One of the important issues in finance is an evaluation of a stochastic memory. Given a stochastic memory in an asset return, it is possible to obtain increased profit on the basis of price-change predictions that contradict the efficient market hypothesis and deny the random walk process on financial returns. Along with this aspect, a long memory existence for hedge ratios of financial instrumentals may have some implication for effectively managing risks. In such a case, it may decrease a reliance on the impact of statrup's financing risk.

Numerous empirical studies for this theme have tested the random walk hypothesis(RWH) for various stock and futures assets. For instance, Grammatikos and Saunders(1983) examine the stability of hedge ratios. Authors of the paper find that the hedge ratios for five major foreign currency futures are unstable over time. With commodity futures assets such as live cattle and corn, McNew and Fackler(1994) support the constant hedge ratios hypothesis. Ferguson and Leistikow(1998) strongly reject the random walk hypothesis, using the OLS regression and the Dickey-Fuller test for currency markets. Such results do mean that time varying dynamic approaches for forecasting hedge ratios is not effective.

On the other hand, Malliaris and Urrutia(1991) address an evidence that the futures hedge ratios do follow a random walk for currencies and equities introduced Standard and Poor’s 500 Index, New York Stock Exchange Index, Deutsche mark, British pound, Japanese yen, and Swiss franc. Perfect and Wiles(1994) also support the finding of Malliaris and Urrutia(1991) as well. The findings imply that hedgers cannot consistently fit on an optimal hedge and so dynamic hedging techniques must be considered. Most of the previous literatures on the RWH for hedge ratios of assets focus on currency markets or commodities, while literatures which investigate hedge ratios of stock markets, if any, are few. This paper tries to contribute to current literatures by examining the RWH of hedge ratios on stock and futures markets. For the end, in this paper, hedge ratios are estimated by time varying methods and the long memory tests for the estimated ratios are conducted through the R/S analysis and GPH Spectral Regression.

This paper collects the stock and futures data from Korea Composite Stock Price Index 200(KOSPI200) and Standard and Poor’s 500(S&P500). Here, the KOSPI200 represents the emerging market and the S&P500 is the most famous stock index in the world. Especially, the Korean futures market has experienced rapid and explosive growth since the opening of the KOSPI200 index on 3rd May 1996. According to the annual report(2006) from World Federation of Exchanges, the notional value for KOSPI200 futures contract is 2,983 billions US dollars in 2005, which is ranked to the 1st position in the Asia-pacific exchanges, except Chicago Mercantile Exchange(CME).

The remainder of the paper is organized as follows. Section 2 describes the primary methodologies for the study. Data and the results of stationary test are presented in section 3. Section 4 is devoted to the empirical results. Finally, section 5 briefly concludes.

Following a unit root test for stock and futures of KOSPI200 and S&P500, this paper conducts a co-integration test on each series. If stock and futures are co-integrated, the paper employs the Vector Error Correction Model(VECM) as the conditional mean equations of the model, shown as equation (1). In the VECM model, ϵ_t = [ϵ_st, ϵ_ft] represents an innovation vector from the stock and futures models respectively and S_t-1-θF_t-1 represents an error correction term.

Engle and Kroner(1995) propose the Bollerslev, Engle, Kroner, and Kraft(BEKK) parameterization. Normally the coefficients of BEKK are needed the exclusively 11 parameters in the conditional variance-covariance structure and H_t is required to be positive definite. The BEKK model can be presented as an eq (2). An eq (3) and an eq (4) are the coefficients matrix for BEKK and diagonal-BEKK respectively. Noticeably, in the diagonal-BEKK model, the non-diagonal elements of A and B are zero.

Normally, a stationary time series has correlation that depends only on the time lag k between time t and t-k observations, and decays to zero as k increases. The stochastic “memory” in the random process is basically measured through a decaying speed. It is said that a process with all observations uncorrelated is a white noise, and the process is the random walk without any long memory. In reference, a short memory process has autocorrelations decaying to zero at a geometric rate whereas a long memory process holds autocorrelations that decay much more slowly, asymptotically following a hyperbolic decay(Crato and Ray, 2000).

Several tests and statistics have been suggested that detect the existence of long memory in each of series. Among them, Geweke and Porter-Hudak(1983)(henceforth GPH) and R/S statistics modified by Lo(1991) are the most famous methods.

Thus this paper uses the two methods, namely, GPH and R/S for determining the existence of long-range dependent process. Firstly, the R/S statistic, then is defined as R_T/S_T where R_T is computed as R_T=max_1<k<T-min_1<j<T and S_T is the standard deviation for sample data. For short memory processes, the value of R_T/S_T converges to n^J with J=1/2. The parameter J is called the Hurst exponent and is related to the long memory parameter d by J=d+1/2

Mandelbrot and Taqqu(1979) prove that a process has long memory when J>1/2. A natural estimate for a series of length n is simply =log(R_T/S_T)/log_n.

The second method for capturing an existence of the long-memory is the GPH spectral regression by Geweke and Porter-Hudak(1983). According to Crato and Ray(2000), “The regression is performed using a set of Fourier frequencies close to zero, where the slope of log spectrum relative to the frequency is dependent directly on the long-memory parameter d. The regression estimator could capture the long memory characteristic of the process without being contaminated in the estimation by short memory correlation in the time series evident at higher frequencies”(Crato and Ray, 2000).

The data sets employed in this study comprise 525 weekly observations on the KOSPI200 and 1141 weekly observations on the S&P500, containing stock index and stock index futures contract. The time periods under study extend from May 3rd, 1996 through December 29th, 2005 for the KOSPI200 and from April 22th, 1982 through February 26th, 2004 for the S&P500. Coakley, Dollery, and Kellard(2008) employ a data set 1995-2005 including a stock index and commodities foreign exchange, and suggested the S&P 500 to be a fractionally integrated process. The prices used is Thursday closing ones(Wednesday and Friday closing prices are alternatively used, given a Thursday closing prices missed). More importantly, to avoid thin trading and expiration effects, this paper uses the nearest contracts, rolling over to next nearest contract prior to expiration month of the current contract.

Let S_t and F_t represent the logarithms of the stock index and stock index futures prices. The daily return on a spot position held from t-1 to t are calculated as R_st=(S_t-S_t-1)×100. Similarly, the actual return on a futures position is as R_ft=(F_t-F_t-1)×100.

A Summary statistics for the data is indicated in table 1. Noticeably, four series show excess kurtosis, implying fatter tails than a normal distribution. This result is backed by the Jarque-Bera statistics. The values of Ljung-Box(hereafter LB) for the return series of KOSPI200 stock and futures are significant at the 1% level. The LB(10)s for squared return series of KOSPI200 and S&P500 are highly significant equally, suggesting the possibility of the presence of autoregressive conditional heteroskedasticity.