Edgeworth expansions and normalizing transforms for inequality measures

Finite sample distributions of studentized inequality measures differ substantially from their asymptotic normal distribution in terms of location and skewness. We study these aspects formally by deriving the second order expansion of the first and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coefficients. In the second part we improve over first-order theory by deriving Edgeworth expansions and normalizing transforms. These normalizing transforms are designed to eliminate the second order term in the distributional expansion of the studentized transform and converge to the Gaussian limit at rate O(n−1). This leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n−3/2). We illustrate our procedure with an application to regional inequality measurement in Côte d'Ivoire

Almost Similar Tests for Mediation Effects and other Hypotheses with Singularities

Testing for mediation effects is empirically important and theoretically interesting. It is important in psychology, medicine, economics, accountancy, and marketing for instance, generating over 90,000 citations to a single key paper in the field. It also leads to a statistically interesting and long-standing problem that this paper solves. The no-mediation hypothesis, expressed as $H_{0}:\theta_{1}\theta_{2}=0$, defines a manifold that is non-regular in the origin where rejection probabilities of standard tests are extremely low. We propose a general method for obtaining near similar tests using a flexible $g$-function to bound the critical region. We prove that no similar test exists for mediation, but using our new varying $g$-method obtain a test that is all but similar and easy to use in practice. We derive tight upper bounds to similar and nonsimilar power envelopes and derive an optimal test. We extend the test to higher dimensions and illustrate the results in a trade union sentiment application

Fractional Matrix Calculus Introduction, Applications and Future Developments

Fractional Calculus gives a generalisation of the common techniques of integration and differentiation. Although the origin of Fractional Calculus lies more than 150 years behind us, it is still quite unknown. Recently however, Phillips (1984,1985) used fractional calculus to derive the distributions of the Stein-Rule and SUB estimator and Knight (1986b) used it to find the moments of the 2SLS estimator. This paper gives an introduction to fractional calculus by showing the ideas behind the definitions, giving examples and some applications. At the end an attempt is made to show the relevance of this calculus for the different fields of Econometrics and to put it in a more general mathematical perspective

Exact geometry of explosive autoregressive models

This paper derives exact expressions for statistical curvature and related geometric quantities in the first order autoregressive models with stable and unit roots, as well as explosive roots larger than unity. We develop a method for deriving exact moments of arbitrary order in general autoregressive models. The covariance of the minimal sufficient statistic is an application of this method. Of particular interest is the Efron curvature which is continuous and bounded in finite samples, but increases rapidly when the autoregressive parameter changes from stable to explosive values, which has important inferential consequences. The initial value effect is also quantified exactly. We also include results for the Efron curvature in the pure stationary case with stochastic initial value for comparison, extending and correcting results from a previous discussion paper.Differential geometry, statistical curvature, exact distribution theory, exact mgf, unit root, explosive time series, conditional inference, curved exponential models

Exact Geometry of Autoregressive Models

This paper derives exact expressions for the statistical curvature and related geometric quantities in the first order autoregressive models. We present a method that combines the algebra of differential and difference operators to simplify the problem, and to obtain results valid for all sample sizes. The exact covariance matrix for the sufficient statistic is also derived.Differential geometry, statistical curvature, time series, curved exponential models, exact distribution theory

Curved Exponential Models in Econometrics

Curved exponential models have the property that the dimension of the minimal sufficient statistic is larger than the number of parameters in the model. Many econometric models share this feature. The first part of the paper shows that, in fact, econometric models with this property are necessarily curved exponential. A method for constructing an explicit set of minimal sufficient statistics, based on partial scores and likelihood ratios, is given. The difference in dimension between parameterand statistic and the curvature of these models have important consequences for inference. It is not the purpose of this paper to contribute significantly to the theory of curved exponential models, other than to show that the theory applies to many econometric models and to highlight some multivariate aspects. Using the methods developed in the first part, we show that demand systems, the single structural equation model, the seemingly unrelated regressions, and autoregressive models are all curved exponential models.

Edgeworth expansions and normalizing transforms for inequality measures

Finite sample distributions of studentized inequality measures differ substantially from their asymptotic normal distribution in terms of location and skewness. We study these aspects formally by deriving the second-order expansion of the first and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coefficients. In the second part we improve over first-order theory by deriving Edgeworth expansions and normalizing transforms. These normalizing transforms are designed to eliminate the second-order term in the distributional expansion of the studentized transform and converge to the Gaussian limit at rate O(n-1). This leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n-3/2). We illustrate our procedure with an application to regional inequality measurement in Côte d'Ivoire.Generalized Entropy inequality measures Higher- order expansions Normalizing transformations

Improved tests for Mediation

Testing for a mediation effect is important in many disciplines, but is made difficult – even asymptotically – by the influence of nuisance parameters. Classical tests such as likelihood ratio (LR) and Wald tests have very poor size and power properties in some parts of the parameter space, and many attempts have been made to produce improved tests, with limited success. In this paper we show that augmenting the critical region of the LR test can produce a test with much improved behaviour everywhere. In fact, we first show that there exists a test of this type that is (asymptotically) exact for certain test sizes α, including the common choices α = .01, .05, .10. This is evidently an important result, but we also observe that the critical region of this exact test has some undesirable properties. Thus, we then go on to show that there is a very simple class of augmented LR critical regions which provides tests that, while not exact, are very nearly so, and which avoid the issues inherent in the exact test. We suggest an optimal member of this class, and provide the tables needed to implement it. Although motivated by a simple two-equation linear model, the results apply to any model structure that reduces to the same testing problem asymptotically. A short application of the method to an entrepreneurial attitudes study is included for illustration