Forecast combination for euro area inflation : a cure in times of crisis? 1972 / October 2016.
Date: 2016-11-09
Institutions: European Central Bank
Subjects: economic cycle, economic forecasting, euro area, inflation, monetary crisis

Kirstin Hubrich, Frauke Skudelny 

Working Paper Series 

Forecast combination 
for euro area inflation: 
a cure in times of crisis? 

No 1972 / October 2016 

Note: This Working Paper should not be reported as representing the views of the European Central Bank 
(ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB. 

 
 
 
Abstract 

The period of extraordinary volatility in euro area headline inflation starting in 2007 raised the 

question  whether  forecast  combination  methods  can  be  used  to  hedge  against  bad  forecast 

performance  of  single  models  during  such  periods  and  provide  more  robust  forecasts.  We 

investigate this issue for forecasts from a range of short-term forecasting models. Our analysis 

shows  that  there  is  considerable  variation  of  the  relative  performance  of  the  different  models 

over  time.  To  take  that  into  account  we  suggest  employing  performance-based  forecast 

combination methods, in particular one with more weight on the recent forecast performance. 

We  compare  such  an  approach  with  equal  forecast  combination  that  has  been  found  to 

outperform  more  sophisticated  forecast  combination  methods  in  the  past,  and  investigate 

whether it can improve forecast accuracy over the single best model. The time-varying weights 

assign weights to the economic interpretations of the forecast stemming from different models. 

We also include a number of benchmark models in our analysis. The combination methods are 

evaluated for HICP headline inflation and HICP excluding food and energy. We investigate how 

forecast accuracy of the combination methods differs between pre-crisis times, the period after 

the  global  financial  crisis  and  the  full  evaluation  period  including  the  global  financial  crisis 

with  its  extraordinary  volatility  in  inflation.  Overall,  we  find  that  forecast  combination  helps 

hedge  against  bad  forecast  performance  and  that  performance-based  weighting  outperforms 

simple averaging. 

Keywords: Forecasting, euro area inflation, forecast combinations, forecast evaluation 

JEL Classification: C32, C52, C53, E31, E37 

ECB Working Paper 1972, October 20161 
 
Non-technical summary 

In  this  paper  we  compare  the  forecast  performance  of  different  forecast  combination  methods 

with that of a range of short-term inflation forecasting models to evaluate in how far the latter 

can be used to hedge against bad forecast performance in particular episodes, such as the global 

financial  crisis  in  2007/2008.  The  suite  of  models  that  we  consider  comprises  an  individual 

equation framework (an update of Benalal et al. (2004), as also described in ECB (2010)) which 

is  based  on  individual  equations  using  a  number  of  explanatory  variables;  a  Bayesian  Vector 

Autoregressive  (BVAR)  model  for  the  five  main  components  of  HICP  (see  Giannone  et  al. 

(2014),  and  also  ECB  (2010));  Vector  Autoregressive  (VAR)  models  of  the  respective 

component  of  the  Harmonized  Index  of  Consumer  Prices  (HICP),  with  consumer  goods  PPI, 

unit labour costs, oil prices, non-oil commodity prices and the nominal effective exchange rate 

(VAR_C);  and  a  VAR  of  all  5  HICP  components  and  the  aggregate  HICP  (VAR_U).  These 

models  are  compared  with  a  range  of  benchmark  models,  including  an  Autoregressive  (AR) 

model, an ARIMA model, a random walk with drift (RWWD) and a random walk as in Atkeson 

and  Ohanian  (2001)  (RWAO).  In  addition,  we  compare  them  with  two  univariate  models  that 

might capture potential non-linearities, an MA(1) and a STAR model. 

We  use  models  that  provide  forecasts  of  the  five  main  components  of  euro  area  HICP  on  a 

monthly  basis  for  the  period  1990(1)  to  2014(6)  and  most  are  updated  at  least  on  a  quarterly 

basis. We analyse the results for the full forecast period, but also for the pre-crisis and the post-

crisis period and present the evaluation of the aggregation of the component forecasts. We carry 

out  our  forecast  combination  and  evaluation  based  on  every  third  month,  as  in  the  quarterly 

projections  of  the  Eurosystem.  We  aggregate  the  disaggregate  forecasts  to  headline  HICP 

inflation  forecasts  by  using  the  weights  that  would last  have  been  available  and  known  to  the 

forecaster in real time. 

Three  different  combination  approaches  are  included  in  our  forecast  model  comparison:  the 

simple average, using equal weights; a performance-based forecast combination using the root 

mean squared error for a rolling 2-year window of the most recent past to weight the different 

forecasts;  and  a  performance-based  forecast  combination  as  above,  with  geometrically 

(backwards) decaying weights, i.e. recent performance is given more weight than performance 

longer ago. 

We find that the best model for forecasting differs depending on whether the overall HICP or the 

HICP  excluding  food  and  energy  is  considered,  and  which  period  and  forecast  horizon  is 

studied.  Therefore  we  conclude  that  performance-based  forecast  combination  helps  to  hedge 

against bad forecast performance of some of the models in some situations, even though in the 

presence of large shocks or crises it does not necessarily improve over the best forecast model 

since  the  forecast  accuracy  of  all  models  might,  for  example,  be  biased  in  the  same  way. 

Performance-based forecast combination appears to be useful when the models included in the 

set of models exhibit very different forecast performance over time.  

ECB Working Paper 1972, October 20162Forecast  combination  for  the  full  sample  period  typically  improves  forecast  accuracy  over  the 

autoregressive benchmark model for core and headline inflation, and often improves over single 

multivariate  models.  The  forecast  accuracy  gain  of  combinations  is  largest  for  inflation 

excluding  energy  and  food  for  the  full  sample.  We  also  find  that  performance-based  forecast 

combination improves significantly over the simple average for our application. 

Investigating combination weights and their development over time, we find significant changes 

in the weights. Moreover, there appears to be more pronounced evidence for a structural break in 

the forecast model performance around the time of the recent global financial crisis, in particular 

for longer horizons. This time-variation in the weights can be seen as capturing non-linearities in 

the underlying economic relationships. For instance, the importance of certain variables, such as 

oil  prices  or  labour-market  variables,  might  change  over  time.  In  episodes  of  more  volatile 

inflation, a multivariate model allowing for feedback effects between inflation and its predictors 

might  improve  forecast  accuracy.  The  time-variation  in  the  weights  assigned  to  the  forecasts 

from  different  single  models  can  help  to  interpret  the  combination  forecast  and  improve  the 

forecasting models. 

Overall, we conclude from our evidence taking into account RMSFE, forecast accuracy tests and 

turning point predictions that for euro area inflation, first, performance-based combination tends 

to  outperform  simple  averages,  and,  second,  that  performance-based  forecast  combination 

protects against bad forecasts from single models, thereby making the forecast more robust. 

ECB Working Paper 1972, October 20163 
 
 
 
 
 
 
 
 
 
 
 
1. 

Introduction 

The period of high volatility in euro area headline inflation starting from around 2007 made it 

extremely difficult to forecast inflation, even in the short term. Chart 1 shows the evolution of 

forecasts for annual inflation in 2009 from different institutions and private forecasters. This was 

a  year  where  it  was  particularly  difficult  to  forecast  inflation,  since  the  volatility  of  headline 

inflation  was  particularly  high.  The  first  forecasts  represented  in  the  chart  have  been  made  in 

January  2008  and  were  subsequently  revised  over  time  at  different  time  intervals  (mostly 

monthly or quarterly). The last forecast for annual inflation in 2009 was published in October 

2009. While the outcome of annual inflation in 2009 was 0.3% (the red dot end-2009), it took 

until mid-2009 for forecasts of different institutions and private forecasters to come close to this 

number.  Also,  the  different  forecasts  appear  quite  diverse.  This  raises  the  question  whether 

forecast combination methods can be used to hedge against bad forecast performance of single 

models during such periods and provide more robust forecasts.  

The  increasing  number  of  different  models  used  for  short-term  inflation  forecasting  pose  a 

challenge  on  how to  extract  and  summarise the  most  important information  from  the  different 

forecasts in real time. We investigate whether forecast combination methods can help summarise 

the  forecasts  from  many  different  models  in  a  meaningful  and  accurate  point  forecast.  Our 

analysis  shows  that  there  is  considerable  variation  of  the  relative  performance  of  the  different 

models over time, and this variation can be utilised in a performance-based forecast combination 

method,  in  particular  one  with  more  weight  on  the  recent  forecast  performance.  Forecast 

combination with time-varying weights might also be viewed as an approximation of underlying 

non-linearities.  For  example,  in  an  environment  of  relatively  stable  inflation,  a  simple 

autoregressive  model  might  work  very  well,  while  in  episodes  of  more  volatile  inflation,  a 

multivariate model allowing for feedback effects or a model including conditioning information 

may improve forecast accuracy for inflation.  

We  investigate  how  a  forecast  combination  approach  with  time-varying  performance-based 

combination  weights  compares  with  equal  forecast  combination,  which  has  been  found  to 

outperform  more  sophisticated  forecast  combination  methods  in  the  past,  and  whether  it  can 

improve forecast accuracy over the single best model. We employ a range of single equation and 

vector  autoregressive  models  built  to  forecast  HICP  components  and  HICP  headline  inflation 

and also include a number of benchmark models in our analysis. The combination methods are 

evaluated for the Harmonized Index of Comsumer Price (HICP) headline inflation and inflation 

in  HICP  excluding  food  and  energy  in  terms  of  forecast  accuracy  to  investigate  the  source  of 

good forecast performance and forecast failures in more detail. We investigate to what extent the 

forecast  accuracy  of  the  combination  methods  differs  between  normal  times  before  the  global 

financial  crisis  and  the  post-crisis  period  as  well  as  over  the  full  sample  including  the  global 

financial crisis with its extraordinary volatility in inflation.  

ECB Working Paper 1972, October 20164 
 
Chart 1 

Forecasts for annual inflation in 2009 

Eurosystem range
Eurozone barometer
Consensus
SPF
OECD
IMF
European Com.
Data

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Note: The chart shows annual inflation forecasts (y-axis) forecasted at different points in time (x-axis) by a number of 
institutions (see legend). The published ranges are shown for the Eurosystem Macroeconomic Projections. 

In addition, we provide further evidence on the comparison of a direct forecast of all inflation 

items in comparison with forecasting the disaggregate components of inflation and aggregating 

those forecasts and new insights into the role of combinations in that context. This evidence is 

presented in the Appendix. 

The  paper  is  organised  as  follows:  Section  2  presents  some  theoretical  and  empirical 

considerations to motivate forecast combination and discusses the forecast combination methods 

employed in this paper. Section 3 presents the data and the models used for forecasting inflation 

as well as the methods of evaluation.  Section 4 presents the results of the forecast evaluation for 

the individual models and for the forecast combination methods. The time-varying performance 

based forecast combination weights are discussed. Finally, concluding remarks are presented. 

2. 

Forecast combination 

2.1. 

Motivation for forecast combination 

The  literature  on  forecast  combinationhas  recommended  combining  forecasts  in  the  following 

situations to get more accurate forecasts: 

•  models are misspecified in some dimension: forecast combination is more robust against 

misspecification bias  

• 

• 

information  sets  underlying  the  forecasts  are  sufficiently  different  and  combining 

information sets might not provide the best forecast 

forecasts from different models will be differently affected by structural breaks  

Combining  has  often  been  argued  to  be  a  useful  hedging  strategy  against  structural  breaks. 

Hendry and Clements (2002) argue that this might be the case in the presence of an (unknown) 

break  after  the  estimation period,  in  particular  when a  shift  occurs  in  the  intercept  of  a  single 

variable or a shift of two correlated predictor variables occurs in the opposite direction. Stock 

and  Watson  (2004)  show  empirically  that  combined  forecasts  tend  to  outperform  individual 

ECB Working Paper 1972, October 20165 
 
forecasts and that simple averages are best in terms of point forecast accuracy. Also, Aiolfi and 

Timmermann (2006) have shown that an equal-weight combination is best in many situations.   

In  this  study,  our  motivation  to  use  a  forecast  combination  approach  is  threefold:  First,  to 

investigate whether forecast combination can be a useful hedging strategy against bad forecast 

performance of single models (see also ECB monetary policy strategy 2003) in the presence of 

high  inflation  volatility  and  potential  structural  shifts  during  the  recent  global  financial  crisis; 

second,  to  take  into  account  potential  non-linearities  and  time-variation  in  the  performance  of 

single  models  and  investigate  whether  that  improves  forecast  accuracy  over  equal-weight 

combination schemes; and third, to improve over single forecasts, even though that is not always 

possible  and  depends  on  the  particular  situation.  The  performance-based  combination  weights 

can potentially also be used in real time to inform the forecaster how well the different models 

have  performed  recently  relative  to  each  other.  They  also  allow  evaluating  the  relative 

importance of the underlying economic interpretations of the forecast from the different models. 

A  combination  issue  separate  from  the  one just  discussed  (that  focuses  on  the  combination  of 

forecasts  of  the  same  variable  of  interest),  is  whether  combining  forecasts  of  different 

disaggregate  variables  to  forecast  the  aggregate  (“indirect  forecast”  of  the  aggregate)  is 

preferable to combining disaggregate information in a model for the aggregate and use this for 

forecasting (“direct forecast” of the aggregate). We present some new results on forecasting the 

aggregate versus aggregating component forecasts in the context of the models employed in this 

study in Appendix IV. 

2.2. 

Forecast combination methods  

In  this  paper,  we  include  three  different  combination  approaches  in  our  forecast  model 

comparison:  

•  Simple average (equal weights); 

•  Performance-based  forecast  combination  using  the  RMSE  for  a  rolling  2-year 

window of the most recent past to weight the different forecasts; 

•  Performance-based 

forecast  combination  as  above,  with  geometrically 

(backwards)  decaying  weights,  i.e.  recent  performance  is  given  more  weight 

than performance longer ago. 

Forecast  combination  weights  are  calculated  based  on  forecast  accuracy  measured  in  terms  of 

the inverse of the mean squared forecast errors (MSFEs) of the respective models relative to the 

sum  of  the  MSFEs  of  all  models  (see  also  Timmermann,  2006).  A  key  feature  of  the 

performance-based forecast weights using a rolling window is that they vary over time for each 

of the models, so that the resulting combined forecasts capture potential non-linearities and time-

variation in the performance of single models.  

Optimal  least-squares  weights,  as  discussed,  for  example,  in  Timmermann  (2006)  have  some 

appeal  from  a  theoretical  point  of  view  due  to  their  optimality  properties.  In  fact, 

ECB Working Paper 1972, October 20166Genre et al (2010)  find  for  the  euro  area  survey  of  professional  forecasters  that  optimal  least 

squares  weighting  combination  performs  well  in  comparison  with  other  forecast  combination 

methods. However, optimal least squares weights are known to depend on the precision of the 

estimate of the variance-covariance matrix of the forecast errors, which might be quite imprecise 

in short forecast evaluation samples, such as we have in our study. On the same grounds, we do 

not consider combination methods that are based on estimated non-linear schemes either. 

One  alternative  combination  method  is  performance-based  weighting,  -which  we  have  chosen 

for  our  context  -  where  the  weights  of  the  individual  models  are  based  on  past  RMSE 

performance and therefore are not sensitive to estimation problems in small samples.  

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The schemes we are using also include time-varying weights that give more weight to the recent 

performance  of  the  models.  Such  combination  schemes  are  sensible  when  a  time-invariant 

Gaussian  distribution  for  the  forecasts  and  realisation  is  not  a  good  approximation  which  is 

particularly  the  case  during  the  recent  global  financial  crisis  and  the  euro-area  sovereign  debt 

crisis.  Our  performance-based  combination  method  that  gives  more  weight  to  recent 

performance  has  some  resemblance  with  the  Bayesian  approach  of  dynamic  model  averaging 
that has recently been suggested by Koop and Korobilis (2010).4  

3. 

Data and forecast models 

3.1. 

HICP inflation and its components 

The data in this study include headline HICP for the euro area as well as its breakdown into five 

subcomponents: unprocessed food, processed food, industrial goods, energy and services prices. 

The data are of monthly frequency, starting in 1990(1) until 2014(6). Chart 2 shows the period 

of relatively stable headline inflation during most of the 2000s, and the sudden rise in volatility 

beginning in summer 2007. 

These  data  correspond  to what  are  used  in  the  regular  analysis  of  HICP  developments  for  the 

euro  area  as  provided,  for  example,  in  the  commentary  of  the  ECB  Economic  Bulletin.  The 

breakdown helps the understanding of recent developments and also helps forecasting aggregate 

HICP developments. It has indeed been shown that in the short run, disaggregated forecasts tend 

to  be  more  accurate  than  direct  forecasts  of  the  aggregate  (see  Hubrich  (2005),  Benalal  et  al. 

(2004)). 

4 Koop and Koribilis (2010) refer to Raftery, Karny and Ettler (2010) and Smith and Miller (1986) who argue that to 
include  a  “forgetting  factor”  that  imposes  an  exponential  decay  for  past  observations  is  a  sensible  and  not  too 
restrictive approximation in the context of dynamic model averaging. 

ECB Working Paper 1972, October 20167 
     
 
 
Chart 2 

HICP  headline  inflation  and  components,  HICP  excluding 

food and energy 

headline

excl. energy

excl. food & energy

energy
unprocessed food
processed food

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3.2. 

Forecasting models for inflation 

We  use  models  that  provide  forecasts  of  the  five  main  components  of  euro  area  HICP  on  a 

monthly basis and most are updated at least on a quarterly basis. We present the evaluation of 
the  aggregation  of  the  component  forecasts.5  The  suite  of  models  that  we  consider  (see 

Appendix II for a detailed description) comprises an individual equation framework (an update 

of Benalal et al. (2004), as also described in ECB (2010)) which is based on individual equations 

using a number of explanatory variables; a Bayesian Vector Autoregressive (BVAR) model for 

the five main components of HICP (see Giannone et al. (2014), and also ECB (2010)); and VAR 

models of the respective component of the HICP, with consumer goods PPI, unit labour costs, 

oil  prices,  non-oil  commodity  prices  and  the  nominal  effective  exchange  rate  (VAR_C). 

Forecasts  using  these  models  are  based  on  a  real  time  database  (see  Appendix  II  for  details): 
they  condition  on  assumptions  as  produced  in  the  Eurosystem  projections.6  The  conditioned 

VAR (VAR_C) complements the model set since it takes into account only the real side of the 

5  We  comment  also  on  the  direct  forecast  of  aggregate  inflation  in  comparison  with  the  aggregated  component 

forecast in Appendix IV. 

6 We take the mid-point of the published ranges as the relevant point forecast. 

ECB Working Paper 1972, October 20168 
economy  and  some  other  important  predictors  of  inflation  –  like  the  BVAR  –  but  not  the 

components of HICP inflation.  

In addition, we estimate a VAR of all 5 HICP components and the aggregate HICP (VAR_U). 

This  model  does  not condition  on  the  assumptions in  the  projections since  it  does  not include 

any of those variables. We include this model in our forecast combination scheme because it is a 

relatively  simple  multivariate  model  that  takes  into  account  the  interdependences  and  second-

round effects between the different components of inflation, even though it does not include the 

real side of the economy as does the BVAR.  

We  compare  these  models  with  two  simple  univariate  non-linear  models  that  might  capture 

potential non-linearities, an MA(1) and a STAR  model. The constant parameter MA(1) model 

from  a  rolling  estimation  window  is  included  as  an  approximation  for  an  Unobserved 

Component  Model  with  stochastic  volatility,  as  suggested  by  Stock  and  Watson  (2007).  The 

Smooth Transition Autoregressive (STAR) Model is a generalisation of the 2-regime threshold 

autoregressive model and allows the estimation of the – potentially smooth - transition from one 

regime  to  the  other.  This  model  is  well-suited  for  forecasting  variables  with  asymmetric 
behaviour  (see  e.g.  Teräsvirta,  2006)78.  Pre-crisis  studies  have  presented  mixed  results  on  the 

forecast performance of non-linear models. However, so far there is little evidence on how these 

models  fare  during  the  crisis.  We  also  consider  a  number  of  benchmark  models  in  our  study, 

including an AR, an ARIMA model, a random walk with drift (RWWD) and a random walk as 

in Atkenson and Ohanian (2001) (RWAO). 

The lag length for the AR and the VARs is based on the Bayesian information criterion.9. The 

estimation period for those models is determined by the availability of the component data. The 

first estimation sample is 1992(1) to 2000(1) with the first one-step ahead forecast for 2000(2). 

The sample is extended recursively until 2014(06). The models are re-estimated for each sample 

of the recursively expanding estimation window. The forecasts for the BVAR, the VAR, and the 

individual  equation  framework  are  set  up  so  as  to  reflect  the  data  situation  prevailing  at  the 

respective, and take into account real-time data for real variables and labour market variables, as 

well  as  for  expectations regarding  exchange  rates (flat  profile)  and  oil  prices (futures).  In  that 

sense, all the models included in the forecast combination are based on the same information set. 

We  carry  out  our  forecast  combination  and  evaluation  based  on  every  third  month,  as  in  the 

quarterly  projections  of  the  Eurosystem.  We  aggregate  the  disaggregate  forecasts  to  headline 

HICP inflation forecasts by using the weights that would last have been available and known to 

the forecaster in real time. 

7 Note that the STAR model is re-estimated only once a year while the other models are re-estimated each month. 
8

 Multivariate nonlinear models are beyond the scope of this paper; see for example Hubrich and Teräsvirta (2013) for 
a  recent  review  of  thresholds  and  smooth  transitions  in  vector  autoregressive  models.  Forecast  combinations  with 
time-varying  weights  might  be  viewed  as  an  alternative  to  modelling  nonlinearities  with  multivariate  nonlinear 
models. 
9  Note  that  we  find  that  models  with  the  lag  selection  based  on  the  Bayesian  information  criterion  provided  more 

accurate forecasts than when using the Akaike criterion 

ECB Working Paper 1972, October 20169It is interesting to evaluate whether the combined forecast outperforms the forecasts of some or 

all  of  the  single  forecasting  tools  and  which  combination  method  does  best.  In  addition,  in  a 

policy institution, it will be useful to combine forecasts from different inflation forecasting tools 

to  get  a  single  inflation  forecast  (e.g.  in  a  monthly  inflation  note  or  other  briefing).  The 

individual  models then  might  still  be  important to facilitate  the interpretation  of  the  combined 

inflation forecast. 

3.3. 

Forecast evaluation 

The  evaluation  of  the  above-mentioned  models  is  done  in  terms  of  root  mean  squared  error 

(RMSE). We also present comparisons in terms of mean absolute values (MAD) to investigate 

the robustness of our results. The evaluation period for the HICP starts with forecasts produced 

in  February  2002 and uses  one forecast  per  quarter  until  and including  May  2007  for the pre-

crisis period, from May 2010 to June 2014 for the post-crisis period, and from February 2002 to 

June 2014 for the full period.  

We  perform  a  number  of  forecast  accuracy  tests  to  evaluate  the  statistical  significance  of  the 

differences  in  forecast  accuracy  between the  models.  These tests include  the  Diebold-Mariano 
(DM) test (Diebold and Mariano, 1995; see also West, 1996) for non-nested models.10 We also 
investigate the ability of the models to predict turning points, where turning points are defined as 

in Canova (2007). A downward turn in inflation occurs when inflation has been below the value 

at  the  peak  for  two  periods  before  the  peak  and  one  period  after.  An  upward  peak  is  defined 

analogously. In addition to the test of statistical significance, we also provide graphical evidence 

on  the  economic  significance  of  the  differences  in  the  forecasts  and  how  the  relative 

performance of the models changes over time.  

In  particular,  we  evaluate  whether  different  methods  of  forecast  combination  improve 

significantly  over  the  benchmark  model  in  terms  of  the  root  mean  squared  forecast  error 

(RMSE) and help to forecast turning points. We also investigate whether any of the combination 

methods provides significantly better forecasts than the other methods. 

4. 

Results 

The  discussion  of  the  results  is  structured  as  follows:  We  first  compare  univariate  benchmark 

and  non-linear  models  for  HICP  headline  inflation  and  HICP  inflation  excluding  food  and 

energy. Second, we compare conditional models and VARs for the same period and variables. In 

a  third  step,  we  present  the  results  on  the  relative  forecast  accuracy  of  the  different  forecast 

combination methods for the three above-mentioned periods. For the three different combination 

10 Note that the DM test is applied here since the benchmark direct AR forecast model of aggregate inflation is not 
nested in the alternative aggregated component models. In further analyses not presented here we have compared 
the  forecasts  from  the  direct  forecast  of  aggregate  inflation  based  on  single  models  with  the direct  AR  forecast 
using the Clark and West (2007) test for pairwise nested model comparisons and the maximum t-test suggested by 
Hubrich and West (2010) for a small model sets; see also Granziera, Hubrich and Moon (2014) for an overview of 
multiple forecast comparison tests. The results did not change the overall conclusions of the paper. 

ECB Working Paper 1972, October 201610methods  –  that  is  simple  average,  rolling  two  year  window,  and  geometrically  declining 
performance-based combination - we include two different model sets11:  

-  Combination 1: individual equation framework, BVAR, VAR_U, VAR_C, STAR, AR – 

here  we  include  the  conditional  models,  the  non-linear  model  (STAR)  and  the 

benchmark model. 

-  Combination 2: individual equation framework, BVAR, ARIMA, STAR, AR – here we 

only include the four best performing models and the benchmark model. 

Chart 3 

Headline HICP forecasts – 6 months ahead 

data

Ind. equ.

VAR_U_sub

VAR_C

BVAR

LSTAR

5

4

3

2

1

0

-1
Jan02

Jan04

Jan06

Jan08

Jan10

Jan12

Jan14

To give a first impression of the forecasts of the different models, Chart 3 shows 6-month ahead 

forecasts  of  the  individual  equation  framework,  the  BVAR,  the  VARs  and  of  the  STAR.  All 

models were lagging in predicting the timing and the size of the increase in headline inflation in 

2008 and the decline thereafter. The differences across models are small and typically happen at 

particular episodes, such as in 2004 or at the end of the sample. 

When excluding food and energy, the differences across models appear to be somewhat stronger. 

Chart 4 shows that the two VARs have overpredicted inflation over most of the sample, while 

the individual equation framework and the BVAR had no systematic over- or underprediction. 

This is related to an upward bias in the forecasts for services inflation in the former two models.  

11  A  third  combination  was  also  investigated,  which  included  the  same  models  as  combination  1,  but  the  worst  2 
models (corresponding to 20%) were excluded. These excluded models were the RWAO and the MA1.The results 
from  this  scheme  were  very  similar  than  the  results  for  the  second  combination  scheme,  and  are  therefore  not 
presented here. 

ECB Working Paper 1972, October 201611 
 
 
 
 
Chart 4 

HICP excluding food and energy forecasts – 6 months ahead 

data

Ind. equ.

VAR_U_sub

VAR_C

BVAR

LSTAR

3.5

3

2.5

2

1.5

1

0.5

Jan02

Jan04

Jan06

Jan08

Jan10

Jan12

Jan14

4.1. 

Univariate benchmark and non-linear model 

Chart 5 presents on the left-hand side the RMSEs of univariate and non-linear models relative to 

the AR benchmark model for total HICP inflation and on the right hand side for HICP inflation 

excluding food and energy. The first two charts cover the full period, the second two the pre-

crisis  period  and  the  third  two  the  post  crisis  period  (see  Annex  I  for  the  full  results).  A 

number/bar  below  one  indicates  an  improvement  of  the  respective  model  over  the  AR 

benchmark.  

For  total  inflation,  we  find  that  the  RW_AO  performs  significantly  worse  than  the  AR,  in 

particular  in  the  pre-crisis  period  and  over  longer  horizons  also  in  the  post-crisis  period.  The 

MA1 based on a rolling estimation window, the random walk with drift (UM) and the ARIMA 

model also do not systematically outperform the AR. The STAR model shows a somewhat better 

forecast  performance  than  the  AR  over  the  full  sample  mainly  on  account  of  the  post-crisis 

period.  

ECB Working Paper 1972, October 201612 
 
 
Chart 5 

Univariate benchmark and non-linear models 

  Headline inflation 

Excl. energy and food 

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Note: RWAO – random walk as in Atkinson and Ohanian (2001), UM – random walk with drift, RIMA – 
ARIMA  model,  MA1  –  moving  average  model  of  order  1,  LSTAR  –  logistic  smooth  transition 
autoregressive model (STAR) 

ECB Working Paper 1972, October 201613 
 
 
 
 
 
 
 
 
 
We  have  investigated  how  the  slope  coefficient  of  the  logistic  function  in  the  STAR  model 

changes over time. We find that there is a substantial increase in the slope coefficient in 2008 for 

both  headline  inflation  and  inflation  excluding  energy  and  food,  indicating  a  rather  abrupt 

change in regime. This might explain the somewhat higher forecast accuracy of the STAR model 
in  comparison  with  the  univariate  benchmark  models  in  the  post-crisis  period.12  Finally,  the 
ARIMA has a very similar forecast performance as the AR.  

Overall, this confirms that the AR model is a useful benchmark model for forecasting headline 

inflation. 

For HICP excluding food and energy, the results are quite different (see Chart 5 right-hand side). 

While the UM and the MA1 have forecast errors which are sometimes twice as large as the ones 

of  the  benchmark,  the  ARIMA  performs  better  than  the  benchmark  over  all  horizons.  The 

RWAO  model  cannot  beat  the  benchmark.  The  STAR  model  performs  similar  or  better  over 

longer  horizons  and  for  all  horizons  in  the  post-crisis  period.  When  considering  the  pre-crisis 

period, the STAR performs worse than the benchmark up to 3 steps ahead and is better 12 steps 

ahead.  

We draw two conclusions from this first part of the analysis: 

-  First, we take the AR model as a benchmark model for the combination forecasts.  

-  Second,  we  only  include  the  AR  and  the  STAR  model  in  addition  to  the  individual 

equations  and  VAR  models  in  the  combination  scheme  1,  since  those  were  the 

univariate model with the best performance for headline inflation. Combination scheme 

2 also includes the ARIMA, as that performed particularly well for inflation excluding 

energy  and  food.  We  have  carried  out  the  forecast  comparison  also  for  a  combination 

scheme where we deleted only the 20 % worst models (2 models in our case), but that 

did not change much the results in comparison to the combination scheme 2. 

One  reason  why  the  AR  benchmark  is  difficult  to  beat  may  be  a  change  in  the  sum  of  the 

coefficients  over time,  possibly  making  this  model  more robust and  adaptive  to  large  shifts  in 

inflation such as observed during the crisis. We estimated the AR recursively and found indeed 

an increase in the sum of the coefficients for headline inflation around the crisis time. However, 

we could not find the same pattern for inflation excluding food and energy.  

4.2. 

Conditional models and VARs 

In  this  section,  we  compare  different  multivariate  models.  An  important  difference  between 

these  models  is  that  the  individual  equation  framework,  the  BVAR,  and  the  VAR_C  are 

conditioned on assumptions regarding exchange rates, oil prices, non-oil commodity prices and 
on interpolated annual variables as used in the Eurosystem projections,13 while the VAR_U is  

12 In the past, the record of this kind of model for forecasting has been mixed. So it is interesting that we find some 
improvement of the STAR forecasts over the benchmark AR forecasts in some situations. These findings indicate 
that it might be worthwhile to investigate further the potential of non-linear models, in particular multivariate, for 
forecasting. 

13 We take the mid-point of the published ranges as the relevant point forecast. 

ECB Working Paper 1972, October 201614Chart 6 

Results for conditional models and VARs 

  Headline inflation 

Excl. energy and food 

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Note:  Ind.equ.  –  Individual  equations,  BVAR  –  Bayesian  VAR,  VAR_U_sub  –  unconditional  VAR  with 

disaggregate subcomponents of inflation, VAR_C – conditional VAR  

ECB Working Paper 1972, October 201615 
 
 
 
 
 
 
 
 
 
unconditional as it only includes the HICP and its subcomponents as endogenous variables, i.e. 

no variable that is part of the assumptions. Therefore, forecast errors of the conditional models 

are partly a result of errors in the assumptions. However, the assumptions can also help yielding 

better forecasts. 

Chart 6 shows the RMSE relative to the benchmark AR model of the conditional models and the 

VARs for total HICP inflation (left-hand side) and for HICP inflation excluding food and energy 

(right-hand side) for the full period, the pre-crisis period and the post-crisis period.  

For the full period, most models perform similarly to or worse than the AR benchmark for 1 step 

ahead forecasts of headline inflation. Beyond that horizon, the individual equations and, though 

to a lesser extent, the BVAR performed somewhat better. This seems to be mainly related to the 

post-crisis  period  where  these  models  show  a  somewhat  better  forecast  performance. 

Interestingly, the VAR_C has difficulties in forecasting 1 step ahead while it performs similarly 

to or better than the AR over the other horizons. All models perform worse than the AR in the 

pre-crisis  period  for  the  12  step  ahead  horizon,  with  particularly  large  differences  for  the 

individual equations, the BVAR and the unconditional VAR_U. 

For inflation excluding food and energy (i.e. for the less volatile component of HICP), two of the 

VAR models, VAR_U and VAR_C, cannot beat the AR in any of the situations, with very large 

differences  compared  to  the  AR  in  particular  beyond  the  1  step  ahead  horizon.  The  BVAR 

outperforms the AR model only for the 12-months ahead horizon for all samples. The individual 

equations  have  a  similar  performance  as  the  benchmark  over  the  full  period.  The  subsamples 

show that this model performs better over all horizons than the AR during the pre-crisis period, 

while the AR yields lower forecast errors in the post-crisis period. Note that the HICP excluding 

food and energy consists of the services component and of non-energy industrial goods, which 

are  two  components  with  relatively  stable  prices.  This  result  shows  that  the  relatively  weak 

forecast performance of the individual equations for headline inflation in the pre-crisis period is 

mainly  related  to the  difficulty  in  forecasting  energy  and  food inflation.  Meanwhile,  the VAR 

models seem to perform better for the volatile components than for the more stable ones. For the 

VAR_U, this could be related to the fact that the past movements of all sub-components of the 

HICP explain the current movement for each of them individually – meaning that core inflation 

(services and non-industrial goods price inflation) are also explained by food and energy prices 

which might not be appropriate. For the VAR_C, it could reflect the benefit of using commodity 

prices as explanatory variable for energy and food prices. 

One reason why the individual-equation framework performs relatively better in most situations 

than  the  VAR_C  (which  also  includes  variables  other  than  the  HICP  components)  is  that  the 

individual-equation  framework  equations  for the  two  components  of  the  HICP  excluding  food 

and energy, namely non-energy industrial goods and services, also include taxes. For the latter, 

no  change  was  assumed  when  the  different  forecasts  were  run.  Whether  a  model  forecast  is 

conditional  on  the  Eurosystem  projection  assumptions  seems  to  matter  somewhat  as  the 

unconditional VAR_U – the only of the four models which is not conditional - tends to have the 

highest relative RMSE.  

ECB Working Paper 1972, October 201616In addition to the RMSE, we have also computed the mean absolute deviation (MAD) to assess 

the forecast performance of the individual models with a measure that is robust to outliers (see 

Appendix  I  third  and  fourth  table).  Overall,  the  results  are  similar  to  those  obtained  with  the 

RMSE. The results are very similar across the univariate models. The errors for the BVAR are a 

bit higher than for the other models, while those for the individual equations and the conditional 

VAR are somewhat smaller. Meanwhile, the results for inflation excluding energy and food are 

somewhat better for the BVAR, while the two VAR models perform less well.  

4.3. 

Combination Results 

When  comparing  the  different  combination  schemes  for  the  full  period  for  HICP  headline 

inflation  (see  first row  left-hand  side of  Chart 7, the combination forecasts  perform  somewhat 

better  than  the  AR  benchmark  model.  The  differences  between  the  combination  methods  are 

rather small. This result is, however, mainly driven by the volatile components, as the forecast 

performance  for  the  different  combinations  differs  when  looking  at  HICP  excluding  food  and 

energy inflation (see first row left-hand side of Chart 7). It appears that the second combination 

scheme, which drops the VAR_C and the VAR_U and adds the ARIMA compared with the first 

combination scheme, works better than the first combination scheme and also the AR for core 

inflation. The simple average yields higher forecast errors than the two other weighting schemes, 

in  particular  for  the  first  model  combination.  Taking  geometrically  declining  weights  or  MSE 

weights improves the forecast performance over the AR benchmark.  

Looking  at  the  pre-crisis  period,  for  headline  inflation  none  of  the  combination  schemes 

outperforms the AR, while all of them yield lower forecast errors in the post-crisis period. Here 

again, the differences across models are minor, with the exception of the 12 step ahead forecast 

where the second combination works worse for the pre-crisis sample. This is consistent with the 

finding that the VAR models perform relatively better when including the volatile components. 

The  opposite  holds  true  for  core  inflation,  where  the  combination  schemes  outperform  the 

benchmark  in  the  pre-crisis  period  but  mostly  not  in  the  post-crisis  period.  The  second 

combination scheme again works somewhat better than the first, with a much stronger difference 

for the simple average. This is consistent with the finding above that most models outperform 

the AR in the post-crisis period for headline inflation, but not in the pre-crisis period. It could 

indicate that the AR is a relatively good model to forecast in particular the volatile component 

over a period which was mainly characterised by a trend increase in commodity prices, while the 

individual  models  which  partly  include  commodity  prices  work  better  in  a  period  with  higher 

volatility of commodity prices.  

These results are also in line with the mean absolute deviation, i.e. the combination results are 

similar  to  or  better  than  the  best  models,  and  confirm  that  forecast  combination  helps  hedge 
against the worst forecast performance.14 

14 We also compared forecasting the aggregate directly and aggregating the component forecasts (‘indirect’ forecasts) 
for  the  two  different  inflation  aggregates  we  consider  in  this  paper.  When  looking  at  the  results  for  the 
combination methods, we find that forecast accuracy is very similar between the direct and the indirect approach, 
both for headline and core inflation. We present the results in Appendix IV. 

ECB Working Paper 1972, October 201617Chart 7 

Results forecast combinations 

Headline inflation 

Excl. energy and food 

Simple 1
Geom. decl. 1
MSE weighted 2

MSE weighted 1
Simple 2
Geom. decl. 2

1

3

6

12

Forecast horizon

Simple 1
Geom. decl. 1
MSE weighted 2

MSE weighted 1
Simple 2
Geom. decl. 2

1

3

6

12

Forecast horizon

Simple 1
Geom. decl. 1
MSE weighted 2

MSE weighted 1
Simple 2
Geom. decl. 2

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1.2

1.0

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1.6

1.4

1.2

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Geom. decl. 1
MSE weighted 2

MSE weighted 1
Simple 2
Geom. decl. 2

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Forecast horizon

Simple 1
Geom. decl. 1
MSE weighted 2

MSE weighted 1
Simple 2
Geom. decl. 2

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Simple 1
Geom. decl. 1
MSE weighted 2

MSE weighted 1
Simple 2
Geom. decl. 2

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ECB Working Paper 1972, October 201618 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Time-varying Weights in Model combinations  

As  the  RMSEs  are  average  numbers  over  time,  it  is  useful  to  provide  additional  insight  by 

looking at the developments of the performance-based combination weights over time. All charts 

on the performance based weights for the headline (aggregate) inflation and the HICP excluding 

food  and  energy  for  both  combination  schemes  display  considerable  variation  of  the  weights 

over  time.  We  choose  to  display  the  weights  for  combination  scheme  2  with  backward 

geometrically declining weights since this scheme improved the forecast accuracy more than the 

other combination schemes. 

Looking  at  the  performance-based  combination  weights  in  Chart  8  shows that  there  are larger 

changes  in  the  relative  weights  starting  with  the  increased  volatility  in  inflation  in  mid-2007. 

Even though there are weight changes before, the change in 2007 seems to be somewhat broader 

across different horizons and models and more pronounced, in particular for the 6- and 12-month 

horizons. We find that both the rolling estimation window scheme with equal weights within the 

estimation  window  (not  shown  here)  and  the  backward  geometrically  declining  combination 

weights scheme (within the rolling window) giving recent forecasts more weight than those in 

the past, exhibit very similar performance. Also, we found similar developments of the relative 

forecast performance over time represented by the weights for both combination methods. Chart 

8 shows that the time-variation in the performance of the models is much more pronounced for 

inflation excluding  energy  and  food than  for headline  inflation. This is  presumably  due  to  the 

fact that for headline inflation the forecast performance of all models is negatively affected by 

the  volatile  components  of  inflation,  while  for  inflation  excluding  energy  and  food  the  model 

choice plays a more important role for the forecast accuracy.  

We  also  carried  out  Bai-Perron  tests  for  unknown,  multiple  breakpoints  for  the  performance 

weights of each of the models in combination scheme 2. The graphs with the significant break 

dates  for  the  weights  of  the  respective  forecasting  methods  and  horizons  are  presented  in 

Appendix III. The results show a large number of structural breaks in the weights of individual 

models  across  all  forecast  horizons,  models  and  the  two  aggregate  inflation  measures 

considered.  The  break  dates  in  the  weights  become  more  synchronised  when  the  changes  are 

more  pronounced,  as  with  longer  forecast  horizons.  This  becomes  particularly  apparent  for 

headline inflation (‘hlagg’) for a forecast horizon of 12 months where the break dates resulting 

from the Bai-Perron test are most synchronised across different methods in 2007.  

Considering  more  closely  the  time-varying  performance  of  the  different  models  for  inflation 

excluding energy and food for a 12-month horizon, we find that in 2007/2008 the BVAR gets 

substantially less weight than in other periods, while the Individual-Equations and the LSTAR 

get  more  weight.  This  might  indicate  that  the  interaction  between  the  different  inflation 

component equations in the BVAR is less important during this time, and that other factor might 

have  played  a  larger  role.  This  provides  an  example  how  the  analysis  of  the  time  variation  in 

forecast  performance  helps  to  interpret  the  resulting  combined  forecasts  and  improve  the 

forecasting models 

ECB Working Paper 1972, October 201619 
Chart 8 

Performance based weights, Combination 2 

  Headline inflation 

Excl. energy and food 

1

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Note: AR – Autoregressive model, VAR_C – conditional VAR, VAR_U_sub – unconditional VAR with disaggregate 

subcomponents  of  inflation,  LSTAR  –  logistic  smooth  transition  autoregressive  model,  BVAR  –  Bayesian  VAR, 

Ind.equ. – Individual equations 

4.4. 

Forecast Accuracy Tests 

To evaluate the statistical significance of the improvement in forecast accuracy of the different 

models  and  model  combinations  we  carry  out  the  Diebold-Mariano  (DM)  test  (Diebold  and 

ECB Working Paper 1972, October 201620 
 
 
 
 
 
 
 
 
 
 
 
 
Mariano,  1996,  see  also  West,  1996)  for  non-nested  forecast  model  comparisons  for  headline 
inflation and inflation excluding energy and food.15 

In the following, we present the results from the Diebold-Mariano (DM) tests for combination 

scheme  2,  since  that  scheme  performed  better  in  terms  of  forecast  accuracy  gains  than 

combination scheme 1. 

For  the  full  sample  period  (see  Tables  1a  and  1b),  the  combinations  perform  best  in  terms  of 

RMSFE and are  often significant, in particular for those with geometrically declining weights. 

For inflation excluding energy and food this is true for all horizons, while for headline inflation 

there is only significant improvement for horizons 3 and 6 months. 

 Table 1a Full sample, headline inflation, combination scheme 2 – relative RMSFE, 
DM test, turning point prediction 

Horizon

AR (benchmark)

1
t-stat 
adj

RMSFE 
rel.
0.16

UM
MA1
RIMA
RWAO
VAR_C
VAR_U_sub
LSTAR
BVAR
STIP

Simple avg
MSE Weighted avg
Geom Decl avg 

1.07
1.03
1.03
1.06
1.18
1.05
1.04
1.03
1.03

1.00
1.01
1.01

-1.18
-0.78
-1.22
-0.72
-2.19
-0.76
-0.92
-0.81
-0.46

0.49
0.43
0.39

3
t-stat 
adj

RMSFE 
rel.
0.35

1.06
1.05
1.00
1.10
1.04
1.03
0.99
1.00
0.99

0.98
0.98
0.98

-0.59
-0.21
0.10
-0.77
0.03
-0.76
0.90
0.99
0.47

1.77
1.78
1.78

DC

0.81

0.71
0.77
0.78
0.76
0.72
0.78
0.77
0.77
0.78

0.76
0.76
0.77

6
t-stat 
adj

RMSFE 
rel.
0.56

1.06
1.05
0.99
1.21
1.03
1.03
0.96
0.99
0.95

0.97
0.97
0.96

-0.94
-0.83
-0.16
-1.30
0.15
-0.88
0.48
0.66
0.38

1.48
1.50
1.52

DC

0.72

0.68
0.67
0.72
0.71
0.76
0.68
0.70
0.74
0.74

0.73
0.73
0.73

12
t-stat 
adj

RMSFE 
rel.
0.88

1.04
1.04
1.02
1.44
1.01
1.03
0.96
1.02
0.96

0.98
0.98
0.98

-1.22
-1.22
-1.53
-2.34
-0.28
-2.30
-0.47
-0.81
-0.42

0.34
0.41
0.43

DC

0.70

0.66
0.65
0.68
0.66
0.71
0.66
0.64
0.73
0.73

0.70
0.70
0.70

DC

0.70

0.76
0.78
0.72
0.63
0.77
0.78
0.72
0.73
0.76

0.72
0.73
0.73

Note: The different models and their abbreviations are described in Section 3.2. The table presents relative RMSFEs, 
except for the AR benchmark model where the absolute RMSFE is reported. The t-statistic unadjusted for estimation 
uncertainty  is  reported,  which  is  the  usual  Diebold-Mariano  statistic  and  the  critical  value  for  DM  test  from  the  t-
distribution  is  1.282  at  a  10%  significance  level.  The  accuracy  of  turning  point  predictions  are  evaluated  by  DC 
(directon of change or turning point) counting the fraction of times that a change in the right direction was predicted. 
Combination 2 includes individual equation framework, BVAR, ARIMA, STAR, AR. Green shaded (medium grey) 
values mark the models with RMSFE smaller than the AR benchmark, orange coloring (light greay) indicates those 
models  that  have  significantly  lower  RMSFE.  Blue  shading  (dark  grey)  indicates  the  best  performance  in  terms  of 
turning points.   

15 Note that the benchmark model is the aggregate AR model, while the alternative model we consider in this table are 

the aggregated model forecasts of the components. Further results are presented in Appendix IV. 

ECB Working Paper 1972, October 201621 
 
 
 
 
 
 
 
 
Table  1b  Full  sample,  inflation  excl.  energy  and  food,  combination  scheme  2  – 
relative RMSFE, DM test, turning point prediction 

Horizon

AR (benchmark)

1
t-stat 
adj

RMSFE 
rel.
0.08

UM
MA1
RIMA
RWAO
VAR_C
VAR_U_sub
LSTAR
BVAR
Ind. equ.

1.28
1.28
0.99
1.11
1.40
1.26
1.26
1.13
1.03

Simple avg 2
1.01
MSE Weighted avg 2 1.00
1.00
Geom Decl avg 2

Note: See Table 1. 

-2.87
-2.68
1.63
-0.79
-3.13
-3.65
-1.21
-1.71
0.91

0.98
1.24
1.22

3
t-stat 
adj

RMSFE 
rel.
0.11

1.77
1.74
0.99
1.07
1.75
1.67
1.16
1.09
1.03

0.96
0.95
0.95

-4.07
-4.19
-1.13
-1.09
-4.28
-5.63
-1.58
-1.68
0.29

1.08
1.18
1.35

DC
0.79

0.70
0.69
0.80
0.77
0.70
0.69
0.75
0.74
0.82

0.83
0.83
0.81

6
t-stat 
adj

RMSFE 
rel.
0.18

2.10
2.07
0.97
1.13
1.94
1.99
1.04
1.02
1.00

0.90
0.90
0.88

-4.29
-4.25
-1.28
-1.99
-3.97
-4.70
-1.08
-0.53
0.70

1.67
1.74
2.17

DC
0.76

0.57
0.58
0.75
0.74
0.59
0.62
0.70
0.73
0.74

0.77
0.75
0.75

12
t-stat 
adj

RMSFE 
rel.
0.37

1.94
1.93
0.92
1.14
1.79
1.87
0.87
0.87
0.99

0.84
0.82
0.79

-3.79
-3.73
-1.28
-2.17
-3.32
-3.30
-0.02
0.79
1.03

2.66
3.02
3.38

DC
0.79

0.46
0.46
0.80
0.72
0.46
0.53
0.75
0.75
0.75

0.80
0.80
0.81

DC
0.68

0.45
0.45
0.78
0.39
0.43
0.45
0.76
0.76
0.62

0.80
0.79
0.82

Table  2a  Pre-crisis  sample,  headline  inflation,  combination  scheme  2  –  relative 
RMSFE, DM test, turning point prediction  

Horizon

AR (benchmark)

1
t-stat 
adj

RMSFE 
rel.
0.14

UM
MA1
RIMA
RWAO
VAR_C
VAR_U_sub
LSTAR
BVAR
STIP

Simple avg
MSE Weighted avg
Geom Decl avg 

Note: See Table 1. 

0.97
0.96
1.02
1.00
1.20
1.06
1.06
1.01
0.99

0.99
0.99
0.99

0.54
0.83
-1.52
-0.82
-3.29
-0.79
-1.42
-0.68
-0.87

-0.80
-0.90
-0.94

3
t-stat 
adj

RMSFE 
rel.
0.26

0.95
0.95
1.00
1.02
0.94
1.00
0.98
1.00
0.98

0.97
0.97
0.97

1.73
1.71
-1.31
-0.47
1.36
-0.77
0.15
-0.15
-0.84

-0.03
-0.06
-0.05

DC

0.79

0.71
0.74
0.79
0.74
0.72
0.77
0.75
0.75
0.79

0.75
0.75
0.75

6
t-stat 
adj

RMSFE 
rel.
0.32

0.93
0.94
0.99
1.06
0.91
1.05
0.97
0.98
1.01

0.96
0.96
0.96

1.87
1.52
-1.96
-1.80
1.32
-1.87
-1.31
-1.32
-1.66

-0.30
-0.31
-0.31

DC

0.65

0.68
0.67
0.68
0.65
0.71
0.68
0.65
0.68
0.70

0.68
0.68
0.68

12
t-stat 
adj

RMSFE 
rel.
0.36

0.83
0.85
1.00
1.04
0.79
0.96
1.02
1.03
1.11

0.98
0.98
0.98

0.70
0.45
-2.32
-3.38
-0.03
-2.64
-1.87
-2.42
-2.05

-1.32
-1.32
-1.32

DC

0.67

0.65
0.58
0.60
0.52
0.65
0.58
0.58
0.62
0.68

0.65
0.67
0.67

DC

0.61

0.72
0.74
0.63
0.52
0.69
0.72
0.57
0.59
0.61

0.59
0.59
0.59

ECB Working Paper 1972, October 201622 
 
 
 
 
 
Table 2b Pre-crisis sample, inflation excl. energy and food, combination scheme 2 – 
relative RMSFE, DM test, turning point prediction 

Horizon

AR (benchmark)

1
t-stat 
adj

RMSFE 
rel.
0.06

UM
MA1
RIMA
RWAO
VAR_C
VAR_U_sub
LSTAR
BVAR
Ind. equ.

1.22
1.21
1.00
1.07
1.23
1.41
1.65
1.14
0.98

1.02
Simple avg 2
MSE Weighted avg 2 1.00
1.00
Geom Decl avg 2

Note: See Table 1. 

-0.74
-0.43
1.06
1.10
-1.20
-1.17
-1.13
-0.34
2.09

0.53
1.11
1.17

3
t-stat 
adj

RMSFE 
rel.
0.10

1.66
1.62
1.03
1.13
1.55
1.74
1.38
1.17
0.90

0.97
0.96
0.95

-1.98
-2.01
-1.33
-0.62
-2.25
-2.90
-1.37
-1.50
1.57

0.78
1.03
1.19

DC

0.80

0.77
0.74
0.77
0.75
0.72
0.72
0.69
0.68
0.83

0.79
0.79
0.77

6
t-stat 
adj

RMSFE 
rel.
0.17

1.86
1.82
1.00
1.20
1.69
1.77
1.03
1.09
0.89

0.89
0.89
0.86

-2.11
-2.03
-1.45
-1.36
-1.95
-1.99
-0.81
-0.61
1.58

1.35
1.41
1.70

12
t-stat 
adj

RMSFE 
rel.
0.35

1.81
1.79
0.91
1.17
1.62
1.68
0.75
0.78
0.92

0.79
0.77
0.75

-1.36
-1.29
-1.59
-1.46
-0.85
-0.71
-0.16
0.76
1.71

2.03
2.15
2.27

DC

0.78

0.52
0.53
0.78
0.72
0.57
0.63
0.72
0.70
0.80

0.83
0.82
0.83

DC

0.71

0.67
0.67
0.70
0.68
0.68
0.67
0.70
0.70
0.78

0.75
0.70
0.70

DC

0.72

0.54
0.56
0.78
0.44
0.50
0.56
0.72
0.78
0.72

0.80
0.83
0.82

Table  3a  Post-crisis  sample,  headline  inflation,  combination  scheme  2  –  relative 
RMSFE, DM test, turning point prediction 

Horizon

AR (benchmark)

1
t-stat 
adj

RMSFE 
rel.
0.13

UM
MA1
RIMA
RWAO
VAR_C
VAR_U_sub
LSTAR
BVAR
STIP

Simple avg
MSE Weighted avg
Geom Decl avg 

Note: See Table 1. 

1.16
1.07
1.00
1.01
1.18
1.10
1.05
1.09
1.05

1.04
1.02
1.02

-1.71
-1.58
0.79
-0.76
-1.53
-1.27
0.35
-0.96
-0.47

0.54
0.53
0.53

3
t-stat 
adj

RMSFE 
rel.
0.32

1.10
1.09
1.02
0.93
1.04
1.10
1.02
1.01
0.96

1.00
0.99
0.99

-1.75
-1.86
1.70
-0.72
-0.79
-1.31
1.34
1.29
1.56

1.25
1.28
1.28

DC

0.82

0.70
0.78
0.76
0.78
0.74
0.74
0.76
0.80
0.74

0.76
0.76
0.76

6
t-stat 
adj

RMSFE 
rel.
0.46

1.14
1.12
1.01
1.01
1.01
1.10
1.01
0.96
0.94

0.98
0.97
0.97

-2.08
-2.19
1.54
-1.11
-0.54
-0.05
1.38
1.93
2.30

1.02
1.05
1.07

DC

0.69

0.60
0.58
0.69
0.65
0.73
0.58
0.63
0.71
0.67

0.69
0.69
0.69

12
t-stat 
adj

RMSFE 
rel.
0.78

1.07
1.06
1.01
1.45
0.94
1.04
0.98
0.97
0.89

0.93
0.95
0.95

-1.84
-1.89
-0.04
-2.40
-0.40
-1.60
1.21
0.45
1.46

0.33
0.38
0.40

DC

0.67

0.53
0.58
0.62
0.71
0.69
0.56
0.58
0.73
0.69

0.67
0.67
0.67

DC

0.59

0.69
0.69
0.62
0.59
0.74
0.72
0.72
0.77
0.80

0.69
0.74
0.74

ECB Working Paper 1972, October 201623 
 
 
 
 
Table 3b Post-crisis sample, inflation excl. energy and food, combination scheme 2 
– relative RMSFE, DM test, turning point prediction 

Horizon

AR (benchmark)

1
t-stat 
adj

RMSFE 
rel.
0.09

UM
MA1
RIMA
RWAO
VAR_C
VAR_U_sub
LSTAR
BVAR
Ind. equ.

1.30
1.31
0.97
1.15
1.19
1.42
1.06
1.11
1.07

1.00
Simple avg 2
MSE Weighted avg 2 1.00
1.00
Geom Decl avg 2

Note: See Table 1. 

-3.92
-3.52
0.75
-1.94
-3.07
-4.03
-1.77
-2.04
-0.70

0.89
0.90
0.88

3
t-stat 
adj

RMSFE 
rel.
0.12

1.85
1.83
0.94
1.05
1.54
1.84
0.99
1.06
1.12

0.96
0.96
0.95

-4.85
-5.00
-0.11
-2.00
-4.26
-5.36
-1.30
-0.79
-1.53

0.57
0.60
0.60

DC

0.78

0.68
0.68
0.82
0.76
0.72
0.66
0.82
0.82
0.80

0.86
0.88
0.86

6
t-stat 
adj

RMSFE 
rel.
0.17

2.51
2.48
0.94
1.11
2.07
2.33
1.04
0.97
1.19

0.95
0.95
0.93

-4.98
-5.02
-0.09
-2.35
-4.36
-5.93
-1.01
0.12
-0.77

0.29
0.37
0.57

12
t-stat 
adj

RMSFE 
rel.
0.35

2.36
2.34
0.87
1.14
2.07
2.18
0.98
0.82
1.19

0.89
0.86
0.84

-5.11
-5.12
1.33
-2.26
-4.67
-5.57
0.34
1.63
-0.75

1.13
1.66
2.06

DC

0.82

0.44
0.44
0.89
0.73
0.49
0.47
0.82
0.84
0.78

0.82
0.82
0.82

DC

0.83

0.56
0.56
0.81
0.79
0.60
0.60
0.73
0.83
0.71

0.81
0.81
0.81

DC

0.64

0.44
0.44
0.92
0.36
0.44
0.44
0.82
0.87
0.54

0.90
0.90
0.90

For  the  pre-crisis  period, the  combinations also  significantly  outperform  the  single  models  for 

inflation excluding energy and food over the longer horizon (see Table 2b), with the individual 

equations also significantly improving the forecast performance for most horizons. For headline 

inflation,  the  MA(1)  model,  the  random  walk  with  drift  (UM)  and  the  conditional  VAR 

significantly outperform the AR model (see Table 2a). 

For the post-crisis period, the combinations and the ARIMA and the BVAR model outperform 

the AR model for inflation excluding energy and food over a 12-month horizon only (see Table 

3b), while for headline inflation there is a significant improvement for the combinations and the 

individual equation framework over a 3-month horizon (see Table 3a). The individual equation 

framework also significantly improves over the AR model for the 6- and 12-month horizons. 

Overall,  these  results  suggest  that  forecast  combination  leads  to  significant  improvements  in 

terms  of  RMSFE for core inflation,  and  occasionally also  for  headline  inflation except  for the 

pre-crisis  period.  For  headline  inflation  during  the  pre-crisis  period,  simple  univariate  linear 

models like the RW with drift and the MA(1) are usually significantly better than the AR model, 

while  for  the  post-crisis  period  it  is  the  BVAR  and  the  individual  equation  framework  that 
improve significantly over the AR model.16 This result is in line with results from the literature 
that  during  the  Great  moderation  (pre-crisis)  it  was  very  difficult  to  improve  over  simple 

univariate models since there was not much variation in inflation to be explained (see e.g. Stock 

and  Watson,  2008,  and  Hendry  and  Hubrich,  2011).  On  the  other  hand,  the  combinations 

outperform the AR model over forecast horizons of 3, 6 and 12 months for inflation excluding 

16These test results are confirmed when carrying out the tests proposed by Hubrich and West (2011), which take into 
account  that  a  small  number  of  forecast  models  are  compared  here  instead  of  only  pairwise  comparisons  of 
forecasts by using the appropriate larger critical values. The results are available from the authors upon request. 

ECB Working Paper 1972, October 201624 
 
 
energy and food for the full and the pre-crisis sample. This improvement is significant for 6- and 

12-month  horizons  and  for  the  full  sample  also  for  a  3-  month  horizon.  More  generally,  our 

evidence  suggests  using  combination  schemes,  because  according  to  our  full  sample  results, 

combination forecasts improve significantly over the autoregressive benchmark model for both 

headline inflation and inflation excluding energy and food. For the pre- and post-crisis episodes, 

combinations  perform  at  least  as  well  as  the  benchmark  or  improve  over  it,  and  the 

improvements are often significant. 

We have also carried out DM tests comparing the different combination schemes (see Table 4). 

We  find  that  the  MSE-weighted  combination  scheme  outperforms  the  simple  average  for  a 

forecast horizon of 1 month, but for horizons of 6 and 12 months, the geometrically declining 

weights  combination  forecasts  outperform  the  simple  average.  Comparing  the  MSE-weighted 

combination  scheme  with  the  geometrically  declining  one  (not  presented  in  the  tables,  but 

available upon request), we find that the latter combination scheme outperforms the former only 

for  headline  inflation  and  for  a  1-month  horizon,  but  otherwise  the  two  forecast  combination 

methods are not significantly different from each other. It is plausible that giving higher weight 
to models that do best in the most recent period improves the short-term horizon forecast. 

Table 4: DM test, different forecast combinations vs simple average  

MSE weighted  Geom. decl. weights 

Headline inflation 

Inflation excl. energy and food 

Headline inflation 

Inflation excl. energy and food 

Headline inflation 

h=1 

h=1 

h=6 

h=6 

h=12 

Inflation excl. energy and food 

h=12 

1.745 

-1.185 

-1.237 

0.277 

-2.524 

-1.194 

-1.298 

-0.589 

2.039 

2.906 

2.859 

3.405 

Note:  t-statistics  reported;  forecast  horizons  are  h=1,6  and  12  months,  yellow  shading  indicates  significantly  better 
weighting scheme; benchmark model for this comparison is the simple average; combination scheme 2 is underlying 
this comparison, i.e. the four best forecast models and the AR benchmark are included in the forecast combinations. 

4.5. 

Turning Point Predictions 

The accuracy of the turning point predictions of the different models is evaluated by the fraction 

of times forecasts predicted a change in the right direction, denoted “DC” in the tables. As can 

be seen in Table 1b, over the full sample, for inflation excluding energy and food, the forecast 

combinations perform best in terms of turning point prediction, whereas for headline inflation, 

the  VARs  (including  VAR_C,  VAR_U_SUB,  BVAR)  and  the  individual  equation  framework 

tend to perform best (see Table 1a), where the exact best model depends on the horizon. 

For the pre-crisis period (Table 2a and 2b), the combinations also outperform the single models 

for  the  longer  horizon  in  terms  of  turning  point  prediction  for  inflation  excluding  energy  and 

ECB Working Paper 1972, October 201625 
 
food.  Otherwise,  it  is  the  individual-equation  framework  that  performs  best.  For  headline 

inflation, the best models include the individual-equation framework, VAR_C and MA(1). 

For the post-crisis period (Table 3b), the combination, BVAR and ARIMA models do best for 

inflation  excluding  energy  and  food  in  terms  of  turning  point  prediction,  depending  on  the 

horizon, while for headline inflation, the AR, the VAR_C, the BVAR or the individual equation 

framework provide the most accurate turning-point prediction depending on the horizon. 

Overall, forecast combinations outperform single models for inflation excluding energy and food 

for all horizon over the full sample, while for headline inflation, usually single models prevail, 

but which particular single model performs best changes with the forecast horizon and forecast 

evaluation period considered. Since combinations usually still perform better than many of the 

single models, and there is some variation concerning which model performs best over time and 

horizons, we conclude from the turning-point prediction evidence that combining forecasts still 

hedges against bad performance of some models. 

5. 

Conclusions 

The  extraordinary  volatility  in  inflation  during  the  global  financial  crisis  led  to  large  forecast 

errors  of  inflation  forecasting  models.  This  raised  the  question  whether  forecast  combination 

methods  can  help  to  improve  forecast  performance  and/or  provide  more  robust  forecast 

performance in the presence of large shocks. 

We provide evidence that forecast combination methods have helped in the crisis period for the 

models  considered  here.  We  find  that:  1.  performance-based  combination  methods  tend  to 

perform at least as well and, in a number of cases, significantly better than the AR benchmark 

model;  2.  forecast  combination  methods  also  outperform  a  large  number  of  the  alternative 

models;  3.  the  relative  forecast  performance  is  particularly  good  for  headline  inflation  in  the 

post-crisis  period,  and  for  core  inflation  in  the  pre-crisis  period.  For  the  full  sample,  which 

includes  substantially  different  episodes  in  terms  of  inflation  dynamics,  forecast  combination 

methods exhibit comparatively high forecast accuracy for both headline and core inflation;   4. 

Forecast  combination  for  the  full  sample  period  typically  improves  forecast  accuracy  over  the 

autoregressive benchmark model for core and headline inflation, and often improves over single 

multivariate  models.  The  forecast  accuracy  gain  of  combinations  is  largest  for  inflation 

excluding  energy  and  food  for  the  full  sample;  5.  We  find  that  performance-based  forecast 

combination improves significantly over the simple average for our application..  

We find that the best model for forecasting differs depending on whether the overall HICP or the 

HICP  excluding  food  and  energy  is  considered,  and  which  period  and  forecast  horizon  is 

studied.  Therefore  we  conclude  that  performance-based  forecast  combination  helps  to  hedge 

against bad forecast performance of some of the models in some situations, even though in the 

ECB Working Paper 1972, October 201626 
presence of large shocks or crises it does not necessarily improve over the best forecast model 

since  the  forecast  accuracy  of  all  models  might,  for  example,  be  biased  in  the  same  way. 

Performance-based forecast combination appears to be useful when the models included in the 

set of models exhibit very different forecast performance over time. 

Investigating combination weights and their development over time, we find significant changes 

in the weights. Moreover, there appears to be more pronounced evidence for a structural break in 

the forecast model performance around the time of the recent global financial crisis, in particular 

for longer horizons. This time-variation in the weights can be seen as capturing non-linearities in 

the underlying economic relationships. For instance, the importance of certain variables, such as 

oil  prices  or  labour-market  variables,  might  change  over  time.  In  episodes  of  more  volatile 

inflation, a multivariate model allowing for feedback effects between inflation and its predictors 

might  improve  forecast  accuracy.  The  time-variation  in  the  weights  assigned  to  the  forecasts 

from  different  single  models  can  help  to  interpret  the  combination  forecast  and  improve  the 

forecasting models. 

Overall, we conclude from our evidence taking into account RMSFE, forecast accuracy tests and 

turning point predictions that for euro area inflation, first, performance-based combination tends 

to  outperform  simple  averages,  and,  second,  that  performance-based  forecast  combination 

protects against bad forecasts from single models, thereby making the forecast more robust. 

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Benalal, Diaz Del Hoyo, Landau, Roma and Skudelny (2004), To aggregate or not to aggregate? 

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April. 

ECB Working Paper 1972, October 201627 
 
 
Genre,  V.,  Kenny,  G.,  Meyler,  A.  and  Timmermann,  A.  (2010),  Combining  Expert  Forecasts: 
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Hubrich,  K.  (2005).  Forecasting  euro  area  inflation:  Does  aggregating  forecasts  by  HICP 
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in  Econometrics  “VAR  Models 

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1067 – 1084 

ECB Working Paper 1972, October 201628 
Appendix I 

 Forecast errors (RMSE and MAD)  

RMSE - HICP - full period

RWAO UM
0.17
0.17
0.28
0.28
0.37
0.39
0.45
0.48
0.53
0.58
0.59
0.68
0.66
0.77
0.72
0.88
0.77
0.97
0.82
1.07
0.87
1.16
0.92
1.26
0.92
1.30
0.93
1.34
0.93
1.36
0.93
1.38
0.94
1.39
0.94
1.40

AR
0.16
0.26
0.35
0.42
0.49
0.56
0.62
0.68
0.74
0.79
0.83
0.88
0.89
0.90
0.91
0.91
0.92
0.92

RIMA MA1
0.17
0.17
0.27
0.27
0.37
0.35
0.44
0.42
0.52
0.49
0.59
0.55
0.65
0.62
0.71
0.68
0.77
0.74
0.82
0.80
0.86
0.85
0.91
0.90
0.92
0.91
0.93
0.93
0.93
0.94
0.93
0.95
0.94
0.95
0.94
0.95

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

RMSE - HICP - pre-crisis period

RWAO UM
0.13
0.14
0.20
0.21
0.25
0.27
0.27
0.30
0.29
0.33
0.30
0.34
0.30
0.36
0.31
0.38
0.31
0.38
0.29
0.37
0.29
0.35
0.30
0.37
0.30
0.36
0.30
0.34
0.30
0.34
0.30
0.33
0.30
0.32
0.30
0.33

AR
0.14
0.21
0.26
0.29
0.31
0.32
0.32
0.33
0.33
0.34
0.34
0.36
0.35
0.34
0.34
0.34
0.31
0.29

RIMA MA1
0.13
0.14
0.20
0.22
0.25
0.26
0.28
0.29
0.29
0.31
0.30
0.32
0.31
0.32
0.32
0.33
0.31
0.34
0.30
0.34
0.29
0.34
0.30
0.36
0.30
0.34
0.30
0.33
0.30
0.33
0.30
0.32
0.31
0.32
0.30
0.31

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

RMSE - HICP - post-crisis period

RWAO UM
0.15
0.13
0.26
0.22
0.35
0.29
0.41
0.35
0.47
0.40
0.53
0.47
0.59
0.57
0.65
0.69
0.70
0.79
0.74
0.90
0.78
1.01
0.83
1.13
0.83
1.19
0.84
1.26
0.84
1.32
0.84
1.38
0.84
1.43
0.84
1.48

AR
0.13
0.23
0.31
0.37
0.41
0.46
0.53
0.60
0.65
0.69
0.73
0.78
0.80
0.81
0.83
0.83
0.84
0.85

RIMA MA1
0.14
0.13
0.25
0.24
0.34
0.32
0.41
0.37
0.46
0.42
0.52
0.47
0.58
0.54
0.65
0.61
0.70
0.66
0.74
0.70
0.78
0.73
0.83
0.78
0.83
0.80
0.84
0.81
0.84
0.82
0.84
0.82
0.84
0.83
0.84
0.84

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

Ind. 
equ.
0.17
0.26
0.35
0.41
0.47
0.53
0.59
0.64
0.70
0.75
0.79
0.84
0.85
0.87
0.88
0.88
0.89
0.90

Ind. 
equ.
0.14
0.21
0.26
0.29
0.31
0.31
0.32
0.33
0.33
0.35
0.35
0.37
0.35
0.34
0.33
0.31
0.29
0.29

Ind. 
equ.
0.14
0.23
0.30
0.35
0.39
0.44
0.49
0.55
0.59
0.62
0.65
0.70
0.70
0.72
0.73
0.73
0.74
0.75

LSTAR
0.17
0.26
0.35
0.41
0.47
0.53
0.59
0.65
0.71
0.75
0.80
0.84
0.87
0.88
0.89
0.90
0.90
0.91

LSTAR
0.14
0.21
0.26
0.28
0.30
0.31
0.33
0.34
0.34
0.34
0.34
0.36
0.40
0.40
0.40
0.39
0.39
0.39

LSTAR
0.14
0.24
0.32
0.37
0.42
0.47
0.53
0.59
0.64
0.68
0.72
0.76
0.78
0.79
0.79
0.79
0.79
0.79

VAR_U_ 
sub
0.17
0.27
0.36
0.44
0.51
0.57
0.64
0.70
0.75
0.80
0.85
0.90
0.91
0.92
0.92
0.93
0.93
0.93

VAR_U_ 
sub
0.14
0.21
0.26
0.29
0.32
0.33
0.34
0.35
0.33
0.34
0.34
0.34
0.30
0.29
0.29
0.29
0.29
0.28

VAR_U_ 
sub
0.15
0.26
0.34
0.41
0.45
0.51
0.57
0.64
0.69
0.72
0.76
0.81
0.82
0.83
0.83
0.83
0.83
0.83

VAR_C
0.19
0.27
0.37
0.44
0.51
0.57
0.63
0.69
0.75
0.80
0.84
0.89
0.89
0.90
0.90
0.91
0.91
0.91

VAR_C
0.16
0.21
0.25
0.27
0.29
0.29
0.29
0.29
0.28
0.28
0.28
0.28
0.27
0.27
0.27
0.27
0.26
0.26

VAR_C
0.16
0.24
0.33
0.38
0.42
0.47
0.52
0.58
0.63
0.66
0.69
0.73
0.74
0.75
0.76
0.76
0.77
0.77

MSE 
weight-
ed avg1
0.16
0.26
0.35
0.41
0.48
0.54
0.60
0.66
0.71
0.76
0.80
0.85
0.86
0.88
0.88
0.89
0.90
0.90

MSE 
weight-
ed avg1
0.14
0.20
0.25
0.28
0.30
0.30
0.30
0.31
0.30
0.30
0.30
0.30
0.28
0.27
0.27
0.26
0.25
0.25

MSE 
weight-
ed avg1
0.14
0.24
0.31
0.36
0.40
0.45
0.51
0.57
0.62
0.64
0.67
0.72
0.74
0.75
0.77
0.77
0.78
0.78

Simple 
avg 1
0.16
0.26
0.35
0.42
0.48
0.54
0.60
0.66
0.71
0.76
0.81
0.86
0.87
0.88
0.89
0.90
0.90
0.91

Simple 
avg 1
0.14
0.20
0.25
0.28
0.30
0.30
0.30
0.31
0.30
0.30
0.30
0.31
0.29
0.29
0.28
0.28
0.27
0.26

Simple 
avg 1
0.14
0.24
0.31
0.37
0.41
0.45
0.51
0.57
0.62
0.65
0.68
0.73
0.74
0.76
0.77
0.78
0.78
0.79

Geom 
decl 
avg 1
0.16
0.26
0.35
0.41
0.48
0.54
0.60
0.66
0.71
0.75
0.80
0.85
0.86
0.87
0.88
0.89
0.89
0.89

Geom 
decl 
avg 1
0.14
0.20
0.25
0.28
0.30
0.30
0.30
0.30
0.29
0.29
0.29
0.30
0.27
0.26
0.26
0.25
0.24
0.24

Geom 
decl 
avg 1
0.14
0.24
0.31
0.36
0.40
0.45
0.51
0.57
0.61
0.64
0.67
0.71
0.73
0.74
0.75
0.76
0.77
0.77

MSE 
weight-
ed avg2
0.16
0.26
0.35
0.41
0.48
0.54
0.60
0.66
0.72
0.76
0.81
0.86
0.88
0.89
0.90
0.91
0.91
0.92

MSE 
weight-
ed avg2
0.13
0.21
0.25
0.28
0.30
0.31
0.31
0.32
0.32
0.33
0.33
0.35
0.34
0.34
0.33
0.32
0.31
0.30

MSE 
weight-
ed avg2
0.13
0.23
0.31
0.36
0.40
0.45
0.52
0.58
0.62
0.66
0.69
0.74
0.76
0.78
0.79
0.79
0.80
0.81

Simple 
avg 2
0.16
0.26
0.35
0.41
0.48
0.54
0.60
0.66
0.72
0.77
0.81
0.86
0.88
0.90
0.91
0.91
0.92
0.92

Simple 
avg 2
0.13
0.21
0.25
0.28
0.30
0.31
0.31
0.32
0.32
0.33
0.33
0.35
0.34
0.34
0.33
0.32
0.31
0.31

Simple 
avg 2
0.13
0.23
0.31
0.36
0.40
0.45
0.52
0.58
0.63
0.66
0.70
0.75
0.76
0.78
0.80
0.80
0.81
0.82

Geom 
decl 
avg 2
0.16
0.26
0.35
0.41
0.48
0.54
0.60
0.66
0.71
0.76
0.81
0.86
0.88
0.89
0.90
0.91
0.91
0.92

Geom 
Decl 
avg 2
0.14
0.21
0.25
0.28
0.30
0.31
0.31
0.32
0.32
0.33
0.33
0.35
0.34
0.33
0.33
0.32
0.31
0.30

Geom 
Decl 
avg 2
0.13
0.23
0.31
0.36
0.40
0.45
0.51
0.58
0.62
0.66
0.69
0.74
0.76
0.77
0.79
0.79
0.80
0.80

BVAR
0.17
0.26
0.35
0.42
0.49
0.55
0.61
0.67
0.73
0.78
0.83
0.89
0.92
0.94
0.95
0.97
0.98
0.98

BVAR
0.14
0.20
0.26
0.29
0.31
0.32
0.33
0.33
0.33
0.35
0.37
0.39
0.38
0.37
0.36
0.37
0.38
0.39

BVAR
0.14
0.24
0.32
0.36
0.40
0.45
0.51
0.58
0.63
0.66
0.70
0.76
0.79
0.82
0.86
0.87
0.89
0.91

Note: Average 1: AR, VAR_C, VAR_U_sub, LSTAR, BVAR, individual equations; Average 2: AR, RIMA, LSTAR, BVAR, individual equations

ECB Working Paper 1972, October 201629 
RMSE - HICP excluding food and energy - full period

RWAO UM
0.10
0.09
0.14
0.10
0.20
0.12
0.25
0.14
0.31
0.17
0.37
0.20
0.43
0.23
0.48
0.27
0.54
0.31
0.60
0.34
0.66
0.38
0.72
0.42
0.72
0.43
0.73
0.45
0.73
0.47
0.74
0.48
0.74
0.50
0.74
0.50

AR
0.08
0.09
0.11
0.13
0.15
0.18
0.21
0.24
0.27
0.30
0.33
0.37
0.38
0.40
0.41
0.43
0.44
0.45

RIMA MA1
0.10
0.08
0.14
0.09
0.19
0.11
0.25
0.13
0.30
0.15
0.36
0.17
0.42
0.20
0.48
0.23
0.54
0.26
0.60
0.28
0.65
0.31
0.71
0.34
0.72
0.34
0.73
0.36
0.73
0.37
0.74
0.38
0.74
0.39
0.74
0.40

LSTAR
0.10
0.11
0.13
0.15
0.16
0.18
0.21
0.23
0.25
0.27
0.30
0.32
0.33
0.34
0.35
0.36
0.37
0.38

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

RMSE - HICP excluding food and energy - pre-crisis period

RWAO UM
0.08
0.07
0.12
0.09
0.16
0.11
0.21
0.14
0.26
0.17
0.31
0.20
0.36
0.24
0.42
0.27
0.47
0.30
0.53
0.34
0.58
0.38
0.64
0.41
0.65
0.43
0.66
0.44
0.66
0.45
0.67
0.46
0.68
0.47
0.69
0.47

AR
0.06
0.08
0.10
0.12
0.14
0.17
0.20
0.23
0.26
0.29
0.32
0.35
0.36
0.37
0.38
0.38
0.39
0.40

RIMA MA1
0.08
0.06
0.11
0.08
0.16
0.10
0.20
0.12
0.25
0.14
0.31
0.17
0.36
0.20
0.41
0.23
0.47
0.25
0.52
0.27
0.58
0.29
0.63
0.32
0.65
0.32
0.66
0.33
0.67
0.32
0.67
0.33
0.68
0.33
0.69
0.33

LSTAR
0.10
0.12
0.13
0.15
0.16
0.17
0.19
0.21
0.21
0.23
0.24
0.26
0.27
0.29
0.31
0.33
0.36
0.38

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

Ind. 
equ.
0.08
0.09
0.11
0.13
0.15
0.17
0.21
0.24
0.27
0.30
0.33
0.36
0.37
0.40
0.42
0.43
0.45
0.46

Ind. 
equ.
0.06
0.07
0.09
0.10
0.13
0.15
0.18
0.21
0.23
0.26
0.29
0.33
0.34
0.35
0.36
0.37
0.39
0.40

BVAR
0.09
0.10
0.12
0.14
0.15
0.18
0.20
0.22
0.25
0.27
0.29
0.32
0.32
0.33
0.34
0.35
0.36
0.37

BVAR
0.07
0.09
0.11
0.13
0.16
0.18
0.21
0.23
0.24
0.25
0.26
0.28
0.27
0.26
0.25
0.24
0.25
0.25

VAR_U_ 
sub
0.11
0.15
0.20
0.24
0.29
0.34
0.39
0.45
0.50
0.55
0.61
0.66
0.67
0.68
0.69
0.70
0.71
0.71

VAR_U_ 
sub
0.09
0.13
0.17
0.21
0.25
0.30
0.34
0.39
0.44
0.49
0.54
0.59
0.60
0.61
0.62
0.63
0.64
0.64

VAR_C
0.10
0.13
0.19
0.24
0.29
0.35
0.41
0.46
0.52
0.58
0.63
0.69
0.70
0.70
0.71
0.70
0.70
0.70

VAR_C
0.08
0.11
0.15
0.19
0.24
0.28
0.33
0.38
0.43
0.48
0.52
0.57
0.59
0.59
0.59
0.59
0.59
0.59

MSE 
weight-
ed avg1
0.08
0.10
0.12
0.13
0.15
0.18
0.20
0.23
0.26
0.28
0.31
0.34
0.34
0.36
0.37
0.38
0.39
0.40

MSE 
weight-
ed avg1
0.07
0.08
0.10
0.12
0.14
0.16
0.19
0.22
0.24
0.26
0.28
0.32
0.32
0.33
0.33
0.33
0.34
0.34

Simple 
avg 1
0.08
0.10
0.13
0.15
0.17
0.20
0.24
0.27
0.31
0.34
0.37
0.40
0.41
0.42
0.43
0.44
0.45
0.45

Simple 
avg 1
0.07
0.09
0.11
0.13
0.15
0.18
0.20
0.24
0.26
0.28
0.31
0.34
0.34
0.35
0.36
0.36
0.37
0.38

RMSE - HICP excluding food and energy - post-crisis period

RWAO UM
0.12
0.10
0.16
0.11
0.23
0.13
0.29
0.15
0.35
0.16
0.42
0.19
0.49
0.22
0.56
0.24
0.63
0.29
0.70
0.32
0.76
0.36
0.83
0.40
0.83
0.42
0.84
0.44
0.84
0.47
0.84
0.49
0.85
0.52
0.85
0.53

LSTAR
0.09
0.10
0.12
0.14
0.15
0.18
0.20
0.23
0.26
0.29
0.31
0.35
0.35
0.36
0.38
0.39
0.41
0.42
Note: Average 1: UM, AR, MA1, RIMA, RWAO, VAR_C, VAR_U_sub, LSTAR, BVAR, STIP; Average 2: AR, VAR_C

RIMA MA1
0.12
0.09
0.16
0.10
0.22
0.12
0.29
0.13
0.35
0.14
0.42
0.16
0.49
0.19
0.55
0.20
0.62
0.24
0.69
0.26
0.76
0.28
0.83
0.31
0.84
0.31
0.84
0.33
0.84
0.36
0.85
0.38
0.85
0.40
0.85
0.42

VAR_C
0.11
0.14
0.19
0.24
0.29
0.35
0.41
0.47
0.54
0.60
0.66
0.73
0.74
0.76
0.77
0.77
0.78
0.78

BVAR
0.10
0.11
0.13
0.15
0.15
0.16
0.19
0.20
0.23
0.25
0.26
0.29
0.29
0.30
0.32
0.32
0.34
0.34

AR
0.09
0.11
0.12
0.13
0.14
0.17
0.20
0.23
0.27
0.29
0.32
0.35
0.35
0.38
0.41
0.44
0.47
0.49

VAR_U_ 
sub
0.13
0.17
0.23
0.28
0.33
0.39
0.46
0.51
0.58
0.64
0.70
0.77
0.78
0.79
0.80
0.81
0.82
0.82

Simple 
avg 1
0.09
0.11
0.14
0.17
0.19
0.22
0.26
0.29
0.34
0.37
0.40
0.44
0.45
0.47
0.48
0.49
0.51
0.52

Ind. 
equ.
0.09
0.11
0.14
0.16
0.17
0.20
0.24
0.27
0.32
0.35
0.39
0.42
0.43
0.45
0.47
0.48
0.50
0.52

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

MSE 
weight-
ed avg1
0.09
0.11
0.13
0.15
0.15
0.18
0.21
0.23
0.27
0.29
0.31
0.34
0.34
0.36
0.38
0.39
0.42
0.42

Geom 
decl 
avg 1
0.08
0.10
0.12
0.13
0.15
0.17
0.20
0.22
0.25
0.27
0.29
0.32
0.32
0.34
0.35
0.36
0.38
0.38

Geom 
decl 
avg 1
0.07
0.08
0.10
0.12
0.13
0.16
0.19
0.21
0.23
0.25
0.27
0.31
0.31
0.32
0.32
0.33
0.34
0.34

Geom 
decl 
avg 1
0.09
0.11
0.13
0.15
0.15
0.17
0.20
0.22
0.26
0.28
0.30
0.33
0.33
0.35
0.37
0.37
0.39
0.40

MSE 
weight-
ed avg2
0.08
0.09
0.11
0.12
0.14
0.16
0.18
0.21
0.23
0.25
0.28
0.30
0.30
0.32
0.34
0.35
0.36
0.37

MSE 
weight-
ed avg2
0.06
0.08
0.09
0.11
0.13
0.15
0.17
0.20
0.21
0.23
0.25
0.27
0.27
0.28
0.28
0.29
0.30
0.30

MSE 
weight-
ed avg2
0.09
0.10
0.12
0.13
0.14
0.16
0.19
0.20
0.24
0.26
0.28
0.31
0.31
0.33
0.35
0.36
0.38
0.39

Simple 
avg 2
0.08
0.09
0.11
0.12
0.14
0.16
0.18
0.21
0.24
0.26
0.28
0.31
0.31
0.33
0.34
0.35
0.37
0.37

Simple 
avg 2
0.06
0.08
0.10
0.11
0.13
0.15
0.17
0.20
0.21
0.23
0.25
0.28
0.28
0.29
0.29
0.30
0.31
0.31

Simple 
avg 2
0.09
0.10
0.12
0.13
0.14
0.16
0.19
0.21
0.25
0.26
0.28
0.31
0.31
0.33
0.36
0.37
0.39
0.40

Geom 
decl 
avg 2
0.08
0.09
0.11
0.12
0.13
0.15
0.18
0.20
0.23
0.25
0.27
0.29
0.29
0.31
0.32
0.33
0.34
0.35

Geom 
Decl 
avg 2
0.06
0.08
0.09
0.11
0.12
0.14
0.17
0.19
0.20
0.22
0.24
0.27
0.27
0.28
0.28
0.28
0.29
0.30

Geom 
Decl 
avg 2
0.09
0.10
0.12
0.13
0.14
0.16
0.18
0.20
0.24
0.25
0.27
0.30
0.29
0.31
0.33
0.34
0.36
0.37

ECB Working Paper 1972, October 201630 
MAD - HICP - full period

RWAO UM
0.13
0.13
0.21
0.20
0.29
0.28
0.34
0.34
0.39
0.40
0.43
0.47
0.47
0.54
0.51
0.62
0.55
0.70
0.58
0.76
0.62
0.83
0.66
0.91
0.66
0.95
0.67
0.97
0.67
0.99
0.68
0.99
0.68
1.01
0.68
1.02

AR
0.12
0.20
0.27
0.33
0.37
0.41
0.46
0.51
0.55
0.59
0.62
0.66
0.67
0.68
0.68
0.69
0.69
0.69

RIMA MA1
0.12
0.12
0.20
0.20
0.28
0.27
0.34
0.32
0.39
0.37
0.43
0.41
0.47
0.46
0.51
0.51
0.55
0.55
0.58
0.60
0.62
0.63
0.66
0.67
0.66
0.67
0.67
0.68
0.67
0.69
0.68
0.70
0.68
0.70
0.68
0.71

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

MAD - HICP - pre-crisis period

RWAO UM
0.10
0.11
0.15
0.16
0.19
0.21
0.22
0.25
0.24
0.27
0.24
0.28
0.24
0.30
0.24
0.31
0.24
0.32
0.23
0.31
0.23
0.29
0.24
0.31
0.25
0.30
0.25
0.30
0.25
0.28
0.25
0.26
0.25
0.27
0.25
0.26

AR
0.10
0.17
0.21
0.24
0.26
0.27
0.27
0.27
0.27
0.27
0.27
0.28
0.27
0.27
0.27
0.27
0.25
0.24

RIMA MA1
0.10
0.11
0.15
0.18
0.19
0.22
0.22
0.25
0.24
0.26
0.24
0.26
0.25
0.27
0.25
0.26
0.24
0.27
0.24
0.27
0.24
0.27
0.25
0.28
0.25
0.26
0.25
0.27
0.25
0.27
0.25
0.26
0.25
0.26
0.25
0.25

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

MAD - HICP - post-crisis period

LSTAR
0.13
0.19
0.27
0.32
0.37
0.41
0.45
0.50
0.54
0.58
0.61
0.65
0.67
0.68
0.69
0.69
0.70
0.70

LSTAR
0.11
0.16
0.21
0.23
0.25
0.26
0.27
0.28
0.29
0.28
0.28
0.30
0.31
0.32
0.33
0.34
0.34
0.33

Ind. 
equ.
0.12
0.20
0.27
0.32
0.36
0.39
0.43
0.48
0.52
0.56
0.59
0.63
0.64
0.64
0.64
0.64
0.65
0.65

Ind. 
equ.
0.11
0.17
0.21
0.24
0.25
0.26
0.26
0.26
0.27
0.28
0.28
0.29
0.27
0.26
0.25
0.24
0.23
0.23

BVAR
0.13
0.20
0.27
0.32
0.36
0.40
0.45
0.50
0.54
0.59
0.63
0.69
0.71
0.72
0.72
0.74
0.74
0.75

BVAR
0.11
0.16
0.20
0.24
0.26
0.26
0.27
0.27
0.27
0.29
0.32
0.34
0.33
0.31
0.30
0.31
0.32
0.32

VAR_U_ 
sub
0.13
0.20
0.28
0.34
0.38
0.42
0.46
0.50
0.53
0.57
0.61
0.66
0.66
0.66
0.67
0.67
0.68
0.68

VAR_U_ 
sub
0.11
0.16
0.21
0.25
0.26
0.27
0.26
0.26
0.25
0.25
0.27
0.28
0.25
0.24
0.25
0.24
0.24
0.23

VAR_C
0.14
0.21
0.27
0.33
0.37
0.41
0.45
0.48
0.52
0.55
0.59
0.63
0.63
0.64
0.64
0.64
0.65
0.65

VAR_C
0.13
0.16
0.20
0.22
0.23
0.23
0.22
0.22
0.22
0.22
0.23
0.25
0.23
0.22
0.23
0.23
0.23
0.22

MSE 
weight-
ed avg1
0.12
0.20
0.26
0.32
0.36
0.40
0.44
0.48
0.52
0.55
0.59
0.63
0.64
0.64
0.65
0.65
0.66
0.66

MSE 
weight-
ed avg1
0.11
0.16
0.20
0.23
0.24
0.25
0.24
0.24
0.24
0.24
0.25
0.25
0.24
0.24
0.24
0.23
0.22
0.21

Simple 
avg 1
0.12
0.20
0.26
0.32
0.36
0.40
0.44
0.48
0.52
0.56
0.59
0.63
0.64
0.65
0.66
0.66
0.66
0.66

Simple 
avg 1
0.11
0.16
0.20
0.23
0.24
0.25
0.24
0.24
0.25
0.24
0.24
0.25
0.24
0.24
0.24
0.23
0.22
0.21

RWAO UM
0.12
0.10
0.22
0.17
0.30
0.24
0.35
0.28
0.39
0.32
0.42
0.39
0.46
0.48
0.51
0.58
0.55
0.69
0.59
0.78
0.63
0.88
0.67
1.00
0.67
1.06
0.67
1.12
0.67
1.17
0.67
1.21
0.67
1.24
0.68
1.27

LSTAR
0.11
0.19
0.27
0.31
0.35
0.38
0.43
0.48
0.53
0.58
0.62
0.66
0.68
0.68
0.68
0.68
0.68
0.68
Note: Average 1: UM, AR, MA1, RIMA, RWAO, VAR_C, VAR_U_sub, LSTAR, BVAR, STIP; Average 2: AR, VAR_C

RIMA MA1
0.11
0.10
0.21
0.19
0.30
0.27
0.35
0.31
0.39
0.35
0.42
0.38
0.46
0.44
0.51
0.51
0.55
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.67
0.69
0.67
0.70
0.67
0.70
0.67
0.71
0.67
0.72
0.68
0.73

VAR_C
0.12
0.21
0.28
0.33
0.35
0.38
0.42
0.46
0.50
0.52
0.55
0.58
0.59
0.60
0.61
0.61
0.61
0.61

BVAR
0.11
0.20
0.27
0.31
0.34
0.36
0.43
0.50
0.55
0.59
0.63
0.68
0.71
0.73
0.76
0.78
0.78
0.80

AR
0.10
0.19
0.26
0.31
0.34
0.37
0.44
0.51
0.56
0.60
0.65
0.69
0.70
0.71
0.72
0.73
0.73
0.74

VAR_U_ 
sub
0.11
0.21
0.30
0.35
0.38
0.41
0.45
0.50
0.55
0.57
0.60
0.64
0.66
0.67
0.67
0.67
0.67
0.67

Simple 
avg 1
0.10
0.19
0.27
0.32
0.34
0.37
0.41
0.47
0.51
0.54
0.59
0.63
0.64
0.65
0.66
0.67
0.67
0.67

Ind. 
equ.
0.10
0.19
0.26
0.31
0.33
0.35
0.40
0.46
0.50
0.54
0.58
0.61
0.62
0.62
0.63
0.63
0.63
0.64

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

MSE 
weight-
ed avg1
0.10
0.19
0.27
0.31
0.34
0.37
0.41
0.46
0.51
0.54
0.58
0.62
0.63
0.64
0.65
0.66
0.66
0.66

Geom 
decl 
avg 1
0.12
0.19
0.26
0.32
0.36
0.40
0.43
0.48
0.52
0.55
0.58
0.62
0.63
0.63
0.64
0.64
0.65
0.65

Geom 
decl 
avg 1
0.11
0.16
0.20
0.23
0.24
0.25
0.24
0.24
0.24
0.23
0.24
0.25
0.23
0.23
0.23
0.22
0.21
0.20

Geom 
decl 
avg 1
0.10
0.19
0.27
0.31
0.34
0.36
0.41
0.46
0.51
0.54
0.58
0.61
0.62
0.63
0.65
0.65
0.65
0.65

MSE 
weight-
ed avg2
0.12
0.20
0.26
0.32
0.36
0.40
0.44
0.49
0.53
0.57
0.60
0.65
0.65
0.66
0.67
0.67
0.68
0.68

MSE 
weight-
ed avg2
0.10
0.16
0.20
0.24
0.25
0.26
0.25
0.26
0.26
0.26
0.26
0.27
0.27
0.26
0.26
0.25
0.26
0.24

MSE 
weight-
ed avg2
0.10
0.19
0.26
0.31
0.34
0.36
0.42
0.49
0.53
0.57
0.62
0.66
0.67
0.68
0.69
0.69
0.70
0.70

Simple 
avg 2
0.12
0.20
0.26
0.32
0.36
0.40
0.44
0.49
0.54
0.57
0.61
0.65
0.66
0.67
0.67
0.68
0.69
0.69

Simple 
avg 2
0.10
0.16
0.20
0.24
0.25
0.26
0.25
0.26
0.26
0.26
0.26
0.27
0.27
0.26
0.26
0.25
0.26
0.25

Simple 
avg 2
0.10
0.19
0.26
0.31
0.34
0.36
0.42
0.49
0.54
0.58
0.62
0.66
0.67
0.68
0.69
0.70
0.70
0.71

Geom 
decl 
avg 2
0.12
0.20
0.26
0.32
0.36
0.40
0.44
0.49
0.53
0.57
0.60
0.65
0.65
0.66
0.67
0.67
0.68
0.68

Geom 
Decl 
avg 2
0.10
0.16
0.20
0.24
0.25
0.26
0.25
0.26
0.26
0.26
0.26
0.27
0.27
0.25
0.26
0.25
0.25
0.24

Geom 
Decl 
avg 2
0.10
0.19
0.26
0.31
0.33
0.36
0.42
0.49
0.53
0.57
0.62
0.66
0.66
0.68
0.69
0.69
0.69
0.70

ECB Working Paper 1972, October 201631 
MAD - HICP excluding food and energy - full period

RWAO UM
0.08
0.06
0.12
0.08
0.17
0.10
0.22
0.11
0.27
0.14
0.32
0.16
0.38
0.19
0.43
0.22
0.49
0.25
0.54
0.28
0.60
0.31
0.65
0.35
0.66
0.36
0.66
0.38
0.67
0.40
0.67
0.41
0.68
0.42
0.68
0.43

AR
0.06
0.07
0.09
0.10
0.12
0.15
0.17
0.20
0.23
0.25
0.27
0.30
0.31
0.32
0.33
0.34
0.35
0.36

RIMA MA1
0.07
0.05
0.12
0.07
0.16
0.09
0.21
0.10
0.27
0.12
0.32
0.14
0.37
0.16
0.43
0.19
0.48
0.21
0.54
0.23
0.59
0.25
0.65
0.27
0.66
0.28
0.66
0.29
0.67
0.30
0.67
0.31
0.68
0.32
0.68
0.32

LSTAR
0.07
0.08
0.10
0.12
0.13
0.15
0.17
0.19
0.21
0.22
0.24
0.26
0.27
0.28
0.29
0.29
0.30
0.31

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

MAD - HICP excluding food and energy - pre-crisis period

RWAO UM
0.06
0.05
0.10
0.07
0.14
0.09
0.18
0.11
0.23
0.14
0.28
0.17
0.33
0.19
0.38
0.23
0.44
0.26
0.49
0.28
0.55
0.32
0.60
0.35
0.61
0.36
0.62
0.37
0.63
0.38
0.64
0.38
0.65
0.40
0.65
0.40

AR
0.05
0.06
0.08
0.10
0.12
0.14
0.17
0.20
0.22
0.24
0.27
0.29
0.30
0.31
0.31
0.32
0.32
0.33

RIMA MA1
0.06
0.05
0.10
0.06
0.13
0.08
0.17
0.10
0.22
0.12
0.27
0.14
0.32
0.16
0.37
0.19
0.43
0.21
0.48
0.23
0.54
0.24
0.60
0.26
0.61
0.26
0.62
0.27
0.63
0.27
0.64
0.27
0.65
0.28
0.65
0.28

LSTAR
0.07
0.09
0.10
0.11
0.13
0.14
0.16
0.18
0.19
0.20
0.21
0.23
0.25
0.26
0.26
0.27
0.28
0.28

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

MAD - HICP excluding food and energy - post-crisis period

Ind. 
equ.
0.06
0.07
0.09
0.10
0.12
0.14
0.17
0.20
0.22
0.24
0.27
0.30
0.30
0.32
0.34
0.34
0.36
0.37

Ind. 
equ.
0.05
0.06
0.07
0.08
0.10
0.12
0.15
0.17
0.19
0.21
0.24
0.27
0.28
0.28
0.30
0.31
0.32
0.34

BVAR
0.06
0.08
0.09
0.11
0.12
0.14
0.15
0.17
0.19
0.22
0.23
0.26
0.26
0.27
0.28
0.30
0.31
0.31

BVAR
0.06
0.08
0.09
0.11
0.12
0.14
0.15
0.18
0.18
0.19
0.20
0.22
0.20
0.20
0.20
0.20
0.20
0.21

VAR_U_ 
sub
0.08
0.12
0.16
0.20
0.24
0.29
0.34
0.39
0.44
0.49
0.54
0.59
0.60
0.62
0.63
0.63
0.64
0.65

VAR_U_ 
sub
0.07
0.10
0.13
0.16
0.21
0.25
0.29
0.34
0.39
0.44
0.48
0.54
0.56
0.57
0.58
0.59
0.60
0.61

VAR_C
0.07
0.11
0.15
0.20
0.25
0.30
0.36
0.41
0.46
0.51
0.56
0.61
0.62
0.63
0.63
0.63
0.63
0.63

VAR_C
0.06
0.09
0.13
0.16
0.21
0.26
0.30
0.35
0.40
0.45
0.50
0.54
0.56
0.56
0.57
0.57
0.57
0.57

MSE 
weight-
ed avg1
0.06
0.07
0.09
0.11
0.12
0.15
0.17
0.19
0.22
0.23
0.25
0.28
0.28
0.29
0.30
0.31
0.32
0.33

MSE 
weight-
ed avg1
0.05
0.06
0.08
0.09
0.12
0.14
0.17
0.19
0.20
0.22
0.24
0.26
0.27
0.27
0.28
0.29
0.30
0.30

Simple 
avg 1
0.06
0.08
0.10
0.12
0.14
0.17
0.20
0.23
0.25
0.28
0.30
0.33
0.33
0.34
0.35
0.35
0.36
0.37

Simple 
avg 1
0.05
0.07
0.08
0.10
0.12
0.15
0.17
0.20
0.22
0.24
0.27
0.30
0.30
0.31
0.31
0.32
0.33
0.34

RWAO UM
0.09
0.07
0.14
0.08
0.20
0.10
0.26
0.11
0.32
0.13
0.39
0.15
0.45
0.18
0.52
0.20
0.58
0.24
0.65
0.27
0.71
0.31
0.78
0.35
0.78
0.37
0.79
0.39
0.79
0.42
0.79
0.44
0.80
0.46
0.80
0.48

LSTAR
0.07
0.08
0.10
0.12
0.13
0.14
0.16
0.18
0.22
0.24
0.26
0.29
0.29
0.30
0.31
0.32
0.34
0.35
Note: Average 1: UM, AR, MA1, RIMA, RWAO, VAR_C, VAR_U_sub, LSTAR, BVAR, STIP; Average 2: AR, VAR_C

RIMA MA1
0.09
0.06
0.14
0.07
0.20
0.09
0.26
0.10
0.32
0.11
0.38
0.13
0.45
0.15
0.51
0.17
0.58
0.19
0.64
0.21
0.71
0.23
0.78
0.25
0.78
0.26
0.79
0.28
0.79
0.30
0.79
0.31
0.80
0.33
0.80
0.35

VAR_C
0.08
0.11
0.15
0.20
0.25
0.31
0.37
0.42
0.49
0.55
0.61
0.67
0.68
0.70
0.71
0.71
0.71
0.72

BVAR
0.07
0.08
0.09
0.11
0.11
0.12
0.14
0.15
0.18
0.20
0.21
0.24
0.25
0.26
0.28
0.29
0.30
0.31

AR
0.06
0.08
0.09
0.11
0.11
0.14
0.16
0.19
0.22
0.24
0.27
0.29
0.29
0.31
0.34
0.35
0.38
0.39

VAR_U_ 
sub
0.09
0.14
0.19
0.24
0.29
0.35
0.42
0.47
0.53
0.59
0.65
0.72
0.73
0.74
0.75
0.76
0.77
0.77

Simple 
avg 1
0.07
0.09
0.11
0.14
0.16
0.18
0.22
0.24
0.28
0.31
0.34
0.38
0.37
0.38
0.40
0.41
0.42
0.43

Ind. 
equ.
0.07
0.09
0.10
0.12
0.14
0.16
0.20
0.23
0.27
0.29
0.32
0.35
0.36
0.37
0.39
0.40
0.42
0.43

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18

MSE 
weight-
ed avg1
0.07
0.08
0.10
0.11
0.13
0.14
0.17
0.18
0.22
0.23
0.26
0.28
0.28
0.29
0.30
0.32
0.34
0.35

Geom 
decl 
avg 1
0.06
0.07
0.09
0.10
0.12
0.14
0.16
0.18
0.21
0.22
0.24
0.26
0.27
0.28
0.29
0.29
0.31
0.31

Geom 
decl 
avg 1
0.05
0.06
0.08
0.09
0.11
0.14
0.16
0.19
0.20
0.21
0.23
0.25
0.26
0.27
0.28
0.28
0.29
0.30

Geom 
decl 
avg 1
0.07
0.08
0.10
0.11
0.12
0.14
0.16
0.18
0.21
0.22
0.25
0.27
0.27
0.28
0.29
0.30
0.32
0.33

MSE 
weight-
ed avg2
0.06
0.07
0.08
0.10
0.11
0.13
0.15
0.17
0.19
0.21
0.23
0.25
0.25
0.27
0.28
0.29
0.30
0.30

MSE 
weight-
ed avg2
0.05
0.06
0.08
0.09
0.11
0.13
0.15
0.17
0.18
0.20
0.21
0.23
0.23
0.24
0.25
0.25
0.26
0.26

MSE 
weight-
ed avg2
0.06
0.08
0.09
0.10
0.11
0.12
0.15
0.16
0.20
0.21
0.23
0.25
0.25
0.26
0.28
0.30
0.31
0.33

Simple 
avg 2
0.06
0.07
0.08
0.10
0.11
0.13
0.15
0.17
0.20
0.21
0.23
0.25
0.25
0.27
0.28
0.28
0.30
0.30

Simple 
avg 2
0.05
0.06
0.08
0.09
0.11
0.13
0.15
0.17
0.19
0.20
0.21
0.23
0.23
0.24
0.25
0.25
0.26
0.26

Simple 
avg 2
0.06
0.08
0.09
0.10
0.11
0.12
0.15
0.17
0.20
0.21
0.23
0.25
0.26
0.27
0.28
0.29
0.32
0.33

Geom 
decl 
avg 2
0.06
0.07
0.08
0.09
0.11
0.13
0.15
0.17
0.19
0.20
0.22
0.24
0.24
0.25
0.26
0.27
0.28
0.28

Geom 
Decl 
avg 2
0.05
0.06
0.08
0.09
0.11
0.12
0.14
0.17
0.18
0.19
0.20
0.22
0.23
0.24
0.24
0.25
0.25
0.25

Geom 
Decl 
avg 2
0.06
0.08
0.09
0.10
0.11
0.12
0.14
0.16
0.19
0.20
0.22
0.24
0.24
0.25
0.27
0.28
0.30
0.31

ECB Working Paper 1972, October 201632 
Appendix II 

 Details on forecasting methods for inflation 

1.  Random Walk Models 

The  random  walk  models  include  the  random  walk  with  drift  and  the  random  walk  model 

according to Atkenson and Ohanian (2001). 

2.  AR Models 

The lag selection is done recursively based on the Bayesian Information Criterion. The number 

of lags from which these criteria select for the inflation models ranges from 0 and 6 months. 

3.  STAR Models 

The Smooth Transition Autoregressive (STAR) Model is included in the analysis. This model is 

a generalisation of the 2-regime threshold autoregressive model, that allows the estimation of the 

– potentially smooth - transition from one regime to the other and can help forecasting variables 

with asymmetric behaviour (see e.g. Teraesvirta, 2006). 

4.  VAR Models  

Two VAR model versions are being employed to generate forecasts for all variables contained in 

the  models.  The  first  model  comprises  HICP  components  but  not  the  aggregate  itself  (i.e.  5 

variables:  processed  food,  unprocessed  food,  industrial  goods,  services,  energy)  and  the 

aggregate,  but  does  not  include  any  exogenous  variables  on  which  the  forecasts  could  be 

conditioned. The second model does not include any of the other HICP components, but uses the 

producer prices of consumer goods, a moving average of interpolated monthly unit labour costs, 

oil  prices,  non-oil  commodity  prices  and  the  nominal  effective  exchange  rate  as  explanatory 

variables.  These  variables  are  endogenous  in  the  estimation  of  the  VAR,  but  the  forecasts  are 

conditioned on the assumptions as they are used in the Eurosystem Macroeconomic Projection 

Exercises.  

All  HICP  index  variables  are  modelled  in  month-on-month  log-differences,  i.e.  approximate 

percentage changes. Similar to the AR models, the lag selection is done recursively based on the 

Bayesian  Information  Criterion.  The  number  of  lags  from  which  these  criteria  select  for  the 

inflation models ranges from 0 and 6 months.  

5.  MA(1) 

This model is employed to approximate the unobserved component model by Stock and Watson 

(2007) as these authors suggest. 

6. 

Individual equation framework for inflation: 

The individual equations for inflation (individual equation framework) are an updated version of 

Benalal  et  al.  (2004)  and  are  also  described  in  ECB  (2010).  According  to  these  equations, 

unprocessed food prices are modelled as an AR model with 12 lags with a seasonal dummy for 

January;  the  model  for  processed  food  prices  includes  food  commodity  prices  in  euro  (lag  9), 

ECB Working Paper 1972, October 201633tobacco taxes, a 3 month moving average of unit labour costs (lag 5) and 2 lags of the dependent 

variable; non-energy industrial goods prices are explained by the effective exchange rate (lag 1), 

the  VAT,  GDP,  consumer  goods  producer  prices  (lag  5)  and  lags  5  and  6  of  the  dependent 

variable; energy prices have as explanatory variables oil prices in USD and the USD exchange 

rate (both lags 0 and 1), energy taxes and seasonal dummies for January and November; services 

prices are modelled with a 3 month moving average of compensation per employee (lags 3 and 

9), consumer goods producer prices (lags 5 and 7) and lags 1 to 5 of the dependent variable; the 

direct approach for headline HICP uses a model with a 3 month moving average of unit labour 

costs,  a  3  month  moving  average  of  oil  prices  in  euro  (lags  0  and  3),  the  nominal  effective 

exchange rate (lags 0 to 0), consumer goods producer prices (lags 0 and 1) and lags 1 to 12 of 

the  dependent  variable;  the  direct  equation  for  the  HICP  excluding  food  and  energy  uses  a 

moving  average  of  unit  labour  costs,  the  nominal  effective  exchange  rate  (lag  7),  consumer 

goods producer prices (lags 0 and 1) and lags 2, 5 and 6 of the dependent variable. Consumer 

goods producer prices, which are used as explanatory variable in some equations, are modelled 

with  non-oil  commodity  prices  (lag  6),  the  nominal  effective  exchange  rate  (lag  4),  GDP  and 

lags 1 to 4 of the dependent variable. All equations are estimated in dlogs on seasonal adjusted 

prices. 

6. 

BVAR 

The  BVAR  model  is  the  one  proposed  by  Giannone  et  al.  (2014),  with  priors  following 

Giannone,  Lenza  and  Primiceri  (2015).  It  allows  for  all  possible  interactions  between  main 

components of HICP and their determinants and captures indirect and second-round effects. 

Their  dataset  is  very  close  to  the  one  for  the  individual  equation  approach,  it  is  of  monthly 

frequency, and includes 14 variables. Namely, the 5 components of HICP, PPI consumer goods, 

unit  labour  costs,  GDP,  compensation  per  employee,  oil  price  in  US  dollars,  food  commodity 

prices,  commodity  prices  excluding  food,  EUR/USD  exchange  rate  and  nominal  effective 

exchange rate. 

In  order  to  mimic  the  information  available  to  forecasters  in  real  time,  they  gathered,  to  the 

extent  possible,  39  vintages  of  the  real-time  data  available  for  the  corresponding  quarterly 
Eurosystem/ECB  staff  projection  exercises  from  March  2000  to  September  2009.17  They 
reconstruct the availability of official data, as well as the assumptions on future paths that were 

available  in  real  time  for  almost  all  the  variables  which  are  used  to  condition  the  inflation 

forecast in the individual equation and the conditional BVAR. For the quarterly variables (unit 

labour costs, GDP and compensation per employee), they make use of the historical data, as well 

as  the,  publicly  available, projected  path  of annual  GDP  growth  rates. For  what  concerns  unit 

labour costs and compensation per employee, no particular future projected path is assumed. Oil 

and non-oil commodity prices are based on Bloomberg data on spot and futures prices. As the 

back-data  were  not  complete  for  all  of  the  39  vintages,  they  reconstructed  the  oil  and  non-oil 

commodities real-time database on the basis of spot and futures’ prices available at earlier points 

17 Note that Eurosystem projections were at the time published in form of ranges. In order to obtain point estimates, 

they took the mid-point of the ranges. 

ECB Working Paper 1972, October 201634in  time.  Finally,  exchange  rates  are  unrevised  and  the  assumed  future  path  in  the  forecasting 

exercises in their case is actually flat at the latest observed value at each different data cut-off 

point of the different quarterly projections (implied random walk assumption). Notice that taxes 

also enter the original individual equations. Accordingly, they make use of them in the study of 

the  forecasting  performance  of  the  individual  equations.  However,  in  the  BVARs,  they  have 

chosen to exclude them since they were not found to improve on forecast accuracy. 

ECB Working Paper 1972, October 201635Appendix III  Weight charts with Bai-Perron break dates 

Weights with Bai-Perron Break dates (vertical lines), Combination 2, full forecast sample 

Horizon 1, Headline inflation (hlagg) 

Horizon 1, Inflation excluding energy and food (exagg) 

ECB Working Paper 1972, October 201636 
 
 
Horizon 6, Headline inflatioin (hlagg) 

Horizon 6, Inflation excluding energy and food (exagg) 

ECB Working Paper 1972, October 201637 
 
Horizon 12, Headline inflation (hlagg) 

Horizon 12, Inflation excluding energy and food (exagg) 

ECB Working Paper 1972, October 201638 
 
 
Appendix IV  Forecasting the aggregate versus combining component forecasts 

As  mentioned  in  Section  2.1,  a  combination  issue  separate  from  the  one  just  discussed  that 

focuses on the combination of forecasts of the same variable of interest, is whether combining 

forecasts of different disaggregate variables to forecast the aggregate (“indirect forecast” of the 

aggregate) is preferable to combining disaggregate information in a model for the aggregate and 

use this for forecasting (“direct forecast” of the aggregate). 

This  has  been  investigated  empirically  for  the  euro  area  by  Benalal,  Diaz  Del  Hoyo,  Landau, 

Roma  and  Skudelny  (2004),  and  Hubrich  (2005),  and  has  been  analysed  theoretically,  by 

simulation and empirically for the US in Hendry and Hubrich (2011) Benalal, Diaz Del Hoyo, 

Landau, Roma and Skudelny (2004) find that the direct forecast of the aggregate provides better 

results  than  the  aggregation  of  component  forecasts  for  12  and  18  months  ahead  for  overall 

HICP while for the HICP excluding unprocessed food and energy, the forecast performance was 

better for the aggregation of the component forecast. Hubrich (2005) also found for 12-months 

ahead  forecasts  that  the  direct  forecast  of  the  aggregate  outperformed  the  aggregation  of 

component  forecasts  for  total  HICP,  but  not  for  the  HICP  excluding  unprocessed  food  and 

energy.  

Hendry and Hubrich (2011) derive analytical results and conclude with two main findings: (1) 

Unknown breaks at the forecast origin (location shifts or slope parameter changes) do not affect 

the  relative  forecast  accuracy  of  forecasting  the  aggregate  directly  or  aggregating  component 

forecasts. This is in contrast to the findings in the usual forecast combination literature focusing 

on combining forecasts of the same variable. Of course, the absolute forecast accuracy might be 

affected  substantially.  (2)  Slope  parameter  misspecification  and  estimation  uncertainty  are  the 

main sources of relative forecast error differences.] 

Hendry  and  Hubrich  (2011)  recommend  reducing  estimation  uncertainty  by  (a)  variable 

selection  procedures  (or  Bayesian  shrinkage)  and  (b)  methods  to  combine  disaggregate 

information (e.g. factor models or Bayesian shrinkage). The Bayesian VAR model included in 

this  study  addresses  the  issue  of  estimation  uncertainty  /  curse  of  dimensionality  by  imposing 

prior information on the VARs for forecasting the disaggregates. 

We present some results on forecasting the aggregate versus aggregating component forecasts in 

the context of the models employed in this study below.  In addition to comparing the indirect 

forecasts,  i.e.  the  aggregation  of  the  component  forecasts,  described  in  the  main  text,  we  also 

computed and compared the results with the direct forecasts of headline HICP inflation and of 

HICP excluding food and energy inflation for the different models.We describe the details of the 

forecast experiments and discuss the results. 

For the individual equation framework the direct equation for headline inflation contains up to 

12 lags of the dependent variable, a moving average of unit labour costs (lags 0 to 3), a moving 

average of oil prices (lags 0 to 3), the nominal effective exchange rate (lags 0 to 9) and the PPI 

consumer goods (lags 0 and 1). For the HICP excluding food and energy, the equation includes 

ECB Working Paper 1972, October 201639lags  2,  5  and 6  of the  dependent  variable,  a  moving  average  of  unit labour  costs,  the nominal 

effective exchange rate (lag 7) and the PPI consumer goods (lags 0 and 1). For the other models, 

the same variables are used for the direct approach as for the components based forecasts. 

For the AR models, the direct AR forecast is always clearly better than the indirect forecast of 

aggregate  inflation  for  the  pre-crisis  period,  while  for  the  period  including  the  crisis  period  it 

clearly outperforms the indirect forecast only from a horizon of 7 months onwards and is very 

similar  to  it  before.  For  the  HICP  excluding  food  and  energy,  the  indirect  forecast  performs 

somewhat  better.  However,  as  we  want  to  have  a  consistent  benchmark  across  models  and 

aggregate, our comparisons above were all based on the direct AR forecast. 

Chart 9 shows the RMSE of the indirect (bottom-up) approach relative to the direct approach for 

headline  inflation,  both for  the  pre-crisis  period  and  the  period including  the crisis.  A  number 

below one indicates that the indirect approach has a better forecast performance than the direct 

approach. 

The results are quite different across models. For the individual equation framework, the indirect 

approach  works  somewhat  better  than  the  direct  approach  for  the  full  sample  for  headline 

inflation and, though only up to 6 months ahead, for core inflation. In the two sub-periods, the 

results of the two approaches are very similar for headline inflation while for core inflation the 

indirect  approach  works  better  only  for  short  horizons  in  the  pre-crisis  period  and  cannot 

outperform  the  AR  for  the  post-crisis  period.  The  VAR_C  model  provides  similar  or  better 

results  when  using  the  direct  rather  than  the  indirect  approach  for  all  horizons,  with  larger 

differences for core inflation. The STAR model yields similar or better results for the indirect 

approach  in  particular  over  the  longer  run.  This  is  an  interesting  result  since    Benalal  et  al. 

(2004)  and  Hubrich  (2005)  who  find  that  indirect  approaches  tend  to  outperform  direct 

approaches  in  particular  over  the  short  run  using  linear  models.  The  results  above  show  that 

different reference periods can lead to different results regarding whether it is better to forecast 

components or aggregates.   

We also compared the direct and indirect forecasts of the two different inflation aggregates we 

consider in this paper for the different combination schemes (see Chart 10). When looking at the 

results for the combination  models, we find that forecast accuracy is very similar between the 

direct and the indirect approach, both for headline and core inflation, although for core inflation 

there is some improvement of the indirect over the direct forecast. The difference between the 

periods is rather marginal. 

ECB Working Paper 1972, October 201640 
Chart 9 

Indirect RMSE relative to direct RMSE – individual models 

Headline inflation 

Excl. energy and food 

Ind. equ.

VAR_C

STAR

Ind. equ.

VAR_C

STAR

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Forecast horizon

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Forecast horizon

Ind. equ.

VAR_C

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Forecast horizon

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Forecast horizon

Note:  “indirect  forecast”  of  the  aggregate  (and  the  associated  RMSFE)  refers  to  combining  forecasts  of 

different  disaggregate  variables  to  forecast  the  aggregate;  “direct  forecast”  of  the  aggregate  (and  its 

RMSFE) refers to the aggregate forecasting model. 

ECB Working Paper 1972, October 201641 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chart 10 

Indirect RMSE relative to direct RMSE – combinations 

Headline inflation 

Excl. energy and food 

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ECB Working Paper 1972, October 201642 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Acknowledgements 

We thank Tim van Ark, Marco Buchmann and Paul Körbitz for excellent research assistance and Domenico Giannone and Michele 
Lenza for sharing with us the codes for the Bayesian Vector Autoregressive Model. We thank Hans-Joachim Klöckers, Julian Morgan, 
John Roberts and participants of ESCB workshops for comments. The views expressed in this paper are those of the authors and do 
not necessarily reflect those of the European Central Bank or the Federal Reserve Board. The first draft of the paper was circulated in 
July 2010 when the first author was working at the European Central Bank. 

Kirstin Hubrich 
Federal Reserve Board; email: kirstin.hubrich@frb.gov 

Frauke Skudelny 
European Central Bank; email: frauke.skudelny@ecb.europa.eu 

© European Central Bank, 2016 

Postal address 
Telephone 
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www.ecb.europa.eu 

All rights reserved. Any reproduction, publication and reprint in the form of a different publication, whether printed or produced 
electronically, in whole or in part, is permitted only with the explicit written authorisation of the ECB or the authors.  

This paper can be downloaded without charge from www.ecb.europa.eu, from the Social Science Research Network electronic library or 
from RePEc: Research Papers in Economics. Information on all of the papers published in the ECB Working Paper Series can be found 
on the ECB’s website. 

ISSN  
ISBN  

1725-2806 (pdf) 
978-92-899-2220-3 (pdf) 

DOI 
EU catalogue No 

10.2866/052798 (pdf) 
QB-AR-16-089-EN-N (pdf)