Empirical Likelihood and Bootstrap Inference with Constraints

Empirical likelihood and the bootstrap play influential roles in contemporary statistics. This thesis studies two distinct statistical inference problems, referred to as Part I and Part II, related to the empirical likelihood and bootstrap, respectively. Part I of this thesis concerns making statistical inferences on multiple groups of samples that contain excess zero observations. A unique feature of the target populations is that the distribution of each group is characterized by a non-standard mixture of a singular distribution at zero and a skewed nonnegative component. In Part I of this thesis, we propose modelling the nonnegative components using a semiparametric, multiple-sample, density ratio model (DRM). Under this semiparametric setup, we can efficiently utilize information from the combined samples even with unspecified underlying distributions. We first study the question of testing homogeneity of multiple nonnegative distributions when there is an excess of zeros in the data, under the proposed semiparametric setup. We develop a new empirical likelihood ratio (ELR) test for homogeneity and show that this ELR has a $\chi^2$-type limiting distribution under the homogeneous null hypothesis. A nonparametric bootstrap procedure is proposed to calibrate the finite-sample distribution of the ELR. The consistency of this bootstrap procedure is established under both the null and alternative hypotheses. Simulation studies show that the bootstrap ELR test has an accurate nominal type I error, is robust to changes of underlying distributions, is competitive to, and sometimes more powerful than, several popular one- and two-part tests. A real data example is used to illustrate the advantages of the proposed test. We next investigate the problem of comparing the means of multiple nonnegative distributions, with excess zero observations, under the proposed semiparametric setup. We develop a unified inference framework based on our new ELR statistic, and show that this ELR has a $\chi^2$-type limiting distribution under a general null hypothesis. This allows us to construct a new test for mean equality. Simulation results show favourable performance of the proposed ELR test compared with other existing tests for mean equality, especially when the correctly specified basis function in the DRM is the logarithm function. A real data set is analyzed to illustrate the advantages of the proposed method. In Part II of this thesis, we investigate the asymptotic behaviour of, the commonly used, bootstrap percentile confidence intervals when the parameters are subject to inequality constraints. We concentrate on the important one- and two-sample problems with data generated from distributions in the natural exponential family. Our attention is focused on quantifying asymptotic coverage probabilities of the percentile confidence intervals based on bootstrapping maximum likelihood estimators. We propose a novel local framework to study the subtle asymptotic behaviour of bootstrap percentile confidence intervals when the true parameter values are close to the boundary. Under this framework, we discover that when the true parameter is on, or close to, the restriction boundary, the local asymptotic coverage probabilities can always exceed the nominal level in the one-sample case; however, they can be, surprisingly, both under and over the nominal level in the two-sample case. The results provide theoretical justification and guidance on applying the bootstrap percentile method to constrained inference problems. The two individual parts of this thesis are connected by being referred to as {\em constrained statistical inference}. Specifically, in Part I, the semiparametric density ratio model uses an exponential tilting constraint, which is a type of equality constraint, on the parameter space. In Part II, we deal with inequality constraints, such as a boundary or ordering constraints, on the parameter space. For both parts, an important regularity condition in traditional likelihood inference, that parameters should be interior points of the parameter space, is violated. Therefore, the respective inference procedures involve non-standard asymptotics that create new technical challenges

Prices and Portfolio Choices in Financial Markets: Theory, Econometrics, Experiments

Many tests of asset-pricing models address only the pricing predictions, but these pricing predictions rest on portfolio choice predictions that seem obviously wrong. This paper suggests a new approach to asset pricing and portfolio choices based on unobserved heterogeneity. This approach yields the standard pricing conclusions of classical models but is consistent with very different portfolio choices. Novel econometric tests link the price and portfolio predictions and take into account the general equilibrium effects of sample-size bias. This paper works through the approach in detail for the case of the classical capital asset pricing model (CAPM), producing a model called CAPM+ε. When these econometric tests are applied to data generated by large-scale laboratory asset markets that reveal both prices and portfolio choices, CAPM+εis not rejected

Semiparametric inference on the means of multiple nonnegative distributions with excess zero observations

The final publication is available at Elsevier via https://dx.doi.org/10.1016/j.jmva.2018.02.010 © 2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/A non-standard, but not uncommon, situation is to observe multiple samples of nonnegative data which have a high proportion of zeros. This is the so-called excess of zeros situation and this paper looks at the problem of making inferences about the means of the underlying distributions. Under the semiparametric setup, proposed by Wang et al. (2017), we develop a unified inference framework, based on an empirical likelihood ratio (ELR) statistic, for making inferences on the means of multiple such distributions. A chi-square-type limiting distribution of this statistic is established under a general linear null hypothesis about the means. This result allows us to construct a new test for mean equality. Simulation results show favorable performance of the proposed ELR when compared with other existing methods for testing mean equality, especially when the correctly specified basis function in the density ratio model is the logarithm function. A real data set is analyzed to illustrate the advantages of the proposed method.Natural Sciences and Engineering Research Council of Canada (Grants RGPIN-2014-05424; RGPIN-2015-06592)Fundamental Research Funds for the Central Universities (Grants 20720181043; 20720181003

Demand Dispersion, Metonymy and Ideal Panel Data

In a generic competitive economy with constant returns production and "increasing dispersion," market demand satisfies the weak axiom of revealed preference and equilibrium is unique. Increasing dispersion requires, roughly, that when the households' incomes rise slightly their demand vectors move apart. We show how to test for it using panel data with fixed relative prices under a "structural stability" hypothesis due to Hildenbrand and Kneip (1999). We also show how to test for it using cross section data if the households' demand functions and incomes are independently distributed, or under a much weaker condition called "dispersion metonymy." We show that this weaker condition is untestable---even with ideal panel data that allow a direct test of increasing dispersion. Thus, cross section tests of increasing dispersion rely on an assumption that is not potentially falsifiable.Aggregation, Weak Axiom, Increasing Dispersion, Cross Section, Structural Stability

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Infinite Density at the Median and the Typical Shape of Stock Return Distributions

Statistics are developed to test for the presence of an asymptotic discontinuity (or infinite density or peakedness) in a probability density at the median. The approach makes use of work by Knight (1998) on L_1 estimation asymptotics in conjunction with non-parametric kernel density estimation methods. The size and power of the tests are assessed, and conditions under which the tests have good performance are explored in simulations. The new methods are applied to stock returns of leading companies across major U.S. industry groups. The results confirm the presence of infinite density at the median as a new significant empirical evidence for stock return distributions.Asymptotic leptokurtosis, Infinite density at the median, Least absolute deviations, Kernel density estimation, Stock returns, Stylized facts

Dynamic Seemingly Unrelated Cointegrating Regression

Multiple cointegrating regressions are frequently encountered in empirical work as, for example, in the analysis of panel data. When the equilibrium errors are correlated across equations, the seemingly unrelated regression estimation strategy can be applied to cointegrating regressions to obtain asymptotically ecient estimators. While non-parametric methods for seemingly unrelated cointegrating regressions have been proposed in the literature, in practice, specification of the estimation problem is not always straightforward. We propose Dynamic Seemingly Unrelated Regression (DSUR) estimators which can be made fully parametric and are computationally straightforward to use. We study the asymptotic and small sample properties of the DSUR estimators both for heterogeneous and homogenous cointegrating vectors. The estimation techniques are then applied to analyze two long-standing problems in international economics. Our first application revisits the issue of whether the forward exchange rate is an unbiased predictor of the future spot rate. Our second application revisits the problem of estimating long-run correlations between national investment and national saving.

Generic inference on quantile and quantile effect functions for discrete outcomes

Quantile and quantile effect functions are important tools for descriptive and inferential analysis due to their natural and intuitive interpretation. Existing inference methods for these functions do not apply to discrete random variables. This paper offers a simple, practical construction of simultaneous confidence bands for quantile and quantile effect functions of possibly discrete random variables. It is based on a natural transformation of simultaneous confidence bands for distribution functions, which are readily available for many problems. The construction is generic and does not depend on the nature of the underlying problem. It works in conjunction with parametric, semiparametric, and nonparametric modeling strategies and does not depend on the sampling scheme. We apply our method to characterize the distributional impact of insurance coverage on health care utilization and obtain the distributional decomposition of the racial test score gap. Our analysis generates new, interesting empirical findings, and complements previous analyses that focused on mean effects only. In both applications, the outcomes of interest are discrete rendering existing inference methods invalid for obtaining uniform confidence bands for quantile and quantile effects functions.https://arxiv.org/abs/1608.05142First author draf

Geometrically stopped Markovian random growth processes and Pareto tails

Many empirical studies document power law behavior in size distributions of economic interest such as cities, firms, income, and wealth. One mechanism for generating such behavior combines independent and identically distributed Gaussian additive shocks to log-size with a geometric age distribution. We generalize this mechanism by allowing the shocks to be non-Gaussian (but light-tailed) and dependent upon a Markov state variable. Our main results provide sharp bounds on tail probabilities, a simple equation determining Pareto exponents, and comparative statics. We present two applications: we show that (i) the tails of the wealth distribution in a heterogeneous-agent dynamic general equilibrium model with idiosyncratic investment risk are Paretian, and (ii) a random growth model for the population dynamics of Japanese municipalities is consistent with the observed Pareto exponent but only after allowing for Markovian dynamics