Yannick Malevergne

Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities.

We provide definitive results to close the debate between Eec khout (2004, 2009) and Levy (2009) on the validity of Zipf’s law, which is the special Pareto law with tail exponent 1, to describe the tail of the distribution of U.S. city sizes. Because the origin of th e disagreement between Eeckhout and Levy stems from the limited power of their tests, we perform the uniformly most powerful unbiased test for the null hypothesis of the Pareto distribution against the l ognormal. Thep-value and Hill’s estimator as a function of city size lower threshold confirm indubitabl y that the size distribution of the 1000 largest cities or so, which include more than half of the tota l U.S. population, is Pareto, but we rule out that the tail exponent, estimated to be 1.4 ± 0.1, is equal to1. For larger ranks, the p-value becomes very small and Hill’s estimator decays systematically with decreasing ranks, qualifying the lognormal distribution as the better model for the set of smaller citie s. These two results reconcile the opposite views of Eeckhout (2004) and Levy (2009). We explain how Gibr at’s law of proportional growth underpins both the Pareto and lognormal distributions and s tress the key ingredient at the origin of their difference in standard stochastic growth models of ci ties (Gabaix 1999, Eeckhout 2004). JEL classification: D30, D51, J61, R12.

Imitation and contrarian behaviour: hyperbolic bubbles, crashes and chaos

Abstract Imitative and contrarian behaviours are the two typical opposite attitudes of investors in stock markets. We introduce a simple model to investigate their interplay in a stock market where agents can take only two states, bullish or bearish. Each bullish (bearish) agent polls m ‘friends’ and changes her opinion to bearish (bullish) if (i) at least mρ hb (mρ bh ) among the m agents inspected are bearish (bullish) or (ii) at least mρ hh >mρ hb (mρ bb >mρ bh ) among the m agents inspected are bullish (bearish). The condition (i) ((ii)) corresponds to imitative (antagonistic) behaviour. In the limit where the number N of agents is infinite, the dynamics of the fraction of bullish agents is deterministic and exhibits chaotic behaviour in a significant domain of the parameter space {ρ hb ,ρ bh ,ρ hh ,ρ bb ,m}. A typical chaotic trajectory is characterized by intermittent phases of chaos, quasi-periodic behaviour and super-exponentially growing bubbles followed by crashes. A typical bubble starts initially by growing at an exponential rate and then crosses over to a nonlinear power-law growth rate leading to a finite-time singularity. The reinjection mechanism provided by the contrarian behaviour introduces a finite-size effect, rounding off these singularities and leads to chaos. We document the main stylized facts of this model in the symmetric and asymmetric cases. This model is one of the rare agent-based models that give rise to interesting non-periodic complex dynamics in the ‘thermodynamic’ limit (of an infinite number N of agents). We also discuss the case of a finite number of agents, which introduces an endogenous source of noise superimposed on the chaotic dynamics.

Collective origin of the coexistence of apparent random matrix theory noise and of factors in large sample correlation matrices

Through simple analytical calculations and numerical simulations, we demonstrate the generic existence of a self-organized macroscopic state in any large multivariate system possessing non-vanishing average correlations between a finite fraction of all pairs of elements. The coexistence of an eigenvalue spectrum predicted by random matrix theory (RMT) and a few very large eigenvalues in large empirical correlation matrices is shown to result from a bottom–up collective effect of the underlying time series rather than a top–down impact of factors. Our results, in excellent agreement with previous results obtained on large financial correlation matrices, show that there is relevant information also in the bulk of the eigenvalue spectrum and rationalize the presence of market factors previously introduced in an ad hoc manner.

From rational bubbles to crashes

We study and generalize in various ways the model of rational expectation (RE) bubbles introduced by Blanchard and Watson in the economic literature. Bubbles are argued to be the equivalent of Goldstone modes of the fundamental rational pricing equation, associated with the symmetry-breaking introduced by non-vanishing dividends. Generalizing bubbles in terms of multiplicative stochastic maps, we summarize the result of Lux and Sornette that the no-arbitrage condition imposes that the tail of the return distribution is hyperbolic with an exponent μ<1. We then outline the main results of Malevergne and Sornette, who extend the RE bubble model to arbitrary dimensions d: a number d of market time series are made linearly interdependent via d×d stochastic coupling coefficients. We derive the no-arbitrage condition in this context and, with the renewal theory for products of random matrices applied to stochastic recurrence equations, we extend the theorem of Lux and Sornette to demonstrate that the tails of the unconditional distributions associated with such d-dimensional bubble processes follow power laws, with the same asymptotic tail exponent μ<1 for all assets. The distribution of price differences and of returns is dominated by the same power-law over an extended range of large returns. Although power-law tails are a pervasive feature of empirical data, the numerical value μ<1 is in disagreement with the usual empirical estimates μ≈3. We then discuss two extensions (the crash hazard rate model and the non-stationary growth rate model) of the RE bubble model that provide two ways of reconciliation with the stylized facts of financial data.

On the power of generalized extreme value (GEV) and generalized Pareto distribution (GPD) estimators for empirical distributions of stock returns

Using synthetic tests performed on time series with time dependence in the volatility with both Pareto and Stretched-Exponential distributions, it is shown that for samples of moderate sizes the standard generalized extreme value (GEV) estimator is quite inefficient due to the possibly slow convergence toward the asymptotic theoretical distribution and the existence of biases in the presence of dependence between data. Thus, it cannot distinguish reliably between rapidly and regularly varying classes of distributions. The Generalized Pareto distribution (GPD) estimator works better, but still lacks power in the presence of strong dependence. Applied to 100 years of daily returns of the Dow Jones Industrial Average and over one years of five-minutes returns of the Nasdaq Composite index, the GEV and GDP estimators are found insufficient to prove that the distributions of empirical returns of financial time series are regularly varying, because the rapidly varying exponential or stretched exponential distributions are equally acceptable.

Gibrat’s Law for Cities: Uniformly Most Powerful Unbiased Test of the Pareto Against the Lognormal

We provide definitive results to close the debate between Eeckhout (2004, 2009) and Levy (2009) on the validity of Zipf’s law, which is the special Pareto law with tail exponent 1, to describe the tail of the distribution of U.S. city sizes. Because the origin of the disagreement between Eeckhout and Levy stems from the limited power of their tests, we performthe uniformly most powerful unbiased test for the null hypothesis of the Pareto distribution against the lognormal. The p-value and Hill’s estimator as a function of city size lower threshold confirm indubitably that the size distribution of the 1000 largest cities or so, which includemore than half of the total U.S. population, is Pareto, but we rule out that the tail exponent, estimated to be 1.4 ± 0.1, is equal to 1. For larger ranks, the p-value becomes very small and Hill’s estimator decays systematically with decreasing ranks, qualifying the lognormal distribution as the better model for the set of smaller cities. These two results reconcile the opposite views of Eeckhout (2004) and Levy (2009). We explain how Gibrat’s law of proportional growth underpins both the Pareto and lognormal distributions and stress the key ingredient at the origin of their difference in standard stochastic growth models of cities (Gabaix 1999, Eeckhout 2004).

Volatility fingerprints of large shocks: Endogeneous versus exogeneous

Finance is about how the continuous stream of news gets incorporated into prices. But not all news have the same impact. Can one distinguish the effects of the Sept. 11, 2001 attack or of the coup against Gorbachev on Aug., 19, 1991 from financial crashes such as Oct. 1987 as well as smaller volatility bursts? Using a parsimonious autoregressive process with long-range memory defined on the logarithm of the volatility, we predict strikingly different response functions of the price volatility to great external shocks compared to what we term endogeneous shocks, i.e., which result from the cooperative accumulation of many small shocks. These predictions are remarkably well-confirmed empirically on a hierarchy of volatility shocks. Our theory allows us to classify two classes of events (endogeneous and exogeneous) with specific signatures and characteristic precursors for the endogeneous class. It also explains the origin of endogeneous shocks as the coherent accumulations of tiny bad news, and thus unify all previous explanations of large crashes including Oct. 1987.

Value-at-Risk-efficient portfolios for a class of super- and sub-exponentially decaying assets return distributions

Abstract Using a family of modified Weibull distributions encompassing both sub-exponentials and super-exponentials to parametrize the marginal distributions of asset returns and their multivariate generalizations with Gaussian copulas, we offer exact formulae for the tails of the distribution P(S) of returns S of a portfolio of arbitrary composition of these assets. We find that the tail of P(S) is also asymptotically a modified Weibull distribution with a characteristic scale χ function of the asset weights with different functional forms depending on the super- or sub-exponential behaviour of the marginals and on the strength of the dependence between the assets. We then treat in detail the problem of risk minimization using the Value-at-Risk and expected shortfall which are shown to be (asymptotically) equivalent in this framework.

Investigating Extreme Dependences: Concepts and Tools

We investigate the relative information content of six measures of dependence between two random variables

Higher-Moment Portfolio Theory

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