Understanding the source of multifractality in financial markets

In this paper, we use the generalized Hurst exponent approach to study the multi- scaling behavior of different financial time series. We show that this approach is robust and powerful in detecting different types of multiscaling. We observe a puzzling phenomenon where an apparent increase in multifractality is measured in time series generated from shuffled returns, where all time-correlations are destroyed, while the return distributions are conserved. This effect is robust and it is reproduced in several real financial data including stock market indices, exchange rates and interest rates. In order to understand the origin of this effect we investigate different simulated time series by means of the Markov switching multifractal (MSM) model, autoregressive fractionally integrated moving average (ARFIMA) processes with stable innovations, fractional Brownian motion and Levy flights. Overall we conclude that the multifractality observed in financial time series is mainly a consequence of the characteristic fat-tailed distribution of the returns and time-correlations have the effect to decrease the measured multifractality

"A Generalized SSAR Model and Predictive Distribution with an Application to VaR"

The asymmetrical movements between the downward and upward phases of the sample paths of time series have been sometimes observed. By generalizing the SSAR (simultaneous switching autoregressive) models, we introduce a class of nonlinear time series models having the asymmetrical sample paths in the upward and downward phases. We show that the class of generalized SSAR models is useful for estimating the asymmetrical predictive distribution given the present and past information. Applications to the prediction based on the predictive median and the estimation of the VaR (value at risk) in financial risk management are discussed.

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.

Statistical inference for Poisson time series models

There are many nonlinear econometric models which are useful in analysis of financial time series. In this thesis, we consider two kinds of nonlinear autoregressive models for nonnegative integer-valued time series: threshold autoregressive models and Markov switching models, in which the conditional distribution given historical information is the Poisson distribution. The link between the conditional variance (i.e. the conditional mean for the Poisson distribution) and its past values as well as the observed values of the Poisson process may be different according to the threshold variable in threshold autoregressive models, and to an unobservable state variable in Markov switching models in different regimes. We give a condition on parameters under which the Poisson generalized threshold autoregressive heteroscedastic (PTGARCH) process can be approximated by a geometrically ergodic process. Under this condition, we discuss statistical inference (estimation and tests) for PTGARCH models, and give the asymptotic theory on the inference. The complete structure of the threshold autoregressive model is not exactly specific in economic theory for the most financial applications of the model. In particular, the number of regimes, the value of threshold and the delay parameter are often unknown and cannot be assumed known. Therefore, in this research, the performance of various information criteria for choosing the number of regimes, the threshold value and the delay parameters for different sample sizes is investigated. Tests for threshold nonlinearity are applied. The characteristics of Markovian switching Poisson generalized autoregressive hetero-scedastic (MS-PGARCH) models are given, and the maximum likelihood estimation of parameters is discussed. Simulation studies and applications to modelling financial counting time series are presented to support our methodology for both the PTGARCH model and the MS-PGARCH model.There are many nonlinear econometric models which are useful in analysis of financial time series. In this thesis, we consider two kinds of nonlinear autoregressive models for nonnegative integer-valued time series: threshold autoregressive models and Markov switching models, in which the conditional distribution given historical information is the Poisson distribution. The link between the conditional variance (i.e. the conditional mean for the Poisson distribution) and its past values as well as the observed values of the Poisson process may be different according to the threshold variable in threshold autoregressive models, and to an unobservable state variable in Markov switching models in different regimes. We give a condition on parameters under which the Poisson generalized threshold autoregressive heteroscedastic (PTGARCH) process can be approximated by a geometrically ergodic process. Under this condition, we discuss statistical inference (estimation and tests) for PTGARCH models, and give the asymptotic theory on the inference. The complete structure of the threshold autoregressive model is not exactly specific in economic theory for the most financial applications of the model. In particular, the number of regimes, the value of threshold and the delay parameter are often unknown and cannot be assumed known. Therefore, in this research, the performance of various information criteria for choosing the number of regimes, the threshold value and the delay parameters for different sample sizes is investigated. Tests for threshold nonlinearity are applied. The characteristics of Markovian switching Poisson generalized autoregressive hetero-scedastic (MS-PGARCH) models are given, and the maximum likelihood estimation of parameters is discussed. Simulation studies and applications to modelling financial counting time series are presented to support our methodology for both the PTGARCH model and the MS-PGARCH model

The Markov-Switching Multifractal Model of asset returns: GMM estimation and linear forecasting of volatility

Multifractal processes have recently been proposed as a new formalism for modelling the time series of returns in insurance. The major attraction of these processes is their ability to generate various degrees of long memory in different powers of returns - a feature that has been found in virtually all financial data. Initial difficulties stemming from non-stationarity and the combinatorial nature of the original model have been overcome by the introduction of an iterative Markov-switching multifractal model in Calvet and Fisher (2001) which allows for estimation of its parameters via maximum likelihood and Bayesian forecasting of volatility. However, applicability of MLE is restricted to cases with a discrete distribution of volatility components. From a practical point of view, ML also becomes computationally unfeasible for large numbers of components even if they are drawn from a discrete distribution. Here we propose an alternative GMM estimator together with linear forecasts which in principle is applicable for any continuous distribution with any number of volatility components. Monte Carlo studies show that GMM performs reasonably well for the popular Binomial and Lognormal models and that the loss incurred with linear compared to optimal forecasts is small. Extending the number of volatility components beyond what is feasible with MLE leads to gains in forecasting accuracy for some time series. --Markov-switching,Multifractal,Forecasting,Volatility,GMM estimation

The Markov-switching multi-fractal model of asset returns: GMM estimation and linear forecasting of volatility

Multi-fractal processes have recently been proposed as a new formalism for modelling the time series of returns in finance. The major attraction of these processes is their ability to generate various degrees of long memory in different powers of returns - a feature that has been found in virtually all financial data. Initial difficulties stemming from non-stationarity and the combinatorial nature of the original model have been overcome by the introduction of an iterative Markov-switching multi-fractal model in Calvet and Fisher (2001) which allows for estimation of its parameters via maximum likelihood and Bayesian forecasting of volatility. However, applicability of MLE is restricted to cases with a discrete distribution of volatility components. From a practical point of view, ML also becomes computationally unfeasible for large numbers of components even if they are drawn from a discrete distribution. Here we propose an alternative GMM estimator together with linear forecasts which in principle is applicable for any continuous distribution with any number of volatility components. Monte Carlo studies show that GMM performs reasonably well for the popular Binomial and Lognormal models and that the loss incured with linear compared to optimal forecasts is small. Extending the number of volatility components beyond what is feasible with MLE leads to gains in forecasting accuracy for some time series. --Markov-switching,Multifractal,Forecasting,Volatility,GMM estimation

Risk management and extreme scenario development using multiple regime switching approaches : a thesis presented in partial fulfilment of the requirements for the degree of Master of Business Studies in Finance at Massey University

Over the last twenty-five years, there have been an increasingly large number of extreme events in the financial markets. This includes market crashes and natural disasters that have led to extremely large losses and claims. Extreme event risk affects all aspects of risk assessment modeling and management. Traditional risk measurement methods focus on probability of laws governing average of sums, and do not focus on the tails of distribution. The investigation concerns the characterization and development of extreme markets scenarios for use in risk measurement and capital adequacy determination frameworks. The first part of the investigation concerns the development of event timelines that can be used for characterizing whether a period of time should be considered normal or extreme market conditions or regimes. The time lines have allowed the identification of the different times when the markets were calm and when the markets were turbulent. They assist in building scenarios, and also to identify the scenarios for decomposition of data to model the different regions, either the tail or the center of the distribution using the mentioned regime switching models. The information from the event time line can be used to define scenarios in a stress testing context. In this investigation, extreme value analysis, which is an extension of the standard VaR techniques, useful in measuring extreme events has been used, which fits density functions by placing more weights in the tails than the normal Gaussian distribution and model the upper and lower tail of an underlying distribution. Extreme value distribution functions including "fat tailed" will be fitted to the tails of critical market factors to model the extreme market events that are not given appropriate probability of occurrence under normal conditions. The Hill estimator, which is recognized as the consistent estimator for empirical analysis is used for calculating the tail index parameter for EVT modeling, However, it has to be noted that the Hill estimator is efficient when the underlying distribution is fat-tailed as compared to the gaussian, where the tail index estimates tend to go to infinity. The performance of Extreme value theory estimation technique with multiple regimes on real and simulated financial time series for efficient results, compared to the standard VaR techniques has been studied. In this investigation, multiple regime switching approach has been used to identify regimes and measure risk accordingly. It is assumed that the center of the returns distribution is normally distributed with 90 percent of the data in the in the center region and each tail contains 5 percent of the data. Three regime switching models have been used in this analysis which includes, the Unconditional LT-C-RT (Left tail - Center - Right Tail) transfers, the 3 State Regime Markov Transition Model and the Geometric Time in Trail Model. The regime switching models are modeled using the following procedures: 1) The Unconditional LT-C-RT (Left Tail - Center - Right Tail) model is an IID model (Independent and Identically Distributed) model and has a simple Bernoulli approach where the market is in a normal state with probability P or an abnormal state with probability 1 - p . The transition between states is independent of the last state. 2) A Markov chain approach where the next state of the market is a function of the current state. That there are the following transitions possible: 2.1) Normal to normal 2.2) Normal to abnormal 2.3) Abnormal to abnormal 2.4) Abnormal to normal 3). The Geometric Time in Tail model is a hybrid Bernoulli approach where the markets stays in a given state based on a duration model and when the duration in a given states has expired, the sampling of the next state using a independent Bernoulli approach similar to approach one. This implies that the after the market has stayed in a given regime for the sample duration time, it can stay in the current regime with probability p or leave the regime with probability 1 - p. The sample duration can be based on the exponential distribution for continuous time and the geometric distribution for discrete time such as daily movements. Tail index estimation results using EVT indicate the presence of fat tails in equity data and the results of Value-at-Risk (VaR) and Expected Shortfall (ES) are considerably similar for the three regime switching models. The comparison of results from the multiple regime switching models to the one region distribution results, which serve as the base case prove the efficiency of using this approach for a better risk measure

Penalized likelihood based tests for regime switching in autoregressive models

In this thesis, we are mainly concerned with the basic methodological issue to test for regime switching in various Markov-switching autoregressive models. To this end, we develop some penalized likelihood based tests which neglect the dependence structure in the latent process. We derive the asymptotic distribution of the corresponding test statistics under the hypothesis. Finally, we apply our methods to financial and macroeconomic time series

Heterogeneous Agent Models in Economics and Finance, In: Handbook of Computational Economics II: Agent-Based Computational Economics, edited by Leigh Tesfatsion and Ken Judd , Elsevier, Amsterdam 2006, pp.1109-1186.

This chapter surveys work on dynamic heterogeneous agent models (HAMs) in economics and finance. Emphasis is given to simple models that, at least to some extent, are tractable by analytic methods in combination with computational tools. Most of these models are behavioral models with boundedly rational agents using different heuristics or rule of thumb strategies that may not be perfect, but perform reasonably well. Typically these models are highly nonlinear, e.g. due to evolutionary switching between strategies, and exhibit a wide range of dynamical behavior ranging from a unique stable steady state to complex, chaotic dynamics. Aggregation of simple interactions at the micro level may generate sophisticated structure at the macro level. Simple HAMs can explain important observed stylized facts in financial time series, such as excess volatility, high trading volume, temporary bubbles and trend following, sudden crashes and mean reversion, clustered volatility and fat tails in the returns distribution.