Determining and Comparing Multivariate Distributions: An Application to AORD and GSPC with their related financial markets

Abstract

Many real world applications are associated with more than one variable and hence, identifying multivariate distributions associated with real world problems portrays great importance today. Many studies can be found in the literature in this aspect and most of them are associated with two variables/dimensions and the maximum dimension of multivariate distribution found in the literature is four. Different optimization techniques have been used by researchers to find multivariate distributions in their studies. Numerical methods can be identified as more preferable than analytical methods when the dimension of the problem is high. The main objective of this study is to identify the multivariate distribution associated with the return series of Australian all ordinary index (AORD) and those of the related financial markets and compare it with the multivariate distribution of return series of the US GSPC index and its related financial markets. No research were found in the literature which were aimed at finding aforesaid multivariate distribution and comparisons. Moreover no evidence found for identifying a multivariate distribution with six dimensions. Five financial markets: Amex oil index, Amex gold index, world cocoa index, exchange rate of Australian dollar to United States dollar and US GSPC index were found to be associated with AORD. Hence the attempt was to derive the multivariate distribution of return series of AORD and these five return series and therefore the optimization problem of the study is a six dimension problem which associated with forty three parameters need to be estimated. A local optimization technique and a global optimization technique were used to estimate the parameters of the multivariate distribution. Results exhibit that the parameter estimates obtained from the global optimization technique are better than the parameter estimates obtained from the local optimization technique. The multivariate distribution of return series of AORD and related financial markets is central, less peaked and have fat tails. A comparison was done with another multivariate distribution of a return series of a leading stock market index: GSPC and return series of its associated financial markets and found that both distributions are alike in shape. Two periods were identified in the AORD series and found that the shape of the multivariate distribution of one period is similar to the shape of the multivariate distribution of full data set while the shape of the multivariate distribution of the other period is dissimilar to that of full data set.

Techniques to produce and evaluate realistic multivariate synthetic data

“…This decoupling is similar to the objective of Copula modeling that follows from Sklar’s work 39 , 40 . Copula modeling allows specification of marginal pdfs and the correlation structure independently 41 ; in this approach, the marginals must be specified accurately and finding analytical solutions for d > 4 is difficult 42 . In contrast with Copula modeling, which is flexible, the covariance (or correlation) structure in our approach is fixed by the normal form and empirically derived; the marginals were forced to normality rather than specified.…”