Stock market speculative bubbles:
the case of Visegrad countries
Oleg Deev1
, Veronika Kajurová2
, Daniel Stavárek3
Abstract: Conventional theory of speculative bubbles describes stock bubbles as
stock prices that exceed their fundamental value because current owners believe that
the stocks can be resold at an even higher price in the future. We employ a special
methodological technique examine the presence of the phenomenon of stock market
bubbles in the Visegrad group countries (Czech Republic, Hungary, Poland, and
Slovakia) and selected developed European stock markets. The methodology is
based on the examining of residuals of VAR fundamentals with exclusion of ARCH
effects. The presence of bubbles is studied by regime switching tests and Hurst
persistence tests. Although we examine the bubbles presence over various time
periods we found almost no evidence of speculative bubbles across the markets.
Keywords: stock bubble, regime switching test, Hurst persistence test
JEL Classification: G14, G15
AMS Classification: 91G70
1 Introduction
The phenomenon of asset bubbles has been known and studied for centuries, but there is still no common
framework on how to detect or predict the formation of a bubble. Since 1980s bubbles are investigated with
application of time series econometric analysis. However, the use of such mathematical apparatus raises the
question of whether econometric tests can truly detect a bubble or just discover an error in the market evaluation
of assets. The majority of empirical research (such as Bohl [4] or Nasseh and Strauss [15]) examines the
existence of stock market bubbles using traditional unit root tests of price-dividend ratios of US highly
capitalized companies, for which long-term data are available. Other studies were also conducted on markets, for
which the history of dividend payments exists. Unfortunately, there are only few papers investigating the
occurrence of asset bubbles in emerging markets, especially considering the fact that in the last twenty years
those economies were the subjects of large financial inflows and the data on dividends are of limited use.
Emerging market studies, primarily focused on China or countries of MENA region (such as Jahan-Parvar and
Waters [10], Lehkonen [13] or Ahmed et al [3]), reveal inconsistent results. We can find several references
concerning the stock bubbles in Visegrad group countries within some of published papers (such as Kizys and
Pierdzioch [11]. The study dealt with the collapse of stock markets in the Czech Republic, Poland and Hungary
during the financial crisis and if it was due to international linkages of deteriorating fundamentals or international
spillovers of speculative bubbles; Hanousek and Novotný [8] performed an extensive analysis of price
jump for emerging stock market indexes from the CEE Visegrad region), but not overall research focused on the
stock market bubbles in this region. Clearly new methodological approaches and more research in the area are
needed. Furthermore, the individual analysis of the possibility of stock market bubbles in smaller financial
markets in Europe, including the Czech Republic and other central European countries, is of particular interest.
The main aim of this paper is to examine the presence of stock market bubbles in the central European
countries, namely in the Czech Republic, Hungary, Poland, and Slovakia. In order to prove the statistical
significance of proposed methodology, the results are compared to the outcomes of research obtained from
selected European developed markets, such as Germany, Austria, France and the United Kingdom. The presence
of the 2007-2009 market turmoil brought by the global financial crisis is addressed by dividing the long-time
horizon data into three periods: before, during and after the financial crisis.
Identifying stock bubbles is a challenging task not only in terms of time, but also in terms of distinguishing
the fundamental and non-fundamental determinants. Since the fundamental value is not directly observable, it
must be estimated. On the other hand, it is difficult to confirm the existence of a bubble with a particular
1
Masaryk University, Faculty of Economics and Administration, Department of Finance, Lipová 41a, 602 00
Brno, oleg@econ.muni.cz.
2
Masaryk University, Faculty of Economics and Administration, Department of Finance, Lipová 41a, 602 00
Brno, vkajurova@seznam.cz.
3
Silesian University in Opava, School of Business Administration in Karviná, Department of Finance,
Univerzitní nám. 1934/9, 733 40 Karviná, stavarek@opf.slu.cz.
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certainty, since the determination of the fundamental value is not a trivial task [12]. The majority of valuation
models include dividend payments in their calculations, which values are not available in our inquiry. Therefore,
following the methodology of Ahmad et al [1], [2], we assume, that changes in dividends are reflected in the
market prices, and abstract from dividends, basing our analysis only on the stock index returns.
2 Data
For the purposes of our study high-frequency daily data are preferred, taking into consideration the market
environment of advanced information technology and rapid general information sharing. Daily data capture
speedy information as both short run and long run dynamic linkages play role in the bubble formation. We
employ the data from the main stock indices representing the chosen markets (basic characteristics of stock
exchanges is given in Table 1), weighted indices of the profitability (long-term interest rates) of 10-year
government bonds for each country (as calculated in Bloomberg), and the MSCI world index, summarizing the
developments of the global stock markets. Returns of the variables as their first log differences are used.
Tick
symbol
Country
Number
of listed
stocks
Market
capitalization,
mln. USD
Market
capitalization
as % of GDP
Market
turnover,
mln. USD
Market
turnover as
% of GDP
Market
liquidity
%
PX Czech Rep (CZ) 16 43 055.6 22.4 14 082.5 7.3 29.4
BUX Hungary (HU) 48 27 708.4 21.5 26 466.1 20.6 94.5
WIG Poland (PL) 569 190 234.9 40.5 77 463.9 16.5 47.6
KSM Slovakia (SK) 90 4 149.6 4.8 173.7 0.2 3.9
ATX Austria (AU) 86 67 682.8 17.9 48 117.4 12.7 79.4
DAX Germany (DE) 571 1 429 706.7 43.6 1 405 037.1 42.8 103.0
CAC France (FR) 901 1 926 488.3 75.3 1 467 073.7 57.3 75.3
UKX UK 2056 3 107 037.9 137.4 3 006 680.0 132.9 101.9
Sources: The World Bank (World Development Indicators)
Table 1 Stock market characteristics of selected stock exchanges at the end of 2010
The study period is similar for all countries. The total sample period is divided into three sub-periods
according to clearly observed trends in the prices’ movements. The sub-periods are the pre-crisis period (May
2004 – July 2007), the crisis period (August 2007 – March 2009) and the post-crisis period (April 2009 – March
2012).
Figure 1 Daily returns (in %) of stock market indices during the whole period
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The beginning of the study period is set as an accession date of the Visegrad group countries to the European
Union. We chose the crisis period to not start with the Lehman Brothers bankruptcy and major panic on the
markets; instead we would like to capture prior anticipations on the markets, when the 2007 banking crisis
changed the comfort expectations with a fear of it becoming a sovereign debt crisis. Fluctuations of index returns
are illustrated in Figure 1.
3 Methodology
To estimate the fundamental value of index returns we apply the methodology based on the VAR modeling. The
VAR model with index returns, government bonds’ interest rate and world index returns is employed for each
country:
‫ݎ‬௧ = ‫ܣ‬ଵ‫ݎ‬௧ିଵ + ⋯ + ‫ܣ‬௣‫ݎ‬௧ି௣ + ‫݅ܿݏ݉ܤ‬௧ + ‫ܾܥ‬௧ + ߝ௧ (1)
where ‫ݎ‬௧ are index returns, ݉‫݅ܿݏ‬௧ are returns of the MSCI world index, ܾ௧ are 10-year government bond interest
rates and ߝ௧	is an error term.
To exclude the phenomenon of changing volatility in the time series we should remove autoregressive
conditional heteroskedasticity (ARCH) effects from VAR residual series. According to Engle [6] the nonlinear
variance dependence measure of ARCH is:
ߝ௧ =	ߜ௧ߤ௧ (2)
ߜ௧
ଶ
= ߙ଴ + ∑ ߙଵ
௡
௜ୀଵ ߝଵି௜
ଶ
(3)
with ߤ	is independent and identically distributed (i.i.d.) variable and ߙଵ	is a coefficient for chosen lags.
Two tests for the identification of stock market speculative bubbles are used. First, we employ the Hurst
persistence test to find the existence of long-term linear dependence (memory) in the stock market volatility.
Second, we perform the rescale range test. The method based on the Hurst persistence approach is also called
rescaled range (R/S) analysis because the significance test breaks the sample into sub-subsamples and then
estimates a Chow test on the null that the sub-periods possess identical slopes [1]. This test was developed by
Hurst [9] and was firstly implemented in economic analysis by Mandelbrot [14]. Using the R/S analysis, the
Hurst exponent H is estimated from the VAR residual series:
ሺܴ ܵ⁄ ሻ௡ =	
ଵ
ௌሺ௡ሻ
ൣmaxଵஸ௧ஸ௡ ∑ ൫ߝሺ‫ݐ‬ሻ − ߝ̅ሺ݊ሻ൯௡
௧ୀଵ − minଵஸ௧ஸ௡ ∑ ൫ߝሺ‫ݐ‬ሻ − ߝ̅ሺ݊ሻ൯௡
௧ୀଵ ൧ (4)
where ܵ௡	is the standard deviation estimation and ߝ̅ሺ݊ሻ	is the sample mean of the return time series:
εതሺnሻ =	ሺ1 n⁄ ሻ ∑ εሺnሻ୬ (5)
R/S is then described as:
ሺܴ/ܵሻ௡ = ቀ
௡
ଶ
ቁ
ு
(6)
Hurst exponent allows us to reveal the behavior of stock market efficiency over time [16]. If 0 < H < 0.5, it
denotes an anti-persistent behavior, which means that positive trends in one period tend to become negative and
vice versa. If 0.5 < H < 1, a persistent behavior is indicated in stock market behavior, that is, positive trends in
one period tend to continue being positive and vice versa. If H is close to 0.5, it indicates a random walk in data,
meaning that market returns are independent. Estimated Hurst exponents are then used to compute F-values for
the Chow test to examine its statistical significance.
Second test to detect bubbles in stock market time series is the regime-switching test introduced by Hamilton
[7]. The approach of Engle and Hamilton [5] is utilized to test the null hypothesis of no bubbles:
ߝ௧ = ‫݀݊݁ݎݐ‬௧ + ‫ݖ‬௧ (7)
where ‫ݖ‬௧ is the white noise and ‫݀݊݁ݎݐ‬௧ = ߤଵ + ߤଶ‫ݏ‬௧ (8)
with ‫ݏ‬ = 1	being a positive trend and ‫ݏ‬	 = 0 being a negative trend. Moreover, we let:
ܲ‫ܾ݋ݎ‬ሾ‫ݏ‬௧ = 1	‫ݏ‬௧ିଵ = 1ሿ = ‫,݌‬ ܲ‫ܾ݋ݎ‬ሾ‫ݏ‬௧ = 0	‫ݏ‬௧ିଵ = 1ሿ = 1 − ‫݌‬ (9)
ܲ‫ܾ݋ݎ‬ሾ‫ݏ‬௧ = 0	‫ݏ‬௧ିଵ = 0ሿ = ‫,ݍ‬ ܲ‫ܾ݋ݎ‬ሾ‫ݏ‬௧ = 1	‫ݏ‬௧ିଵ = 0ሿ = 1 − ‫ݍ‬ (10)
The null hypothesis of no trend is given by ‫݌‬ = 1 − ‫ݍ‬ and the Wald test statistic calculated as:
௣	ିሺଵି௤ሻ
௩௔௥ሺ௣ሻା௩௔௥ሺଵି௤ሻା௖௢௩௔௥ሺ௣,ଵି௤ሻ
(11)
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The Wald test statistic evaluates how close the unrestricted estimates come to satisfying the restrictions under the
null hypothesis.
The results of both tests allows us perceiving asset bubbles in the chosen Eastern European countries with a
certain degree of confidence, since tests unveils different characteristics of the same time series.
4 Empirical findings
Both tests’ results indicate the same situation of no bubbles in stock markets of studied countries (with one
exception of Slovakia from the results of Hurst persistence test). There is no significant difference in persistence
of stock returns in the highly developed European countries and the Visegrad countries, except for Slovakia. For
the majority of cases, Hurst exponent values are not significantly different from its average of 0.5 (see Table 2).
Stock index prices follow random walk and do not show any speculation developments.
However, Hurst exponent values for Slovakia highlight irregular market dynamics, which probably disclose
the overall inefficiency of the market rather than the existence of price bubbles. The statistical significance of
Hurst persistence tests is verified by the Chow test, F-values of which are above its critical values, hence the null
hypothesis of no persistence in the time series is rejected.
Estimated Hurst exponents of residuals
Full sample Pre-crisis period Crisis period Post-crisis period
PX 0.526192 0.571909 0.508225 0.504513
BUX 0.517590 0.568348 0.418661 0.533226
SKSM 0.653698 0.731891 0.612539 0.531983
WIG20 0.498979 0.496172 0.406587 0.464584
ATX 0.543151 0.557359 0.463424 0.534663
DAX 0.484378 0.466146 0.452908 0.590430
CAC 0.550297 0.486435 0.448344 0.549227
UKX 0.452292 0.515526 0.482410 0.511554
F-values for Chow test
Full sample Pre-crisis period Crisis period Post-crisis period
PX 59.093300 60.659000 60.193500 60.400200
BUX 59.603600 56.668500 65.820900 58.693000
SKSM 52.023100 48.131800 54.246500 58.766000
WIG20 60.722600 60.903800 66.618200 62.857500
ATX 58.100000 57.293400 62.944600 58.608400
DAX 61.615200 62.757900 63.608500 55.434000
CAC 57.686500 61.499000 63.898900 57.762200
UKX 63.623400 59.738100 61.763500 59.977100
Critical value F = 4.61
Source: Authors’ calculations based on data from Bloomberg
Table 2 Hurst exponents and related Chow test results
Based on the Hurst exponent values in the sub-periods, the Visegrad stock markets appear to be more volatile
than the developed markets in the later periods (except for Poland), supposedly indicating the presence of
growing financial inflows. From the results of the Hurst persistence test, the global financial crisis might be seen
as a stabilizing mechanism updating the upturning and downgrading market forces (for example, through
changing the trading trends in France and Germany). Less than 0.5 values of Hurst exponents in the crisis period
signify the decline of asset prices in all observed markets.
Table 3 reports results of regime switching tests. The null hypothesis of no trend in all investigated stock
market returns is rejected. Estimated critical value for rejecting the null hypothesis is in all cases lower than the
values of the Wald test statistics.
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Full sample Pre-crisis period Crisis period Post-crisis period
PX 819.360 568.246 136.253 304.379
BUX 1123.30 760.251 138.439 441.899
SKSM 791.087 510.217 113.721 265.227
WIG20 1191.91 692.992 236.185 385.211
ATX 707.994 394.937 147.353 290.520
DAX 444.307 480.623 43.4170 136.398
CAC 1552.93 961.337 351.834 780.026
UKX 447.418 324.963 119.101 128.735
Critical value χχχχ2
(1) = 3.84
Source: Authors’ calculations based on data from Bloomberg
Table 3 Wald test results
5 Conclusions
We found no evidence of stock market bubbles neither in the countries of the Visegrad group, nor in the
developed European countries. However, taking into consideration the limitations of the proposed methodology,
we could not declare with the full certainty that asset bubbles are not present in those markets. If tests have not
proved the existence of bubbles, they at least have identified the substantial volatility. Further search of relevant
methodology is needed, while tests should be performed not only on market indices, but also on chosen stocks
and industry indices.
Acknowledgements
Support of the Czech Science Foundation within the project GAČR 403/11/2073 „Procyclicality of financial
markets, asset price bubbles and macroprudential regulation“ is gratefully acknowledged.
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