| Summary: | 碩士 === 國立臺灣大學 === 統計碩士學位學程 === 104 === With the development of financial markets, market participants manage his or her own portfolio with a great diversity. Besides the risks, what investors concern the most is the asset returns. In the past decades, one of the widely-discussed topics of financial research is the probabilistic structure on asset returns. During mid-20 century, financial analysts and researchers found that all the past research based on Gaussian assumption is fallacious and noticed that this may underestimate the potential of financial risks. In spite of the well-known fact that asset returns are not normally distributed, some researchers and practitioners still maintain the normal inferences and henceforth ignore the information on asymmetry and heavy tail.
According to Mandelbrot (1963) and Fama (1965), stressing on the use of stable distributions, we would like to conduct the empirical study on financial index returns by means of comparisons with the inferences based on normal distribution and other heavy-tailed distributions, such as Laplace distribution and stable Paretian distribution. In this study, from Bloomberg, we collected the closing price of the last trading day (PX_1D_CLOSE) of 6 financial indices (TWSE, HSI, NKY, SPX, INDU and DEM/US) and transformed them into log returns. Then we carried out two-fold analyses: marginal and conditional perspective. In marginal aspect, the returns were fitted by univariate normal, Gumbel, Laplace and (asymmetric and symmetric) stable Paretian models and we compare the results by their own goodness of fits ; whilst in conditional part, reconsidering the temporal dependency into data, the returns constitute time series naturally and therefore we fit the log returns by combination of homoscedastic part (ARMA) and heteroscedastic part (GARCH).
According to finance literature, we fit ARMA(1,1)-GARCH(1,1) with normal, Laplace and stable innovations, which can be compared with goodness of fit. Above all the methods, the statistical inferences based on asymmetric stable Paretian distribution are usually better than the ones based on normality.
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