




































© 2017 Nicolaus Copernicus University. All rights reserved.  

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.2017.010  Vol. 17 (2017) 161−176 

Submitted November 29, 2017  ISSN (online) 2450-7067 

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

Barbara Będowska-Sójka
*
  

Evaluating the Accuracy of Time-varying Beta.  
The Evidence from Poland  

A b s t r a c t. This paper empirically investigates various approaches to model time-varying 

systematic risk on the Polish capital market. A plenty of methods is examined in the 

developed markets and the Kalman filter approach is usually indicated as the best method for 

estimation of time-varying beta. However, there exists a gap in the studies for the emerging 

markets. In the paper we apply weekly data of fifteen stocks listed on the Warsaw Stock 

Exchange from banking and informatics sector. The sample starts at the beginning of 2001 

and ends in 2015 including the hectic crisis period. We estimate beta within few competing 

approaches: two MGARCH models, BEKK and DCC, unobserved component model, and 

static beta from linear regression. All beta estimates are compared in the securities market line 

framework. We find that unobserved component beta together with beta from DCC model 

have higher predictive accuracy than beta from BEKK model or static beta. The beta 

estimates are positively correlated within the industry and negatively correlated for stocks 

from different sectors. Finally, the prediction of beta coefficients are more accurate for stocks 

from banking sector than for IT companies. 

K e y w o r d s: BEKK; DCC; Kalman filter; MGARCH; time-varying beta. 

J E L Classification: G15; Q47. 

                                                           
* Correspondence to: Barbara Będowska-Sójka, Poznań University of Economics and 

Business, Faculty of Informatics and Electronic Economy,  al. Niepodległości 10, 61-857 

Poznań, Poland, e-mail: barbara.bedowska-sojka-ue.poznan.pl. 

This paper was presented at Forecasting Financial Markets and Economic Decision Making 

FindEcon 2016 in Łódź, 18th OxMetrics User Conference 2016 in London and Econometric 

Research in Finance Workshop 2016 in Warsaw. I would like to thank two anonymous 

reviewers for their valuable comments and suggestions. The usual disclaimer applies.  
1 The other possible specifications are presented in Kurach and Stelmach (Kurach and 

This paper was presented at Forecasting Financial Markets and Economic Decision Making 

FindEcon 2016 in Łódź, 18th OxMetrics User Conference 2016 in London and Econometric 

Research in Finance Workshop 2016 in Warsaw. I would like to thank two anonymous 

reviewers for their valuable comments and suggestions. The usual disclaimer applies.  

https://orcid.org/0000-0001-5193-8304


Barbara Będowska-Sójka 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

162 

Introduction 

According to capital asset pricing model, CAPM, the beta as the measure of 

the systematic risk, is the only risk factor important for investors (Andersen, 

et al., 2006). The estimation and prediction of beta value is relevant for 

investment decisions as well as for measuring the performance of fund 

managers (e.g. through the Treynor ratio), Betas are also strongly required in 

asset pricing, portfolio selection, asset allocation and risk management 

(Choudhry and Wu, 2008). Moreover, investors need a good approximation 

of beta as it allows to measure the cost of capital. The important question in 

the literature is how to estimate true latent beta coefficient. 

 In the classic CAPM, we assume that beta is a constant measure of 

systematic risk. However, a vast of the literature devoted to examination of 

the beta stability shows a considerable evidence against this assumption 

(Andersen et al., 2006; Brooks et al., 1998; Faff et al., 2000; Huang and 

Litzenberger, 1988; Jagannathan and Wang, 1996; Menchero et al., 2016). 

The static CAPM loses in favor of the conditional version of CAPM with 

time-varying betas (Campbell et al., 1997), which accounts for the fact that 

betas and expected returns vary over business cycle and “depend on the 

nature of the information available at any given point in time” (Jagannathan 

and Wang, 1996). 

 A great number of empirical studies use different approaches to estimate 

beta, all having some advantages and drawbacks. The estimates from 

different models usually differ substantially. The most common approaches 

are: linear regression where beta is assumed to be stable (Dębski, et al., 

2014), multivariate GARCH models (MGARCH), that are employed to 

estimate the time-varying beta (Brooks et al., 1998), realized betas derived 

from realized volatility introduced by Andersen and Bollerslev (1998) and 

developed in further works (Andersen et al., 2001; Andersen et al., 2010; 

Andersen et al., 2006; Hajizadeh et al., 2012), or beta estimated in the rolling 

window within a linear regression. Another and very promising approach to 

estimate time-varying beta is the Kalman filter technique (Brooks et al., 

1998; Kurach and Stelmach, 2014; Lie et al., 2000). In few works the 

Kalman filter is found to perform better than the GARCH specifications 

(Brooks et al., 1998; Lie et al., 2000).  

 Several papers examine the beta coefficients on the Warsaw Stock 

Exchange. Dębski et al. (2014) study the impact of sampling frequency of 

the beta estimates, Kurach and Stelmach (Kurach and Stelmach, 2014) focus 

on different behavior of sector beta that are estimated with Kalman filter 



Evaluating the Accuracy of Time-varying Beta. The Evidence from Poland 

 
DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

163 

approach, while Dębski et al. (2016) examine the stability of the beta 

parameters over bull and bear market for 134 largest companies. 

 The purpose of our paper is to compare beta coefficients obtained from 

different parametric methods. We focus on data of weekly frequency for 

fifteen stocks quoted on the Warsaw Stock Exchange in two sectors, banking 

and IT. According to FTSE Russell Country Classification the Warsaw 

Stock Exchange is perceived as an advanced emerging market (FTSE, 2016; 

Wyman, 2016). The stocks in our sample are listed through the relatively 

long period of time as for the non-developed market. In the paper we 

estimate time-varying beta from MGARCH models and use Kalman filter 

for unobserved component model. We examine how methods, that have been 

already used in the developed markets, work when applied to data from the 

Warsaw Stock Exchange. Two step agenda is used: first we calculate time-

varying beta coefficients with different methods for each stock separately, 

and second we employ a procedure to asses which estimate is the closest to 

the true unobservable beta. This examination is done in the framework of 

Securities Market Line, and as such is similar to approach presented in 

Choudhry and Wu (2008).  

 We find that beta estimates from Kalman filter together with beta 

estimates from DCC models have the highest accuracy. In our sample the 

beta estimates within one industry are highly positively correlated, while 

beta estimates from different sectors are characterized by negative 

correlation coefficients. We also find that in-sample errors are lower in case 

of banking companies when comparing to IT stocks.   

 The rest of the paper is as follows: in Section 1 the data are described, 

Section 2 is devoted to model specification, in Section 3 we show how the 

different beta estimates are compared, in Section 4 the empirical results are 

described and Section 5 concludes.  

1. Data 

Our sample data consists of prices of 15 stocks quoted on the WSE 

constantly from the beginning of the 2000 till the end of 2015. They are 

representing two sectors: banking and informatics. These two sectors have 

the biggest number of stocks representatives quoted constantly in the whole 

period of the study. The price data are obtained from Stooq database 

(www.stooq.pl), and they are adjusted for dividends and splits. These stocks 

are listed together with their full names, tickers, industry, size category and 

capitalization in Table 1. Generally, banks are big companies, whereas IT 

stocks belongs to different size groups. For the approximation of the market 



Barbara Będowska-Sójka 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

164 

portfolio we use WIG index which comprises all companies listed on the 

Warsaw Stock Exchange (WSE) Main List. In the study we also use risk free 

rate that is calculated as a mid-quote of 1 month WIBOR and WIBID rates 

(that are the Polish counterparts of LIBOR and LIBID rates),  

Table 1. The list of the stocks included in the sample 

Company name Ticker Sector 
Firm size 
(category) 

Firm size 
(in mln euro) 

HANDLOWY SA BHW Banking big 2256 
ING SA ING Banking big 4746 

MBANK SA MBK Banking big 3204 
MILLENIUM SA MIL Banking big 1423 

PEKAO SA PEO Banking big 7464 
BZ WBK SA BZW Banking big 7088 

BOS SA BOS Banking medium 152 
ASSECOPOL SA ACP Informatics big 1012 
CDPROJEKT SA CDR Informatics big 1134 
COMARCH SA CMR Informatics big 320 
SYGNITY SA SGN Informatics small 13 

ELZAB SA ELZ Informatics medium 54 
MACROLOGIC SA MCL Informatics small 16 

SIMPLE SA SME Informatics small 7 
LARK SA LRK Informatics very small 3 

Note: The companies on the WSE are categorized according to their size measured by 

capitalization in the following manner: ‘big’ stands for capitalization higher than 250mln of 

euro, ‘medium’ is in the interval (50mln, 250mln), ‘small’ is in the interval (5mln, 50mln) and 

‘very small’ stands for capitalization lower than 5mln euro. The last column reports 

capitalization of stocks at the end of 2016. 

 The daily stock prices, index values and WIBID/WIBOR rates are 

aggregated into weekly data based on the last observation in the week. The 

whole sample consists of 887 weekly returns. In the further work we use the 

percentage logarithmic returns. The calculations and graphics are done in 

OxMetrics STAMP7 (Koopman et al., 2006), PcGive (Doornik and Hendry, 

2006) and G@RCH (Laurent, 2013). 

2. Beta Estimation  

In this section we briefly describe models that are used in the empirical part. 

The common factor for obtained measures is a time-varying feature of beta 

coefficient. We use two MGARCH specifications and Kalman filter in 

unobserved component models. 

  



Evaluating the Accuracy of Time-varying Beta. The Evidence from Poland 

 
DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

165 

2.1. MGARCH Specifications: Scalar BEKK and DCC Models 

Within multivariate GARCH models the conditional beta are obtained. After 

preliminary estimations we consider two multivariate GARCH 

specifications: scalar BEKK model (Engle and Kroner, 1995), and dynamic 

conditional correlation model, DCC (Andersen et al., 2006; Engle, 2002). 

Each of these models is bivariate model with two equations, one for stock 

return, iR , and one for market portfolio return, MR : 

),0(~

...

...

1

22110

22110

ttt

MtMtMMtMMMt

ititiitiiit

D

eRRR

eRRR

He 















 (1) 

 In the general BEKK(p,q) model the conditional covariance matrix, Ht, 

is described in the following way: 








 
p

j

jjtj

q

i

iititit

11

'''' EHEDeeDCCH  (2) 

where Di and Ej are identity matrix multiplied by scalars. In our approach 

1and1  pq .  

 In the dynamic conditional correlation DCC model of Engle (Engle, 

2002) the following specification is used: 

 
















P

j

jtjitit

Q

i

i

P

j

j

Q

i

it

tttt

tttt

1111

1*1*

)'()1( QzzSQ

QQQR

DRDH



 (3) 

where ),...,,(diag 2/12/1

2

2/1

1 Nttt hhhD , kth  is the conditional variance 

described with univariate GARCH models, tz  is the vector of standardized 

kte , 
2/1/ ktktkt hez  , tR  is a matrix of time-varying correlation coefficients 

of tz , and 
*

tQ  is diagonal matrix in which elements are square roots of 

diagonal elements of matrix tQ . 

 In our study in both MGARCH models, BEKK and DCC model, in the 

conditional mean equations (eq.1) ARMA(1,0) specification are considered. 

In BEKK model the conditional variance equations are modeled with 



Barbara Będowska-Sójka 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

166 

GARCH(1,1), whereas in DCC model the conditional variance is modeled 

with GJR-GARCH(1,1), that accounts for possible leverage effect (Andersen 

et al., 2007). In both models the conditional distribution of the model error 

terms is assumed to be Student t.   

 As the conditional beta for a stock i is described as 

)var(/),cov( MtMtitit RRR , both estimates of conditional covariance and 

conditional variance come directly from MGARCH models fitted to the 

returns of an individual stock iR  and returns of a market index MR , a proxy 

for the market portfolio. Both the conditional variance and the conditional 

covariance are provided in the matrix Ht. Finally, the time-varying beta 

series are calculated from the conditional covariances in MGARCH models. 

2.2. Unobserved Component Model 

The second approach used in the paper is a state-space representation where 

the Kalman Filter is used. We apply the unobserved component model, UC, 

that is considered as a multiple regression model with time-varying 

coefficients. This specification is based on the theory of structural time 

series models presented in Harvey (1989), In the general form a time series 

is viewed as being decomposable into trend, seasonal, and cycle 

components. In the UC model it is assumed that near and far distant 

observations should not be given equal weight. For our purpose we use 

a local level model (random walk) with drift in the following way (Harvey 

1989)
1
: 

TtRR ttMtittit ...,,1),0(~, 2   NID , (4) 

Tttttiit ...,,1),0(~, 2
1,    NID  (5) 

where NID  denotes Normally and Independently Distribute, and t  and t  

are independent variables. Equation (4) is a measurement equation, whereas 

equation (5) is the transition equation. Within this specification, any shock to 

asset’ beta is persistent. The Kalman filter allows to obtain the time-varying 

beta.   

 It should be noted that the specification of the UC model for weekly data 

might raise some concerns as this model assumes inter alia that the 

conditional variance of returns is homoscedastic (has no ARCH effect 

described in e.g. Bollerslev et al. (1994). However, this effect is pretty well 

                                                           
1 The other possible specifications are presented in Kurach and Stelmach (Kurach and 

Stelmach, 2014) or Będowska-Sójka (2015). 



Evaluating the Accuracy of Time-varying Beta. The Evidence from Poland 

 
DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

167 

captured by BEKK and DCC models. Thus, if the weekly data are 

characterized by the ARCH effect, than the UC model is not appropriate
2
.  

 Finally we also obtain static beta from the ordinary least square 

regression as a benchmark for the comparisons of time-varying beta 

estimates. 

3. Comparisons of Different Betas  

There is no obvious benchmark for unobservable beta. Therefore some 

proxies must be introduced when attempting to establish the relative 

dominance of one method over another. In a similar manner to Choudhry 

and Wu (2008) after calculation of different beta measures, we compare 

them on the basis of the fit to the Securities Market Line, SML:  

)( ftMtitftit rRrR    (6) 

where ftr  is a risk-free rate of return.  

 With the estimates of time-varying betas, one easily calculates in-sample 

theoretical returns based on the market return and the risk-free rate of return 

that are actually observed. We assess the relative accuracy of time-varying 

beta estimates by comparing the theoretical return with the actual returns and 

calculating the residual,  :  

ititit RR ˆ  (7) 

where itR  is the actual return at time t, itR̂  is the theoretical return of stock  

i according to the SML, and  Tt ,...,1  stands for the consecutive weeks.   

 The comparison of beta estimates are based on the standard forecasting 

error measures: the lower the errors, the better in-sample beta 

approximation. We use two standard errors used most frequently in 

empirical studies (Menchero et al., 2016; Wang, 2009): Mean Absolute 

Error, 



T

t
titi RR

T
MAE

1
,,

ˆ1
 and Mean Square Error, 





T

t
titi RR

T
MSE

1

2
,, )ˆ(

1
. Out of these two measures, MAE is less sensitive to 

outliers. We also consider median Relative Absolute Error, mRAE, 

calculated as a median of the distribution of ratios b
ti

b
tititi RRRR ,,,,

ˆ/ˆ  , 

                                                           
2 A possible solution to this issue was proposed by Rockinger and Urga (2001). 



Barbara Będowska-Sójka 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

168 

where b stands for benchmark model (the OLS beta in our case), The last 

error measure is perceived in the literature as a robust comparative measure 

of performance (Armstrong and Collopy, 1992).  

4. Empirical Results 

We estimated four type of models for each of fifteen stocks. Due to the 

highly volatile measures of conditional beta within the first period, we 

excluded from the analysis first 52 observations and thus analysis starts from 

2001. In Figure 1 we show beta estimates from different methods for two 

stocks, each representing different sector: BHW (banking) and CDR 

(informatics), The remaining graphics are available on request. While the 

overall dynamic of beta is similar across different approaches, beta estimates 

from unobserved component model seem to be most smoothed and therefore 

most stable, whereas the beta estimates from both MGARCH models are 

highly volatile.  

 

Figure 1. The estimates of beta for BZW and CDR 
Note: Beta estimates shown in the figure are the following: BEKK stands for conditional beta from 
MGARCH scalar BEKK models, DCC stands for conditional beta from MGARCH DCC model and UC 

stands for time-varying beta from unobserved component model. 



Evaluating the Accuracy of Time-varying Beta. The Evidence from Poland 

 
DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

169 

 We also calculate correlation coefficients for different beta estimates 

across the sample and find that these correlations are positive, medium 

strong and statistically significant. In case of both MGARCH (BEKK and 

DCC) models the correlation coefficient is on average 0.63, for BEKK and 

UC models the correlation accounts for 0.68, while for DCC and UC models 

the correlation is equal to 0.59.  

 Table 2 reports the mean, minimum and maximum values of beta 

estimates from different methods. The differences in means obtained on the 

basis of different methods are not significantly different from each other. All 

OLS beta coefficients are significantly different from zero. In four out of 

fifteen cases OLS beta is not statistically different from 1. The minimum and 

maximum beta estimates show high variability within the sample.  

Table 2. Time Varying Beta Estimates for Polish Banking and Informatics Stocks 

 
BEKK DCC UN  

OLS beta 

 
Mean Min Max Mean Min Max Mean Min Max 

BHW 0.7814 –0.0887 1.5478 0.7485 0.2697 1.4857 0.8161 0.0970 1.2811 0.7670 
ING 0.6881 –0.0638 1.7841 0.7248 0.4115 1.6610 0.7732 0.4315 1.1505 0.7954 
MBK 1.2074 0.5514 2.0564 1.1883 0.7394 2.5867 1.2191 0.7114 1.8901 1.2638 
MIL 1.2189 0.4439 2.0981 1.2388 0.6987 1.8624 1.2698 0.9030 1.8001 1.3310 
PEO 1.1592 0.3874 1.7685 1.1189 0.5660 2.0766 1.1496 0.3451 1.5487 1.1367 
BZW 0.9399 –0.2699 1.8446 1.0181 0.2614 2.0313 1.0453 0.1442 1.6265 1.0875 
BOS 0.2717 –0.4900 1.1031 0.2742 –0.0956 0.6591 0.3536 0.0576 0.5768 0.3271 
ACP 0.9367 –0.0665 2.2527 1.0216 0.3337 2.5907 1.0055 0.4907 2.0023 1.0496a 

CDR 0.9909 –0.6677 2.6589 1.0989 0.5225 2.2277 1.1818 0.9375 1.9225 1.2213 
CMR 0.8684 0.0688 1.8595 0.9418 0.3789 2.1970 0.9473 0.2939 2.1340 1.0616a 

SGN 0.9736 –0.0142 2.2289 1.0125 0.2495 2.6802 0.9917 0.3650 2.1843 0.9963a 

ELZ 0.4080 –0.5330 1.7864 0.4407 0.1423 0.7134 0.4427 0.3515 0.6608 0.4612 
MCL 0.5738 –0.2633 1.8075 0.6836 0.2787 1.3665 0.6460 0.1155 1.2151 0.6734 
SME 0.4905 –0.0393 1.6444 0.5250 0.1732 0.8998 0.6088 0.2196 1.2768 0.6607 
LRK 0.7959 –1.5195 4.4349 0.9401 0.3165 3.3889 0.9343 0.2236 1.6165 0.9578a 

Note: Mean is the arithmetic average of conditional beta obtained from BEKK, DCC and Kalman filter 
models, Min and Max stand respectively for the minimum and maximum value of the conditional beta 

estimates in the sample. OLS best stands for the point estimates of beta from the linear regression. All 
OLS beta are statistically different from 0; letter a is a subscript used to show the estimates that are not 

statistically significantly different from 1.  

 To evaluate beta estimates, three different measures of errors based on 

in-sample fit to SML (eq. 6) are employed. Table 3 reports the error 

measures. In most cases the lowest value is observed for time-varying beta 

estimates from unobserved component model: MAE is the lowest in all 

cases, whereas the mRAE is the lowest in twelve out if fifteen cases. Next is 

the BEKK model with fourteen lowest forecasting errors, twelve in MSE and 

two mRAE. The linear beta obtains the lowest error measures in three cases, 



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170 

all of them are MSE errors. The DCC model has got the only one lowest 

mRAE.  

Table 3. The measures of in-sample forecasting errors for different beta estimates 

 MSE MAE mRAE MSE MAE mRAE MSE MAE mRAE 

 
BHW ING MBK 

BEKK 0.0322 0.0239 0.9816 0.0329 0.0229 0.9898 0.0359 0.0271 0.9853 
DCC 0.0336 0.0241 0.9971 0.0344 0.0230 0.9915 0.0373 0.0273 0.9940 
UC 0.0339 0.0236 0.9643 0.0344 0.0225 0.9873 0.0375 0.0264 0.9838 
OLS 0.0328 0.0243 1 0.0334 0.0232 1 0.0362 0.0277 1 

 MIL PEO BZW 
BEKK 0.0468 0.0338 0.9954 0.0273 0.0209 0.9996 0.0326 0.0240 0.9884 
DCC 0.0499 0.0341 0.9978 0.0288 0.0207 1.0000 0.0335 0.0242 0.9937 
UC 0.0505 0.0334 0.9893 0.0287 0.0203 0.9920 0.0335 0.0233 0.9812 
OLS 0.0488 0.0347 1 0.0278 0.0208 1 0.0321 0.0253 1 

 
BOS ACP CDR 

BEKK 0.0413 0.0278 0.9983 0.0447 0.0318 0.9768 0.0725 0.0494 0.9973 
DCC 0.0426 0.0274 0.9971 0.0495 0.0321 0.9830 0.0761 0.0491 0.9994 
UC 0.0423 0.0274 0.9992 0.0496 0.0311 0.9753 0.0751 0.0490 0.9984 
OLS 0.0418 0.0277 1 0.0481 0.0324 1 0.0747 0.0496 1 

 
CMR SGN ELZ 

BEKK 0.0429 0.0308 0.9862 0.0550 0.0388 0.9998 0.0607 0.0396 0.9990 
DCC 0.0454 0.0308 0.9876 0.0579 0.0387 0.9940 0.0610 0.0394 1.0007 
UC 0.0454 0.0302 0.9767 0.0578 0.0380 0.9932 0.0603 0.0393 0.9986 
OLS 0.0443 0.0316 1 0.0566 0.0388 1 0.0602 0.0395 1 

 
MCL SME LRK 

BEKK 0.0679 0.0453 0.9996 0.0740 0.0504 0.9978 0.0821 0.0568 0.9909 
DCC 0.0701 0.0451 1.0002 0.0779 0.0503 0.9993 0.0747 0.0565 0.9979 
UC 0.0690 0.0448 0.9969 0.0778 0.0501 0.9949 0.0740 0.0558 0.9939 
OLS 0.0688 0.0454 1 0.0770 0.0509 1 0.0733 0.0571 1 

Note: OLS stands for beta obtained from the linear regression estimated for the whole sample, 

BEKK stands for conditional beta from MGARCH scalar BEKK models, DCC stands for 

conditional beta from MGARCH DCC model and UC stands for time-varying beta from 

unobserved component model with random walk. The lowest errors are underlined. 

 

 If the predictive accuracy of betas across the sectors are compared, we 

notice that beta estimates in banking sector have almost 1.7 lower average 

MSE than the beta estimates in informatics sector. As the companies in the 

banking sector are generally bigger than these from IT, the beta estimates for 

big stocks seem to be more accurate than estimates for medium and small 

stocks.  

 In Figure 2 we show the beta estimates from unobserved component 

model for all stocks. The behaviour of betas is similar within the industries: 

in case of stocks from banking sector (from BHW to BOS) the beta estimates 

increase over the whole sample period and specially in the beginning of 



Evaluating the Accuracy of Time-varying Beta. The Evidence from Poland 

 
DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

171 

financial crisis in 2008. With respect to informatics stocks (from ACP to 

LRK), beta estimates generally decrease. Some exceptions in the overall 

tendency are recognized, e.g. beta in LRK is changing up and down as well 

as beta of BZW. The correlation matrix showing the interdependencies 

between beta coefficients estimated from UC model is presented in the 

Appendix. On the one hand, the correlation matrix shows that within 

banking sector the correlations between beta estimates are in majority 

positive and statistically significant. One exception is PEO, which shows 

negative or statistically insignificant coefficients with five out of six stocks. 

In case of IT sector the situation is similar – the correlations for stocks from 

this industry are positive and statistically significant with one exception, 

LRK, where for four out of six stocks the correlations are negative. Those 

two stocks, PEO and LRK, are respectively the biggest and the smallest in 

the sample. On the other hand, the correlations between stocks from two 

different sectors are in most cases negative and significant.  

 

Figure 2. The time-varying beta estimates from the UC model 
Note: Beta estimates shown in the figure are from unobserved component model. Each graph presents the 
beta coefficients for a single stock. 

 As the differences between the predictive errors are rather small we 

decide to use the statistic that allows for comparing accuracy of the 

examined methods. We employ modified Diebold-Mariano statistic 



Barbara Będowska-Sójka 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

172 

(henceforth modDM) (Harvey et al., 1997), that examines if the forecasting 

precision differs significantly across the methods used. It is found to perform 

much better than the original Diebold-Mariano test for different forecast 

horizons, as well as in cases when the forecast errors are autocorrelated or 

have non-normal distribution (Choudhry and Wu, 2009), We calculate 

modified Diebold-Mariano statistic for MSE. These statistics are calculated  

in pairs, in which forecast errors come from one of the MGARCH models, 

Kalman filter or the OLS regression beta. In table 4 we show the results of 

the Diebold-Mariano statistic for each stock and each pairs of models 

(model1 and model2) separately. We reject the null of equal predictive 

accuracy at the 5% level. The statistic has a Student t distribution with 1T

degrees of freedom, where T is a number of observations. In case of the 

negative values the errors from model 2 are lower than from model1, in case 

of positive values the opposite holds. 

Table 4. The comparison of predictive accuracy of models used in the study – the 

modified Diebold-Mariano statistics 

model1 BEKK BEKK BEKK DCC DCC UC 
model2 DCC UC OLS beta  UC OLS beta OLS beta 

BHW –7.4207 –7.4223 –7.3172 –1.2862 2.5595 2.7477 
ING –4.7595 –4.7775 –4.5831 –0.7200 3.6377 3.2197 
MBK –0.3156 –0.4811 0.2765 –0.9381 3.5396 3.1355 
MIL –4.1499 –4.5043 –3.7641 –0.4977 3.5911 2.0957 
PEO 3.3009 3.2579 4.0393 0.6722 3.2991 2.0614 
BZW –1.4723 –1.5255 –0.5930 –0.2289 5.9932 3.9648 
BOS –9.8146 –9.4839 –9.7841 0.6072 2.3638 1.5911 
ACP –4.8505 –4.8918 –4.6438 –0.7892 3.4923 3.8068 
CDR –7.9989 –7.8383 –7.8351 1.4541 2.2466 1.2660 
CMR –5.7058 –5.4700 –5.4087 –0.1598 3.8289 2.6390 
SGN –8.0543 –7.9966 –7.8688 0.7554 2.8694 2.4824 
ELZ –8.9527 –8.8164 –8.7934 1.7822 2.0485 0.3529 
MCL –8.5644 –8.8225 –8.7245 1.8249 2.8946 1.5217 
SME –9.0150 –8.8529 –9.0097 0.5598 2.8379 1.9662 
LRK –10.4690 –10.0370 –9.9295 0.6386 2.6865 2.4141 

Note: The table provides the modified Diebold-Mariano statistics. We compare MSE from two non-

nested models, model1 and model2, for each stock separately. The negative values of the statistic indicate 
that predictive accuracy of model2 is better than of model1. The positive values of statistics indicate the 

opposite. The statistics in grey font have p-values higher than α=0.05 – in such cases the predictive 

accuracy of two models is equal.  

 Based on the results of modified DM statistics we find that for our 

sample of stocks both beta from dynamic conditional correlation (DCC) 

models and unobserved component (UC) models provide better beta 

forecasts than BEKK models. Surprisingly, even point OLS estimate of beta 



Evaluating the Accuracy of Time-varying Beta. The Evidence from Poland 

 
DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

173 

provides better forecasts than BEKK model. This result is consistent for all 

stocks with minor exceptions: first, in case of PEO stock BEKK model 

offers better forecasts than DCC and UC models. This company is one of the 

biggest and more liquid among those listed on the WSE. Second, in case of 

MBK and BZW the differences between errors are not statistically 

significant. Additionally, both DCC models and UC models have greater 

predictive accuracy than the point beta estimates, although in case of UC 

model the significant difference is observed only in 12 out of 15 stocks. We 

do not find any difference between forecasting accuracy of DCC model and 

UC model, although the latter model is usually presented as a winner in beta 

predictive horse races (Brooks et al., 1998; Choudhry and Wu, 2008; Lie et 

al., 2000). However, the result of the Diebold-Mariano tests might depend on 

the selection of the loss function, that is MSE. In case of stocks listed on the 

Warsaw Stock Exchange, which is the developed emerging market, the beta 

coefficients from Kalman filter method gives as good forecasts as from 

MGARCH DCC model.     

Conclusion 

In the modern investment theory time-varying beta concept replaces the 

static one. In the paper we compare differently estimated beta coefficient in 

securities market line framework for the stocks from the banking and IT 

sectors. These stocks have been listed constantly on the Warsaw Stock 

Exchange over the period 2000–2015. First of all, we find that contrary to 

the previous results presented in the literature (Dębski et al., 2016) the 

estimated beta coefficients in our sample are time-varying within the given 

period. We consider two MGARCH model specifications, DCC and BEKK 

model, the Kalman filter technique and the estimates from linear regression 

models.  

 Our results show that beta estimated from the unobserved component 

models brings the most stable and smoothed betas. The estimates from 

BEKK models have often the lowest mean square error, while the estimates 

from Kalman filter in most cases offer the lowest mean absolute error as well 

as median relative absolute error. However, the comparison of predictive 

accuracy of methods used in the study shows that the errors obtained from 

the SML with beta estimated from BEKK models are significantly bigger 

than from the other models. The beta estimates of both DCC model and UC 

model fit better in terms of the SML than the estimates of beta from ordinary 

least squares. We do not find the evidence for statistical difference between 

the predictive accuracy of DCC model and UC models. Not surprisingly the 



Barbara Będowska-Sójka 

DYNAMIC ECONOMETRIC MODELS 17 (2017) 161–176 

174 

correlation coefficients for beta estimates within a sect or are positive, 

whereas  between sectors are negative. Finally, when betas for stocks from 

different industries are compared, in-sample forecasts are more accurate for 

stocks from the banking sector than for stocks from the informatics sector. 

 A natural extension of the presented study is to consider out-of-sample 

predictive accuracy of individual models for beta estimation. The results of 

such empirical exercise might be of practical interest for different groups of 

market professionals. 

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Appendix 

Table 5 The Spearman correlation matrix of beta coefficients from Kalman filter 

approach 

 
BHW ING MBK MIL PEO BZW BOS ACP 

BHW 1.00 0.67 0.66 0.57 -0.32 -0.06 0.98 -0.79 
ING 0.67 1.00 0.83 0.77 0.03 0.30 0.67 -0.78 
MBK 0.66 0.83 1.00 0.87 0.04 0.27 0.67 -0.68 
MIL 0.57 0.77 0.87 1.00 -0.14 0.32 0.57 -0.65 
PEO -0.32 0.03 0.04 -0.14 1.00 0.19 -0.27 0.31 
BZW -0.06 0.30 0.27 0.32 0.19 1.00 -0.09 -0.36 
BOS 0.98 0.67 0.67 0.57 -0.27 -0.09 1.00 -0.77 
ACP -0.79 -0.78 -0.68 -0.65 0.31 -0.36 -0.77 1.00 
CDR 0.08 -0.19 0.00 0.16 -0.35 -0.59 0.10 0.27 
CMR -0.73 -0.27 -0.25 -0.08 0.15 0.01 -0.73 0.55 
SGN -0.51 -0.62 -0.65 -0.47 -0.21 -0.53 -0.52 0.69 
ELZ -0.79 -0.90 -0.83 -0.78 0.04 -0.34 -0.80 0.87 
MCL -0.94 -0.71 -0.68 -0.65 0.35 -0.12 -0.92 0.87 
SME -0.98 -0.68 -0.70 -0.59 0.27 0.00 -0.98 0.81 
LRK 0.19 0.26 0.31 0.28 -0.08 -0.26 0.23 -0.08 

 
ACP CDR CMR SGN ELZ MCL SME LRK 

BHW -0.79 0.08 -0.73 -0.51 -0.79 -0.94 -0.98 0.19 
ING -0.78 -0.19 -0.27 -0.62 -0.90 -0.71 -0.68 0.26 
MBK -0.68 0.00 -0.25 -0.65 -0.83 -0.68 -0.70 0.31 
MIL -0.65 0.16 -0.08 -0.47 -0.78 -0.65 -0.59 0.28 
PEO 0.31 -0.35 0.15 -0.21 0.04 0.35 0.27 -0.08 
BZW -0.36 -0.59 0.01 -0.53 -0.34 -0.12 0.00 -0.26 
BOS -0.77 0.10 -0.73 -0.52 -0.80 -0.92 -0.98 0.23 
ACP 1.00 0.27 0.55 0.69 0.87 0.87 0.81 -0.08 
CDR 0.27 1.00 0.23 0.54 0.15 -0.08 -0.02 0.55 
CMR 0.55 0.23 1.00 0.53 0.41 0.66 0.77 0.31 
SGN 0.69 0.54 0.53 1.00 0.74 0.58 0.57 0.27 
ELZ 0.87 0.15 0.41 0.74 1.00 0.85 0.80 -0.22 
MCL 0.87 -0.08 0.66 0.58 0.85 1.00 0.94 -0.21 
SME 0.81 -0.02 0.77 0.57 0.80 0.94 1.00 -0.12 
LRK -0.08 0.55 0.31 0.27 -0.22 -0.21 -0.12 1.00 

Note: The table reports Spearman rank correlations for pairs of beta coefficients estimated with 

unobserved component model. The bolded values are statistically significant at α=0.05.  


