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

Rational forecasts or social opinion dynamics? Identification of interaction effects in a business climate survey

This paper develops a methodology for estimating the parameters of dynamic opinion or expectation formation processes with social interactions. We study a simple stochastic framework of a collective process of opinion formation by a group of agents who face a binary decision problem. The aggregate dynamics of the individuals' decisions can be analyzed via the stochastic process governing the ensemble average of choices. Numerical approximations to the transient density for this ensemble average allow the evaluation of the likelihood function on the base of discrete observations of the social dynamics. This approach can be used to estimate the parameters of the opinion formation process from aggregate data on its average realization. Our application to a well-known business climate index provides strong indication of social interaction as an important element in respondents' assessment of the business climate. --Business climate,Business cycle forecasts,Opinion formation,Social interactions

The multi-fractal model of asset returns : its estimation via GMM and its use for volatility forecasting

Multi-fractal processes have been proposed as a new formalism for modeling 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 to characterize virtually all financial prices. Furthermore, elementary variants of multi-fractal models are very parsimonious formalizations as they are essentially one-parameter families of stochastic processes. The aim of this paper is to provide the characteristics of a causal multi-fractal model (replacing the earlier combinatorial approaches discussed in the literature), to estimate the parameters of this model and to use these estimates in forecasting financial volatility. We use the auto-covariances of log increments of the multi-fractal process in order to estimate its parameters consistently via GMM (Generalized Method of Moment). Simulations show that this approach leads to essentially unbiased estimates, which also have much smaller root mean squared errors than those obtained from the traditional ?scaling? approach. Our empirical estimates are used in out-of-sample forecasting of volatility for a number of important financial assets. Comparing the multi-fractal forecasts with those derived from GARCH and FIGARCH models yields results in favor of the new model: multi-fractal forecasts dominate all other forecasts in one out of four cases considered, while in the remaining cases they are head to head with one or more of their competitors. --multi-fractality , financial volatility , forecasting

Detecting multi-fractal properties in asset returns : the failure of the scaling estimator

It has become popular recently to apply the multifractal formalism of statistical physics (scaling analysis of structure functions and f(a) singularity spectrum analysis) to financial data. The outcome of such studies is a nonlinear shape of the structure function and a nontrivial behavior of the spectrum. Eventually, this literature has moved from basic data analysis to estimation of particular variants of multi-fractal models for asset returns via fitting of the empirical t(q) and f(a) functions. Here, we reinvestigate earlier claims of multi-fractality using four long time series of important financial markets. Taking the recently proposed multi-fractal models of asset returns as our starting point, we show that the typical ?scaling estimators? used in the physics literature are unable to distinguish between spurious and ?real? multi-scaling of financial data. Designing explicit tests for multi-scaling, we can in no case reject the null hypothesis that the apparent curvature of both the scaling function and the Hölder spectrum are spuriously generated by the particular fattailed distribution of innovations characterizing financial data. Given the well-known overwhelming evidence in favor of different degrees of long-term dependence in the powers of returns, we interpret this inability to reject the null hypothesis of multi-scaling as a lack of discriminatory power of the standard approach rather than as a true rejection of multi-scaling in financial data. However, the complete ?failure? of the multi-fractal apparatus in this setting also raises the question whether results in other areas (like geophysics) suffer from similar short-comings of the traditional methodology. --

Financial power laws: Empirical evidence, models, and mechanism

Financial markets (share markets, foreign exchange markets and others) are all characterized by a number of universal power laws. The most prominent example is the ubiquitous finding of a robust, approximately cubic power law characterizing the distribution of large returns. A similarly robust feature is long-range dependence in volatility (i.e., hyperbolic decline of its autocorrelation function). The recent literature adds temporal scaling of trading volume and multi-scaling of higher moments of returns. Increasing awareness of these properties has recently spurred attempts at theoretical explanations of the emergence of these key characteristics form the market process. In principle, different types of dynamic processes could be responsible for these power-laws. Examples to be found in the economics literature include multiplicative stochastic processes as well as dynamic processes with multiple equilibria. Though both types of dynamics are characterized by intermittent behavior which occasionally generates large bursts of activity, they can be based on fundamentally different perceptions of the trading process. The present chapter reviews both the analytical background of the power laws emerging from the above data generating mechanism as well as pertinent models proposed in the economics literature. --

A model of the topology of the bank – firm credit network and its role as channel of contagion

This paper proposes a stochastic model of a bipartite credit network between banks and the non-bank corporate sector that encapsulates basic stylized facts found in comprehensive data sets for bank-firm loans for a number of countries. When performing computational experiments with this model, we find that it shows a pronounced non-linear behavior under shocks: the default of a single unit will mostly have practically no knock-on effects, but might lead to an almost full-scale collapse of the entire system in a certain number of cases. The dependency of the overall outcome on firm characteristics like size or number of loans seems fuzzy. Distinguishing between contagion due to interbank credit and due to joint exposures to counterparty risk via loans to firms, the later channel appears more important for contagious spread of defaults.The research reported in this paper has been initiated during a visit at the Financial Stability Assessment Division of the European Central Bank during September 2013. The author is extremely grateful for the hospitality of the Division as well as very stimulating discussions with Christoffer Kok, Grzegorz Halaj, Mattia Montagna, Michael Wellman, Reiner Franke, Philipp Kolberg and Shu-Heng Chen. Helpful comments from the audience of various seminar and workshop presentations are also gratefully acknowledged. I am also very thankful for the research assistance of Ricardo Giglio. The paper has also gained from the comments and suggestions of two anonymous reviewers. This report is part of a research initiative found by the Leibniz Community. The research leading to these results has also received funding from the Spanish Ministry of Science and Innovation (ECO2011-23634), from Universitat Jaume I (P1.1B2012-27) and from the European Union Seventh Framework Programme (FP7/2007-2013) under Grant agreement no. 612955

A noise trader model as a generator of apparent financial power laws and long memory

In various agent-based models the stylized facts of financial markets (unit-roots, fat tails and volatility clustering) have been shown to emerge from the interactions of agents. However, the complexity of these models often limits their analytical accessibility. In this paper we show that even a very simple model of a financial market with heterogeneous interacting agents is capable of reproducing these ubiquitous statistical properties. The simplicity of our approach permits to derive some analytical insights using concepts from statistical mechanics. In our model, traders are divided into two groups: fundamentalists and chartists, and their interactions are based on a variant of the herding mechanism introduced by Kirman [1993]. The statistical analysis of simulated data points toward long-term dependence in the auto-correlations of squared and absolute returns and hyperbolic decay in the tail of the distribution of raw returns, both with estimated decay parameters in the same range like those of empirical data. Theoretical analysis, however, excludes the possibility of ‘true’ scaling behavior because of the Markovian nature of the underlying process and the boundedness of returns. The model, therefore, only mimics power law behavior. Similarly as with the phenomenological volatility models analyzed in LeBaron [2001], the usual statistical tests are not able to distinguish between true or pseudo-scaling laws in the dynamics of our artificial market --Herd Behavior,Speculative Dynamics,Fat Tails,Volatility Clustering