Rustam Ibragimov

Rank-1/2: A Simple Way to Improve the Ols Estimation of Tail Exponents

Despite the availability of more sophisticated methods, a popular way to estimate a Pareto exponent is still to run an OLS regression: log(Rank) = a − b log(Size), and take b as an estimate of the Pareto exponent. The reason for this popularity is arguably the simplicity and robustness of this method. Unfortunately, this procedure is strongly biased in small samples. We provide a simple practical remedy for this bias, and propose that, if one wants to use an OLS regression, one should use the Rank −1 / 2, and run log(Rank − 1 / 2) = a − b log(Size). The shift of 1 / 2 is optimal, and reduces the bias to a leading order. The standard error on the Pareto exponent ζ is not the OLS standard error, but is asymptotically (2 / n)1 / 2ζ. Numerical results demonstrate the advantage of the proposed approach over the standard OLS estimation procedures and indicate that it performs well under dependent heavy-tailed processes exhibiting deviations from power laws. The estimation procedures considered are illustrated using an empirical application to Zipf’s law for the United States city size distribution.

T-Statistic Based Correlation and Heterogeneity Robust Inference

We develop a general approach to robust inference about a scalar parameter when the data is potentially heterogeneous and correlated in a largely unknown way. The key ingredient is the following result of Bakirov and Sz´ekely (2005) concerning the small sample properties of the standard t-test: For a significance level of 5% or lower, the t-test remains conservative for underlying observations that are independent and Gaussian with heterogenous variances. One might thus conduct robust large sample inference as follows: partition the data into q ≥ 2 groups, estimate the model for each group and conduct a standard t-test with the resulting q parameter estimators. This results in valid inference as long as the groups are chosen in a way that ensures the parameter estimators to be asymptotically independent, unbiased and Gaussian of possibly different variances. We provide examples of how to apply this approach to time series, panel, clustered and spatially correlated data.

Regression Asymptotics Using Martingale Convergence Methods

Weak convergence of partial sums and multilinear forms in independent random variables and linear processes to stochastic integrals now plays a major role in nonstationary time series and has been central to the development of unit root econometrics. The present paper develops a new and conceptually simple method for obtaining such forms of convergence. The method relies on the fact that the econometric quantities of interest involve discrete time martingales or semimartingales and shows how in the limit these quantities become continuous martingales and semimartingales. The limit theory itself uses very general convergence results for semimartingales that were obtained in work by Jacod and Shiryaev (2003). The theory that is developed here is applicable in a wide range of econometric models and many examples are given. One notable outcome of the new approach is that it provides a unified treatment of the asymptotics for stationary autoregression and autoregression with roots at or near unity, as both these cases are subsumed within the martingale convergence approach and different rates of convergence are accommodated in a natural way. The approach is also useful in developing asymptotics for certain nonlinear functions of integrated processes, which are now receiving attention in econometric applications, and some new results in this area are presented. The paper is partly of pedagogical interest and the conceptual simplicity of the methods is appealing. Since this is the first time the methods have been used in econometrics, the exposition is presented in some detail with illustrations of new derivations of some well-known existing results, as well as some new asymptotic results and the unification of the limit theory for autoregression.

Portfolio Diversification and Value At Risk Under Thick-Tailedness

This paper focuses on the study of portfolio diversification and value at risk analysis under heavy-tailedness. We use a notion of diversification based on majorization theory that will be explained in the text. The paper shows that the stylized fact that portfolio diversification is preferable is reversed for extremely heavy-tailed risks or returns. However, the stylized facts on diversification are robust to heavy-tailedness of risks or returns as long as their distributions are moderately heavy-tailed. Extensions of the results to the case of dependence, including convolutions of α-symmetric distributions and models with common shocks are provided. †The results in this paper constitute a part of the authors dissertation ‘New majorization theory in economics and martingale convergence results in econometrics’ presented to the faculty of the Graduate School of Yale University in candidacy for the degree of Doctor of Philosophy in Economics in March, 2005. The results were originally contained in the work circulated in 2003–2005 under the titles ‘Shifting paradigms: On the robustness of economic models to heavy-tailedness assumptions’ and ‘On the robustness of economic models to heavy-tailedness assumptions’.

COPULA-BASED CHARACTERIZATIONS FOR HIGHER ORDER MARKOV PROCESSES

In this paper, we obtain characterizations of higher order Markov processes in terms of copulas corresponding to their finite-dimensional distributions. The results are applied to establish necessary and sufficient conditions for Markov processes of a given order to exhibit m -dependence, r -independence, or conditional symmetry. The paper also presents a study of applicability and limitations of different copula families in constructing higher order Markov processes with the preceding dependence properties. We further introduce new classes of copulas that allow one to combine Markovness with m -dependence or r -independence in time series.

Short Communications: On an Exact Constant for the Rosenthal Inequality

Let

Characterizations of joint distributions, copulas, information, dependence and decoupling, with applications to time series

\xi_1\lz \xi_n

EFFICIENCY OF LINEAR ESTIMATORS UNDER HEAVY-TAILEDNESS: CONVOLUTIONS OF [alpha]-SYMMETRIC DISTRIBUTIONS

be independent random variables having symmetric distribution with finite pth moment,

The exact constant in the rosenthal inequality for random variables with mean zero

2 < p < \ iy

Inference with Few Heterogeneous Clusters

. It is shown that the precise constant