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http://www.dem.umk.pl/dem 

D Y N A M I C  E C O N O M E T R I C  M O D E L S  
DOI: http://dx.doi.org/10.12775/DEM.2017.006  Vol. 17 (2017) 97−114 

Submitted October 31, 2017  ISSN (online) 2450-7067 

Accepted December 28, 2017 ISSN (print) 1234-3862 

Adrian Burda, Błażej Mazur , Mateusz Pipień
*
 

Forecasting EUR/PLN Exchange Rate: the Role  
of Purchasing Power Parity Hypothesis  

in ESTVEC Models 

A b s t r a c t. The purpose of this paper is to verify empirical consequences of imposing 

various forms of purchasing power parity (PPP) within a class of smooth transition vector 

error correction models (ESTVECM) for analysis of EUR/PLN exchange rage. Empirical 

importance of exponential smooth transition functions is confronted with the linear error-

correction mechanism. A class of competing models for recursive samples are compared by 

the likelihood ratio test, information criteria, and out of sample forecast accuracy measures. 

K e y w o r d s: PPP; ESTVECM; cointegration; exchange rate forecasting 

J E L Classification: C32; F31; F37  

1. Introduction 

Purchasing power parity (PPP) is one of the oldest theories regarding ex-

change rate. Its empirical importance has been investigated intensively for 

decades. Rogoff (1996) and Officer (1982) point out that PPP theory had 

                                                 
* Correspondence to: Adrian Burda, e-mail: adrian.marek.burda07@gmail.com; Błażej 

Mazur, Department of Econometrics and Operations Research, Cracow University of Eco-

nomics, Rakowicka 27, 31-510 Kraków, e-mail: blazej.mazur@uek.krakow.pl, Mateusz 

Pipień, Department of Econometrics and Operations Research, Cracow University of Eco-

nomics, Rakowicka 27, 31-510 Kraków, e-mail eepipien@cyf-kr.edu.pl. 
 This research was supported by the National Science Center, Poland, based on decision 

number UMO-2014/13/N/HS4/03593, and by the funds granted to the Faculty of Manage-

ment, Cracow University of Economics within the framework of the subsidy for the mainte-

nance of research potential. 

https://orcid.org/0000-0001-5096-5175


Adrian Burda, Błażej Mazur, Mateusz Pipień 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

98 

been initially articulated by the scholars of Salamanca school in Spain in the 

sixteenth century. Modern analyses and formulations of PPP theory could be 

dated on early 20’s of the 20th century (e.g. Cassel, 1918). Nowadays PPP 

constitutes theoretical foundations of many long-run exchange rate models, 

as e.g. sticky-prices monetary models (Dornbusch, 1976), flexible-price 

monetary model (e.g. Johnson, 1978) or asset pricing models (Lucas, 1982). 

Consequently, the PPP hypothesis is also embedded in larger macroeconom-

ic models, as e.g. in DSGE models for small open economy or two econo-

mies (e.g. Ca‘Zorzi et al., 2017; Senbeta, 2011). Furthermore, PPP could be 

useful in forecasting exchange rate, in particular in the long-run (e.g. 

Ca‘Zorzi et al., 2016). 

Despite of wide applications of PPP theory, empirical evidence is far 

from being conclusive (e.g. Arize et al., 2015; Kelm, 2013, p. 68). While 

most of the existing literature do not confirm or even reject PPP hypothesis 

in the short-run (Arize et al., 2015), for the long-run results may vary. For 

example Chang et al. (2010), Wang (2000) or Pappel (1997) reject long-run 

PPP, while Arize et al. (2004; 2015), Cheung et al. (2004), Lothian & Taylor 

(1996) found support for PPP in terms of cointegrating relationships or real 

exchange rate for most of their samples. However, conclusion from most of 

the existing research should be treated with caution due to its methodological 

drawbacks (for discussion see e.g., Kelm, 2013, pp. 58–67). Among PPP 

studies, positive results are reported quite frequently, when the smooth tran-

sition (STR) approach is utilized, either to test stationarity of the real ex-

change rate (e.g. Kapetanios et al., 2003; Sollis, 2009; McMillan, 2009; 

Kelm, 2013) or to verify cointegration relationship (e.g. Gefang, 2008). 

The aim of this study is to utilize STR cointegration framework to inves-

tigate PPP hypothesis. The research is conducted for EUR/PLN exchange 

rate as an interesting example of emerging market currency. The methodolo-

gy of this research is motivated by the work of Gefang (2008), though the 

paper includes a number of extensions. Firstly, we allow for STR mechanism 

separately in selected components of the VECM model. This gives oppor-

tunity to find optimal specification, preferably linking parsimony of 

parametrization and good explanatory power. Furthermore, in this research 

strong-form PPP is considered, while Gefang (2008) investigated only weak-

form PPP. Moreover, we conduct in-sample and out-of-sample analysis. 

The research sample covers the period between January 1999 and Febru-

ary 2016, while out of sample forecasts are tested until February 2017. On 

the one hand, it takes into account only period with one exchange rate re-

gime, while on the other hand it has heterogeneous features as in this period 



Forecasting EUR/PLN Exchange Rate: the Role of Purchasing Power Parity… 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

99 

Poland accessed to European Union, and global financial crisis as well as 

Euro Area sovereign debt crisis took place. 

The article is organized as follows. Firstly, we provide a brief summary 

of PPP theory, methods of its verification and existing empirical literature. In 

section 3 we introduce the models used here, while section 4 contains empir-

ical comparison of different models in terms of model fit and out of sample 

forecast performance. 

2. Purchasing Power Parity – General Concept and Empirical 
Importance 

The PPP theory in all variants is rooted in the “law of the one price” 

(LOP). It states that for any good i: 

      
  

   
  (1) 

where     is the domestic-currency price of good i at time t,    
  is the foreign 

currency price of good i at time t and     is the equilibrium exchange rate at 

time t, defined as home price of foreign currency. The relationship (1) could 

be expressed also for price indices: 

         
 
      

        
   

     
  

  
   (2) 

where    is the weight of price of good i in domestic and foreign price indi-

ces and n denotes the number of goods in domestic and foreign price indices.  

Deviations from LOP for one item does not mean deviation from LOP for 

the whole price index, as aggregations and weighing may compensate impact 

of this breaches; see Wdowiński (2010). Thus the strong (or strict) form of 

PPP
1
 states:  

  
  

      
  (3) 

or, in the log-linear form:  

     
  

                
  .  (4) 

Hence, the real exchange rate Q can be written as: 

             
  

               
     , (5) 

where    is a constant value. 

Empirical testing of existence of the strict PPP and law of the one price 

is based on assumptions that market works perfectly and any deviations from 

                                                 
1 Alternatively, PPP models could be distinguished between satisfying and non-satisfying 

the long-term homogeneity restriction. 



Adrian Burda, Błażej Mazur, Mateusz Pipień 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

100 

the aforementioned relationships (4) or (5) are not persistent. Thus two em-

pirical strategies could be applied. In the first one, testing of unit root for the 

logarithm of the exchange rate is utilized. Stationarity of the real exchange 

rate is interpreted as empirical support of existence of the strong-form of 

PPP. The second approach bases on estimation of the following long run 

equation: 

     
  

                   
     , (6) 

where    is a stationary error term and restriction           is subject 

to analysis. 

Economic assumptions that guarantee the existence of the strong-form of 

PPP are rather restrictive. For instance, on has to assume costless spatial 

arbitrage (e.g. no quotas, import tariffs, transportation costs), no measure-

ment errors, or information costs (Arize et al., 2015), which implicitly is 

equivalent to existence of rational expectations. Further conditions men-

tioned in the literature are: non-existence of pricing-to market (PTM) strate-

gies, lack of nominal rigidities and lack of entry-barriers to international 

markets (Kelm, 2013, p. 28). Consequently, three general approaches have 

been developed in the literature to relax the assumption.  

In the first approach, the restriction:           in (6) is relaxed 

and only the symmetry restriction        is analysed. It reflects presence 

of transportation costs, other trade barriers, measurement problems (e.g. 

Taylor, 1988; Arize et al., 2015).  

Second approach is connected with conclusion from Dumas (1992), 

Uppal (1993), Sercu et al. (1995), O’Connel & Wei (1997) and more recent-

ly Pavlidis et al. (2011) emphasizes the role of transaction costs: 

      
  

                 
  .  (7) 

Therefore two regimes can be identified. The first (so-called inner regime) is 

defined by condition                 , while the second regime (called 

outer regime) is obtained in the case                       . Within 

the inner regime the real exchange rate can be described as I(1) process. 

Within the outer regime real exchange rate should return toward inner re-

gime. Empirical testing should take into accounts existence of three regimes: 

the inner regime and two – positive and negative outer regimes. Consequent-

ly different variants of the threshold autoregressive (TAR) or smooth transi-

tion autoregressive (STAR) models are applied as a natural generalization of 

the linear scheme. Similarly as in more restrictive cases above, two strategies 

utilizing TAR or STAR approach has been developed. The first and the most 

popular one, allows for testing the unit root hypothesis of logarithm of the 



Forecasting EUR/PLN Exchange Rate: the Role of Purchasing Power Parity… 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

101 

exchange rate against TAR/STAR alternatives (e.g. Kapetanios et al., 2003; 

Sollis, 2009; McMillan, 2009; Månsson & Sjölander 2014). The second ap-

proach, which is also applied in this study, is connected with verification of 

the PPP hypothesis in the multivariate framework, within Threshold or 

Smooth Transition Vector Error Correction frameworks  

(TVECM/STVECM), both strong-form PPP (e.g. Wu & Chen, 2008; Naka-

gawa, 2010), as well as weak-form PPP (e.g. Gefang, 2008). 

Third approach is connected with a reformulation of (6) resulting in: 

     
  

                      
        

        

                                   
            

           (8) 

where      ,      ,       – are vectors of long, medium and short run deter-

minants of nominal exchange rate (except PPP),                  
  – 

vector of equilibrium coefficients,        ,         (see Kelm & Bęza-

Bojanowska, 2005; Kelm, 2010). Restriction            =        sup-

ports PPP hypothesis; see Kelm (2013, p. 27). 

The empirical literature where PPP hypothesis is analysed is vast. In this 

paper only key papers are mentioned. Firstly the enormous short-term vola-

tility of the real exchange rate with the extremely slow rate at which shocks 

appear to damp out still misses satisfactory and sound explanation (PPP puz-

zle, see Rogoff,1996) and that the nominal rigidities and market frictions 

are still unable to explain why real exchange rates (RER) deviate from the 

PPP level and high estimates of RERs’ half-lives (3–5 years), see Kelm 

(2017). Secondly, allowing for smooth transition mechanism, seems to solve 

at least partially “PPP puzzle” (e.g. Schnatz, 2007, Norman, 2010). On the 

other hand, both theoretical and empirical soundness of testing procedures in 

many cases could be doubtful (e.g. Kelm, 2013, p. 90–92). For example 

Månsson & Sjölander, (2014), and Emirmahmutoglu & Omay, (2014) em-

phasized weak power of KSS (Kapetanios, Shin & Snell) and AKSS (aug-

mented KSS) tests. Thirdly, verification of the PPP within the cointegration 

framework might be difficult if the number of cointegrating vectors differs 

from one. 

Empirical results of PPP testing for the Polish Zloty depend on the sam-

ple used in the research and on the choice of the price indices. In particular if 

the sample starts before 1999, the strong-form PPP is not supported, due to 

characteristics of the exchange rate regimes in Poland before 1999 (as de 

facto Polish Zloty became free-floating). More specifically, both fixed ex-

change rate regime (before October 1991) and crawling peg (before May 

1995) represented the nominal anchor feature (Kokoszczyński, 2001). In 



Adrian Burda, Błażej Mazur, Mateusz Pipień 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

102 

terms of price indices, one should note that production price index (PPI) in 

manufacturing seems to be the best proxy for tradables prices (see Kelm, 

2013, p. 141). 

Hence in prevailing part of the literature, strong-form PPP for the Polish 

Zloty is not supported, since the research sample contains data from mid 

1990s (e.g. Rubaszek & Serwa; 2009; Wdowiński 2010; Chang & Tzeng, 

2011) or even include the whole transition period (Arize et al., 2015). Fur-

thermore, in many studies (Arize et al., 2015), at least for some specifica-

tions (Rubaszek & Serwa, 2009; Wdowiński; 2010) several forms of weak-

form PPP (as proportionality restrictions) were rejected. Recently for sam-

ples including only free floating exchange rate regime period the weak-form 

PPP is confirmed or not rejected quite frequently. Surprisingly, the strong-

form PPP is supported only exceptionally. Kelm (2013, p.188) does not re-

ject symmetry restriction in one of specification of VECM with I(2) vari-

ables and rejects null hypothesis about unit rot against ESTAR process for 

the real exchange rate. However both cases are controversial – as in VECM 

with I(2) variables p-value for symmetry restriction is only 0.118 and inter-

pretation of model is not straightforward as in VECM with only I(1) vari-

ables. Furthermore, results of unit root test against ESTAR are driven by 

abnormal observations, mainly from the year 2008 (Kelm, 2013, p.172–173).  

3. Econometric Framework Utilizing Nonlinear Cointegration 

The econometric framework applied here is designed to fulfill assump-

tions of existence of transaction costs. In the multivariate framework, 

Threshold VECM (TVECM) or Exponentially Smooth Transition VECM 

(ESTVECM) are relevant. In these cases the dynamics of the adjustment 

changes across regimes (inner and outer), while in the simple linear VECM 

adjustment is described by a linear function of the magnitude of the devia-

tions from the long run equilibrium. The driving forces of the regime 

changes are governed by the observed deviations from the equilibrium 

through the transition function. In a TVECM, the regime changes are as-

sumed to be discrete, whereas in an ESTVECM, the regimes change 

smoothly. 

The final ESTVECM specification used in the research is consistent with 

Gefang (2008), with some modifications – allowing for testing the strong-

form PPP and more flexible specification. 

Let                         
   , where  ,  , and    are respectively 

EUR/PLN exchange rate, domestic price index and foreign price index. As-



Forecasting EUR/PLN Exchange Rate: the Role of Purchasing Power Parity… 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

103 

suming that cointegration relationship is common among regimes, the EST-

VECM is described for         as follows:  

                                    
  

 
   

                       +Dt Z+h=1p Yt h h z)+εt (9) 

where            , the error time ε  is Gaussian white noise process 

with         and: 

          
          
         

  . 

Finally, the deterministic term    contains intercept only. The dimen-

sions of    and  
  
  are [    , the dimension of   is [     ,while the di-

mensions of   and    are [    , with r denoting the cointegration rank. In 

this research it is assumed that    . If strong-form PPP holds, we have: 

          . 
In (9) changes of regimes are driven by past deviations from the equilib-

rium relationships and dynamics of changes of regimes are captured by the 

exponential smooth transition function proposed by Teräsvirta (1994): 

                      ), (10) 

where the transition variable          is the cointegrating combination 

between ln(S), ln(P) and ln(  ) at time t–d
2
, c denotes the equilibrium level 

of cointegration relationship and γ is the smoothness parameter. Higher γ 

induces faster transition. This function has symmetric U shape, which illus-

trative example for the case with c = 0 is presented on Figure 1. 

Formulas (9) and (10) allow for a set of models, varying the order of the 

autoregressive process, lag length of the transition variable and presence of 

nonlinearity in loading coefficients, autoregressive process and deterministic 

terms. In this research we assume that    ,    , while different variants 

of non-linearity in particular element of ESTVECM are allowed in (9). 

These variants of nonlinearities could be described by zero restrictions in     
  ,  

  
 . The set of competing specifications are presented in Table 1. 

                                                 
2 When strict PPP holds it could be interpreted as real exchange rate at time t-d. 



Adrian Burda, Błażej Mazur, Mateusz Pipień 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

104 

 

Figure 1. Illustrative example for the exponential smooth transition function with 

different values of γ 

Table 1. The set of competing specifications 

Models 
ESTVEC 

Full 

ESTVEC 

α 

ESTVEC 

α,ξ 

ESTVEC 

α,Г 

ESTVEC 

Г,ξ 

ESTVEC 

Г 

ESTVEC 

ξ 

Parameters 
under zero 
restrictions 

none       
     

                    
  

4. Empirical Results 

We used monthly data of the nominal exchange rate EUR/PLN (monthly 

average) and PPI manufacturing indices in Poland and Euro area. The full 

sample covers the period from January 1999 to February 2016 and hence the 

total number of observations, including starting values, is 206. We also per-

formed recursive estimation and prediction on the basis of expanding win-

dows, starting from the smallest sample covering the period from January 

1999 to February 2006 and ending up on the sample from January 1999 to 

February 2016. 

We estimated parameters of all competing models on the basis of Maxi-

mum Likelihood estimator. The strong form PPP is tested with the use of LR 

statistics. The model under the null hypothesis is obtained by the following 

restriction: 

          .  

We compared the forecasting performance of models on the basis of 

MAE and RMSE summaries. Also Diebold and Mariano (1995) test (DM) 



Forecasting EUR/PLN Exchange Rate: the Role of Purchasing Power Parity… 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

105 

was applied to check significance of differences of generated series of fore-

casts from the analogous series obtained on the basis of the Random Walk 

(RW) strategy. The DM procedure was applied together with small sample 

corrections (HLN) proposed by Harvey, Leybourne and Newbold (1997). 

Table 2 contains model comparison results for VEC specifications and sev-

eral generalisations towards smooth transition mechanism. We present loga-

rithmic values of the likelihood calculated at ML estimates (LL) and AIC, 

BIC and HIC scores, respectively. Estimation is conducted for unrestricted 

cases and alternatively with PPP restriction imposed. The likelihood infer-

ence clearly indicates superiority of the smooth transition mechanism against 

simple VEC construct. The greatest data support, measured by LL value, 

received Full ESTVEC model and some limited parameterisations with 

nonlinearities in Г and ξ separately and jointly Г, ξ. However these models 

seem to be too heavily parameterised and are penalised substantially by in-

formation criteria scores. Among unrestricted specifications, smooth transi-

tion mechanism in parameters ξ seems to be an optimal compromise result-

ing with good data fit and parsimony. 

Analyses conducted within a class of models with PPP restriction im-

posed show again superiority of nonlinear mechanisms. The full ESTVECM 

model receives again the highest LL value, but it is rejected by information 

scores as unparsimonious. Among restricted cases the smooth transition 

construct in parameters α is supported by the data and receives the best in-

formation score. Also the case with the smooth transition mechanism in pa-

rameters Г receives attention. 

In the next step we analysed statistical significance of parameters of 

ESTR function and restriction that guarantees PPP effect. On the basis of 

expanding data window described above we performed a sequence of appro-

priate LR tests. On Figure 2 and 3 we present fractions of analysed subsam-

ples where ESTR coefficients were significant (Figure 2) and where strong 

form of PPP restriction was not rejected (Figure 3). 

Decisive and more importantly time invariant inference about signifi-

cance of underlying nonlinearities is impossible for heavy parameterised 

models. Full ESTVEC model and these limitations with smooth transition 

parameters imposed on pairs of groups of parameters – α and Г, Г and ξ, α 

and ξ perform worse and there are subsamples that do not support statistical 

significance of ESTR coefficients. The model ESTVEC-ξ, performs the best 

receiving in all analysed subsamples statistically significant ESTR construct 

at 0.1 and 0.05 significance level.  

Rejection of the strong form of PPP are analysed in subsamples on Fig-

ure 3. In this element of analyses simple linear VECM is also considered. 



Adrian Burda, Błażej Mazur, Mateusz Pipień 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

106 

We report surprisingly good performance of VECM specification and also 

relatively worse results given models with smooth transition component. 

Table 2.  Log-likelihood and information criteria obtained for all competing specifi-

cations in case of the whole sample 

 
Model LL AIC BIC HIC 

Unrestricted specifications 

VEC 2267.2 –4500.3 –4443.9 –4477.5 

ESTVEC Full 2282.3 –4496.6 –4383.8 –4450.9 

ESTVEC α 2274.8 –4505.6 –4432.6 –4476.1 

ESTVEC α,ξ 2277.8 –4505.6 –4422.7 –4472.1 

ESTVEC α,Г 2280.0 –4498.0 –4395.1 –4456.4 

ESTVEC Г,ξ 2282.6 –4503.2 –4400.3 –4461.6 

ESTVEC Г 2283.4 –4510.9 –4418.0 –4473.3 

ESTVEC ξ 2280.6 –4517.1 –4444.1 –4487.6 

Models with strong-form 
PPP restriction imposed 

VEC 2265.1 –4500.3 –4450.5 –4480.1 

ESTVEC Full 2280.5 –4497.1 –4390.9 –4454.1 

ESTVEC α 2272.6 –4505.1 –4438.8 –4478.3 

ESTVEC α,ξ 2273.3 –4500.6 –4424.3 –4469.8 

ESTVEC α,Г 2277.8 –4497.7 –4401.5 –4458.8 

ESTVEC Г,ξ 2278.9 –4499.7 –4403.5 –4460.8 

ESTVEC Г 2277.9 –4503.7 –4417.5 –4468.8 

ESTVEC ξ 2267.4 –4494.7 –4428.4 –4467.9 

 
Figure 2. Fraction of subsamples, where ESTR coefficients are significant on given 

significance level, according to LR test 

We also compare competing specifications with respect to the forecast-

ing power. For the whole set of observations we generated 1, 3, 6 and 12 

month-ahead forecasts. In Table 3 we put RMSE values normalised to RW 



Forecasting EUR/PLN Exchange Rate: the Role of Purchasing Power Parity… 

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107 

strategy. We also present p-values (in italics) of the HLN-DM test against 

RW for quadratic loss function. Simple VECM specification exhibits much 

better forecasting performance in the short term. For one month-ahead case 

none of analysed smooth transition models generate better forecasts. Also in 

case of this horizon all competing specifications are much better than Ran-

dom Walk and generate lower RMSE. However only simple VECM and 

ESTVECM- ξ generate forecasts significantly different than RW case at the 

significance level of 0.1. In case of unrestricted models specification EST-

VEC with smooth transition mechanism in parameters ξ outperform other 

models for long term forecasting. For the case of 12 months ahead forecasts 

simple VECM is worse than RW and differences in point forecasts are statis-

tically insignificant. For a set of restricted models we report relatively good 

forecasting performance in smooth transition class except the full case and 

cases with ST mechanism in α, jointly α,Г and ξ. VECM model produces 

much better forecasts than ST class and outperforms RW case.  

 
Figure 3. Fraction of subsamples, where strong-form PPP restrictions are not re-

jected given significance level, according to LR test 

Analysing statistical significance of differences in forecasting perform-

ance (against RW) on the basis of HLN-DM test, in the set of unrestricted 

models, we can only be sure that data support poor forecasting performance 

in case of some ESTVEC specifications. Simple VECM model generates 

better forecasts that RW, but with differences to RW being statistically in-

significant in case of all analysed horizons. In case of restricted models only 

the case of one month horizon exhibit significant differences from RW strat-



Adrian Burda, Błażej Mazur, Mateusz Pipień 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

108 

egy in all cases. The strongest data support in favour of significance of dif-

ferences is attached to simple VECM model. 

Table 3. RMSE for the whole sample and p-values (in italics) for HLN-DM test 

against RW for quadratic loss function 

Relative RMSE (RW=1) – log(FX) unrestricted models 

horizon VECM 
ESTVEC  

α 
ESTVEC  

ξ 
ESTVEC 

Г 
ESTVEC 

α,ξ 
ESTVEC 

Г,ξ 
ESTVEC 

α,Г,ξ 
ESTVEC 

Full 

1m. 0.917 0.958 0.933 0.948 0.962 0.992 0.961 0.983 

 0.1 0.3 0.1 0.2 0.3 0.5 0.3 0.4 

3m. 0.955 1.049 0.965 0.996 1.055 1.068 1.102 1.065 

 0.3 0.7 0.4 0.5 0.7 0.7 0.8 0.8 

6m. 0.997 1.142 0.937 0.965 1.117 1.080 161.531 1.153 

 0.5 1.0 0.2 0.3 0.9 0.2 0.2 1.0 

12m. 1.014 1.265 0.889 0.921 1.201 76.448 >1e+10 1.321 

 0.7 0.9 0.06 0.1 0.9 0.8 0.8 1.0 

Relative RMSE (RW=1) – log(FX) restricted models 

horizon VECM 
ESTVEC  

α 
ESTVEC  

ξ 
ESTVEC 

Г 
ESTVEC 

α,ξ 
ESTVEC 

Г,ξ 
ESTVEC 

α,Г,ξ 
ESTVEC 

Full 

1m. 0.900 0.918 0.916 0.900 0.925 0.931 0.928 0.933 

 0.07 0.1 0.1 0.07 0.1 0.1 0.1 0.1 

3m. 0.923 0.969 0.950 0.923 0.956 0.946 0.962 0.969 

 0.3 0.4 0.3 0.3 0.4 0.3 0.4 0.4 

6m. 0.949 0.997 0.958 0.949 0.946 0.932 1.036 0.969 

 0.3 0.5 0.3 0.3 0.3 0.3 1.0 0.4 

12m. 0.936 1.049 0.944 0.936 0.989 0.913 176.208 1.015 

 0.3 0.9 0.3 0.3 0.5 0.3 0.8 0.5 

The forecasting performance of both (restricted and unrestricted) class of 

models for subsamples is presented in Tables 4 and 5. Table 4 presents 

RMSE relative to RW for a group of unrestricted specifications, while Table 

5 shows results for models with strong PPP restriction imposed. The se-

quence of subsamples were split into two sets, the first one covers the sub-

samples ending ad 2011:03 to 2016:02 and the second one contains these 

ending at 2011:03 to 2016:02. Just like in case of the whole sample, we re-

port in all Tables p-values of the HLN-DM test against RW for the quadratic 

loss function. Again, in case of a set of unrestricted models ESTVECM with 

smooth transition function in ξ provides much better forecasts for longer 

horizon in both series of subsamples. In short term forecasting this model 

also beats simple VECM in second set of subsamples (2011:03 – 2016:02). 

Simple VECM performs relatively better than RW only in case of first series 

of subsamples. 



Forecasting EUR/PLN Exchange Rate: the Role of Purchasing Power Parity… 

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109 

Analysing results presented in Table 5 we report relatively good fore-

casting power of ESTVECM specification in the long term. Also simple 

VECM provide the best forecasts in case of one month ahead and three 

month-ahead horizon. 

Table 4. RMSE for the sequential forecasting comparison out of sample periods for 

unrestricted models and p-values (in italics) for HLN-DM test against RW 

for quadratic loss function 

Relative RMSE (RW=1) – log(FX) –for subsamples: 2006:03–2011:02 

horizon VECM 
ESTVEC  

α 
ESTVEC  

ξ 
ESTVEC 

Г 
ESTVEC 

α,ξ 
ESTVEC 

Г,ξ 
ESTVEC 

α,Г,ξ 
ESTVEC 

Full 

1m. 0.904 0.948 0.926 0.945 0.955 1.005 0.959 0.983 

 0.1 0.3 0.2 0.2 0.3 0.5 0.3 0.4 

3m. 0.917 1.022 0.937 0.972 1.022 1.069 1.120 1.037 

 0.3 0.6 0.3 0.4 0.6 0.7 0.8 0.6 

6m. 0.988 1.126 0.929 0.954 1.087 1.091 174.947 1.137 

 0.4 1.0 0.2 0.3 1.0 0.8 0.8 1.0 

12m. 1.010 1.249 0.886 0.910 1.172 79.682 >1e+10 1.314 

 0.6 1.0 0.04 0.1 1.0 0.9 0.9 1.0 

Relative RMSE (RW=1) – log(FX) –for subsamples: 2011:03–2016:02 

horizon VECM 
ESTVEC  

α 
ESTVEC  

ξ 
ESTVEC 

Г 
ESTVEC 

α,ξ 
ESTVEC 

Г,ξ 
ESTVEC 

α,Г,ξ 
ESTVEC 

Full 

1m. 0.955 0.979 0.946 0.954 0.977 0.953 0.965 0.982 

 0.3 0.4 0.2 0.2 0.4 0.2 0.3 0.4 

3m. 1.085 1.136 1.060 1.079 1.165 1.066 1.031 1.163 

 0.8 0.9 0.8 0.9 0.9 0.8 0.6 1.0 

6m. 1.056 1.225 0.971 1.022 1.271 1.009 1.188 1.241 

 0.8 1.0 0.4 0.6 1.0 0.5 1.0 0.9 

12m. 1.056 1.434 0.892 1.014 1.487 1.013 >5e+5 1.385 

 0.7 0.9 0.2 0.5 0.8 0.5 0.8 0.8 

However, only a few analysed cases generate forecasts significantly dif-

ferent compared to the RW-based ones. Relatively good-performing simple 

VECM model for one month ahead horizon as well as ESTVEC-ξ and EST-

VEC-Г for 12 months ahead differs significantly from the RW case for unre-

stricted models and the first set of subsamples; see Table 4. In this case the 

best forecasting performance, obtained for 12 month horizon in ESTVEC-ξ 

model receives also the strongest data evidence against RW forecasts as 

tested by HLN-DM quadratic score. 

We report not so strong statistical significance in case of restricted mod-

els; see Table 5. Only in case of short term forecasts p-values are not greater 

than 0.1 making weak data evidence in favour of significant differences of 

generated forecasts from RW case. 



Adrian Burda, Błażej Mazur, Mateusz Pipień 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

110 

Table 5. RMSE for the sequential forecasting comparison out of sample periods for 

models with strong-form PPP restriction and p-values (in italics) for HLN-

DM test against RW for quadratic loss function 

Relative RMSE (RW=1) – log(FX) –for subsamples: 2006:03–2011:02 

horizon VECM 
ESTVEC  

α 
ESTVEC  

ξ 
ESTVEC 

Г 
ESTVEC 

α,ξ 
ESTVEC 

Г,ξ 
ESTVEC 

α,Г,ξ 
ESTVEC 

Full 

1m. 0.886 0.905 0.907 0.886 0.909 0.927 0.919 0.919 

 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 

3m. 0.882 0.928 0.909 0.882 0.900 0.912 0.921 0.921 

 0.2 0.3 0.3 0.2 0.3 0.3 0.3 0.3 

6m. 0.928 0.973 0.924 0.928 0.896 0.892 1.032 0.925 

 0.2 0.4 0.3 0.2 0.2 0.2 0.9 0.3 

12m. 0.905 1.038 0.894 0.905 0.955 0.859 183.662 0.981 

 0.2 0.8 0.2 0.2 0.3 0.2 0.9 0.4 

Relative RMSE (RW=1) – log(FX) –for subsamples: 2011:03–2016:02 

horizon VECM 
ESTVEC  

α 
ESTVEC  

ξ 
ESTVEC 

Г 
ESTVEC 

α,ξ 
ESTVEC 

Г,ξ 
ESTVEC 

α,Г,ξ 
ESTVEC 

Full 

1m. 0.938 0.954 0.943 0.938 0.963 0.939 0.955 0.969 

 0.2 0.3 0.2 0.2 0.3 0.2 0.3 0.3 

3m. 1.062 1.108 1.087 1.062 1.134 1.062 1.102 1.128 

 0.8 0.9 0.9 0.8 1.0 0.9 0.9 1.0 

6m. 1.067 1.130 1.136 1.067 1.186 1.134 1.062 1.188 

 0.8 1.0 1.0 0.8 1.0 1.0 0.9 1.0 

12m. 1.246 1.175 1.399 1.246 1.300 1.386 1.131 1.337 

 0.8 1.0 0.9 0.8 1.0 1.0 0.9 1.0 

Conclusions 

In this paper we investigated PPP hypothesis for the Polish Zloty in 

a multivariate dynamic econometric model. We confront the explanatory 

power of relatively broad class of models utilizing certain nonlinear 

cointegration concepts. We depart from a simple VECM framework into the 

STVECM class and verify empirical importance of generalizations (and 

hence the strength of statistical evidence in favor of PPP hypothesis). 

The in-sample results of model comparison by information criteria 

(based on full dataset) deliver no clear-cut conclusions, though in general the 

strongest support is allocated to two specifications: a linear VECM model 

with PPP restriction and an unrestricted ESTVEC ξ specification. 

For a more detailed analysis we consider an out-of-sample expanding-

window recursive forecasting experiment with two verification windows, 

one for the period 2006–2011 and one for 2011–2016. The results are the 

following: in short forecast horizons (1–3 months ahead), the imposition of 



Forecasting EUR/PLN Exchange Rate: the Role of Purchasing Power Parity… 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 97–114 

111 

PPP improves the forecasting performance across all the specifications and 

verification windows under consideration. The winning specification is a 

linear VEC model (with PPP). However, for longer forecasting horizons  

(6–12 months-ahead), the results are more complicated. In general, 

ESTVECM specifications involving smooth transition mechanism for the ξ  

parameters provide the best performance. There are though important differ-

ences between the two verification periods. In the first period, the 

ESTVECM-ξ model with PPP imposed is an overall winner for the  

12-months ahead forecasts, although some other models involving smooth 

transition for ξ (with or without PPP) also offer similar performance. In the 

second period, the imposition of PPP restrictions results in a drop in long-

horizon forecasting performance of the models mentioned above. In general, 

all the winning specifications mentioned above outperform simple RW-type 

forecasts. 

Hence, our main results are the following. On the one hand, for the 

short-term prediction, a linear VECM with PPP seems to be a preferable 

tool. On the other hand, as longer horizons are involved, the need for nonlin-

ear dynamics (as in ESTVECM-ξ) becomes more evident. As to the validity 

of PPP, the empirical support (based on long-horizon performance) is how-

ever time-inhomogeneous – it seems that more recent observations provide 

evidence against the PPP restrictions of the form considered here. 

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Weryfikacja hipotezy parytetu sił nabywczej dla kursu walutowego 
EUR/PLN w ramach wektorowych modeli korekty błędu  

z funkcją wygładzonego przejścia (ESTVECM) 

Z a r y s  t r e ś c i. Celem artykułu jest ocena empirycznych konsekwencji narzucenia hipote-

zy PPP w formie mocnej (ang. strong-form) dla kursu EUR/PLN przy wykorzystaniu wybra-

nych modeli kointegracji nieliniowej, to jest modeli ESTVEC. Zasadność wykładniczej funk-

cji przejścia dla mechanizmu korekty błędu jest testowana w odniesieniu do liniowego mode-

lu VEC. Konkurencyjne modele są porównywane zarówno pod względem dopasowania we-

wnątrz próby, jak i zdolności predyktywnych. Wyniki wspierają mechanizm wygładzonego 

przejścia w składniku deterministycznym. Żaden z modeli ESTVECM nie generuje systema-

tycznie lepszych prognoz niż liniowy model VECM 

S ł o w a  k l u c z o w e Parytet siły nabywczej; ESTVECM; kointegracja; Prognozowanie  

kursu walutowego.  


