The 'co-movement' connection between two time series.

Pirtea, Gabriel Marilen ; Ioan, Roxana ; Dima, Bogdan 等

1. INTRODUCTION

The latest approaches concerning the capital markets are interdisciplinary researches based upon mathematical or even physics related concepts. Many trading systems are built starting from the assumption that the evolutions of certain equities are--at some level--related with each other.

This is how the concepts of correlation (such as Pearson's linear correlation coefficient and the rank correlation coefficients Kendall and Spearman), cross correlation, regression (such as the famous Sharpe Model) or cointegration (Engel & Granger, 1987) found countless utilization within this complex field.

Still, there are many situations when the correlation coefficients fail to capture the connection between the price series, for different reasons (the main reason being that the Pearson coefficient captures only linear relations and the rank correlation coefficients do not always match the nature and structure of the price series data, often giving contradictory results between the two of them). The regressions don't always describe perfectly capital market models, the accuracy depending on how well all the dependent and independent variables are identified, measured and settled (Gujarati, 2003). The best known procedure of capturing such connections is the Johansen cointegration model. From a theoretical point of view, the model escapes the linearity restriction, having the possibility to capture both linear and nonlinear relations. Still, this model often gives ambiguous results and its main shortcomings are that it only can be used for nonstationary series and is fitted only for long-term analysis (Johansen, 1991). It is a well known fact that the price series' evolution can encounter stationary cycles.

2. THEORETICAL CONCEPTS

Trying to follow this base line, this paper attempts to find and prove the existence of a new type of connection between stock prices (considered to be common time series), without imposing the nonstationarity condition, without having the long run limitation and without having to know the exact type of equation that defines the relation between the analyzed time series. Taking into consideration all the mentioned limitations and restrictions of the existing methods, this paper attempts to create a restriction free method, suitable for capturing the connections between two time series.

We will further call this connection between two time series "co-movement". This relation will be said to exist between two time series if when within the first time series there is a jump which deviates the series from its historical variance, in the second time series appears a correction which determines the second variance to change in such a manner that a certain combination or ratio between the two time series' variances to remain inside a defined confidence interval. This combination or ratio between the two variances will be the point zero in our following reasoning.

In order to define this combination between two time series, we will use the concept of covariance, which in statistical terms is a measure of how much two time series tend to vary together. One of the most important properties of covariance is the fact that according to its formula, the covariance of two independent time series is zero. The covariance between to time series (X and Y), each of them consisting of n elements will be computed as follows:

cov(X,Y) = [n.summation over (i=1)] ([x.sub.i] - [bar.x]([y.sub.i] - [bar.y]/n (1)

where: cov(X,Y) = the covariance between time series X and time series Y;

[bar.x] = the mean of time series X;

[bar.y] = the mean of time series Y.

To resume all the above statements, we can say that for the co-movement relation to exist, the series of covariances (calculated between the time series X and Y, on a moving window basis) will have to maintain within a certain confidence interval.

Firstly, the maintenance of the covariance's series within a certain confidence interval implies the relative maintenance over time of a certain variance of the studied series, which leads to the concept of stationarity (Maddala, 1992). So, in the first place, for the co-movement relation to exist between two series, we will have to prove the stationarity of the covariance's series.

Secondly, in terms of distribution, for a sufficiently large number of observations, the covariance's distribution will have a shape resembling to the normal distribution (for a sufficiently large number of observations the Central Limit Theorem (CLT) applies). But the above mentioned condition that the covariance's series has to maintain within a certain confidence interval, leads to the conclusion that the normal resembling distribution will have very few extreme values (as there are very few or even no cases when the covariance's series outruns the mentioned confidence interval). The fact that there are very few extreme values, as well as the resemblance with the Gaussian distribution will lead to the conclusion that the resulted distribution will tend to agglomerate near its mean value (for a sufficiently long period of time the Law of Large Numbers (LLN) applies)--the phenomenon is graphically described in Fig.1--keeping the Gaussian shape, but altering the kurtosis parameter of a normal distribution, becoming, in fact, a normal shaped leptokurtic distribution.

So, in order to strengthen the demonstration, we will complete the stationarity request with a second request regarding the kurtosis of the covariance's series. This fourth standardized moment will tend to exceed the kurtosis measured on a normal distribution (which is proven to be equal to the value of three).

[FIGURE 1 OMITTED]

We will compute the kurtosis of the time series using the following formula (Newbold et. al., 2003):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

3. NUMERICAL RESULTS

Further, we will try to verify the veracity of the theoretical concepts presented above, by giving an example which to use two historically "connected" stock market time series. In order to have a situation which to be fitted for the Central Limit Theorem, as well as for the Law of Large Numbers, we have to consider a sufficiently large number of observations. Taking into consideration the two criteria mentioned above (large number of observations, as well as historically connected time series), we chose to analyze the "co-movement" of the Dow Jones Industrial Average Index (DJIA) and the S&P 500 Index of the American Stock Market. The analyzed period of time is between March 1989 and March 2009 (daily data). This leads to a number of about 5000 daily observations within the covariance time series.

First of all, we tested the stationarity of DJIA and S&P500, using the Augmented Dickey Fuller test (ADF). The null hypothesis of the ADF test is that the tested series has a unit root. The results are presented in Tab.1. The results indicate that DJIA does not have a unit root at any of the significance levels considered, whereas for S&P500 the null hypothesis is rejected at 5% and 10%, but accepted at 1%. In conclusion, one can state that the analyzed time series are stationary, at least for some periods of time.

In order to prove that the "co-movement concept" overcomes the time horizon shortcoming of the cointegration, we analyzed our time series both for short-term and for long- term time frames. Tab.2 presents the ADF test's results for different time frames, for a window from a week (5 days) to two years (500 days). By comparing the statistical values with the critical ones and by taking into consideration the probability that the null hypothesis to be true, we draw the conclusion that our covariance time series is a I(0) process. So, the first condition of our test is fulfilled.

Moving forward to the second condition, we will analyze the kurtosis of the covariance time series. For the different time frames taken into consideration, the results are also presented in Tab.2. We notice that the kurtosis maintains above the value of three (the normal distribution standard value for kurtosis), confirming our theory. So, the second condition imposed is now accomplished, too.

4. CONCLUSIONS

The main advantage of the method is that of not being bound to the determination of a mathematical relation between the time series. The "co-movement" relation that we described in this paper tries to capture not only long-term connections between two time series, but also short-term ones. The numerical results presented above tend to confirm the time invariance property of the "co-movement" procedure. The proposed method has the advantage to be an easily applicable process, which doesn't require many calculations or high theoretical difficulties. Not having a difficult mathematical theory makes the procedure very suitable to be used by economists. The main disadvantage is that for the method to be properly applied, the user would have to use a large number of observations (in order to meet the CLT and LLN requirements).

As for further research, we will try to check if our method would be fitted for high-frequency data. The reason for our further goal is that our method's main utilization will be within a trading system and it's a well known fact that a trading system is considered to be "robust" by specialists in the capital market area, when it is suitable for any time frame (from high frequency data--several seconds or 1 minute to weekly or even monthly data).

5. REFERENCES

Engle, R. F.; Granger C. W. J. (1987). Co-integration and Error Correction: Representation, Estimation, and Testing, Econometrica, Vol. 55, No.1, (March 1987), pg. 251-276, ISSN: 0012-9682

Gujarati, D. (2003). Basic Econometrics, McGraw Hill, ISBN 0071123423,9780071123426

Johansen, S. (1991). Estimation and Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive Models, Econometrica, Vol.59, No.6, (November 1991), pg. 1551-1580, ISSN: 0012-9682

Maddala, G.S. (1992). Introduction to Econometrics, MacMillan Publishing Company, ISBN 0-02-374545-2, New York

Newbold P., Carlson W., Thorne B. (2003). Statistics for Business and Economics, Prentice Hall, ISBN 0-13-048728-7, New Jersey Tab. 1. The ADF test for DJIA and S&P500 Symbol DJIA S&P500 ADF Statistical value -4,209036 -3,472886 Critical value at 1% level -3,959865 -3,959865 Critical value at 5% level -3,410699 -3,410699 Critical value at 10% level -3,127135 -3,127135 Probability 0,0043 0,0423 Tab. 2. The ADF test for different window lengths Nr of days ADF Stat. Critical value Probability Kurtosis val. at 1% level 5 -6,52 -3,96 0 95,04 10 -7,48 -3,96 0 160,5 15 -8,05 -3,96 0 132,95 20 -8,67 -3,96 0 109,51 45 -7,62 -3,96 0 66,84 60 -7,03 -3,96 0 59,93 90 -6,4 -3,96 0 48,91 125 -9,09 -3,96 0 31,69 250 -7,57 -3,96 0 33,87 375 -6,23 -3,96 0 26,34 500 -4,35 -3,96 0 13,54