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Rapid stock market growth without real economic back-up has led to the 2015 Chinese Stock Market Crash with thousands of stocks hitting the down limit simultaneously multiple times. The authors provide a detailed analysis of structural breaks in heavy-tailedness and asymmetry properties of returns in Chinese A-share markets due to the crash using recently proposed robust approaches to tail index inference. The empirical analysis points out to heavy-tailedness properties often implying possibly infinite second moments and also focuses on gain/loss asymmetry in the tails of daily returns on individual stocks. The authors further present an analysis of the main determinants of heavy-tailedness in Chinese financial markets. It points out to liquidity and company size as being the most important factors affecting the returns’ heavy-tailedness properties. At the same time, the authors do not observe statistically significant differences in tail indices of the returns on A-shares and the coefficients on factors affecting them in the pre-crisis and post-crisis periods.

Heavy-tailed distributions present a tough setting for inference. They are also common in industrial applications, particularly with internet transaction datasets, and machine learners often analyze such data without considering the biases and risks associated with the misuse of standard tools. This chapter outlines a procedure for inference about the mean of a (possibly conditional) heavy-tailed distribution that combines nonparametric analysis for the bulk of the support with Bayesian parametric modeling – motivated from extreme value theory – for the heavy tail. The procedure is fast and massively scalable. The work should find application in settings wherever correct inference is important and reward tails are heavy; we illustrate the framework in causal inference for A/B experiments involving hundreds of millions of users of eBay.com.

We describe a new Bayesian estimation algorithm for fitting a binary treatment, ordered outcome selection model in a potential outcomes framework. We show how recent advances in simulation methods, namely data augmentation, the Gibbs sampler and the Metropolis-Hastings algorithm can be used to fit this model efficiently, and also introduce a reparameterization to help accelerate the convergence of our posterior simulator. Conventional “treatment effects” such as the Average Treatment Effect (ATE), the effect of treatment on the treated (TT) and the Local Average Treatment Effect (LATE) are adapted for this specific model, and Bayesian strategies for calculating these treatment effects are introduced. Finally, we review how one can potentially learn (or at least bound) the non-identified cross-regime correlation parameter and use this learning to calculate (or bound) parameters of interest beyond mean treatment effects.

The variance targeting estimator (VTE) for generalized autoregressive conditionally heteroskedastic (GARCH) processes has been proposed as a computationally simpler and misspecification-robust alternative to the quasi-maximum likelihood estimator (QMLE). In this paper we investigate the asymptotic behavior of the VTE when the stationary distribution of the GARCH process has infinite fourth moment. Existing studies of historical asset returns indicate that this may be a case of empirical relevance. Under suitable technical conditions, we establish a stable limit theory for the VTE, with the rate of convergence determined by the tails of the stationary distribution. This rate is slower than that achieved by the QMLE. The limit distribution of the VTE is nondegenerate but singular. We investigate the use of subsampling techniques for inference, but find that finite sample performance is poor in empirically relevant scenarios.

Popular understandings of the financial crisis tend to focus on the rents extracted by elite personnel in the financial sector. Professional discussions, however, have addressed the faulty assumptions underlying theory and practice – in particular, the assumption that returns to financial assets follow the Gaussian distribution, in the face of much empirical evidence that these have power law distributions with far higher kurtosis. It turns out that the power law tails of returns to financial assets are also a feature of the distribution of company rates of profit, a discovery that stems from proposals to ‘dissolve’ the traditional transformation problem by abandoning the condition of a uniform rate of profit and instead considering its distribution.Marx himself was aware of the importance of considering the distributional properties of economic variables, based on his reading of Quetelet. In fact, heavy-tailed distributions characterise a wide range of variables in capitalist economies, the best-known probably being the Paretian tail component in distributions of income and wealth. Nor is this simply an empirical fact – such distributions emerge readily from a range of agent-based simulations.Capitalist economies are, in a particular technical sense, complex self-organising systems perpetually on the brink of crisis. This modern understanding is prefigured in Marx’s discussion of how the compulsive character of social relations emerges from the atomistic exercise of human free will in commercial society. The developing literature of probabilistic Marxism successfully applies these insights to the wider fields of econophysics and complexity, demonstrating the continuing relevance of Marx’s thought.

Purpose – This chapter provides an overview of issues related to analysing crash data characterised by excess zero responses and/or long tails and how to overcome these problems. Factors affecting excess zeros and/or long tails are discussed, as well as how they can bias the results when traditional distributions or models are used. Recently introduced multi-parameter distributions and models developed specifically for such datasets are described. The chapter is intended to guide readers on how to properly analyse crash datasets with excess zeros and long or heavy tails.

Methodology – Key references from the literature are summarised and discussed, and two examples detailing how multi-parameter distributions and models compare with the negative binomial distribution and model are presented.

Findings – In the event that the characteristics of the crash dataset cannot be changed or modified, recently introduced multi-parameter distributions and models can be used efficiently to analyse datasets characterised by excess zero responses and/or long tails. They offer a simpler way to interpret the relationship between crashes and explanatory variables, while providing better statistical performance in terms of goodness-of-fit and predictive capabilities.

Research implications – Multi-parameter models are expected to become the next series of traditional distributions and models. The research on these models is still ongoing.

Practical implications – With the advancement of computing power and Bayesian simulation methods, multi-parameter models can now be easily coded and applied to analyse crash datasets characterised by excess zero responses and/or long tails.

A new multivariate heavy-tailed distribution is proposed as an extension of the univariate distribution of Politis (2004). The properties of the new distribution are discussed, as well as its effectiveness in modeling ARCH/GARCH residuals. A practical procedure for multi-parameter numerical maximum likelihood is also given, and a real data example is worked out.

The applicability and shortcomings of a well-defined cost-estimating process to forecasting resources required for developing and fielding innovative technologies are examined. Whereas the process itself provides a suitable approach for estimating the cost of any program, investment is required for collecting historical data on analogous programs to serve as the foundation for the estimating methodologies. Particular challenges in costing innovation are summarized. An appropriate form of the cost probability distribution for research programs is offered.

Application of financial risk models in the emerging markets poses special challenges. A fundamental challenge is to accurately model the return distributions which are particularly fat tailed and skewed. Value-at-Risk (VaR) measures based on the Extreme Value Theory (EVT) have been suggested, but typically data histories are limited, making it hard to test and apply EVT. The chapter addresses issues in (i) modeling the VaR measure in the presence of structural breaks in an economy, (ii) the choice of stable innovation distribution with volatility clustering effects, (iii) modeling the tails of the empirical distribution, and (iv) fixing the cut-off point for isolating extreme observations. Pakistan offers an instructive case since its equity market exhibits high volatility and incidence of extreme returns. The recent Global Financial Crisis has been another source of extreme returns. The confluence of the two sources of volatility provides us with a rich data set to test the VaR/EVT model rigorously and examine practical challenges in its application in an emerging market.