56 results found
Brown D, Ibragimov R, 2019, Sign tests for dependent observations, Econometrics and Statistics, Vol: 10, Pages: 1-8, ISSN: 2452-3062
New sign tests for testing equality of conditional distributions of two (arbitrary) adapted processes as well as for testing conditionally symmetric martingale-difference assumptions are introduced. The analysis is based on results that demonstrate that randomization over ties in sign tests for equality of conditional distributions of two adapted sequences produces a stream of i.i.d. symmetric Bernoulli random variables. This reduces the problem of evaluating the critical values of the tests to computing the quantiles or moments of Binomial or normal distributions. Similar properties also hold under randomization over zero values of signs of a conditionally symmetric martingale-difference sequence.
Chen Z, Ibragimov R, 2019, One country, two systems? The heavy-tailedness of Chinese A- and H- share markets, Emerging Markets Review, Vol: 38, Pages: 115-141, ISSN: 1566-0141
Chinese A- and H– share markets operate in different institutional environments (emerging/developing v.s. developed) and thus may have different tail risk properties. This paper focuses on the analysis of heavy-tailedness properties of these two markets using recently developed robust inference methods. The equality of tail indices of returns for A and H dual-listed companies cannot be rejected, and some A- and H– share returns may have infinite second moments. Their heavy-tailedness properties did not change significantly with respect to the 2008 financial crisis and the date when the corresponding company starts to be dual-listed.
Ibragimov M, Ibragimov R, Kattuman P, et al., 2018, Income inequality and price elasticity of market demand: the case of crossing Lorenz curves, Economic Theory, Vol: 65, Pages: 729-750, ISSN: 0938-2259
This paper extends Ibragimov and Ibragimov (Econ Theory 32:579–587,2007) in which the effect of changes income inequality on the price elasticity ofmarket demand is characterized for the class of income distribution changes occurringthrough non-intersecting Lorenz curve shifts. We derive sufficient conditionsfor increase/decrease in price elasticity of market demand, under general changes inincome distribution, allowing Lorenz curves to intersect as they shift. We conclude bydrawing out implications of different types of tax policy changes for demand elasticity.
Ibragimov R, Jaffee D, Walden J, 2018, Equilibrium with monoline and multiline structures, Review of Finance, Vol: 22, Pages: 595-632, ISSN: 1573-692X
We study a competitive market for risk-sharing, in which risk-tolerant providers of risk protection, who face frictional costs in holding capital, offer coverage over a range of risk classes to risk-averse agents. We distinguish monoline and multiline industry structures and characterize when each structure is optimal. Markets for which the risks are limited in number, asymmetric or correlated will be served by monoline structures, whereas markets characterized by a large number of essentially independent risks will be served by many multiline firms. Our results are consistent with observed structures within insurance, and also have general implications for the financial services industry.
Gu Z, Ibragimov R, 2018, The "Cubic Law of the Stock Returns" in emerging markets, JOURNAL OF EMPIRICAL FINANCE, Vol: 46, Pages: 182-190, ISSN: 0927-5398
Excess volatility in main emerging and developed stock markets is carefully analysed in this study. Tail distribution of returns of both stock market index and individual stocks is evaluated and compared with the theoretical distribution found by Gabaix et al. (2003, 2006). For stock market index, recursive and rolling estimation are used. In recursive estimation, we find that all the developed markets obey “the Cubic Law of the Stock Returns”, while most of the emerging countries exhibit heavier tail with a tail index lower than 3 at 95% significance level. In rolling estimation, the tail index in the developed markets does not stabilise around 3, and after 2008 financial crisis, all the developed markets and most emerging ones suffer a drop in the tail index. For individual stocks, the tail distributions of stock returns, trading volume, and the number of trades in each emerging country behave quite differently from the theoretical model by Gabaix et al. (2006), especially the stock returns.
Ankudinov A, Ibragimov R, Lebedev O, 2017, Heavy tails and asymmetry of returns in the Russian stock market, Emerging Markets Review, Vol: 32, Pages: 200-219, ISSN: 1566-0141
The paper presents the robust estimates of tail indices for financial returns and returns asymmetry in the Russian stock market. We also investigate the relation between individual characteristics of companies and the degree of heavy-tailedness and asymmetry of returns. According to our estimates, the degree of heavy-tailedness is strongly related to the liquidity of stocks and the company size. At the same time, no significant effects on estimates of the tail indices of sectoral affiliation, cross-listing, adding into quotation lists, state ownership are revealed. As for the influence of the above-mentioned factors on the asymmetry of returns, the statistical reliability of relevant models is rather low. However, certain indicators of the asymmetry are observed for medium-sized regional companies, the majority showing a heavier right tail compared to the left tail. We also discuss the implications of our findings for managerial decisions and economic modeling. Our results may be useful for risk-managers, financial regulators, investors and policy-makers.
Pinelis I, Peña VHDL, Ibragimov R, et al., 2017, Inequalities and Extremal Problems in Probability and Statistics, Publisher: Academic Press, ISBN: 9780128098189
The book enables the reader to grasp the importance of inequalities and how they relate to probability and statistics.
Ibragimov M, Ibragimov R, 2017, Heavy tails and upper-tail inequality: the case of Russia., Empirical Economics, Vol: 54, Pages: 823-837, ISSN: 0377-7332
Motivated, in part, by the recent surge of interest in robust inequality measurement, cross-country inequality comparisons, applications of heavy-tailed distributions and the study of global and upper-tail inequality, this paper focuses on robust analysis of heavy-tailedness properties and inequality in the upper tails of income distribution in Russia, as measured, mainly, by its tail indices. The study is based on recently developed approaches to robust inference on the degree of heavy-tailedness and their implications for the analysis of upper-tail inequality discussed in the paper. Among other results, the paper provides robust estimates of heavy-tailedness parameters and tail indices for Russian income distribution and their comparisons with the benchmark values in developed economies reported in the previous literature. The estimates point out to important similarity between heavy-tailedness properties of income distribution and their implications for the analysis of upper-tail income inequality in Russia and those in developed markets.
Ankudinov A, Ibragimov R, Lebedev O, 2017, Sanctions and the Russian stock market, Research in International Business and Finance, Vol: 40, Pages: 150-162, ISSN: 0275-5319
The article presents the robust estimates of extreme movements and heavy-tailedness properties for Russian stock indices returns before and after sanctions were introduced. The obtained results show that almost for all sectoral indices there was a statistically significant increase in volatility. At the same time there is not enough evidence of structural breaks in heavy-tailedness, though some indications of heavier both right and left tails in the post-imposition period can be observed for some indices. However, we cannot with complete certainty directly link the increase in heavy-tailedness with the imposed sanctions. The latter to a considerable extent could be caused by higher country-specific risks due to geopolitical tensions as well as oil prices volatility. Whatever is the cause, any increases in heavy-tailedness can have grave consequences for corporate management, economic modeling and financial stability analysis.
Ibragimov M, Ibragimov R, 2017, Unemployment and output dynamics in CIS countries: Okun's law revisited, Applied Economics, Vol: 49, Pages: 3453-3479, ISSN: 0003-6846
Okun’s law is a well-known relationship between the change in the unemployment rate and output growth. The main objective of this article is to provide a rigorous econometric analysis of Okun’s law for several CIS countries using different models and theoretically justified econometric methods. The traditional approach to Okun’s law estimation using OLS regressions does not account for possible endogeneity of regressors and the implied inconsistency of the estimates obtained. These problems point out to incorrectness of applications of the standard OLS estimation techniques. Our study addresses these issues by using econometrically justified instrumental variable regression methods. The article provides the results and discussions on practical use of Okun’s relationships for evaluation of average effects of economic growth on the unemployment rate, and vice versa; importance of accounting for confidence intervals in applications of Okun’s models to economic development analysis and cross-country comparisons and evaluation of effects of crises and other structural shocks on the economies considered. We also discuss in detail the results of formal econometric tests and economic motivation for validity of instrumental variables used in the study. The formal econometric tests, together with economic arguments, allow us to determine the most appropriate Okun-type models for each of the CIS countries under consideration.
Ibragimov R, Prokhorov A, 2017, Heavy Tails and Copulas: Topics in Dependence Modelling in Economics and Finance, Publisher: World Scientific, ISBN: 978-981-4689-79-3
This book offers a unified approach to the study of crises, large fluctuations, dependence and contagion effects in economics and finance. It covers important topics in statistical modeling and estimation, which combine the notions of copulas and heavy tails — two particularly valuable tools of today's research in economics, finance, econometrics and other fields — in order to provide a new way of thinking about such vital problems as diversification of risk and propagation of crises through financial markets due to contagion phenomena, among others. The aim is to arm today's economists with a toolbox suited for analyzing multivariate data with many outliers and with arbitrary dependence patterns. The methods and topics discussed and used in the book include, in particular, majorization theory, heavy-tailed distributions and copula functions — all applied to study robustness of economic, financial and statistical models, and estimation methods to heavy tails and dependence.
Ibragimov R, Lentzas G, 2017, Copulas and long memory, Probability Surveys, Vol: 14, Pages: 289-327
Ibragimov R, Prokhorov A, 2016, Heavy tails and copulas: limits of diversification revisited, Economics Letters, Vol: 149, Pages: 102-107, ISSN: 0165-1765
We show that diversification does not reduce Value-at-Risk for a large class of dependent heavy tailed risks. The class is characterized by power law marginals with tail exponent no greater than one and by a general dependence structure which includes some of the most commonly used copulas.
Ibragimov R, 2016, Inference with few heterogeneous clusters, Review of Economics and Statistics, Vol: 98, Pages: 83-96, ISSN: 0034-6535
Suppose estimating a model on each of a small number of potentially heterogeneous clusters yields approximately independent, unbiased and Gaussian parameter estimators.We make two contributions in this set-up. First, we show how to compare a scalar parameter of interest between treatment and control units using a two-sample t-statistic, extending previous results for the one-sample t-statistic. Second, we developa test for the appropriate level of clustering, which tests the null hypothesis that clustered standard errors from a much finer partition are correct. We illustrate the approach by revisiting empirical studies involving clustered, time series and spatiallycorrelated data.
Brown DJ, Ibragimov R, Walden J, 2015, Bounds for path-dependent options, Annals of Finance, Vol: 11, Pages: 433-451, ISSN: 1614-2446
We develop new semiparametric bounds on the expected payoffs and prices of European call options and a wide range of path-dependent contingent claims. We first focus on the trinomial financial market model in which, as is well-known, an exact calculation of derivative prices based on no-arbitrage arguments is impossible. We show that the expected payoff of a European call option in the trinomial model with martingale-difference log-returns is bounded from above by the expected payoff of a call option written on an asset with i.i.d. symmetric two-valued log-returns. We further show that the expected payoff of a European call option in the multiperiod trinomial option pricing model is bounded by the expected payoff of a call option in the two-period model with a log-normal asset price. We also obtain bounds on the possible prices of call options in the (incomplete) trinomial model in terms of the parameters of the asset’s distribution. Similar bounds also hold for many other contingent claims in the trinomial option pricing model, including those with an arbitrary convex increasing payoff function as well as for path-dependent ones such as Asian options. We further obtain a wide range of new semiparametric moment bounds on the expected payoffs and prices of path-dependent Asian options with an arbitrary distribution of the underlying asset’s price. These results are based on recently obtained sharp moment inequalities for sums of multilinear forms and U-statistics and provide their first financial and economic applications in the literature. Similar bounds also hold for many other path-dependent contingent claims.
Ibragimov M, Ibragimov R, Walden J, 2015, Heavy-tailed distributions and robustness in economics and finance, Publisher: Springer, ISBN: 978-3-319-16877-7
This book focuses on general frameworks for modeling heavy-tailed distributions in economics, finance, econometrics, statistics, risk management and insurance. A central theme is that of (non-)robustness, i.e., the fact that the presence of heavy tails can either reinforce or reverse the implications of a number of models in these fields, depending on the degree of heavy-tailed ness. These results motivate the development and applications of robust inference approaches under heavy tails, heterogeneity and dependence in observations. Several recently developed robust inference approaches are discussed and illustrated, together with applications.
Ibragimov R, 2014, On the robustness of location estimators in models of firm growth under heavy-tailedness, Journal of Econometrics, Vol: 181, Pages: 25-33, ISSN: 0304-4076
Focusing on the model of demand-driven innovation and spatial competition over time in Jovanovic and Rob (1987), we study the effects of the robustness of estimators employed by firms to make inferences about their markets on the firms’ growth patterns. We show that if consumers’ signals in the model are moderately heavy-tailed and the firms use the sample mean of the signals to estimate the ideal product, then the firms’ output levels exhibit positive persistence. In such a setting, large firms have an advantage over their smaller counterparts. These properties are reversed for signals with extremely heavy-tailed distributions. In such a case, the model implies anti-persistence in output levels, together with a surprising pattern of oscillations in firm sizes, with smaller firms being likely to become larger ones next period, and vice versa. We further show that the implications of the model under moderate heavy-tailedness continue to hold under the only assumption of symmetry of consumers’ signals if the firms use a more robust estimator of the ideal product, the sample median.
Ibragimov M, Ibragimov R, Kattuman P, 2013, Emerging markets and heavy tails, Journal of Banking and Finance, Vol: 37, Pages: 2546-2559, ISSN: 1872-6372
Emerging countries are held to be subject to more frequent and more pronounced external and internal shocks than their developed counter-parts. This suggests that key variables pertaining to their markets, including their exchange rates, will be marked by greater likelihood of extreme observations and large fluctuations. We focus on the hypothesis that compared to developed country exchange rates, emerging country exchange rates will be more pronouncedly heavy-tailed. We find support for the hypothesis using recently proposed robust tail index estimation methods which, in particular, perform well under heavy-tailed dependent GARCH processes that are often used for modeling exchange rates. According to the estimation results reported in the paper, variances may be infinite for several emerging country exchange rates. Tail index values ζ = p ∈ (2.6, 2.8) appear to be at the dividing boundary between the two sets of countries: while the moments of order p ∈ (2.6, 2.8) are finite for most of the developed country exchange rates, they may be (or are) infinite for most of the emerging country exchange rates. We also study the impact of the on-going financial and economic crisis, and find that heavy-tailedness properties of most exchange rates did not change significantly with the onset of the crisis. At the same time, some foreign exchange markets have experienced structural changes in their heavy-tailedness properties during the crisis.
Ibragimov R, Walden J, 2011, Value at risk under dependence and heavy-tailedness: Models with common shocks, Annals of Finance, Vol: 7, Pages: 285-318, ISSN: 1614-2446
This paper presents an analysis of diversification and portfolio value at risk for heavy-tailed dependent risks in models with multiple common shocks. We show that, in the framework of value at risk comparisons, diversification is optimal for moderately heavy-tailed dependent risks with common shocks and finite first moments, provided that the model is balanced, i.e., that all the risks are available for portfolio formation. However, diversification is inferior in balanced extremely heavy-tailed risk models with common factors. Finally, in several unbalanced dependent models, diversification is optimal, even though there is extreme heavy-tailedness in common shocks or in idiosyncratic parts of the risks. Analogues of the obtained results further hold for efficiency comparisons of linear estimators in random effects models with dependent and heavy-tailed observations.
Ibragimov R, Jaffee D, Walden J, 2011, Diversification disasters, JOURNAL OF FINANCIAL ECONOMICS, Vol: 99, Pages: 333-348, ISSN: 0304-405X
Gabaix X, Ibragimov R, 2011, Rank-1/2: A simple way to improve the OLS estimation of tail exponents, Journal of Business and Economic Statistics, Vol: 29, Pages: 24-39, ISSN: 0735-0015
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.
Ibragimov R, Walden J, 2010, Optimal Bundling Strategies Under Heavy-Tailed Valuations, MANAGEMENT SCIENCE, Vol: 56, Pages: 1963-1976, ISSN: 0025-1909
Ibragimov R, Mueller UK, 2010, t-Statistic based correlation and heterogeneity robust inference, Journal of Business and Economic Statistics, Vol: 28, Pages: 453-468, ISSN: 0735-0015
We develop a general approach to robust inference about a scalar parameter of interest when the data is potentially heterogeneous and correlated in a largely unknown way. The key ingredient is the following result of Bakirov and Székely (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 of interest. This results in valid and in some sense efficient inference when 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.
Ibragimov R, Jaffee D, Walden J, 2010, Pricing and Capital Allocation for Multiline Insurance Firms, JOURNAL OF RISK AND INSURANCE, Vol: 77, Pages: 551-578, ISSN: 0022-4367
Ibragimov R, 2010, On functions not preserving majorization pre-ordering and their applications, UZBEK MATHEMATICAL JOURNAL, Pages: 64-71
Ibragimov M, Ibragimov R, Karimov J, 2010, Uzbekistan population forecast, Bulletin of Tashkent University of Information Technologies
Choros B, Ibragimov R, Permiakova E, 2010, Copula estimation, WORKSHOP ON COPULA THEORY AND ITS APPLICATIONS, Editors: Durante, Haerdle, Jaworski, Rychlik, Publisher: Springer, Pages: 77-92
Ibragimov M, Ibragimov R, 2010, Measurement of economic progress, INTERNATIONAL ENCYCLOPEDIA OF STATISTICAL SCIENCE, Publisher: Springer
Ibragimov R, 2009, Copula-based characterizations for higher order Markov processes, ECONOMETRIC THEORY, Vol: 25, Pages: 819-846, ISSN: 0266-4666
Ibragimov R, Jaffee D, Walden J, 2009, Nondiversification traps in catastrophe insurance markets, Review of Financial Studies, Vol: 22, Pages: 959-993, ISSN: 0893-9454
We develop a model for markets for catastrophic risk. The model explains why insurance providers may choose not to offer insurance for catastrophic risks and not to participate in reinsurance markets, even though there is a large enough market capacity to reach full risk sharing through diversification in a reinsurance market. This is a “nondiversification trap.” We show that nondiversification traps may arise when risk distributions have heavy left tails and insurance providers have limited liability. When they are present, there may be a coordination role for a centralized agency to ensure that risk sharing takes place.
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