.  It  takes  some  time  after  the  event  for  the  trend 
estimates  to  catch  up  with  the  actual  movements  in  GDP.  However,  the  multivariate  approach  does  not 
suffer from this problem, although revisions are in fact often larger as there are more persistent differences 
between the real-time and final estimates than for the direct and indirect approaches. 

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Globalisation, output gaps and inflation density forecasting 

2 

2.6.2  FORECASTING INFLATION IN A GLOBALISED WORLD 

We now consider the ability of these output gaps to forecast Euro area inflation. An extension will consider 
their ability to forecast national inflation series. We consider various measures of inflation. First we look at 
HICP  and  import  price  inflation,  and  the  data  refer  to  the  EA16.  These  are  final  vintage  data;  although 
revisions  to  HICP  inflation  are  minimal.  Revisions  to  the  HICP  are  concentrated  in  January  2003,  when 
Germany, the Netherlands and Portugal moved their weight reference period from 1995 to 2000. The HICP 
are  published  in  seasonally  unadjusted  form.  In  contrast  the  GDP  deflator  are  published  in  seasonally 
adjusted  form  in  real-time.  Following  O’Reilly  & Whelan  (2005)  in  their  analysis  of  Euro area  inflation  we 
therefore  focus  on  the  GDP  deflator.  However,  importantly  extending  O’Reilly  & Whelan  (2005)  who  use 
the ECB’s AWM data our data are real-time. (We found our results to be robust to consideration of HICP 
inflation (seasonally adjusted).) 

HICP inflation data for the Euro area are published monthly by Eurostat at t+15 days. In recent months they 
have also released a Flash estimate at the end of the month, so at about t-1 days. The deflator data are 
published with the national accounts at about t+65 days. 

Figure 5  plots  quarterly  HICP  (seasonally  unadjusted) and  import price  inflation;  we  see  that  import price 
inflation is much more volatile. While both inflation series trend downwards from the early 1980s, thereafter 
HICP inflation is pretty stable at around 2%, while import price inflation exhibits some big swings. 

We  also  consider  real-time  GDP  deflator  and  import  price  deflator  data  (seasonally  adjusted).15 Here  the 
data  refer  to  an  evolving  concept  of  the  Euro  area.16 Figure  6  plots  both  the  first-release  and  the  final 
estimate, as of Nov 2010. The figure indicates that the real-time data track the general tendency of the final 
data,  but  there can  be  differences.  A  notable  one  is in  the  recession,  when in  real-time  the  GDP  deflator 
suggested there was deflation but this was subsequently revised away. 

15 The  GDP  deflator  and  import  price  deflator  data  are  in  fact  taken  from  the  Euro  Area  Real-Time  Database 
(RTDB)  which  is  an  experimental  dataset  that  consists  of  vintages,  or  snapshots,  of  time  series  of  several 
variables, based on series reported in the ECB’s Monthly Bulletin (MoBu). The underlying data are from Eurostat. 
16 Since February 2009: Euro 16. June 2008 to January 2009: Euro 15. June 2007 to June May 2008: 
Euro 13. January 2001 to May 2007: Euro  12. 

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Globalisation, output gaps and inflation density forecasting 

2 

Figure 5: HICP and import price inflation 

We use 40 out-of-sample data vintages containing successive first releases of GDP, starting in May 2001 
and  ending  in  November  2010,  over  which  to  evaluate  the  quality  of  our  estimates.  The  first  vintage, 
contains quarterly GDP data from 1980q1 until 2000q4. The last vintage contains GDP data from 1980q1 to 
2010q3. These GDP data are then related to the real-time deflator data such that the first vintage of GDP 
and deflator data, containing data up to 2000q4, are used to forecast inflation in 2001q1. Since the GDP 
and import price deflator data are available at about 65-70 days after the end of the quarter, and the Euro 
area GDP data are published from t+45 to about t+60 days (for the slower countries), this means we are 
producing about a one-quarter ahead inflation forecast. 

Our objective is to forecast the first release value for the deflator. 

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27 

 HICP inflation Import Price Inflation 20  15  10  5  0  −5  1980 1985 1990 1995 2000 2005 2010  
 
 
 
 
 
 
 
 
 
Figure 6: The first and final release for the Euro area GDP deflator and the import price deflator 

Globalisation, output gaps and inflation density forecasting 

2 

2.7  AR and Output Gap Ensembles 

2.7.1  FORECASTING THE GDP DEFLATOR 

As a benchmark we run an AR ensemble, which involves estimating AR(p) models with p = 1, ..., 4 and we 
take both EW and RW combinations. Since the weights in the RW ensemble can change over time, this AR 
ensemble is quite flexible and able to capture changes in the persistence of inflation, for example. 

We then consider four output gap ensembles. 

1.  OG  Euro:  The  first  considers  4  aggregate  estimates  of  the  Euro  area  cycle  only.  Specifically  it 
considers 2 indirect estimates of the  Euro area output  gap based on both ARMA(1,1) and AR(8) 
models and 2 direct estimates of the Euro area output gap based on both ARMA(1,1) and AR(8) 
models. 

2.  OG  Agg:  The  second  considers  11  aggregate  estimates  of  the  Euro  area  and  global  cycle  only. 
Specifically it considers 2 indirect estimates of the Euro area output gap based on both ARMA(1,1) 
and AR(8) models; 2 direct estimates of the Euro area output gap based on both ARMA(1,1) and 
AR(8)  models;  direct  estimates  of  the  global  output  gap  based  on  both  ARMA(1,1)  and  AR(8) 
models;  estimates  of  the  US  output  gap  based  on  both  ARMA(1,1)  and  AR(8)  models;  indirect 
estimates  of the Euro area output gap based on estimation of a VAR(2) model with either global 

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28 

 4 GDP deflator: First release GDP deflator: Final release  3  2  1  0  2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 10  GDP import price deflator: First release GDP import price deflator: Final release 0 −10 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011  
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

output,  US  output  or  the  oil  price  as  an  exogenous  I(1)  regressor  and  the  cointegrating  rank 
imposed at r=0, as suggested by the BIC. 

3.  Disag: The third augments this set of output gap estimates with disaggregate estimates based on 
consideration of each of the 9 countries individually. Again both ARMA(1,1) and AR(8) models are 
considered. In total this makes 29 different output gap estimates are considered, so given the lag 
uncertainty (p=1,...,4), 116 component models are considered. (We did also try a variant where we 
dropped  the  4  Euro area  cycles so  that  Disag contained  only  global cycles  or multivariate-based 
Euro  cycles;  this  made  little  difference;  e.g.  the  log  score  increased  from  -1.257  to  -1.262 
indicating that the 4 Euro area cycles in Disag are not affecting performance in any material way). 

4.  DisagM: The fourth augments Disag with consideration of the VAR based multi-variate estimates 
of  the  Euro  area  cycle  for  r=1,...,9.  Uncertainty  about  the  number  of  common  trends  in  the  Euro 
area is thereby accommodated. 

For each ensemble we consider weights that are equal, based on the CRPS and on the log score. 

Before  turning  to  evaluation  of  the  predictive  densities  from  these  four  ensembles  we  look  in  Table  2  at 
their  RMSE.  Inspection  of  the  RMSE  results  indicates  that  there  is  not  much  to  choose  between  the 
different  ensembles.  There  appear  to  be  modest  gains  to  consideration  of  output  gap  information  when 
forecasting inflation relative to the AR benchmark. 

Table 2: RMSE 

RMSE 

EW 

CRPS 

RW 

OG Euro  OG Agg  OG Disag  OG DisagM  AR 

Import/ oil price 

GDP defl 

0.915 

0.902 

0.872 

0.95 

0.953 

0.999 

0.938 

0.929 

0.887 

0.936 

0.932 

0.9 

0.959 

0.965 

0.962 

0.951 

0.969 

0.966 

EW 

Import price defl 

5.072 

4.935 

4.856 

4.833 

4.767 

CRPS 

RW 

EW 

CRPS 

RW 

HICP 

5.113 

5.168 

5.64 

5.953 

5.79 

6.078 

5.932 

6.184 

4.847 

4.798 

0.48 

0.468 

0.464 

- 

- 

- 

Table  3  then  evaluates  the  densities  by  reporting  their  average  logarithmic  scores  over  the  evaluation 
period. It shows that the Disag ensemble, with RW, is preferred. Consideration of the national data appears 
to help, since Disag produces a higher log score than the Agg ensemble which considers  Euro area data 
only. But accommodating uncertainty about the cointegrating rank in the VAR model does not deliver gains. 
It can pay, consistent with Occam’s Razor, to restrict attention to a subset of the models. 

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Globalisation, output gaps and inflation density forecasting 

2 

Table 3: Log Score statistics: GDP deflator 

AR EW 

AR CRPS 

AR RW 

AR1 

AR2 

AR3 

AR4 

OG  Euro EW 

OG Euro CRPS 

OG  Euro RW 

OG Agg EW 

OG Agg CRPS 

OG Agg RW 

OG  Disag EW 

OG Disag CRPS 

OG  Disag RW 

OG DisagM EW 

OG  DisagM CRPS 

OG DisagM RW 

-1.378 

-1.385 

-1.381 

-1.433 

-1.487 

-1.372 

-1.355 

-1.298 

-1.287 

-1.262 

-1.304 

-1.295 

-1.29 

-1.312 

-1.299 

-1.257 

-1.309 

-1.298 

-1.275 

Prop 

1 

1 

1 

0.5 

0.5 

1 

1 

0.813 

0.938 

1 

0.932 

0.977 

0.977 

0.922 

0.948 

0.991 

0.938 

0.951 

0.991 

 Globalisation and density forecasts 

30 

 
 
 
 
 
  
  
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

Table  4  shows  that  the  densities from  the  ensembles  which  consider the  output  gap  in  at  least  one form 
appear well-calibrated according to the pits tests. It is only the AR models and their ensemble which fail at 
least one of the pits tests at a 95% significance level; this appears to be in the tail. 

Table 4: PIT statistics: GDP deflator 

KLIC  LR2 

tail 

tail 

LR3 

AD 

wallis  LB1 

LB2 

LB3 

0.045  0.179  0.061 

0.47 

0.31 

1.208  0.933 

0.229 

0.95 

0.732 

0.05 

0.15 

0.025 

0.365 

0.283 

1.05 

0.899 

0.416 

0.884  0.723 

0.053  0.131  0.008 

0.329 

0.248 

1.04 

0.899 

0.496  0.837 

0.74 

0.068  0.076  0.119  0.418 

0.064  1.863  0.768 

0.041 

0.846 

0.36 

0.079 

0.05 

0.017  0.895 

0.082  1.509  0.565 

0.091  0.954  0.574 

0.048  0.164  0.009 

0.372 

0.3 

1.086 

0.86 

0.385  0.783  0.622 

0.05 

0.15 

0.017  0.315  0.261  1.112 

0.86 

0.559  0.709  0.737 

AR EW 

AR CRPS 

AR RW 

AR1 

AR2 

AR3 

AR4 

OG  Euro EW 

0.009  0.723  0.416  0.523 

0.405  0.634  0.515 

0.157  0.889  0.577 

OG Euro CRPS 

0.003 

0.885 

0.484 

0.507  0.591 

0.498  0.342 

0.203  0.762  0.478 

OG  Euro RW 

OG Agg EW 

0.002  0.917  0.555 

0.906 

0.668  0.364  0.718 

0.225  0.629  0.497 

0.011  0.652  0.355  0.528  0.435  0.754  0.666 

0.122  0.962  0.572 

OG Agg CRPS 

0.005  0.827  0.388 

0.508 

0.65 

0.608  0.214 

0.165  0.918  0.473 

OG Agg RW 

0.004  0.857  0.381 

0.508 

0.885  0.581  0.515 

0.136  0.887  0.374 

OG  Disag EW 

0.014  0.586  0.292  0.514  0.491 

0.832 

0.899 

0.131 

0.94 

0.643 

OG Disag CRPS 

0.008  0.749 

0.31 

0.488 

0.742  0.709  0.565 

0.159 

0.888 

0.58 

OG  Disag RW 

0.007  0.773  0.342  0.531 

0.87 

0.589  0.815 

0.12 

0.882  0.568 

OG DisagM EW 

0.015  0.558  0.476 

0.489 

0.497  0.873  0.815 

0.152  0.916 

0.69 

OG  DisagM CRPS 

0.006  0.798  0.239  0.444  0.875  0.782  0.242 

0.187 

0.73 

0.72 

OG DisagM RW 

0.008  0.733  0.272  0.439  0.722  0.778 

0.306 

0.238 

0.806  0.696 

Table 5 shows that results are robust to focusing attention on the tail of the forecast density for inflation. 

Table 5: PIT statistics: GDP deflator 

GDP 
Deflator 

Tail 
score 

AR 

OG 
Euro 

OG 
Disag 

OG agg 

OG 
DisagM 

Import/ 
oil price 

EW 

RW 

-1.33 

-1.417 

-1.476 

-1.449 

-1.404 

-1.242 

-1.244 

-1.179 

-1.236 

-1.198 

CRPS 

-1.259 

-1.343 

-1.296 

-1.313 

-1.211 

-1.331 

-1.182 

-1.214 

AR1 

AR2 

AR3 

AR4 

-1.603 

-1.591 

-1.228 

-1.197 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

- 

 Globalisation and density forecasts 

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Globalisation, output gaps and inflation density forecasting 

2 

Figure 7 shows that in the Disag ensemble the German output gap has most explanatory power for  Euro 
area  inflation.  But  if  we  focus  on  the  aggregate  ensemble,  which  excludes  the  country-level  output  gap 
estimates, in Figure 8 we see that prior to the recession the US output gap was the most informative. Over 
the recession itself its informational content increased further, but this abruptly ends in 2009 when both of 
the indirect Euro area output gaps become important. Figure 9 shows that the different VARs do receive a 
weight in the DisagM ensemble of around 0.4, although this does vary over time. 

Figure 7: Weights on the 29 different output gap estimates in the disaggregate Output Gap 
ensemble 

 Globalisation and density forecasts 

32 

 0.7   0.6   0.5 Indirect AR(8)        Direct ARMA US AR(8) VAR US Austria ARMA Ger AR(8) Spa ARMA Fr AR(8) Greece ARMA Net AR(8) Indirect ARMA Global AR(8) US ARMA VAR oil Bel AR(8) Ger ARMA Fin AR(8) Fr ARMA Ita AR(8) Net ARMA Direct AR(8) Global ARMA VAR global Austria AR(8) Bel ARMA Spa AR(8) Fin ARMA Greece AR(8) Ita ARMA 0.4 0.3 0.2 0.1 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011  
 
 
 
 
 
 
 
 
Figure 8: Weights on the 11 different output gap estimates in the aggregate Output Gap ensemble 

Globalisation, output gaps and inflation density forecasting 

2 

 Globalisation and density forecasts 

33 

 0.5 Indirect AR(8)   Direct ARMA   US AR(8) VAR US Indirect ARMA Direct AR(8)   Global AR(8)  Global ARMA US ARMA VAR oil VAR global 0.4 0.3 0.2 0.1 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011  
 
 
 
 
 
 
 
Figure 9: Weights on the multivariate BN cycle in the disagM ensemble; and weight on import 
prices and oil prices when forecasting the GDP deflator 

Globalisation, output gaps and inflation density forecasting 

2 

 Globalisation and density forecasts 

34 

 Weight on multivariate BN cycles in DisaggM ensemble 0.6 0.4 0.2 GDP deflator Import price deflator 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 0.7 Weight on import prices and oil prices when forecasting GDP deflator 0.6  0.5  0.4  0.3  Import prices Oil prices 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011  
 
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

To give a clearer idea of how consideration of the output gap estimates deliver improved inflation forecasts 
in  Figure 10  we  plot  the  probability  of  deflation from  the  recursively  weighted  AR ensemble  and  both  the 
aggregate and disaggregate output gap ensembles. This shows clearly that the two output gap ensembles 
pick  up  the  deflationary  tendencies  ahead  of  the  AR  ensemble  which  unsurprisingly  can  pick  up  the 
deflation only at a lag. 

Figure 10: Probability of deflation from the AR ensemble and the output gap ensembles, alongside 
the (scaled) outturn for Euro area inflation 

2.7.2  CONTROLLING FOR CHANGES IN IMPORT PRICES AND OIL PRICES 

The analysis above establishes that when forecasting Euro area inflation information about the output gap 
does help. But there is at best only weak evidence that global gaps matter. There are gains to considering 
disaggregate  Euro  area  gaps,  specifically  examination  of  the  German  output  gap,  over  and  above 
consideration of only Euro area gaps. But foreign gaps are found to matter only when these disaggregate 
gaps are not considered. 

Previous  work,  such  as  Pain  et  al.  (2006),  has  shown  that  any  increases  in  the  sensitivity  of  inflation  to 
foreign  economic  conditions  might  work  through  import  prices.  Therefore,  we  consider  an  alternative 
ensemble  which  seeks 
import  price  deflator  data  and 
contemporaneous oil price inflation data, since these data unlike the deflator data are published at the end 
of the quarter concerned. 

inflation  using  both 

forecast 

lagged 

to 

 Globalisation and density forecasts 

35 

 0.175 AR OG (11 agg models)   OG (29 disagg models) Inflation 0.6 0.150 0.5 0.125 0.4 0.100 0.3 0.075 0.2 0.050 0.1 0.025  0.000 0.0 −0.025 −0. −0.050 −0. 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011  
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

Table 6 shows that this RW ensemble, which makes no recourse to output gap data, performs almost as 
well as the Disag ensemble with a log score of -1.270 instead of -1.257. It also passes the pits tests. This 
shows that consideration of import/oil price data is informative. Inspection of Figure 9 shows that the RW 
weight on import prices and oil prices fluctuates over time but is always non-zero for both series. 

Table 6: Log Score statistics: GDP deflator using import price and oil price ensemble 

EW 

CRPS 

RW 

-1.304 

-1.281 

-1.270 

0.000 

0.750 

1.000 

1.000 

Table 7: PIT statistics: GDP deflator using import price and oil price ensemble 

KLIC  LR2 

tail 

tail 

LR3 

AD  wallis  LB1 

LB2 

LB3 

EW 

0.017  0.533  0.296  0.474  0.649  0.846  0.515  0.301 

0.925  0.857 

CRPS 

0.01 

0.675  0.315  0.415  0.814  0.744  0.899  0.476 

0.867 

0.846 

RW 

0.009  0.721  0.305  0.395  0.876 

0.69 

0.96 

0.571  0.771  0.834 

Figure 11 and 12 then show the probability of deflation from the AR,  OG Euro and  OG Disag gap based 
and  import/oil  price  ensembles.  It  shows  that  the  AR  ensemble,  as  expected,  is  slow  at  picking  up 
deflationary  pressures.  The  gap-based  forecasts  are  quickest  and  give  the  highest  probabilites;  but  the 
import/oil  price  ensemble  does  perform  much  better  than  the  AR  even  if  not  quite  as  well  as  the  Disag 
ensemble. The OG Euro gap estimate picks up the deflationary tendencies earliest. 

 Globalisation and density forecasts 

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Figure 11: Probability of deflation from the recursively weighted ensembles 

Globalisation, output gaps and inflation density forecasting 

2 

 Globalisation and density forecasts 

37 

 0.6 AR Import/oil price OG Agg 0.4    OG Disag OG DisagM 0.2 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 0.6 AR Import/oil price OG Agg 0.4    OG Disag OG DisagM 0.2 2009 2010 2011  
 
 
 
 
 
 
 
 
Figure 12: Probability of deflation from the recursively weighted ensembles with the Euro OG added 
in too 

Globalisation, output gaps and inflation density forecasting 

2 

 Globalisation and density forecasts 

38 

 
 
 
 
 
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

2.7.3  GRAND ENSEMBLE 

To  try  and  weigh  up  the  relative  informational  content  of  forming  inflation  density  forecasts  using 
autoregressive,  output  gap  or  import/oil  price  inflation  information,  we  take  the  ensembles  from  each  of 
these three systems and combine them at a second step, in what following Garratt et al. (2009) we call a 
grand ensemble. Specifically, we consider the Disag output gap ensemble, as this was the best performing. 

Specifically,  we  take  the  ensemble  density  forecasts  produced  from  expert  AR,  expert  OG  (Disag)  and 
expert M and at a second step seek to combine them. This way each density receives an equal prior weight 
of 1/3. 

Figure  13  plots  the  weights  on  the  three  experts.  We  lose  the  first  observation  as  an  extra  training 
observation. We  see  considerable  variation  in  the  weights,  especially  for  the  optimised  weights.  Ignoring 
the  first  couple  of  years,  to  allow  the  sample  period  over  which  the  average  log  score  is  calculated  to 
increase,  we  see  that  the  AR component  receives  a  minimal  weight.  Interestingly,  during  the  recession it 
does again receive a non-zero weight. But Figure 13 shows that information about both the output gap and 
import/oil price does help deliver improved inflation densities. 

Figure 13: Weights on the AR, Output Gap (OG Disag) and oil/import price (ensemble) experts when 
forecasting inflation in the grand ensemble 

It  is  of  note  that  while  the  weights  in  Figure  13  detail  the  optimal  weight  on  each  expert,  use  of  these 
weights in real-time, due to the fact that we have to wait a quarter to evaluate the densities, would not have 
delivered better density forecasts, as measured by the log score, than the best of the individual experts  - 

 Globalisation and density forecasts 

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Globalisation, output gaps and inflation density forecasting 

2 

the OG expert. The OG expert has a log score of -1.188 over this period, the AR expert of -1.300, and the 
oil  expert  of  -1.210;  and  the  RW  and  optimised  combinations  have  log  scores  of  -1.203  and  -1.215 
respectively.  But  the  weights  in  Figure  13  do  indicate  the  relative  informational  content  of  the  different 
experts, even if in real-time due to publication lags one could not exploit this to produce better forecasts. 

Then to compare and contrast the ability of European and global output gap estimates, we combine the AR 
expert, the oil/import price expert and the OG Euro expert; see Figure 14. 

Figure 14: Weights on the AR, Euro Output Gap (OG) and oil/import price (ensemble) experts when 
forecasting inflation in the grand ensemble 

As before we see that the oil/import price receives the lion’s share of the weight for much of the period. But 
from 2006 we see the weight on the output gap increase. But the weight on the Euro OG does not get as 
high  as  that  for  OG  Disag  indicating  that  there  is  some  extra  informational  content  to  the  foreign 
information,  but  this  is  quite  modest.  Indeed  the  log  score  of  the  optimised  combination  is  -1.244  and for 
the RW combination is -1.202, which are not statistically different from those when we consider OG Disag 
instead of OG Euro. 

Therefore, our analysis shows that domestic inflation over the last decade has clearly been influenced by 
foreign  economic  conditions,  working  principally  through  import  and  oil  prices  rather  than  global  output 
gaps. But the domestic output gap has played a role since late 2006. Consideration of import and oil prices, 
in conjunction with domestic and global output gap data, does help deliver better fitting density forecasts of 
inflation  than  autoregressive  models.  In  contrast  when  point  forecasting,  output  gaps  at  least  do  not  so 
clearly help deliver improved forecasts. 

 Globalisation and density forecasts 

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Globalisation, output gaps and inflation density forecasting 

2 

2.7.4  FORECASTING THE IMPORT PRICE DEFLATOR 

Rather than seek to forecast the GDP deflator we now repeat the exercise above and forecast import price 
inflation.  Table  shows  8  that  there is  far less  between  the  different ensembles  in  terms  of  their  log  score 
statistics. But Table 9 shows that the pits are better for the output gap ensembles than the AR ensemble, 
though there is little between the Disag and DisagM ensembles. 

Table 8: Log Score statistics: Import price inflation 

0.000 

AR EW 

-3.106 

1.000 

AR CRPS 

-3.112 

1.000 

AR RW 

-3.119 

1.000 

AR1 

AR2 

AR3 

AR4 

-3.071 

1.000 

-3.134 

1.000 

-3.163 

0.750 

-3.262 

0.250 

OG  Euro EW 

-3.077 

1.000 

OG Euro CRPS 

-3.109 

1.000 

OG  Euro RW 

-3.103 

0.955 

OG Agg EW 

-2.916 

1.000 

OG Agg CRPS 

-2.975 

1.000 

OG Agg RW 

-2.999 

0.955 

OG  Disag EW 

-2.934 

1.000 

OG Disag 
CRPS 

-3.000 

0.983 

OG  Disag RW 

-3.019 

0.966 

OG DisagM EW 

-2.945 

1.000 

OG  DisagM 
CRPS 
OG DisagM RW 

-3.015 

0.987 

-3.026 

0.978 

 Globalisation and density forecasts 

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Globalisation, output gaps and inflation density forecasting 

2 

Table 9: PIT statistics: Import price inflation 

AR EW 

AR CRPS 

AR RW 

AR1 

AR2 

AR3 

AR4 

KLIC  LR2 

tail 

tail 

LR3 

AD 

wallis  LB1 

LB2 

LB3 

0.052  0.136  0.000  0.460  0.258  0.731  0.565  0.649  0.761  0.850 

0.064  0.087  0.000  0.451  0.158  0.584  0.423  0.625  0.743  0.416 

0.064  0.088  0.000  0.537  0.170  0.614  0.381  0.706  0.717  0.591 

0.055  0.123  0.000  0.429  0.213  0.714  0.768  0.717  0.662  0.513 

0.074  0.061  0.000  0.203  0.114  0.826  0.565  0.573  0.736  0.478 

0.077  0.054  0.000  0.285  0.119  0.786  0.815  0.452  0.810  0.598 

0.156  0.003  0.000  0.497  0.008  0.820  0.515  0.612  0.640  0.981 

OG  Euro EW 

0.026  0.375  0.000  0.753  0.563  0.550  0.960  0.341  0.847  0.743 

OG Euro CRPS 

0.042  0.201  0.001  0.128  0.361  0.608 

0.960  0.642  0.929  0.598 

OG  Euro RW 

0.032  0.296  0.001  0.599  0.474  0.535  0.933  0.391  0.899  0.710 

OG Agg EW 

0.000  0.988  0.026  0.355  0.995  0.618  0.899  0.599  0.706  0.891 

OG Agg EW CRPS  0.002  0.934  0.192  0.194  0.985  0.615  0.565  0.579  0.520  0.636 

OG Agg EW RW 

0.003  0.901  0.232  0.679  0.976  0.725  0.381  0.796  0.581  0.937 

OG  Disag EW 

0.000  0.988  0.011  0.285  0.998  0.579  0.718  0.564  0.757  0.880 

OG Disag CRPS 

0.001  0.950  0.170  0.430  0.991  0.498  0.768  0.627  0.759  0.730 

OG  Disag RW 

0.002  0.940  0.160  0.863  0.982  0.660  0.815  0.843  0.522  0.949 

OG DisagM EW 

0.015  0.558  0.476  0.489  0.497  0.873  0.815  0.152  0.916  0.690 

OG  DisagM CRPS 

0.006  0.798  0.239  0.444  0.875  0.782  0.242  0.187  0.730  0.720 

OG DisagM RW 

0.008  0.733  0.272  0.439  0.722  0.778  0.306 

0.238  0.806  0.696 

Table 10: Tail score for the ensembles: Import price Deflator 

Tail score 

AR 

OG Euro  OG Agg  OG Disag  OG DisagM 

EW 

RW 

-3.692 

-3.675 

-3.572 

-3.283 

-3.598 

-3.409 

CRPS 

-3.667 

-3.596 

-3.374 

-3.247 

-3.365 

-3.344 

-3.315 

-3.436 

-3.403 

AR1 

AR2 

AR3 

AR4 

-3.588 

3.766 

-3.796 

-3.999 

- 

- 

- 

- 

 Globalisation and density forecasts 

42 

 
 
 
 
 
  
 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
 
 
 
Globalisation, output gaps and inflation density forecasting 

2 

Figures  15-16  show  that  the  weights  on  the  competing  output  gap  estimators  are  quite  different  when 
forecasting import price inflation; the global and, to a lesser degree, US output gap now play a big role in 
both ensembles, showing inference is robust. Indeed the global gaps are the most informative output gap 
estimates. So, as we might expect, foreign (global or US) output gaps do affect import/oil price inflation; but 
the evidence above suggests that they do not affect domestic inflation when one controls for disaggregate 
output gap. 

Figure 15: Weights on the 11 different output gap estimates in the aggregate Output Gap ensemble: 
Import Price Deflator 

 Globalisation and density forecasts 

43 

Globalisation, output gaps and inflation density forecasting 

2 

Figure 16: Weights on the 29 different output gap estimates in the disaggregate Output Gap 
ensemble: Import Price Deflator 

 Globalisation and density forecasts 

44 

Globalisation, output gaps and inflation density forecasting 

2 

2.8  Conclusion 

Our analysis using real-time data shows that domestic inflation in the Euro area over the last decade has 
clearly  been  influenced  by  foreign  economic  conditions,  working  principally  through  import  and  oil  prices 
rather  than  global  output  gaps.  But  the  domestic  output  gap  has  played  a  role  since  late  2006. 
Consideration of import and oil prices, in conjunction with domestic and global output gap data, does help 
deliver  better  fitting  density  forecasts  of  inflation  than  autoregressive  models.  Therefore  it  is  important  to 
allow for the process determining inflation to vary over time,  as documented in other studies; but there is 
not clear evidence that it is important to consider global output gaps when forecasting  Euro area inflation. 
This contrasts the widely cited findings of Borio & Filardo (2007). 

 Globalisation and density forecasts 

45 

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Appendix: Real-time data vintages 

3 

3  Appendix: Real-time data vintages 

Our study uses real-time data for GDP for the Euro Area countries (Germany, France, Italy, Netherlands, 
Spain,  Austria,  Belgium,  Luxembourg,  Finland,  Greece,  Portugal,  Ireland,  Slovenia,  Slovakia,  Malta, 
Cyprus,  and  Estonia).  Real-time  data  were  downloaded  from  the  EuroIND  database  of  Eurostat.  This 
section explains how we have prepared the real-time datasets of GDP for our analysis. 

3.1  Real-time GDP data 

Real-time  vintages  of  seasonally  adjusted  quarterly  real  GDP  at  market  price  in  million  of  euros  were 
downloaded from the EuroIND database at Eurostat. Since the 2008Q1 release, seasonally adjusted data 
were  also  adjusted  by  working  days.  Eurostat  discontinued  compilation  of  National  Account  “at  1995 
constant prices” from early 2008, while still offering figures in chain-linked volume with reference year 2000. 
Therefore,  the  set  of  real-time  vintages  we  downloaded  from  EuroIND  database  were  comprised  of  (1) 
vintages compiled “at 1995 constant prices” prior to the release in 2008Q1; (2) vintages compiled as chain-
linked volume series with reference year 2000 since the release in 2008Q1. 

3.1.1  CREATION OF REAL-TIME GDP DATA TRIANGLES 

Real-time  GDP  data  triangles  for  Euro  Area  and  for  the  17  individual  countries  in  the  Euro  Area  were 
created for our analysis. 

We have taken the vintage first released for each quarterly GDP to create our real-time GDP data triangles. 
However,  some  data  availability  issues  that  affect  the  dimension  of  the  real-time  data  triangles  of  first 
release arise: 

1. Vintages  do  not  start  at  the  same  point  in  time.  That  is,  the  beginning  part  of  the  vintages  are

missing. This issue affects the T dimension of the data triangle.

2. The first availability GDP estimate is either 2000Q4 or 2001Q1 for most EA17 countries and also
for EA12 aggregate. But for some other smaller countries, their first available GDP estimate was
for a quarter in more recent years. It  means there are a  smaller number of vintages in their data
triangles, an issue affecting the N dimension.

The  following sections  explain  how  the  real-time  GDP  data  triangles  are constructed  in  the  face  of  these 
two issues. 

Quarterly GDP data triangle for EA aggregate 

Quarterly  GDP  data  for  the  EA12  aggregate  have  vintages  starting  in  1990Q1  the  earliest,  and  in  the 
1995Q1 at the latest. The first available vintage is released for the 2000Q4 estimate, while the last vintage 
available is for 2010Q1 which was published on 4 June 2010. Since then, the EA12 quarterly GDP in the 
EuroIND  database  has  not  been  updated.  Since  we  want  our  analysis  to  cover  the  latest  period,  i.e.  to 
cover the entire year of 2010, vintages for the first release in the other 3 quarters in 2010 are needed. To 
resolve this problem, we append the vintages for the first release of quarterly GDP for EA16 from 2010Q2 
to 2010Q4 to the vintages of the first release of EA12 from 2000Q4 to 2010Q1. 

Moreover,  the  first  release  vintages  in  the  data  triangle  have  two  start  dates.  Earlier  vintages  start  in 
1990Q1 while later vintages started in 1995Q1. We therefore make the assumption that there has been no 
revision to the estimates between 1990Q1 to 1994Q4 since the release of 1994Q4 first vintage, and filled 
up those missing part of vintages using the 1990Q1 to 1994Q4 revised estimated from the 1994Q4 release. 

 Globalisation and density forecasts 

50 

Appendix: Real-time data vintages 

3 

As a result of the above two resolutions, our T × N GDP matrix for EA aggregate thus have 40 first release 
vintages in total (i.e. N = 41), and covers 80 quarters for the period 1990Q1 to 2010Q4 (i.e. T = 80, but only 
the last vintage for 2010Q4 has all 80 observations). 

Quarterly GDP data triangle for individual countries in the Euro Area 

Our aim is to use disaggregate information from the individual countries to nowcast/forecast the aggregate 
Euro  Area  GDP;  there  is  therefore  a  practical  need  to  match  the  dimension  of  individual  countries’  data 
triangle with that of the EA aggregate GDP triangle. The data availability issues mentioned above affects 
the dimensions of these data triangles. 

For the issue that affects the T dimension of the data triangles, we treated it the same way as we did for the 
aggregate  EA  GDP  triangle.  That  is,  to  assume  no  revision  in  historical  estimates  and  filled  up  those 
missing  part  of  the  vintages  with  the  previous  with  available  estimates  for  those  missing  quarters.  As  a 
result, data triangles for most EA12 countries contain vintages all start from 1991Q1, with T = 80 (with only 
the  last  vintage  for  2010Q4  has  all  80  observations).  As  a  result,  data  triangles  for most  EA12  countries 
has  41  vintages  in  total  i.e.  N  =  41.  Those  countries’  the  earliest  start  date  of  the  vintages  is  later  than 
1991Q1 (1995Q1 for Portugal, Luxembourg and Cyprus; 1997Q1 for Ireland; and 2001Q1 for Malta) have 
their real-time GDP matrices with T < 80. Luxembourg, Ireland, Malta, Greece, Estonia, Slovenia, Slovakia 
and Cyprus also have less vintages as their first available vintage was published for the estimate later than 
2000Q4, so these matrices have N < 41. 

 Globalisation and density forecasts 

51 

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Globalisation and density forecasts 
of Euro-area inflation from Phillips 
Curve models
GIAN LUIGI MAZZI AND JAMES MITCHELL 

In this paper we test whether globalisation has affected the relationship 
between the output gap and inflation in the Euro-area. Has it led to 
the breakdown of the traditional Phillips curve relationship and the 
emergence of an open-economy (global) Phillips Curve where global 
output gaps, or other indicators of globalisation, affect domestic 
inflation.

For more information
http://ec.europa.eu/eurostat/

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Main title2015 edition