






































Microsoft Word - 00_tresc.docx


DYNAMIC ECONOMETRIC MODELS 
Vol. 10 – Nicolaus Copernicus University – Toruń – 2010 

Jacek Kwiatkowski 
Nicholas Copernicus University in Toruń 

Unobserved Component Model for Forecasting  
Polish Inflation† 

A b s t r a c t. This paper aims to use the local level models with GARCH and SV errors to predict 
Polish inflation. The series to be forecast, measured monthly, is consumer price index (CPI) in 
Poland during 1992-2008. We selected three forecasting models i.e. LL-GARCH(1,1) with 
Normal or Student errors and LL-SV. A simple AR(2)-SV model is used as a benchmark to assess 
the accuracy of prediction. The presented results indicate, that there is no clear advantage of LL 
models in forecasting Polish inflation over standard AR(2)-SV model, although all the models 
give satisfactory results. 

K e y w o r d s: local level model, inflation, conditional heteroscedasticity. 

1. Introduction 
 Econometric models, both of the structural and a-theoretical ones, are 
widely used to provide forecasts of inflation. In a recent study Stock and 
Watson (2007), found that a local level model with stochastic volatility gives 
the most accurate forecasts of quarterly inflation in the United States. In their 
paper they compare the accuracy of inflation forecasts of wide class of models 
including standard ARIMA time series models, time-varying parameters models 
(TVP) and the Phillips curve-based models. 
 In this paper, we examine several types of inflation forecasts in Poland, 
which are based on time-varying parameters model and subject them to tests for 
accuracy. 
 The paper is organized as follows. Section 2 presents the models which are 
used to forecast monthly inflation in Poland: LL-SV and LL-GARCH with two 

                                                 
† This work was financed from the Polish science budget resources in the years 2009-2011 as 

the research project  no. N N111 431737.  
I would like to thank Grzegorz Szafrański and one anonymous referee for helpful comments 

and suggestions; e-mail: jkwiat@umk.pl. 



Jacek Kwiatkowski 122 

different distributions of the disturbances (Normal, Student's t) and standard 
AR(2)-SV model. In section 3, we compare forecast accuracy of the mentioned 
above models. We use two forecast accuracy measures, namely: the sign test 
and Wilcoxon signed rank test (see Diebold and Mariano, 1995). The predictive 
distribution calculated for the future observables enable us to provide a detailed 
analysis of the inflation forecasts, therefore we also present the predictive 
medians and interquartile range. It is also well known, that Polish monetary 
authorities conduct the policy under inflation targeting regime between 1.5% 
and 3.5% and the inflation prediction is one of the inputs to the Monetary Policy 
Council’s decision-making process. Then, it is worth to consider what the 
posterior probability for the hypothesis is that inflation will stay inside the 
targeting bound and how this posterior probability changes as the forecasts 
horizon grows. Section 4 concludes the paper. 

2. Unobserved Component Model with Time-Varying Conditional 
Variance 

 Stock and Watson (2007) used unobserved component model (LL-SV), 
which is very effective for forecasting inflation: 

ttty εδ += ,    ( )2,0~ tt N σε , Tt ...,,2,1=  (1) 

ttt ηδδ += −1 ,    ( )2,0~ tt N ωη , (2) 

where the irregular and level disturbances, tε  and tη , respectively, are mutually 
independent, tδ  denotes underlying stochastic level. 

 Consider now that tε  and tη  are stationary SV processes, where: 

tirregtirregirregtirreg hh ,1,, ζρ += − , (3)  

tleveltlevelleveltlevel, hh ,1, ζρ += − , (4) 

and ( )2,0~ tt N σε , ( )2,0~ tt N ωη , ( )tirregt h ,
2 exp=σ  , ( )tlevelt h ,

2 exp=ω  ,  

( )1,1)( −∈levelirregρ  and ( )2
)()( ,0~ levelirreglevelirreg N γζ . 

 It’s easy to see that the reduced form of unobserved component model (1)-
(2) is a local level model (LL) with restrictions in parameters: 22 σσ =t  and  

22 ωω =t . The local level model is a well-known model and it has a long 
tradition in economic time series. The literature that considers its properties is 
very extensive and previously interested many authors (see for example Muth, 
1960; Harvey, 1989; West and Harrison, 1989; Durbin and Koopman, 2001; 
Koop, 2003). 



Unobserved Component Model for Forecasting Polish Inflation 

 
123

 Consider the LL model when the disturbances follow Normal GARCH or 
Student-t GARCH. Assuming that each noise is a conditionally Normal process, 
we have: 

( )2,0~ tt N σε  and ( )2,0~ tt N ωη . (5) 

The equivalent Student-t disturbances are denoted as: 

( ) ⎟
⎟
⎠

⎞
⎜
⎜
⎝

⎛

− irreg
tirreg

irreg
t v

v
v

t ,
2

,0~ 2σ
ε  and ( ) ⎟⎟

⎠

⎞
⎜⎜
⎝

⎛

− level
tlevel

level
t v

v
vt ,

2
,0~ 2ω

η , (6) 

where ( )vPat ,,  denotes Student-t density with expectation a , precision P  and 
v  degrees of freedom. 
 For GARCH(1,1) process, the variance of the observation equation (1) and 
state equation (2) varies over time according to (see Bos, 2001): 

tirregt h ,
2 =σ  and tlevelt h ,

2 =ω , (7) 

( )21,1,01,,1, −− ++= tirregirregtirregirregtirreg Eaahbh ε , (8) 

( )21,1,01,,1, −− ++= tlevelleveltlevelleveltlevel Eaahbh η , (9) 

with the parametric constraints that are sufficient for positivity and stationarity 
of the conditional variance: 

)(,1)(,1)(,0 1 levelirreglevelirreglevelirreg aba −−≡ , 

0)(,1 ≥levelirregb , 0)(,1 ≥levelirrega , 1)(1)(,1 <+ levelirreglevelirreg ba .  

 The properties of the LL model, when the disturbances follow GARCH(1,1) 
process, are presented in Pellegrini, Ruiz and Espasa (2007, 2008). 
 The last model is a simple AR(2)-SV model, which is used as a benchmark 
model. It has the following form: 

tttt yyy εδδδ +Δ+Δ+=Δ −− 22110 , ( )2,0~ tt N σε . (10) 

3. Forecasting Inflation 
 In this section, we examined whether the proposed models successfully 
predict the value of the inflation rate. We considered 204=T  monthly 
observations on the logarithm of CPI from January 1992 till December 2008. 
The data set employed in this study consists of logarithmic transformations of 
the original series CPI, computed as ( )tt CPIy ln100= . All data has been 
seasonally adjusted, using the moving average method implemented in  
Eviews 6.  



Jacek Kwiatkowski 124 

 Before starting the analysis, we briefly describe the processes that influence 
inflation in Poland over the past twenty years. During the first years of 90’s the 
Polish economy has been transformed to a market economy. Due to 
marketization and stabilization program there were deep economic and social 
changes including the elimination of the state control of prices and liberalizing 
trade, investment and capital flow (Fallenbuchl, 1994). In 1990-1992 the Polish 
economy was in an early stage of transition and inflationary processes visibly 
accelerated reaching 685.8 % in 1990. Therefore we begin our analysis 
from January 1992 to avoid the unusual effects of hyperinflation. In the years 
1993-1997, monetary policy was focused on neutralizing the powerful 
inflationary forces discernible within the Polish economy and then to achieve 
a further reduction in inflation1. The next years (1998–2004) increased Poland's 
central bank independence and monetary policy was focused on maintaining 
price stability and preparation for integration with the EU. After 2004 the main 
goal of monetary policy was to achieve the Maastricht price stability criterion in 
the coming future. The data previously discussed are presented in Figure 1. 
The vertical line indicates the limit between sample and forecast-period. 

 
Figure 1. The values of logarithms of CPI (seasonally adjusted data). The vertical line 

indicates the limit between sample and forecast-period 
 
 Using Bayes' rule and Monte Carlo techniques, we calculated four 
competing Bayesian models – LL-SV, LL-GARCH(1,1), LL-GARCH(1,1)-
Student and AR(2)-SV – based on dataset ( )ty  for  192...,,1=t . For each model 
and 12 months in 2008, as a result we obtained predictive distributions in the 
following form: 

( )( )i
t

h Myyp ,|192+ , for 4...,,1=i , 12...,,1=h , (11)  

                                                 
1 NBP Annual Report, http://www.nbp.pl/ 

-0.01

 0

 0.01

 0.02

 0.03

 0.04

 0.05

 0.06

 0.07

 0.08

 1992  1994  1996  1998  2000  2002  2004  2006  2008

ln
 C

P
I



Unobserved Component Model for Forecasting Polish Inflation 

 
125

where LL-GARCH(1,1) – 1M , LL-GARCH(1,1)-Student – 2M , LL-SV – 3M  
and finally AR(2)-SV – 4M .  

 Due to number of models and months, the entire procedure should be 
performed four times, giving a total of 48 predictive distributions.  

Table 1. The quantiles of the predictive distribution  

Date 
Model 

LL-GARCH(1,1) LL-GARCH(1,1)-
Student LL-SV AR(2)-SV 

2008M01 
[-0,0584] 

0.3429 
(0.0367, 0.6564) 

0.3535 
(0.1118, 0.6075) 

0.3214 
(0.0894, 0.5398) 

0.2825 
(0.0437, 0.5265) 

2008M02 
[0,4363] 

0.3429 
(-0.0187, 0.7177) 

0.3535 
(0.0791, 0.6499) 

0.3214 
(0.0742, 0.5513) 

0.4195 
(0.1303, 0.7037) 

2008M03 
[0,4509] 

0.3429 
(-0.0662, 0.7587) 

0.3535 
(0.0482, 0.6693) 

0.3214 
(0.0520, 0.5732) 

0.3364 
(0.0242, 0.6593) 

2008M04 
[0,1780] 

0.3429 
(-0.0974, 0.8076) 

0.3535 
(0.0160, 0.7123) 

0.3214 
(0.0443, 0.5923) 

0.3181 
(-0.0413, 0.6756) 

2008M05 
[0,8002] 

0.3429 
(-0.1545, 0.8569) 

0.3535 
(-0.0098, 0.7290) 

0.3214 
(0.0329, 0.6017) 

0.3396 
(-0.0584, 0.7361) 

2008M06 
[0,4702] 

0.3429 
(-0.1728, 0.8888) 

0.3535 
(-0.0348, 0.7577) 

0.3214 
(0.0290, 0.6110) 

0.3399 
(-0.0947, 0.7633) 

2008M07 
[0,8979] 

0.3429 
(-0.2305, 0.9164) 

0.3535 
(-0.0471, 0.7721) 

0.3214 
(0.0128, 0.6237) 

0.3223 
(-0.1432, 0.8031) 

2008M08 
[0,1819] 

0.3429 
(-0.2603, 0.9458) 

0.3535 
(-0.0671, 0.7965) 

0.3214 
(-0.0159, 0.6441) 

0.3221 
(-0.1724, 0.8319) 

2008M09 
[-0,2893] 

0.3429 
(-0.2777, 0.9822) 

0.3535 
(-0.0910, 0.8156) 

0.3214 
(-0.0228, 0.6456) 

0.3245 
(-0.1968, 0.8558) 

2008M10 
[0,2222] 

0.3429 
(-0.3148, 1.0157) 

0.3535 
(-0.1182, 0.8377) 

0.3214 
(-0.0341, 0.6670) 

0.3257 
(-0.2442, 0.8905) 

2008M11 
[0,1811] 

0.3429 
(-0.3148, 1.0157) 

0.3535 
(-0.1182, 0.8377) 

0.3214 
(-0.0341, 0.6670) 

0.3257 
(-0.2442, 0.8905) 

2008M12 
[-0,1699] 

0.3429 
(-0.3944, 1.0672) 

0.3535 
(-0.1724, 0.8619) 

0.3214 
(-0.0571, 0.7093) 

0.3056 
(-0.3194, 0.9236) 

Note: All data are expressed as percentage. 

 Table 1 contains a selection of the corresponding out-of-sample forecasting 
results for the selected models with constant and time-varying mean. The data 
in brackets represent the seasonally adjusted consumer price index. For four 
models and for each month of 2008, the medians and the first and third quartiles 
(on the second line) of the predictive distributions are reported. A forecast will 
be said to be accurate if true value falls within the interquartile range. We can 
observe that during 2008 year the LL-GARCH(1,1) model produces the most 
accurate predictive distribution of CPI. In this case the true values fall within 
the interquartile range in ten cases of twelve. For the other models i.e. the LL-
GARCH(1,1)-Student, LL-SV and AR(2)-SV, true values lie between the first 
and third quartiles only in eight cases of twelve. For the LL-GARCH(1,1) 
model, when inflation forecast is an accurate, the true values of CPI lie mostly 



Jacek Kwiatkowski 126 

between the median and third quartile. In other cases, predictive densities 
underestimate or overestimate the actual observation, both for the models with 
the constant and time-varying mean.    
 It is also worthwhile to check formally the accuracy of point forecasts of the 
mentioned above models. We measure forecast accuracy using two exact finite-
sample test, namely: the sign test and Wilcoxon signed rank test (see Diebold 
and Mariano, 1995). These tests allow us to analyze of statistical differences 
between predictions generated by competing specifications when only a small 
number of forecasts are available. We test the null hypothesis of that the 
forecasting performance of the two different models is equally well (poor). The 
results are summarized in the Table 2, which includes p-values from the sign 
test (the first line) and Wilcoxon signed rank test (in the second line). In our 
case we use quadratic loss function. The observed loss differentials are free of 
serial correlation. 

Table 2.  Results from the sign test and Wilcoxon signed rank test 

Model 
Model 

LL-GARCH(1,1) LL-GARCH(1,1)-
Student LL-SV AR(2)-SV 

LL-
GARCH(1,1) - 0.7744 

0.8501 
1.0000 
0.6772 

1.0000 
0.9697 

LL-
GARCH(1,1)-

Student 
- - 0.7744 

0.7910 
1.0000 
0.9697 

LL-SV - - - 0.7744 
0.1514 

AR(2)-SV - - - - 
Note: The first line denotes p-values from the sign test. The second line includes p-values from the Wilcoxon 
signed rank test  

 According to the results given in the Table 2, we do not reject at 
conventional levels the hypothesis of equal expected quadratic loss. In other 
words there is no significant difference between the accuracy of point forecasts 
of the competing specifications. The standard AR(2)-SV model is not 
a significantly worse (better) predictor of the Polish CPI than the LL model. 
 It is well known, that Polish monetary authorities conduct the policy under 
inflation targeting regime, with the medium and long term target for CPI index 
fixed at 2.5% and with one percentage point of accepted deviation. Therefore it 
is interesting to consider what is the posterior probability of the hypothesis, that 
inflation will stay inside the targeting bound and how this posterior probability 
changes as the forecasts horizon grows. Bayesian methodology provides a direct 
way to predict different scenarios of inflation. The predictive distribution 
depicts the probability of various outcomes for CPI inflation in the future and 
allows to assess uncertainty of monetary policy. Unlike classical (sample-
theory) approach we do not need carry out stochastic simulations since our 



Unobserved Component Model for Forecasting Polish Inflation 

 
127

approach follows from the basic rules of probability. Tables 3 and 4 present 
some characteristics of the probability distribution of the inflation path. 

Table 3.  Posterior probability that inflation will remain within the targeting regime 

Date 

Probability of inflation  

LL-GARCH(1,1) LL-GARCH(1,1)-
Student LL-SV AR(2)-SV 

Within 
(1.5%; 3.5%)  

within 
(1.5%; 3.5%) 

within 
(1.5%; 3.5%) 

Within 
(1.5%; 3.5%) 

2008M01 0.0052 0.0052 0.0104 0.0052 
2008M02 0.0833 0.0365 0.0417 0.0208 
2008M03 0.2083 0.0833 0.0938 0.1198 
2008M04 0.2656 0.1719 0.1771 0.2292 
2008M05 0.3333 0.2240 0.2760 0.3177 
2008M06 0.3385 0.2552 0.3021 0.3438 
2008M07 0.4010 0.3333 0.4479 0.4063 
2008M08 0.3854 0.2813 0.3438 0.3802 
2008M09 0.3906 0.2865 0.3542 0.3750 
2008M10 0.4167 0.3021 0.3958 0.3958 
2008M11 0.4635 0.3385 0.4635 0.4323 
2008M12 0.4688 0.3281 0.4635 0.4271 

Table 4. Posterior probability that inflation will be above the upper bound of targeting 
regime 

Date 

Probability of inflation  

LL-GARCH(1,1) LL-GARCH(1,1)-
Student LL-SV AR(2)-SV 

above 3.5% above 3.5% above 3.5% above 3.5% 
2008M01 0.9948 0.9948 0.9896 0.9948 
2008M02 0.9167 0.9635 0.9583 0.9792 
2008M03 0.7917 0.9167 0.9063 0.8802 
2008M04 0.7344 0.8281 0.8229 0.7708 
2008M05 0.6667 0.7760 0.7240 0.6823 
2008M06 0.6615 0.7448 0.6979 0.6563 
2008M07 0.5990 0.6667 0.5521 0.5938 
2008M08 0.6146 0.7188 0.6563 0.6198 
2008M09 0.6094 0.7135 0.6458 0.6250 
2008M10 0.5833 0.6979 0.6042 0.6042 
2008M11 0.5365 0.6615 0.5365 0.5677 
2008M12 0.5313 0.6719 0.5365 0.5729 

 Tables 3 and 4 include assessment of the risks around central projections for 
prices of consumer goods and services. The predictive results indicate that 
inflation was more likely to be above target in 2008 than below target. The 
predictive probability for the hypothesis, that inflation will stay inside the 
targeting bound ranges from 0.0052 to 0.2656 in the period January–April, from 
0.224 to 0.4479 in the period May–August and from 0.2865 to 0.4688 in the 



Jacek Kwiatkowski 128 

period September–December, whereas the predictive probability of the 
hypothesis that inflation will be above the upper limit of the tolerance band 
(3.5%), for all months and models, ranges from 0.5313 to 0.9948. These 
forecasts are consistent with true values of annual CPI because in 2008, 
according the GUS data2, inflation rose above the upper limit of the tolerance 
band. During first eight months of 2008, CPI inflation showed a rising tendency 
– from 4.0% in January to 4.8% in August. In the period September – December 
we observed decline in the annual growth from 4.3% to 3.3%. According to 
reports published by NBP, the main factor conducing to higher level of inflation 
was the prices of energy commodities in the world market3. Thus, it seems that 
all models have ability to assess correctly the risk associated with CPI inflation. 

4. Conclusions 
 In this paper the local level models are analyzed and compared from point 
of view of their ability to forecast monthly inflation in Poland. The data concern 
the consumer price index and they range from January 1992 till December 
2008. For each model and for each month of 2008 we constructed the predictive 
distributions. According to the sign test and Wilcoxon signed rank test, there is 
no significant difference between the accuracy of point forecasts of the 
competing specifications. Also all models show correctly a rising tendency of 
annual CPI inflation. Analysis of forecast accuracy of competing specifications 
does not lead to decisive conclusion about superiority of any of the considered 
specifications. It seems that we can identify at least two reasons why 
unobserved component model could not satisfactorily predict the inflation.  
 Firstly, it is known that the first differences of local level model display the 
same correlation structure as the IMA(1,1) model – that is, one in which the 
first order autocorrelation is negative for the first difference of series and all 
other autocorrelations are zero (see West and Harrison, 1989). This is a very 
restrictive assumption which is in practice very difficult to obtain. From 
preliminary studies it was known that in our case not only first but also second 
order autocorrelation is negative and statistically significant, which may 
indicate a more complicated correlation structure of analyzed process. For this 
reason we consider a second order autoregressive process.  
 Secondly, the recent publication by Grassi and Proietti (2008) showed the 
strong evidence in favor of the local level model with heteroscedastic 
disturbances only in the core component of the U.S. inflation, whereas the 
transitory component was time invariant. The volatility of the disturbances 
driving only one i.e. core component may improve accuracy of forecasts. 

                                                 
2 Central Statistical Office (GUS), http://www.stat.gov.pl/gus/index_ENG_HTML.htm 
3 Inflation Report, http://www.nbp.pl/ 



Unobserved Component Model for Forecasting Polish Inflation 

 
129

References 
Bos, C. (2001), Time Varying Parameter Models for Inflation and Exchange Rates, WebDOC. 
Diebold, F.X., Mariano, R.S. (1995), Comparing Predictive Accuracy, Journal of Business and 
  Economic Statistics, 13, 253–63. 
Durbin, J., Koopman, S.J. (2001), Time Series Analysis by State Space Methods, Oxford 

University Press, Oxford. 
Fallenbuchl, Z.M. (1994), Foreign Trade in the Process of Transformation in Poland, Atlantic 

Economic Journal, 22, 2, 51–60. 
Grassi, S., Proietti, T. (2008), Has the Volatility of U.S. Inflation Changed and How?, MPRA 

Paper 11453. 
Harvey, A.C. (1989), Forecasting, Structural Time Series Models and the Kalman Filter, 

Cambridge University Press, Cambridge. 
Koop, G. (2003), Bayesian Econometrics, John Wiley & Sons. 
Koop, G., Potter, S. (2001), Are Apparent Findings of Nonlinearity Due to Structural Instability 

in Economic Time Series?, The Econometrics Journal, 4, 1, 37–55. 
Muth, J.F. (1960), Optimal Properties of Exponentially Weighted Forecasts, Journal of the 

American Statistical Association, 55, 299–306. 
Pellegrini, S., Ruiz, E., Espasa, A. (2007), The Relationship Between ARIMA-GARCH and 

Unobserved Component Models with GARCH Disturbances, Working Paper, ws072706. 
Pellegrini, S., Ruiz, E., Espasa, A. (2008), ARIMA-GARCH and Unobserved Component Models 

with GARCH Disturbances: Are their Prediction Intervals Different? Job Market Paper. 
Stock, J.H., Watson, M.W. (2007), Why has U.S. Inflation Become Harder to Forecast? Journal 

of Money, Credit, and Banking, 39, 3–33. 
Stock, J.H., Watson, M.W. (2008), Phillips Curve Inflation Forecasts, Working Paper, 14322. 
West, M., Harrison, J. (1989), Bayesian Forecasting and Dynamic Models, Springer. 

Prognozowanie inflacji w Polsce przy użyciu modelu lokalnego poziomu 

Z a r y s  t r e ś c i.  W artykule przeprowadzono badania dotyczące trafności prognoz 
otrzymanych za pomocą modelu lokalnego poziomu w wersji Stocka i Watsona (2008). 
Rozważono różne postacie tego modelu i zbadano, które z nich dają możliwość uzyskania 
najtrafniejszej prognozy. Badania empiryczne dotyczyły inflacji w Polsce w latach 1992-2008. 
Ostatni rok posłużył do oceny jakości prognoz. Badania przeprowadzono na podstawie wskaźnika 
cen konsumenta CPI. Uzyskane wyniki nie potwierdzają jednoznacznej przewagi modelu 
lokalnego poziomu, w prognozowaniu inflacji, nad standardowym modelem autoregresyjnym. 
Wszystkie modele uzyskały zadowalającą dokładność prognozy. 
S ł o w a  k l u c z o w e: model lokalnego poziomu, prognozowanie, inflacja. 
 



 
 
 




