Like many other researchers, I prefer to study the compound return which is defined as the difference in the logarithmic value of the bid prices. Table 2 shows the summary statistics of the tick-by-tick returns. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy.
In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. The remainder is organized as follows. Section 3 explains the statistical properties of returns and presents the initial findings. It also discusses the estimation results and compares the goodness-of-fit of the two conditional distributions. Section 4 summarizes the concluding remarks. The distribution becomes heavy tailed.
Following Curto et al. We follow the MLE procedure of conditional heteroskedasticity models with Stable distributions presented by Mittnik et al.
The holding periods are one-hour, five-hour, close to open hour , open to open hour , close to close hour and average to average hour changes. The logarithmic returns are calculated for each holding period and the summary statistics related to each period can be seen in Table 1.
Table 1 The values of kurtoses are above the normal value of 3. According to Jarque-Bera statistics, all variables indicate non-normal distributions. Figure 1 According to Figure 1, it can be said that stable distribution fits better than Normal distribution for both series.
The Stable GARCH models indicate that the shape parameters are significant and both are less than 2 based on t-tests indicating heavy tailed pattern, and the skewness parameters are positive and significant. They are preferable models based on the goodness of fits. These results are no surprise since the Stable models take into account the non-normal distribution of the time series estimated.
Conclusions: It is of great importance for those in charge of managing risk to understand how financial asset returns are distributed. Empirical evidence has led many practitioners to reject the normality assumption supporting various heavy-tailed alternatives and it is now commonly accepted that financial asset returns are, in fact, heavy-tailed.
Stable distributions are the probability distributions that allow skewness and heavy tails. Therefore, 3 These assumptions simplify the estimation, but will not alter the properties of the stable distribution.
See, for example, Curto et al. Recent studies show that stable distributions have been used for modeling stock returns, foreign exchange rate changes, commodity-price movements, and real estate returns McCulloch, The return series are gathered for the time period :April 1st May 13th The distributions of the returns are examined. The empirical results show that the foreign exchange changes do not follow the normal distribution and show heavy-tailed behaviors. Another important result is that, when the holding period increases the shape and skewness parameters increase for both periods.
On the whole, the estimated parameters suggest that the normality restriction is misleading and, thus, imposing the normality may cause a bias for financial modeling.
The stable distribution is relatively effective in capturing large changes in exchange rate movements, while the normal distribution screens out the outliers.
One-Hour Changes b. Stable Distributions. Curto, J. Fama, E. The behaviour of stock market prices. Journal of Business Some Properties of Symmetric Stable Distributions. Journal of the American Statistical Association Mandelbrot, B. Marinelli, C. Rachev, S.
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