NONLINEARITY TEST FOR ISTANBUL STOCK EXCHANGE

 

 

 

 

 

Murat  Çinko*

 

 

 

Forecasting issue of financial time series is an important subject of finance. Nonlinearity is the widely accepted feature for the financial time series. Discussions about the market efficiency theory are based on the fact that the return series is a random walk. That means you cannot forecast the prices. When the hypothesis of prediction of the return is rejected then this supports the market efficiency theory which is tested by linear model. On the other hand studies have shown that economic relations are nonlinear. The logic that economic relations are nonlinear is the basis of the nonlinearity tests. If we can prove the relation is nonlinear then we can try to forecast the returns by using the nonlinear model. In this paper three of these tests, which are BDS, Brock et. al (1996), Hinich Bispectrum Test, Hinich (1982), and Lyapunov Exponent test, Nychka et. al. (1992), is applied to the Istanbul Stock Exchange Index. The data is daily closing index between 02/01/1989 and 26/01/2001, which is 2995 observations, obtained by Central Bank of Turkey.  

 

Özet

Bu makalenin temel amacı günlük IMKB getirisinin doğrusal olmayan bir yapıya sahip olup olmadığının ortaya çıkarılmasıdır. Bu amaçla üç tane test yapılmıştır: BDS testi, Hinich Bispectrum testi, NEGM Lyapunov üssü testi. Bu çalışmada kullanılan veriler 02.01.1989 26.01.2001, tarihleri arasında olup toplam  2995 tane gözlemden oluşmaktadır.

 

*Department of Business Administration, Marmara University, Istanbul, Turkey

Marmara University Kuyubası 81040, Istanbul

Tel: + 90 216 345 86 29

Fax: + 90 216 345 86 29

E-mail: mcinko@marmara.edu.tr

 

 

1. Introduction

The chief corollary of the idea that markets are efficient, that prices fully reflect all information, that price movements do not follow any patterns or trends. This means that past price movements cannot be used to predict price movements. Rather, prices follow what is known as a random walk, an intrinsically unpredictable pattern. In fact Fama (1970) defined three types of (informational) capital market efficiency, each of which based on a different notion of exactly what type of information is understood to be relevant. In particular, markets are weak form, semistrong-form, and strong form efficient. The weak form of the Efficient Market Hypothesis (EMH) asserts that all past market prices and data are fully reflected in asset prices. The implication of this is that technical analysis cannot be used to beat the market. The semi-strong form of the EMH asserts that all publicly available information is fully reflected in asset prices. The implication of this is that neither technical nor fundamental analysis can be used to beat the market. The strong form of the EMH asserts that all information - public and private- fully reflected in asset prices. The implication of this is that not even insider information can be used to beat the market. Clearly, strong-form efficiency implies semistrong-form efficiency, which in term implies weak-form efficiency, but the reverse implications do not follow, since a market easily could be weak-form efficient but not semistrong-form efficient or semistrong-form efficient but not strong-form efficient. An immediate and direct implication of an efficient market is that no group of investors should be able to consistently beat the market using a common investment strategy. Recently the efficient markets hypothesis and the notions connected with it have provided the basis for a great deal of research in financial economics. Briefly stated, the hypothesis claims that asset prices are rationally related to economic realities and always incorporate all the information available to the market. This implies the absence of exploitable excess profit opportunities.

Most of the empirical tests are designed to detect linear structure in the financial data - that is, linear predictability is the focus. However, as Campbell, Lo, and MacKinlay (1997) argue many aspects of economic behavior may not be linear. Experimental evidence and causal introspection suggest that investor’s attitudes toward risk and expected return are nonlinear. It is for this reason that interest in deterministic nonlinear chaotic process has in the recent past experienced a tremendous rate of development. In fact, the possible existence of chaos could be exploitable and even invaluable. If, for example, chaos can be shown to exist in asset prices, the implication would be that profitable, nonlinearity-based trading rules exist (at least in the short run and provided the actual generating mechanism is known). Prediction, however, over long periods is all but impossible, due to the sensitive dependence on initial conditions property of chaos.

Many studies show the nonlinearity of financial time series. Hsieh (1993) found that the following financial time series are nonlinear. The data set is consist of daily prices for currency futures contracts traded on the Chicago Merchantile Exchange (CME), British Pound, Deutsche Mark, Japanese Yen, Swiss Frank, from February 22 1985 to March 9 1990 (1275 observation). Abhyankar et al. (1995) found the nonlinearity for the data set consisting of 60.000 minute by minute real time returns on the UK-FTSE 100 index for the first 6 month of 1993. Hsieh (1989) found that the data consist of daily closing bid prices of foreign currencies in terms of US dollars are nonlinear. He used the following financial time series British Pound, Canadian Dollar, Deutsche Mark, Japanese Yen, Swiss Frank. There are a total of 2510 daily observations from January 2 1974 to December 30 1983. Scheinkman et al (1989) found the nonlinearity for the data set consisted of 5200 daily returns (including dividends) on the value-weighted portfolio of the center for research security prices at the University of Chicago. Abhyankar et al. (1997) found the nonlinearity evidence for the data set consists of real-time observations for the period September 1 to November 31 1991 at 1-minute frequency in the case of the FTSE-100, the Deutscher Aktein Index (DAX), and Nikkei-225 and 15-second frequency for the S&P 500 futures. Totally six series are used, four published cash indexes (the FTSE-100, the S&P 500, the DAX, and Nikkei) and two series constructed for futures on the FTSE-100 and S&P 500. Barnett et al. (1997) generate 5 different models of random number series in two categories one is small sample size (380 observation) and large sample size (2000 observation). Model I was fully deterministic, chaotic series. Model II was GARCH process. Model III was a nonlinear moving average process. Model IV was ARCH process. Model V was ARMA process. BDS test was successful 2 out of 5 model in the small sample case and 3 out of 5 is ambiguous. In large sample case, BDS test, was successful for all of the data series. Hinich bispectrum test was successful 3 out of 5 model in the small sample cases and it failed 2 out of 5 cases. In large sample case test is successful in 3 out of five, ambiguous in one case and failed in one case. Liapunov exponent test is successful in all cases.

Frank and Stengos (1989) found low dimensional chaos. They used Correlation dimension test to see whether the data series is chaotic or not. They used daily price for gold beginning of 1975 to June 1986. They also used daily price for silver beginning of 1974 to June 1986.

Brockett, Hinich, and Patterson (1988) use the test for the following data set and found that: first series is both linear and Gaussian but second and third series are both non-linear and non-Gaussian. The first series examined was a sequence of values generated by a random number generator that simulates Gaussian white noise. The series divided into 100 records, each containing 1024 samples. Each record was tested for linearity and Gaussianity. Since the test statistics are one-sided only large positive values are significant. Hence, for this series it is found that accept both linearity and Gaussianity. The results of applying the bispectral tests to ten different common stock series are: daily returns are realizations of nonlinear random process. Although stock-return time series closely resemble white noisy, this evidence in daily suggests a much higher degree of dependence in daily stock returns. The series of stock-prices relatives is decidedly non-Gaussian and nonlinear. Third example, In applying the preceding statistical tests to the analysis of the forward and spot time series for foreign exchange rates, both the original price quotes and the log of the price are examined. For the analysis U.S. dolar and Japanese yen exchange rate are chosen. Two time period are examined, from January 1,1981, to mid-1982 and from December 12,1981, to mid-1983. Daily quotes for rates from The Wall Street Journal are used and it is found that time series is nonlinear and non-Gaussian.

In the next section test statistics are going to be introduced and in section three findings are going to be presented. In the last section conclusion is going to be given.

2. Nonlinearity Tests

In this section each of the test statistics is going to be introduced. There are many tests in the literature for nonlinearity. They are collected under the name of Lagrange Multiplier tests and Portmanteau tests. BDS, Hinich and Lyapunov exponent test are under the second group. BDS test is the most popular one since the test has a power against all type of nonlinearity. Hinich bispectrum test is a frequency domain test and it is based on the fact that linear process has a constant skewness function. If the process is Gaussian then skewness function is constant and also equal to the zero.  Last test is based on the estimation of lyapunov exponent. Posite lyapunov exponent is the operational definition of  chaos.

2.1 BDS test

This test for independence based on estimation of correlation integrals at various dimensions. It has power against virtually all types of linear and non-linear departure.  While the estimation of the BDS statistic is non-parametric, the test statistic asymptotically follows a normal distribution with zero mean and unit variance and therefore lends itself for easy hypothesis testing. BDS test is the most widely used nonlinearity test may be because of no distributional assumption, or easily applied to time series, or theory behind it.

            In principle no distributional assumptions need to be made about the data under the null hypothesis other than that it is independent identically distributed (i.i.d). It can be interpreted as a test for nonlinearity, if the appropriately used in conjunction with Autoregressive Moving Average (ARMA) modeling. In the first step, the best-fitted ARMA (p,q) is determined and fitted to the data, thus eliminating all linearity from the data. Only in the second step is the test applied by running it on the residuals of that ARMA model so that any dependence found in the residuals must be non-linear in nature. Because of this reason, BDS test can be used to produce indirect evidence for non-linearity. The order of the ARMA model is decided by looking at the BIC (Bayesian Information Criteria). Choosing the best ARMA means choosing the lowest BIC value.

 Null hypothesis for the BDS test is data-generating process is IID. The alternative hypothesis is not specified that means the data generating process may be linear nonlinear or chaotic. In practice the test has been used to examine whether fitted model’s errors are IID. Test statistics is  given by Brock et. al. (1996)

   ~ N(0,1).                                                 (1)

where Vm,n(e) is the variance of Tm,n(e)

Tm,n (e) = C (N,m, e) -  C (N,1, e)m

                                   (2)

 

H(z) is the Heavside function which maps the positive arguments into one, and nonpositive arguments into zero.

2.2 Hinich Bispectrum Test

Hinich (1982) developed a test to detect Linearity and Gaussianity of the time series. His approach provides a direct test for Linearity and Gaussianity. Tests statistic has a known asymptotic sampling distribution under both Linearity and Gaussianity.

            If a time series is Gaussian and linear then its third moment is zero.  Hinich test is a test in the frequency domain of flatness of the bispectrum. Bispectrum is a double Fourier transformation of skewness function. If linearity is tested then the null hypothesis will be ‘skewness function is flat’ or ‘lack of third order nonlinear dependence’. Z will denote test statistics for linearity. If Gaussianity will be tested then the null hypothesis will be ‘time series is Gaussian’ or ‘skewness function is flat and equal to zero’. H will denote test statistics for Gaussianity. Flatness of the skewness function is necessary but not sufficient condition for general linearity and Gaussianity. However, flatness of the skewness function is necessary and sufficient condition for third order nonlinearity.

            Autoregressive (AR) or Autoregressive Moving average (ARMA) generation process is the implicit assumption of most of the time series. Let {xt} be of the form:

                                                                                (3)

where e t are independent identically distributed random innovations with E(e (t))=0. If the input is Gaussian then the output is Gaussian and its covariance function completely determines the joint distributions of the process.

The Bispectrum is double Fourier transformation of third order cumulant. The bispectrum, B(w1,w2), gives a measure of the multiplicative nonlinear interaction of frequency components in {xt}. For a real stationary time series with E x(t) =0, the bispectrum is defined as follows:

Bx(w1,w2)=                                   (4)

Assuming that | Cxxx (m,n) | is summable. Given the symmetries of Bx(w1,w2), its principle domain is the triangular set

                                     (5)

                                                   (6)

for all w1 and w2 in W whenever {xt} is linear. If u(t) is Gaussian, m3 = 0 and thus

B (w1,w2)=0.

 

2.3 Lyapunov Exponent Test

Nonlinear dynamical systems can behave in ways that are hard to distinguish from a random process. Nychka et al. (1992) introduce a test statistic for chaotic time series. If a series is bounded and it has a positive Lyapunov exponent then this series is called chaotic. Lyapunov exponents are generalizations to nonlinear systems of the eigenvalues or roots of linear system. There are three possible value for the Lyapunov exponent where Lyapunov exponent denoted by l: (i) l < 0, Lyapunov exponent is less than zero system is convergent (ii) l = 0, Lyapunov exponent is equal to zero indicate that the system is in some sort of steady state mode. (iii) l > 0, Lyapunov exponent is greater than zero indicate a sensitive dependence on initial conditions, i.e. system is chaotic. NEGM is a procedure for testing for chaos by estimating the dominant Lyapunov exponent. As mentioned before, positive Lyapunov exponent for a bounded system is the operational definition of chaos. As is mentioned before Jacobian method is used to calculate Lyapunov exponent where the neural network method has been used to estimate this exponent. In the earlier studies BIC (Bayesian Information Criteria) values were used, but in the more recent version of NEGM approach GCV (Generalized Cross Validation) criteria has been using to determine the order of linear time series process (i.e. the order to find the best combination of L, d, q). As appropriate value of L, d, and q are unknown where q is the number of units in the hidden layer of the neural network, d is the embedding dimension and L is the time delay.

Given a time series Nychka propose to test the Lyapunov exponent by using nonparametric regression. Assume that xt are generated by nonlinear autoregressive model.

                                                                          (7)                                                  

The estimation technique of the functional form of the equation (7) is

                                                                          (8)

where G(u) is the logistic distribution function. This technique finds dominant lyapunov exponent by using nearal network.  

 

3. Data Analysis

The daily closing index of Istanbul Stock Exchange is obtained from the Central Bank of Turkey. The data, which is 2995 observations, is between 02/01/1989 and 26/01/2001. Returns of the ISE composite index are calculated as rt = ln (ISEt / ISEt-1). Before the nonlinearity tests stationarity test should be done and giving the descriptive statistics should be helpful to understand the structure of the data. In table 1 descriptive statistics and normality test Jarque-Bera statistic, which rejects the normality, are given.

 

Table 1 Descriptive Statistics

Observations 2995

Mean                        0.002664

Median                     0.002168

Std. Dev.                  0.032416

Skewness                 0.139312

Kurtosis                    7.976217

Jarque-Bera             3099.870

Probability               0.000000

 

 

In table 2 ADF (Augmented –Dickey Fuller test (1979)) and P-P (Phillips-Peron test (1988)) test statistics are given. Both tests reject the unit root hypothesis. 

 

Table 2 Unit Root Tests Result

ADF Test Statistic

-24.14516

    1%   Critical Value*

-3.4356

 

 

 

    5%   Critical Value

-2.8630

 

 

 

    10% Critical Value

-2.5676

 

PP Test Statistic

-48.76425

    1%   Critical Value*

-3.4356

 

 

 

    5%   Critical Value

-2.8630

 

 

 

    10% Critical Value

-2.5676

 

*MacKinnon critical values for rejection of hypothesis of a unit root.

 

3.1 BDS Test Result

By using the minimum BIC criteria most appropriate ARMA model is found and it is ARMA(6,6). Filtering the original return series by ARMA(6,6) residuals are obtained. BDS test is done to these residuals and result shows that residuals are not independently identically distributed. It is known that dependence is not linear then it is nonlinear. BDS test statistics are given in the table 3. Critical value (for a =0.01) is 2.33. For all of the embedding dimensions BDS test reject the null hypothesis of iidness that means returns are nonlinearly dependent.

 

Table 3 BDS Test Statistics

 

Embedding Dimension

e

2

3

4

5

6

7

8

0.75

14.16

19.81

24.1

28.67

34.56

42.41

52.99

1

14.53

19.55

22.76

25.97

29.76

34.25

39.62

1.25

14.68

19.12

21.54

23.77

26.24

28.91

31.82

1.5

14.84

18.78

20.57

22.15

23.81

25.46

27.08

2

14.81

17.83

18.85

19.63

20.46

21.23

21.77

2.5

15.22

17.24

17.76

18.07

18.54

18.97

19.15

  

3.2 Hinich Bispectrum Test Result

As it is said that Hinich Bispectrum test is a frequency domain approach and two hypotheses can be tested by this test: Linearity and Gaussianity.

 

 

Linearity test:

This test is a test of whether there exist third order nonlinear dependence or not (i.e tests the flatness of skewness function or lack of third order nonlinear dependence). If we reject the null of linearity then the series can be accepted to be nonlinear. However fail to reject the null hypotheses does not mean that series is linear it only says that series is not third order nonlinear. Test statistics is given in the table 4 shows that null hypothesis of linearity cannot be rejected.

Gaussianity test:

In this test we test whether the daily ISE returns are Gaussian or not. In this case, the test statistic, is 1.00 is again smaller than the critical value which implies that the null hypothesis of Gaussianity can not be rejected. That does not mean that ISE returns are Gaussian, this test can reject Gaussainity but Gaussianity cannot be accepted by this test since all Gaussian series have a flat skewness function but inverse is not always correct.

 

Table 4 Result of Hinich Bispectrum Test

Test                      Statistic               Critical value               Conclusion

Linearity test         1.07                    2.55 (a =0.05)        Fail to reject Linearity

Gaussianity test     1.00                                                   Fail to reject Gaussianity

 

3.3 Lyapunov Exponent Test Results

 

GCV rule shows that best combination of (d,L,q) is (3,1,3). The null and the alternative of this test can be written as

H0: Time series is chaotic

H1: Time series is not chaotic

By looking at the estimated Lyapunov exponent for the time series appears to be -0.7480689. Since calculated Lyapunov exponent is less than zero null hypothesis of positive Lyapunov exponent or chaos is rejected. Therefore, based on the NEGM test it can be concluded that the daily ISE return is not chaotic.

4. Conclusion

In this paper three popular tests for nonlinearity are applied on the daily ISE return. We have rejected the presence of linearity for ISE return by the BDS. This can be considered to be significant evidence against linearity since this test is accepted to be powerful tests against any departure from linearity. By NEGM test we could not find evidence on chaos. Furthermore, by Hinich’s bispectrum test we could not find any conclusion that supports a third order nonlinear dependence. It can be concluded that we have found sufficient evidence for nonlinearity in the ISE daily returns but the form of nonlinearity can be any type except choatic or third order nonlinear.

 

 

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