Tianyu He
PhD Candidate, McGill University
Working papers
Measuring inequality for winners and losers: extended Lorenz curves and Gini indices for possibly negative variables (Job market paper)
Measuring inequality is not straightforward, as we face challenges such as selection of variables, distributional features of interest and reference points. The last problem becomes obvious in the presence of negative observations, as common inequality measurements are not defined. The extended inequality measurements have been proposed to incorporate negative values, which, however, have been used essentially as descriptive devices so far. To make these tools formal, we provide a comprehensive study on estimation and asymptotic inference for these extended inequality measurements, including extended Lorenz curves and extended Gini indices. We establish asymptotic normality of the estimators and build confidence intervals for the extended Lorenz curves at a fixed vector of percentiles. To make inference for the entire curves, we prove that the empirical extended Lorenz curves convergence uniformly to Gaussian processes and construct uniform confidence bands for the entire curves by nonparametric bootstrap based on supreme-type test statistics. We also provide asymptotic results of extended Gini indices by U-statistic theory and delta method. We apply our methods to study inequality of household financial incomes in Italy where a non-negligible proportion of financial losses are observed. We observe that the inequality of financial gains is much higher than that of financial losses. Moreover, the traditional inequality metrics, by ignoring negative observations, underestimate the overall inequality level. Through a decomposition of positive Gini index, we find that the inequality of positive financial gains and gross inequality between financial gains and losses are important source of overall inequality. We exploit the usage of extended inequality metrics as dispersion measures by investigating the dispersion of cross-sectional US stock returns where nearly half of observations are negative. We find that dispersion of positive stock returns is larger than that of losses. By decomposing the positive Gini index, we find that more than half of overall dispersion comes from gross dispersion between positive and negative returns. We further find that the dispersion of cross-sectional monthly stock returns is very high from 2000 to 2019 and slightly higher during periods of bear markets.
Statistical inference for comparing extended inequality measures for possibly negative variables
The extended inequality metrics have been proposed recently to incorporate possibly negative variables but essentially used as descriptive devices so far. We make these tools formal by providing estimation and asymptotic inference for comparing extended inequality measures, including extended generalized Lorenz shares, extended Lorenz shares and extended Gini indices for independent samples and completely dependent samples (i.e. matched pairs of observations). We also derive results for partially dependent samples due to the design of household surveys and feature of extended Lorenz curves. We apply our methods to study inequality of household financial income in Italy in 2012. We find that the inequality of financial gains in South Italy is slightly lower than that in North Italy, while the opposite is true for inequality of financial losses and the entire financial incomes. The negative observations have impacts on measuring inequality changes, as the traditional inequality metrics, by ignoring financial losses, yield incorrect results about comparison of overall inequality levels. We also find that the inequality decreased from 2012 to 2014 for entire financial incomes, financial gains and losses. To exploit the potential usage as dispersion measures, we compare dispersion of cross-sectional annual stock returns for large capitalized (large-cap) and small capitalized (small-cap) firms in 2014. The empirical results imply that the small-cap firms have a larger overall dispersion of stock returns than large-cap firms. By examining the monthly data from 2000 to 2019, we find that small-cap firms in general have a larger dispersion than large-cap firms in terms of overall stock returns, positive returns and negative returns.
Exact distribution-free confidence intervals for comparing means of bounded variables with application to poverty measures
(joint with Jean-Marie Dufour)
The comparison of poverty measures is mainly based on asymptotic and bootstrap statistical inferences in the literature. These procedures, however, can be quite unreliable even for relatively large sample sizes. In this paper, we develop finite-sample nonparametric inference methods for the equality of the Foster–Greer–Thorbecke (FGT, 1984) poverty measures. The inference procedures are robust to arbitrary degree of dependence between two samples and allow extremely unbalanced sample sizes. Observing that FGT poverty measures are essentially expectations of bounded variables, we propose exact inference procedures for comparing FGT poverty measures by projection and intersection-union techniques based on improved standardized Kolmogorov-Smirnov statistics and a likelihood-ratio criterion. Our simulation results confirm that asymptotic and bootstrap tests can fail to provide reliable inference, while our methods control the level and exhibit good power performance. Our proposed confidence intervals also yield good coverage rates for the difference of FGT poverty measures. As an illustration, we apply our method to compare poverty levels between urban and rural areas in China in 2013. The empirical results imply that the rural households are poorer than urban households at level of both country and macro-areas (East, Center and West). But we have found mixed evidence about such poverty difference at provincial level.
Identification-robust tests for probit models with endogenous regressors
(joint with Jean-Marie Dufour)
Weak identification is a well-known issue in the context of linear structural models but is less studied in binary outcome models. In this paper, we focus on weak identification in probit models with endogenous regressors and propose the asymptotic maximized Monte Carlo test which is identification-robust. We compare our tests in simulation experiments to generalized minimum distance (MD) robust tests and common asymptotic tests including Wald, Lagrangian multiplier and likelihood ratio (LR) tests based on generalized method of moments (GMM), and likelihood ratio tests based on maximum likelihood estimators (MLE). We find that LR test based on MLE can have large level distortions in the presence of weak identification which is rarely documented in the literature. Meanwhile, our proposed tests control the level regardless of whether the structural parameters are identified. As for power of tests, the simulation evidence suggests that the proposed tests exhibit reasonable power compared with MD type tests and asymptotic tests based on GMM and MLE whose critical values are locally corrected. We finally apply our method to analyze labor force participation of married female.
Working in progress
Exact semi-parametric inference for Lorenz and extended Lorenz shares