1. Introduction
An adequate perception of stochastic processes followed by stock index returns represents the core of the efficient market hypothesis (EMH) theory developed by [
1], which shows that financial markets are informationally efficient. The weak form of this hypothesis asserts that prices already reflect all information and, therefore, in order to prove the EMH, there should be no predictability in stock prices. This means that it is impossible for the investors to achieve abnormally high returns; that is, financial markets are efficient if stock price returns follow a random walk.
Nevertheless, noteworthy studies have shown that stock returns do not follow random walks, being characterized by nonlinearities and chaos (for a review of the literature see [
2]). Consequently, recent developments in investigating the nonlinear dependence and deterministic chaos of financial variables altered the traditional view of their erratic behavior [
3,
4]. If the stock market returns are characterized by nonlinearities and chaos, then the market is inefficient. This means that stock prices do not incorporate all existing information and might be over- or under-evaluated. Thus, the investors can record excess returns or losses.
Even if the nonlinear properties of developed stock markets were well highlighted in the literature [
5], little was done for the Central and Eastern European (CEE) economies. While ref. [
6] proved the presence of a long memory in eight CEE stock markets, more recently, ref. [
7] provided evidence for nonlinear dependencies, nonlinear patterns and chaotic dynamics for the Czech Republic, Hungary, and Poland stock markets. However, ref. [
7] included in the analysis only the 1990s’ period, when CEE countries underwent major changes in their economic and political systems. These changes impacted on the stock markets functioning, and created difficulties associated with low liquidity and high volatility, but also with the lack of hedging opportunities and low price to earnings ratios. Consequently, all this evidence could affect the estimations, as the EMH can hardly be proved in transition markets.
Against this background, the aim of this study is to investigate the presence of nonlinearities and chaos in five CEE stock markets with a focus on the pre- and post-accession period to the European Union (EU). For this purpose, we use daily returns series of BET (Romania), BUX (Hungary), PX (Czech Republic), SAX (Slovak Republic) and WIG20 (Poland) indexes and we cover the period from June 2000 to September 2015, using Datastream statistics. This period is large enough to allow for nonlinear tests estimations, and it is also indicated for comparison between the selected markets, as we report to the same period for all analyzed stock markets.
Another contribution of our paper is represented by the analysis of two stock markets (Romania and Slovak Republic) which are usually not included in the advanced CEE stock markets group. However, following the EU accessions, their behavior is similar to the Czech, Hungarian or Polish stock markets, which recommends their inclusion in the same group of CEE stock markets.
Finally, in addition to tests used by [
7], we also employ the McLeod–Li test for nonlinearities, a commonly used diagnostic tool for the presence of Auto Regressive Conditional Heteroskedasticity (ARCH) effects, and the recent test proposed by [
8]. Further, we employ two different developments of the Lyapunov exponent to test the chaos. We also use the 0–1 test for assessing the chaotic behavior of CEE stock markets. The nonlinearities and chaos tests complement each other. Indeed, chaos is related to nonlinear dynamics of stock markets [
9]. On the one hand, the chaotic dynamics are necessarily nonlinear. On the other hand, nonlinear models can generate much richer types of behavior. We therefore use this battery of tests for two reasons: first, to check the robustness of our findings; second, to compare our findings with those reported by [
7]. A run test for randomness is used as a benchmark for nonlinearities and chaos tests. If this test rejects the hypothesis of random walks, it questions the EMH for the CEE stock markets.
To preview our findings, we discover that CEE stock markets are characterized by a chaotic behavior and nonlinear dynamics of price returns. Further, we discover that the results obtained in the case of the updated sample partially contradicts those reported by [
7] for the 1990s, a result explained by the development of CEE stock markets. However, we also notice a strong heterogeneity of CEE stock markets behavior.
The rest of the paper is structured as follows.
Section 2 presents some stylized facts and a short review of the literature assessing the CEE stock markets’ behavior.
Section 3 describes the methodology while
Section 4 presents the results. The last section concludes.
3. Methodology
An important number of tests were used in the literature for assessing nonlinearities and chaos in financial markets: the McLeod–Li test, the runs test, the variance ratio test, the White test, the Teraesvirta test, the Keenan test, the Tsay test, the Engle LM test, the BDS test, the Lyapunov exponent and the noise titration test. The technical details of these tests are provided by [
3,
52,
53], while ref. [
54,
55,
56,
57] realized a comparison of their efficiency. In this section we provide only a brief description of the characteristics of each retained test in our analysis.
3.1. Runs Test for Randomness
The runs test is a non-parametric test used to decide if a time series follows a random process. It is usually considered a linear test which allows, however, for the identification of nonlinearities in the data series. A run is defined as a series of increasing values or a series of decreasing values. If the randomness assumption is not valid, we can interpret this as a lack of efficiency of stock markets. If a data series is random, in the runs test, the actual number of runs (sequences of positive or negative returns) in the series should be close to the expected number of runs, irrespective of the signs [
2]:
where:
P denotes the number of positive runs, while
N means the number of negative runs.
The variance of runs is given by:
3.2. BDS Test for Independence
The BDS test proposed by [
51] is a test used for independence but also for nonlinear dependences. It tests the null hypothesis that the elements of a time series are independently and identically distributed (
iid):
where:
Wm(
ε) is known as the BDS test,
Cm(
ε) represents the fractions of m-dimensions in the series; and
σm(
ε) is the standard deviation under the null of
iid.
The BDS test rejects the null if the test statistic is large (usually larger than 1.96). If the null is rejected, the residuals contain a nonlinear structure. As in [
7], we use a range of dimensions (
m) from 2 to 7 and four values for the distance (
ε), namely 0.5
σ, 1
σ, 1.5
σ and 2
σ.
3.3. White and Teräsvirta Tests for Neglected Nonlinearities
An alternative way to look for nonlinearities is with the neural network models in which the network output
yt is determined based on input
xt:
where:
β is a column vector connecting the strength from the input to the output layers,
yj is a column vector connecting the strength from the input layer to the hidden unit (
j = 1,
…q),
δj represents a scalar connecting the strength from the hidden unit
j to the output unit, and
is a logistic squashing function.
Ref. [
49] proposes a test for neglected nonlinearity, designed to be more powerful compared with other neural network models. The neural network test by [
49] is based on a test function
h(
xt) which activates the hidden units
. Under the null we have:
so that
where: Γ
j are random column vectors independent of
xt, and
represents a hidden unit activation vector.
Ref. [
49] shows that in the presence of correlation, the network performance improves by including in the model an additional hidden unit with the activation
ψ(
xtΓ
j). The test is thus based on sample correlations of affine network errors:
Different from [
49], ref. [
50] replace
δjψ with a Taylor expansion approximation, in order to solve the linearity testing problem, and to use a score testing framework.
3.4. Keenan and Tsay Tests for Nonlinearities
The null hypothesis of the Keenan test [
47] is that of a linear model against a nonlinear specification. Ref. [
48], building upon [
47], explicitly tests for quadratic serial dependence in the data. It represents, thus, a more general form of the Keenan test.
The [
48] test can be specified as follows. If we have
K = k(
k − 1)/2 column vectors
V1, …,
Vk which contain all possible cross-products
et−iet−j (where
and
), then:
If
represents the projection of
on the orthogonal subspace
(meaning the residuals of the
regression on
), then the parameters
are the Ordinary Least Squares (OLS) estimates of the regression equation:
Note that, if p exceeds K then the projection is unnecessary and the Tsay test is equivalent to a classic F statistic for testing the null that are zero.
3.5. McLeod–Li Test for Nonlinearity
The McLeod–Li test [
46] uses the following statistic to test for nonlinear effects in time series data:
where
(with
k = 0, 1,…,
n − 1) are the autocorrelations of the squared residuals
generated by fitting the model to the data. If the residuals are
iid,
with
m degrees of freedom represents the asymptotic distribution of
Q(m).
3.6. Harvey Test for Nonlinearities
A more powerful test for assessing the nonlinearities in data series is proposed by [
8]. This test does not require an assumption of I(0) behavior of data series as the previous tests do, and represents a simple data-dependent weighted average (
) of two Wald test statistics (computed for an I(0),and an I(1) process respectively). For a time series
, the nonlinearity is assumed to enter through the level of
for an I(0) series, or through the first differences of
if the series is I(1):
If the series is (I0), the null hypothesis of linearity is , while the alternative of nonlinearity is expressed as . The standard Wald statistic for testing these restrictions is given by . The similar applies if is (I1), situation in which the Wald statistic is represented by .
The weights (
and
) are determined by considering the switch between the two efficient Wald statistics, based on an auxiliary test. The new weighted statistic has a standard chi-squared limiting null distribution in both the I(0) and I(1) cases:
where
represents a function which converges in probability to zero when the series
is I(0), and to one when
is I(1).
3.7. The Lyapunov Exponent
The Lyapunov exponent is explicitly used in the literature to test whether a time series is chaotic. In a chaotic system, if an infinitesimal change
appears in the initial conditions, the corresponding change iterated through the system will grow exponentially with the time
t. Technically, the largest Lyapunov exponent is considered the only test explicitly devised for testing chaos and measures the rate at which information is lost from a system [
3]. A process shows chaotic behavior if the maximum Lyapunov exponent is positive [
58].
Considering an infinitesimally small hypersphere of radius
, the maximum Lyapunov exponent is measured by the extent of the deformation as follows [
55]:
where
) represents the length of the
ith principal axis of the ellipsoid at time
t.
The literature recorded several developments of the Lyapunov exponent, based on a neural network, or on a non-neural network. In the first category, we can find the maximum Lyapunov exponent proposed by [
59], which allows for the identification of chaotic dynamics in short noisy systems. In the second category of models, we mention [
60], who propose an alternative way of calculating the largest Lyapunov exponent, which takes advantage of all the available data. We use both tests in our empirical analysis.
3.8. The 0–1 Test for Chaos
The 0–1 test for chaos proposed by proposed by [
61] is based on a Euclidean extension, instead on a phase space reconstruction as the Lyapunov exponent. It tests the chaos in a deterministic dynamical system
, by studding the asymptotic behavior of the translation variables
and
, with
and
representing an arbitrary but fixed frequency. Both
and
remain bounded as
if the system does not exhibit chaos.
Ref. [
61]’s test static is represented by the asymptotic Bravais–Pearson correlation coefficient between
and
:
where
.
5. Conclusions
This paper tests for nonlinearities and chaos in five mature CEE stock markets, using a large battery of tests. Although our results generally contradict the theoretical assumption of linearity and nonchaotic behavior of stock markets, some issues appear. First, most of the selected tests for nonlinearity points to mixed evidence. Second, our results partially contrast from those reported by [
7], which show that mature CEE stock market have a different behavior compared to their behavior during the 1990s and 2000s, when the efficient functioning of the selected markets was questioned. For example, the Runs test’s results underline the presence of randomness for BUX and PX indexes, while the findings of [
7] show the lack of randomness for the same index returns. In addition, the White and Teräsvirta tests highlight a linear behavior of the BUX index returns in our case.
To sum up, our results confirm the findings reported by the existing literature, which point, in general, in favor of nonlinear and chaotic behavior of stock markets that requires adequate forecasting techniques to predict the stock prices behavior. However, our findings partially contrast earlier reported findings for the CEE stock markets behavior in the 1990s. At the same time, our results underline the discrepancies existing between CEE stock markets, which question the idea of CEE stock markets’ increasing integration.