We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

This paper considers the worst-case CVaR in situation where only partial information on the underlying probability distribution is given. It is shown that, like CVaR, worst-case CVaR remains a coherent risk measure. The minimization of worst-case CVaR under mix-ture distribution uncertainty, box uncertainty and ellipsoidal uncertainty are investigated. The application of worst-case CVaR to robust portfolio optimization is proposed, and the corresponding problems are cast as linear programs and second-order cone programs which can be efficiently solved. Market data simulation and Monte Carlo simulation examples are presented to illustrate the methods. Our approaches can be applied in many situations, including those outside of financial risk management.

Abstract. A goodness–of–fit test for two–component homoscedastic and homothetic mixtures of normal distributions is proposed. The tests are based on a weighted L2–type distance between the empirical characteristic function and its population counterpart, where in the latter, parameters are replaced by consistent estimators. Consequently the resulting tests are consistent against general alternatives. When moment estimation is employed and as the decay of the weight function tends to infinity the test statistics approach limit values, which are related to the first nonvanishing moment equation. The new tests are compared via simulation to other omnibus tests for mixtures of normal distributions, and are applied to several real data sets. Keywords. Characteristic function, Goodness-of-fit test, Mixtures of Normal Distributions 1

The Pareto (power-law) wealth distribution, which is empirically observed in many countries, implies rather extreme wealth inequality. For instance, in the U.S. the top 1% of the population holds about 40% of the total wealth. What is the source of this inequality? The answer to this question has profound political, social, and philosophical implications. We show that the Pareto wealth distribution is a robust consequence of a fundamental property of the capital investment process: it is a stochastic multiplicative process. Moreover, the Pareto distribution implies that inequality is driven primarily by chance, rather than by differential investment ability. This result is closely related to the concept of market efficiency, and may have direct implications regarding the economic role and social desirability of wealth inequality. We also show that the Pareto wealth distribution may explain the Lvy distribution of stock returns, which has puzzled researchers for many years. Thus, the Pareto wealth distribution, market efficiency, and the Lvy distribution of stock returns are all closely linked.

The distributional behavior for futures price spread changes is examined through parametric and nonparametric tests on four different commodities: corn and live cattle, and gold and T-bonds with two different sample sizes. Data are examined for selected periods, stable (1992) and unstable (1988). Remarkably different results were found over commodities, time period, and sample size. Actual spread changes for the smaller sample size of gold and T-bonds and of corn produced more normal distributions as intervals were widened from daily to weekly, while all live cattle spreads for actual changes were normally distributed. However, the larger sample size of both gold and T-bonds and the relative spread changes for both corn and live cattle did not converge to a normal distribution. The ‘best fit ’ distribution was tested nonparametrically on all daily spread samples, and the logistic distribution prevailed, which supported the results of nonnormality from parametric distributional tests.

I study a standard Grossman and Stiglitz (1980) noisy rational expectations economy, but relax the usual assumption of the normality of fundamental and supply. My solution approach dispenses with the typical “conjecture and verify ” method and enables me to analytically solve an entire class of previously intractable nonlinear models that nests the standard model. I show how: (1) price jumps and crashes may arise endogenously, purely due to learning effects, (2) observation of the net trading volume of informed and noise traders may be valuable for investors in the economy as it can provide a refinement of the information conveyed by price, (3) the value of acquiring information may be non-monotonic in the number of informed traders, leading to multiple equilibria in the information market, and (4) the relation between disagreement and future returns is ambiguous. In short, many of the results from noisy rational expectations models are not robust. Finally, I introduce monotone likelihood ratio conditions that determine the signs of the various comparative statics, which represents the first demonstration of the importance of the MLRP for comparative statics in this literature. Special thanks to my advisor Christine Parlour for continuous guidance and detailed comments on many