Robust econometric methods: Inference under economic and financial crises and other applications

Abstract

(Based on joint works with Xavier Gabaix, Paul Kattuman, Marat Ibragimov and Ulrich Mueller) The talk focuses on several recently developed approaches to robust inference for heterogeneous, dependent and heavy-tailed data and their applications in economics and finance. The first part of the talk will discuss robust inference on heavy-tailedness using modifications of log-log rank-size regressions with optimal shifts in ranks and correct standard errors, with applications in the analysis of emerging country foreign exchange markets. 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)〗^(1/2)ζ. 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. We provide several applications of the proposed tail index estimation methods, including the analysis of heavy-tailedness for exchange rates in emerging countries. 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 the above recently proposed robust tail index estimation methods. According to the estimation results obtained 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. Motivated, in part, by empirical results on heavy-tailedness and dependence in economic and financial markets, including those discussed in the first part of the talk, its second part will focus on new general approaches to robust inference when the data is potentially heterogeneous and correlated in a largely unknown way. The approach is based on small sample conservativeness properties of the standard t-statistic and Behrens-Fisher statistic for testing equality of means. These properties show that, for commonly used significance levels, the t- and Behrens-Fisher tests remain 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 some number of groups, estimate the model for each group, and conduct standard t- or Behrens-Fisher test with the resulting parameter estimators of interest. This results in valid and in some sense efficient inference with appealing finite sample properties when the groups are chosen in a way that ensures the parameter estimators to be asymptotically independent, unbiased and Gaussian of possibly different variances. The talk will discuss a number of applications of the new robust inference approaches to time series, panel, clustered, spatially correlated and heavy-tailed data. The applications discussed will include robust analysis of structural breaks and treatment effects, inference for heavy-tailed stochastic volatility models and robust econometric analysis of income and wealth distributions, among others.