The Applications of Mixtures of Normal Distributions in Empirical Finance: A Selected Survey

This paper provides a selected review of the recent developments and applications of mixtures of normal (MN) distribution models in empirical finance. Once attractive property of the MN model is that it is flexible enough to accommodate various shapes of continuous distributions, and able to capture leptokurtic, skewed and multimodal characteristics of financial time series data. In addition, the MN-based analysis fits well with the related regime-switching literature. The survey is conducted under two broad themes: (1) minimum-distance estimation methods, and (2) financial modeling and its applications.Mixtures of Normal, Maximum Likelihood, Moment Generating Function, Characteristic Function, Switching Regression Model, (G) ARCH Model, Stochastic Volatility Model, Autoregressive Conditional Duration Model, Stochastic Duration Model, Value at Risk.

A Threshold Stochastic Volatility Model with Realized Volatility

Rapid development in the computer technology has made the financial transaction data visible at an ultimate limit level. The realized volatility, as a proxy for the "true" volatility, can be constructed using the high frequency data. This paper extends a threshold stochastic volatility specification proposed in So, Li and Lam (2002) by incorporating the high frequency volatility measures. Due to the availability of the volatility time series, the parameters estimation can be easily implemented via the standard maximum likelihood estimation (MLE) rather than using the simulated Bayesian methods. In the Monte Carlo section, several mis-specification and sensitivity experiments are conducted. The proposed methodology shows good performance according to the Monte Carlo results. In the empirical study, three stock indices are examined under the threshold stochastic volatility structure. Empirical results show that in different regimes, the returns and volatilities exhibit asymmetric behavior. In addition, this paper allows the threshold in the model to be flexible and uses a sequential optimization based on MLE to search for the "optimal" threshold value. We find that the model with a flexible threshold is always preferred to the model with a fixed threshold according to the log-likelihood measure. Interestingly, the "optimal" threshold is found to be stable across different sampling realized volatility measures.

Continuous Empirical Characteristic Function Estimation of Mixtures of Normal Parameters

This paper develops an e±cient method for estimating the discrete mix- tures of normal family based on the continuous empirical characteristic function (CECF). An iterated estimation procedure based on the closed form objective distance function is proposed to improve the estimation effciency. The results from the Monte Carlo simulation reveal that the CECF estimator produces good finite sample properties. In particular, it outperforms the discrete type of methods when the maximum likelihood estimation fails to converge. An empirical example is provided for illustrative purposes.Empirical characteristic function; Mixtures of normal.

Empirical Evidence of the Leverage Effect in a Stochastic Volatility Model: A Realized Volatility Approach

Increasing attention has been focused on the analysis of the realized volatility, which can be treated as a proxy for the true volatility. In this paper, we study the potential use of the realized volatility as a proxy in a stochastic volatility model estimation. We estimate the leveraged stochastic volatility model using the realized volatility computed from five popular methods across six sampling-frequency transaction data (from 1-min to 60-min). Availability of the realized volatility allows us to estimate the model parameters via the MLE and thus avoids computational challenge in the high dimensional integration.Six stock indices are considered in the empirical investigation. We discover some consistent findings and interesting patterns from the empirical results. In general, the significant leverage effect is consistently detected at each sampling frequency. The volatility persistence becomes weaker at the lower sampling frequency. We also find that the consistent-scaling and "optimal"-weighted realized volatility method proposed by Hansen and Lunde (2005) provide relatively better performances compared to other methods considered.

An Empirical Characteristic Function Approach to VaR under a Mixture of Normal Distribution with Time-Varying Volatility

This paper considers Value at Risk measures constructed under a discrete mixture of normal distribution on the innovations with time-varying volatility, or MN-GARCH, model. We adopt an approach based on the continuous empirical characteristic function to estimate the param eters of the model using several daily foreign exchange rates' return data. This approach has several advantages as a method for estimating the MN-GARCH model. In particular, under certain weighting measures, a closed form objective distance function for estimation is obtained. This reduces the computational burden considerably. In addition, the characteristic function, unlike its likelihood function counterpart, is always uniformly bounded over parameter space due to the Fourier transformation. To evaluate the VaR estimates obtained from alternative specifications, we construct several measures, such as the number of violations, the average size of violations, the sum square of violations and the expected size of violations. Based on these measures, we find that the VaR measures obtained from the MN-GARCH model outperform those obtained from other competing models.Value at Risk; Mixture of Normals; GARCH; Characteristic Function.

Modeling Asymmetric Volatility Clusters Using Copulas and High Frequency Data

Volatility clustering is a well-known stylized feature of financial asset returns. In this paper, we investigate the asymmetric pattern of volatility clustering on both the stock and foreign exchange rate markets. To this end, we employ copula-based semi-parametric univariate time-series models that accommodate the clusters of both large and small volatilities in the analysis. Using daily realized volatilities of the individual company stocks, stock indices and foreign exchange rates constructed from high frequency data, we find that volatility clustering is strongly asymmetric in the sense that clusters of large volatilities tend to be much stronger than those of small volatilities. In addition, the asymmetric pattern of volatility clusters continues to be visible even when the clusters are allowed to be changing over time, and the volatility clusters themselves remain persistent even after forty days.Volatility clustering, Copulas, Realized volatility, High-frequency data.

Asymmetric Stochastic Conditional Duration Model --A Mixture of Normals Approach"

This paper extends the stochastic conditional duration model by imposing mixtures of bivariate normal distributions on the innovations of the observation and latent equations of the duration process. This extension allows the model not only to capture the asymmetric behavior of the expected duration but also to easily accommodate a richer dependence structure between the two innovations. In addition, it proposes a novel estimation methodology based on the empirical characteristic function. A set of Monte Carlo experiments as well as empirical applications based on the IBM and Boeing transaction data are provided to assess and illustrate the performance of the proposed model and the estimation method. One main empirical finding in this paper is that there is a signicantly positive "leverage effect" under both the contemporaneous and lagged inter-temporal de pendence structures for the IBM and Boeing duration data.Stochastic Conditional Duration model; Leverage Effect; Discrete Mixtures of Normal; Empirical Characteristic Function

Comparative Analysis of mRNA Isoform Expression in Cardiac Hypertrophy and Development Reveals Multiple Post-Transcriptional Regulatory Modules

Cardiac hypertrophy is enlargement of the heart in response to physiological or pathological stimuli, chiefly involving growth of myocytes in size rather than in number. Previous studies have shown that the expression pattern of a group of genes in hypertrophied heart induced by pressure overload resembles that at the embryonic stage of heart development, a phenomenon known as activation of the “fetal gene program”. Here, using a genome-wide approach we systematically defined genes and pathways regulated in short- and long-term cardiac hypertrophy conditions using mice with transverse aortic constriction (TAC), and compared them with those regulated at different stages of embryonic and postnatal development. In addition, exon-level analysis revealed widespread mRNA isoform changes during cardiac hypertrophy resulting from alternative usage of terminal or internal exons, some of which are also developmentally regulated and may be attributable to decreased expression of Fox-1 protein in cardiac hypertrophy. Genes with functions in certain pathways, such as cell adhesion and cell morphology, are more likely to be regulated by alternative splicing. Moreover, we found 3′UTRs of mRNAs were generally shortened through alternative cleavage and polyadenylation in hypertrophy, and microRNA target genes were generally de-repressed, suggesting coordinated mechanisms to increase mRNA stability and protein production during hypertrophy. Taken together, our results comprehensively delineated gene and mRNA isoform regulation events in cardiac hypertrophy and revealed their relations to those in development, and suggested that modulation of mRNA isoform expression plays an importance role in heart remodeling under pressure overload