Rustam Ibragimov

Econometric Theory
Applied Econometrics
Financial Economics
Economic Theory
Statistics and Probability Theory

Econometrics
Financial Economics
Mathematical Economics
Statistics and Probability Theory

(Oral) Econometrics (with distinction), Financial Economics (with distinction), 2002
(Written) Microeconomics and Macroeconomics, 2001

New Majorization Theory in Economics and Martingale Convergence Results in Econometrics

Professor Peter C. B. Phillips
Professor Donald W. K. Andrews
Professor Herbert Scarf

May 2005

M. Phil., Economics, Yale University, 2004
M. A., Economics, Central Michigan University, 2000
Ph.D. (Kandidat Nauk), Mathematics, Dissertation: "Estimates for Moments of Symmetric Statistics",
     Institute of Mathematics of Uzbek Academy of Sciences, 1997
M.S., Mathematics, Tashkent State University, Tashkent, 1996, Graduated with Distinction
High School Diploma, High School No. 233, Tashkent, Uzbekistan, 1991, Graduated with a Gold Medal

Annual Cowles Prize, 2004
C.A. Anderson Fellowship, 2003-2004
Dissertation Fellowship, Yale University, 2003-2004
Cowles Foundation Summer Prize, 2000-Present
University Fellowship, Yale University, 2000-Present
Outstanding Thesis and Dissertation Award, Central Michigan University, 2000
E. Muskie Fellowship in Economics, 1998-2000
National Ulugbek Award for the Support of Talented Youth, Uzbekistan, 1996
Graduation from the Department of Mathematics at Tashkent State University with distinction, 1996
Second Prize, National Students' Olympiad on Mathematics, Uzbekistan, 1994
First Prize, National Students' Olympiad on Mathematics, Uzbekistan, April 1993
Third Prize, National Students' Olympiad on Mathematics, Uzbekistan, April 1992
High School graduation: Gold medal, 1991
Second Prize, 29th National High School Students' Mathematical Olympiad, Uzbekistan, 1991

Teaching Assistant, Econometrics III, Yale University, Spring 2005
Teaching Assistant, Econometrics and Data Analysis, Yale University, Fall 2004
Teaching Assistant, Econometrics and Data Analysis, Yale University, Spring 2003
Teaching Assistant, Introduction to Probability and Statistics, Yale University, Fall 2002
Math & Science Tutor, Yale College, 2003-Present, Tutored all undergraduate Economics, Math & Statistics
     courses
Instructor, Department of Mathematics, Central Michigan University, 1999-2000, Courses taught: Introduction
     to Statistics (Undergraduate course), Mathematical Statistics II (Graduate course)
Visiting Research Fellow, Department of Statistics, Columbia University, 1999 (Collaborator: Victor H. de la
     Peņa)
Research Associate, School of Business, Michigan State University – Center for International Business
     Education and Research, 1999
Senior Specialist, Cabinet of Ministers of the Republic of Uzbekistan, Tashkent, Uzbekistan, 1997 - 1998
Senior Research Associate, Tashkent State University, Department of Mathematics, Division of Probability
     Theory and Mathematical Statistics, Tashkent, Uzbekistan, 1996 - 1997
Instructor, International Business School at Tashkent State Economics University, Tashkent, Uzbekistan,
     1996 -1997.

"Copula-based dependence characterizations and modeling for time series".

"Political risk, gradualism and state power:  the supply-side determinants of mass privatization in Uzbekistan" (with A. Cecen).

Referee for "Econometric Theory" and "Statistics and Probability Letters"

The dissertation provides a unified approach to the study of a number of important problems in economic theory, mathematical finance and econometrics using new majorization theory and martingale convergence methods. The dissertation has two parts. The first part develops a unified approach to the analysis of several models in economics that depend on the majorization properties of convolutions of distributions. The main results show that many economic models are robust to heavy-tailedness assumptions as long as the distributions entering these assumptions are not too thick-tailed. But the implications of these models are reversed for distributions with very long-tailed densities. The second part of the dissertation presents a new and conceptually simple method for obtaining weak convergence of partial sums and multilinear forms to stochastic integrals, thereby providing for the first time a completely unified treatment of the asymptotics for stationary autoregression and autoregression with roots at or near unity.

I. On the robustness of economic models to heavy-tailedness assumptions.

Many economic models have a structure that depends on majorization properties of linear combinations of random variables. The majorization relation is a formalization of the concept of diversity in the components of vectors. Over the past decades, majorization theory, which focuses on the study of this relation and functions that preserve it, has found applications in disciplines ranging from statistics, probability theory and economics to mathematical genetics, linear algebra and geometry. This part of the dissertation provides a unified approach to the analysis of the majorization properties of linear combinations of random variables. It further studies the robustness of these majorization properties and the implications of a number of important economic models to heavy-tailedness assumptions. I show, in particular, that majorizations for log-concavely distributed signals established by Proschan (1965) continue to hold for random variables with not too thick-tailed densities. More precisely, the tails of distributions of linear combinations of not too thick-tailed random variables continue to exhibit the property of Schur-convexity, as in the case of log-concave distributions. However, the majorization properties are reversed for very long-tailed distributions, in which case Schur-convexity of the tails is replaced by their Schur-concavity. This is the first probabilistic result that shows that majorization properties of log-concave densities are reversed for a wide class of distributions and is the key to reversals of properties of many economic models built upon the popular log-concavity assumption. One should emphasize here that, although log-concave distributions have many appealing properties that have been utilized in a number of works in economics, they cannot be used in the study of thick-tailedness phenomena since any log-concave density is extremely light-tailed: in particular, its tails decline at least exponentially fast and all its moments exist.

In a series of applications of the main probabilistic results, I study robustness of monotone consistency of the sample mean, value at risk (VaR) analysis and a model of demand-driven innovation and spatial competition as well as that of optimal bundling strategies for a multiproduct monopolist in the case of an arbitrary degree of complementarity or substitutability among the goods. The following list summarizes some of the main results.

(i) I show that the sample mean exhibits monotone consistency in the case of data from not too thick-tailed populations. Thus, an increase in the sample size always improves performance of the sample mean. In addition, VaR is a coherent measure of risk if distributions of risks are not very heavy-tailed. However, coherency of the value at risk is always violated even in the case of independence if distributions of risks are very thick-tailed. Moreover, in the case of not very-heavy tailed risks, diversification of a portfolio decreases riskiness of the portfolio's return in the sense of (first-order) stochastic dominance. However, diversification of a portfolio of very thick-tailed risks always leads to an increase in riskiness of the return on the portfolio. I also obtain sharp bounds on the VaR of the returns on portfolios of risks with long-tailed returns.

(ii) Another application that is explored in depth concerns growth theory for firms that invest in information about their markets. I present a study of robustness of the model of demand-driven innovation and spatial competition over time with log-concavely distributed signals developed by Jovanovic and Rob to heavy-tailedness assumptions. The implications of the model remain valid for not too heavy-tailed distributions of consumers' signals. However, again these properties are reversed for very thick-tailed signals.

(iii) Using the main majorization results, I develop a framework that allows one to model the optimal bundling problem of a multiproduct monopolist providing interrelated goods with an arbitrary degree of complementarity or substitutability. Characterizations of optimal bundling strategies are derived for the seller in the case of long-tailed valuations and tastes for the products. I show that if the goods provided in a Vickrey auction are substitutes and bidders' tastes for the objects are not very heavy-tailed, then the monopolist prefers separate provision of the products. However, if the goods are complements and consumers' tastes are very thick-tailed, then the seller prefers providing the products on a single Vickrey auction. I also obtain characterizations of optimal bundling strategies for a monopolist who provides complements or substitutes for profit-maximizing prices to buyers with heavy-tailed tastes.

(iv) Some extensions of the above results to the case of dependence are obtained, including convolutions of a-symmetric and spherical distributions and models with common shocks which are of great importance in economics and finance.

II. Regression Asymptotics Using Martingale Convergence Methods (joint with Peter C. B. Phillips).

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. This part of the dissertation develops a new and conceptually simple method for obtaining such forms of convergence, while at the same time integrating this new asymptotic theory with that for the stationary time series case. 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) using the triplet of semimartingale characteristics. The theory that is developed here is applicable in a wide range of econometric models and many examples are given.

The most notable contribution of the new approach is that it succeeds in unifying the treatment of the asymptotics for stationary autoregression and autoregression with roots at or near unity. Both these cases are subsumed within the martingale convergence approach and the different rates of convergence that apply in the various cases 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.

In addition to their scientific interest, the results in this part of the dissertation are likely to be of pedagogical interest providing a conceptual simplicity and integration of the asymptotic theory for stationary and nonstationary cases that is inherently appealing and absent in existing proofs. This is the first time these particular martingale methods have been used in econometrics. So the approach is presented with many illustrations of how well-known results can be derived in the new way, how new asymptotic results can be derived for nonlinear cases, and how the unification of the limit theory for autoregression is accomplished.