In this paper, we study the dual long memory property of the Korean stock market. For this purpose, the ARFIMA–FIGARCH model is applied to two daily Korean stock price indices (KOSPI and KOSDAQ). Our empirical results indicate that long memory dynamics in the returns and volatility can be adequately estimated by the joint ARFIMA–FIGARCH model. We also found that the assumption of a skewed Student-tt distribution is better for incorporating the tendency of asymmetric leptokurtosis in a return distribution.
Long memory dynamics are important indicators for determining non-linear dependence in the conditional mean and variance of financial time series. Unlike the knife-edge distinction between stationary process I(0)I(0) of autoregressive moving average (ARMA) model and non-stationary process I(1)I(1) of autoregressive integrated moving average (ARIMA) model, the fractionally integrated process I(d)I(d) or ARFIMA model is characterized by the autocorrelation function which decays at a hyperbolic rate [1], [2] and [3].
The prior literature on long memory in the conditional mean and variance had evolved independently, as the phenomena appear distinct [4], [5] and [6]. However, long memory phenomena are often observed in both the conditional mean and variance at the same time. Based on this idea, the empirical studies have focused on the dual long memory property in the conditional mean and variance [7], [8] and [9].
Many statistical approaches have identified the probability form of financial distributions [10], [11], [12], [13] and [14]. The distributional properties of financial data returns are usually characterized by leptokurtic Lévy distribution [15] and [16], power-law stability [17] and [18], scaling law [10], [12], [19], [20], [21], [22], [23], [24] and [25]. More recent work [26] and [27] has employed ARCH–GARCH models with the assumption of non-Gaussian distribution innovations for measuring and controlling financial risks.
The primary focus of this paper is to investigate the dual long memory property in the returns and volatility of Korean stock market using the ARFIMA–FIGARCH model. The ARFIMA–FIGARCH model can provide a useful way of analyzing the relationship between the conditional mean and conditional variance of a process exhibiting the long memory property, simultaneously. Additionally, this paper also considers the distributional properties of stock returns using the normal and skewed Student-tt distributions. Because the residuals suffer from excess kurtosis and skewness, implying the assumption of normal distribution is not suitable for capturing asymmetry and tail-fatness in a return distribution.
The rest of this paper is organized as follows. Section 2 discusses the ARFIMA–FIGARCH model and estimated densities, i.e. normal and skewed Student-t distribution innovations. Section 3 provides the statistical characteristics of sample data, and the estimation results of the ARFIMA model, FIGARCH model, and the ARFIMA–FIGARCH model. The final section, Section 4, will contain some concluding remarks.
This article has examined the dual long memory property in the Korean stock market. For this purpose, we applied the joint ARFIMA–FIGARCH model to the returns and volatility of the Korean stock market.
The ARFIMA–FIGARCH model provides the robustness of long memory test results, in contrast to the ARFIMA model or the FIGARCH model. The dual long memory model can provide a better explanation for long memory dynamics in both the conditional mean and variance. In addition, it is apparent that the assumption of skewed Student-t distribution captures the asymmetry and tail-fatness in the KOSPI and KOSDAQ returns.
