analysis of the polish stock market indices based on garch ......denotes the return from the...

Vedecký časopis FINANČNÉ TRHY, Bratislava,

Derivat 2015, ISSN 1336-5711, 4/2015

Analysis of the Polish stock market indices

based on GARCH-in-mean models

Krzysztof DRACHAL

Abstract

The aim of this research is to present the result of application of GARCH-in-mean (also know

as GARCH-M) models. Over seventeen years period, starting in 1998, was analysed. Various

frequencies were considered, i.e., daily, weekly and monthly. It was speculated that this kind

of model could shed some light on the problem of a size risk premium for the Polish stock

exchange. Unfortunately, the selected methodology did not give the expected conclusions. On

the other hand, it can be concluded that there is no size risk premium on the Polish stock

market, as far as it could be detected by GARCH-M. This is somehow in agreement with

other cited researches. Yet, for a few time series some weak evidence of a risk premium was

found.

Keywords

GARCH, risk premium, size premium, stocks, volatility

JEL Classification

C22, G12, G17

Introduction

One of the very important problem for the investor in the stock market is a matter of risk

modelling. It is known that a variety of securities has different risk levels. It is assumed that

safe securities (such as, for example, bonds) are characterized by a relatively low rate of

return. Whereas securities offer relatively high-risk but also a high returns.

This regularity is well known in the literature and the theory of economics [19]. In the

literature, however, one can also find the so-called concept of premiums for the size of the

company. It is expected that small companies generate somewhat higher risk than large

capitalization companies [8, 9, 34]. Therefore, it seems that it is worth checking this

relationship for the Warsaw Stock Exchange. This research is based on GARCH-M model.


Derivat 2015, ISSN 1336-5711, 4/2015

Literature review

GARCH model was presented in the article of Bollerslev in 1986 as a generalization of the

ARCH model [3, 7]. It constitutes a wide class of models. In particular, it is said that the

variable x follows GARCH-M process, if

xt = a0 + b . ht + et ,

and et = ut √ht , where ut follows the generalized normal distribution and

ht = c0 + c1 . (et-1)

2 + d1

. ht-1 .

GARCH class models are very useful in finance and economics, especially if the

analysed time series are of a high frequency type [1, 11]. GARCH-M models can be used for

modelling the risk. Namely, the estimate of a parameter b in the above equation (if it is

statistically significant) can be interpreted in the following way. The increase in the

conditional variance (i.e., a measure of the risk) results in the increase in the expected rate of

return. It is assumed that the variable x is a rate of return.

The simplest form of the above model is, if ut follows the standard normal distribution.

However, if one is interested in leptokurtic distributions some generalization is preferred. The

Reader interested in this topic for Polish stock market and in its efficiency should consult

some other literature (for example, [20] and [21]).

Of course, these models can have a more complex variance equation. However, higher

complication of the model does not lead to relatively better result in many cases [17, 25]. On

the other hand, Nelson [26] pointed out that the simple GARCH model has some significant

disadvantages. Among other things, its definition excludes the existence of a negative

correlation between the future conditional variance and the current value of the modelled rate

of return. In addition, the estimated coefficients often do not meet theoretical assumptions (for

example, they tend to be negative). GARCH model has therefore been expanded to include

various modifications.

For time series from the Polish stock exchange GARCH models, taking into the

account the asymmetry of the empirical forecast error has been examined, among others, by

Małecka [24] and Rozkrut [30]. The problem of limits of parameters for the data from the

Polish stock exchange was discussed by Galin [15].

Fiszeder and Kwiatkowski [12] analysed selected 28 companies from the Warsaw

Stock Exchange and came to the conclusion that in the case of stock market indices GARCH

model describes the variability conditional variance the best. On the other hand Bartkowiak


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[2] analysed only the assets related to the option market on the Warsaw Stock Exchange

(WSE). Their results are in part consistent with those from the developed markets. On the

other hand, received concerns suggest that it is still worth to study Polish market, as it

continues to grow and develop. In this sense, even the “stylized facts” are worth noting.

Płuciennik [28] pointed out that the use of models with autoregression results in a

considerably better results than the simple GARCH models for the WIG 20 index. GARCH

models were also explored in the context of the Polish stock exchange by Filipowicz [10],

Karkowska [22], Doman [6], and others.

Finally, it seems interesting to reflect on whether the larger companies on the WSE are

characterized by slightly lower risk compared to the market average [5, 8, 9]. In this context,

certain model of Fama and French is sometimes analysed [8, 9, 13]. This issue is, however,

debatable.

Foye, Mramor and Pahor [13] analysed the markets of the so-called new European

Union members and found that the model of Fama and French has some disadvantages. Thus,

they have proposed some modifications. Yet, another modification was proposed by

Czapkiewicz and Wojtowicz [4] based on the analysis of the data from WSE for the years

2003-2012 using monthly data. On the other hand, Słoński and Kwiatkowski [32] recognized

the three-factor model as sufficient to describe Polish companies. They also claimed that the

premium for the size is important for the smallest companies.

However, Sekula [31] based on the data from the years 2002 to 2010 questioned the

relationship between market capitalization of the companies and their rates of return.

Moreover, he noted that the premium for the size was reversed, i.e., companies with medium

and large capitalization generated higher returns than companies with low capitalization.

Similarly, doubts on the existence of a premium for the size appeared during the analysis of

the data from developed markets [18]. Van Dijk [33] conducted a fairly extensive comparative

analysis and came to the conclusion that the premium for the size fades from approx. 1980s.

Similarly, its existence is disputed by Paschall [27]. For the markets of the Central and

Eastern Europe doubts to its existence were presented by Konieczka and Zaremba [23].

Methodology

The data were obtained from the data base Stooq.pl (http://stooq.pl). Five indices were

analysed: WIG – all stock index, WIG20 – consisting of blue chips, WIG BANKI – consisting

http://stooq.pl/


Derivat 2015, ISSN 1336-5711, 4/2015

of banks, mWIG40 – representing medium sized companies and sWIG80 – representing small

companies. The daily data were obtained from the period between 31/12/1997 and 7/7/2015.

Indices were chosen from the available ones as to cover at least ten year period and to have

the same number of observations. As a result, these five indices and 4390 observations for

each index were able to be obtained.

All calculations were done in R [28]. Everywhere 5% significance level was assumed.

The aim was to estimate the following, aforementioned, equation

xt = a0 + b . ht + et ,

and et = ut √ht , where ut follows the generalized normal distribution and

ht = c0 + c1 . (et-1)

2 + d1

. ht-1 ,

where xt denotes the return from the selected index. This can be done in R, for example, with

a help of rugarch package [15]. Logarithmic daily returns were computed and ordinary

weekly and monthly returns. It was assumed that a week consists of 5 subsequent

observations and a month – 21 observations.

The idea was to estimate the parameter b in the above equation, and if found

statistically significant, to compare it for different time series. It was expected that this

parameter would be smaller for wig20 returns series and bank return series in comparison to

wig returns series. It was also expected that wig80 returns series would give the highest value

for estimate of the parameter b, etc. As the risk premium (expressed by the parameter b)

should diminish with the average size of listed companies.

Results

Fig. 1 presents logarithmic daily returns from all the considered indices. A volatility clustering

seems to be evident. However, ARCH-LM test was performed for every time series. Its results

and that of augmented Dickey-Fuller test for stationarity are presented in Tab.1.

It can be seen that for all time series and both tests the null hypotheses should be

rejected. Therefore, one can assume that all the return time series are stationary and there are

significant ARCH effects.

As a result, it is quite reasonable to perform the analysis of the GARCH type models.

Unfortunately, out of 15 evaluated models, only in 4 of them the parameter b was statistically

significant. In particular, they are the daily and monthly model for sWIG80 and weekly

models for WIG BANKI and mWIG40 (see Tab. 2, Tab. 3, Tab. 4 and Tab. 5).


Derivat 2015, ISSN 1336-5711, 4/2015

Fig. 1: Logarithmic daily returns

Source: Own estimation in R

Tab.1: ADF and ARCH-LM tests results

p-values

ADF ARCH-LM

WIG 0.01 0.00

WIG20 0.01 0.00

WIG BANKI 0.01 0.00

mWIG40 0.01 0.00

sWIG80 0.01 0.00

Source: Own estimation in R

In case of daily logarithmic returns only the model for sWIG80 gave statistically

significant parameter b. Yet, for the same index – but ordinary monthly returns – this

parameter is also statistically significant, however, smaller. Notice, that in case of daily

frequency the parameter is much larger than the unit. It means that the reaction form returns


Derivat 2015, ISSN 1336-5711, 4/2015

on volatility change is high. In case of monthly frequency – the parameter is less than the unit.

For mWIG40 the estimation is a bit surprising, because the estimated parameter is

negative. This would mean that an investor is penalized for holding a risky instrument. Yet, a

similar findings were reported by Gabrisch and Orlowski [14]. Finally, for WIG BANKI the

parameter is the smallest amongst the presented models, less than the unit and positive.

Indeed, comparing the models for sWIG80 (small companies) and WIG BANKI

(banks) it can be stated that small companies are characterised by higher risk premium than

banks (which can be assumed as stable companies).

Another interesting problem is to consider the structural stability of the parameter b in

time. Indeed, a refitting was done for the models reported in Tab. 2, Tab. 3, Tab. 4 and Tab. 5.

The first estimation was done based on the period including 750 session days (approximately

3 years). The refits were done after every new 125 session days (approximately half a year).

The outcomes are presented in Fig. 2, Fig. 3, Fig. 4 and Fig. 5.

For the model reported in Fig. 2 the parameter b is quite stable in time. However, a

high uncertainty in its estimation is present at the beginning of the analysed period and around

2007, when the global financial crisis started. The Nyblom stability test gives the individual

statistics 0.7785 (under 0.47 critical value). Therefore, it confirms that the parameter is not

stable.

For the model reported in Fig. 3 the parameter b varies from the values lower than the

unit, then exceeds the unit and finally, becomes lower than the unit again. However, it is

always positive. The Nyblom stability test gives the individual statistics 1.8322 (under 0.47

critical value). Therefore, it confirms that the parameter is not stable.

For the model reported in Fig. 4 the parameter b starts form the positive value, but

systematically decreases to the certain negative value. It is negative for the most of the

analysed period. The Nyblom stability test gives the individual statistics 0.3399 (under 0.47

critical value). Therefore, it confirms that the parameter is stable.

For the model reported in Fig. 5 the parameter b starts from values higher than the

unit, but decreases to certain positive value, lower than the unit. The Nyblom stability test

gives the individual statistics 0.3060 (under 0.47 critical value). Therefore, it confirms that the

parameter is stable.

It can be seen that the parameter for sWIG80 (small companies) is more volatile. For

example, the direction of the movements changes in time. The parameter for mWIG40 and

WIG BANKI (medium companies and banks) the parameter oscillates around certain time


Derivat 2015, ISSN 1336-5711, 4/2015

trend.

Finally, it should be emphasised that ARCH-LM and Ljung-Box tests for standardized

residuals provided evidence that the constructed models are not good. All models suffer from

the problem of autocorrelation. Fortunately, the models for mWIG40 and WIG BANKI have

no remaining ARCH effects in residuals. Yet, the models for sWIG80 still have some

remaining ARCH effects.

Tab.2: Estimation of GARCH-M for daily logarithmic returns from sWIG80

estimate std. error t value p-value

a0 0.000078 0.000189 0.411323 0.680836

b 5.587139 1.579953 3.536269 0.000406

c0 0.000002 0.000002 0.945457 0.344425

c1 0.120141 0.032749 3.668483 0.000244

d1 0.868351 0.033857 25.647496 0.000000

shape 1.339742 0.045279 29.588833 0.000000 Source: Own estimation in rugarh in R

Tab.3: Estimation of GARCH-M for monthly ordinary returns from sWIG80


a0 -0.002103 0.000916 -2.295856 0.021684

b 0.748130 0.142866 5.236585 0.000000

c0 0.000205 0.000014 14.372725 0.000000

c1 0.573478 0.019047 30.109272 0.000000

d1 0.222899 0.020079 11.101208 0.000000


Tab.4: Estimation of GARCH-M for weekly ordinary returns from mWIG40


a0 0.006786 0.000532 12.761338 0.000000

b -1.930421 0.398550 -4.843607 0.000001

c0 0.000066 0.000006 11.174302 0.000000

c1 0.626267 0.029444 21.269553 0.000000

d1 0.344832 0.022411 15.386625 0.000000



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Tab.5: Estimation of GARCH-M for weekly ordinary returns from WIG BANKI


a0 0.001091 0.000709 1.539269 0.123739

b 0.598315 0.298263 2.006000 0.044856

c0 0.000235 0.000020 11.614925 0.000000

c1 0.684918 0.037822 18.109184 0.000000

d1 0.194012 0.033251 5.834770 0.000000


Fig.2: Refitting of parameters for daily logarithmic returns from sWIG80

Source: Own estimation in rugarh in R

Fig.3: Refitting of parameters for monthly ordinary returns from sWIG80


Derivat 2015, ISSN 1336-5711, 4/2015


Fig.4: Refitting of parameters for weekly ordinary returns from mWIG40


Fig.5: Refitting of parameters for weekly ordinary returns from WIG BANKI


Derivat 2015, ISSN 1336-5711, 4/2015


Yet, also another variations of the models were evaluated. They were not reported in

full details here, however, due to the clarity and simplicity of this short report. The change

was to compute weekly and monthly logarithmic returns, but based on aggregated data (to

weeks and months, where one month is assumed to be equal to 4 weeks).

This resulted in no autocorrelation and ARCH effects in standardized residuals. Yet,

the parameter b was significant only for sWIG80. For weekly aggregation its estimate was

2.907407. According to the Nyblom stability test this parameter is stable.

Conclusions

Some weak evidence was found that small companies are characterised by the positive risk

premium. However, its stability cannot be definitely decided. For medium sized companies

this premium occurred to be negative (and stable). Also, some relatively small risk premium

(also stable) was found for banks. However, no such a premium was found for all stocks or

big companies. Yet, the whole research was based on GARCH-in-mean methodology.

Therefore, some other approach can lead to other interpretation.

Finally, it should be mentioned that even the models, which had the significant


Derivat 2015, ISSN 1336-5711, 4/2015

parameters have serious drawbacks. These drawbacks exclude the possibility of making the

definite and strong conclusions. In particular, they are connected with ARCH effects and

autocorrelation in residuals. The best diagnostic was obtained for weekly logarithmic returns

from sWIG80 (small companies) based on week-frequency data.

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Contact:

Krzysztof Drachal

Faculty of Economic Sciences,

University of Warsaw

Poland

[email protected]

http://www.r-project.org/

analysis of the polish stock market indices based on garch ......denotes the return from the...

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