Using Survey Sampling Algorithms For Exact Inference in Logistic Regression

Several exact inference procedures for logistic regression require the simulation of a 0-1 dependent vector according to its conditional distribution, given the sufficient statistics for some nuisance parameters. This is viewed, in this work, as a sampling problem involving a population of $n$ units, unequal selection probabilities and balancing constraints. The basis for this reformulation of exact inference is a proposition deriving the limit, as $n$ goes to infinity, of the conditional distribution of the dependent vector given the logistic regression sufficient statistics. It is proposed to sample from this distribution using the cube sampling algorithm. The interest of this approach to exact inference is illustrated by tackling new problems. First it allows to carry out exact inference with continuous covariates. It is also useful for the investigation of a partial correlation between several 0-1 vectors. This is illustrated in an example dealing with presence-absence data in ecology.     

Performance Measures in Dose-Finding Experiments

In the first phase of pharmaceutical development, and assuming that the probability of positive response increases with dose, the main statistical goal is to estimate a percentile of the dose–response function for a given target value $\mathrm{\Gamma }$. We compare the Maximum Likelihood and centred isotonic regression estimators of the target dose and we discuss several performance criteria to assess inferential precision, the amount of toxicity exposure and the trade-off between them for a set of some exemplary adaptive designs. We compare these designs using graphical tools. Several scenarios are considered using simulation, including the use of several start-up rules, the change of slope of the dose-toxicity function at the target dose and also different theoretical models, as logistic, normal or skew-normal distribution functions.     

Threshold estimation for continuous three-phase polynomial regression models with constant mean in the middle regime

This paper considers a continuous three-phase polynomial regression model with two threshold points for dependent data with heteroscedasticity. We assume the model is polynomial of order zero in the middle regime, and is polynomial of higher orders elsewhere. We denote this model by ${ℳ}_2$, which includes models with one or no threshold points, denoted by ${ℳ}_1$ and ${ℳ}_0$, respectively, as special cases. We provide an ordered iterative least squares (OiLS) method when estimating ${ℳ}_2$ and establish the consistency of the OiLS estimators under mild conditions. When the underlying model is ${ℳ}_1$ and is $(d_0-1)$th-order differentiable but not $d_0$th-order differentiable at the threshold point, we further show the $O_p(N^{-1/(d_0+2)})$ convergence rate of the OiLS estimators, which can be faster than the $O_p(N^{-1/(2d_0)})$ convergence rate given in Feder when $d_0\ge 3$. We also apply a model-selection procedure for selecting ${ℳ}_{\kappa }$; $\kappa =0,1,2$. When the underlying model exists, we establish the selection consistency under the aforementioned conditions. Finally, we conduct simulation experiments to demonstrate the finite-sample performance of our asymptotic results.     

A bivariate infinitely divisible law for modeling the magnitude and duration of monotone periods of log‐returns

The focus of this article is modeling the magnitude and duration of monotone periods of log‐returns. For this, we propose a new bivariate law assuming that the probabilistic framework over the magnitude and duration is based on the joint distribution of (X,N), where N is geometric distributed and X is the sum of an identically distributed sequence of inverse‐Gaussian random variables independent of N. In this sense, X and N represent the magnitude and duration of the log‐returns, respectively, and the magnitude comes from an infinite mixture of inverse‐Gaussian distributions. This new model is named bivariate inverse‐Gaussian geometric ( in short) law. We provide statistical properties of the model and explore stochastic representations. In particular, we show that the is infinitely divisible, and with this, an induced Lévy process is proposed and studied in some detail. Estimation of the parameters is performed via maximum likelihood, and Fisher's information matrix is obtained. An empirical illustration to the log‐returns of Tyco International stock demonstrates the superior performance of the law compared to an existing model. We expect that the proposed law can be considered as a powerful tool in the modeling of log‐returns and other episodes analyses such as water resources management, risk assessment, and civil engineering projects.     

Estimation in monotone single‐index models

Single‐index models are popular regression models that are more flexible than linear models and still maintain more structure than purely nonparametric models. We consider the problem of estimating the regression parameters under a monotonicity constraint on the unknown link function. In contrast to the standard approach of using smoothing techniques, we review different “non‐smooth” estimators that avoid the difficult smoothing parameter selection. For about 30 years, one has had the conjecture that the profile least squares estimator is an ‐consistent estimator of the regression parameter, but the only non‐smooth argmin/argmax estimators that are actually known to achieve this ‐rate are not based on the nonparametric least squares estimator of the link function. However, solving a score equation corresponding to the least squares approach results in ‐consistent estimators. We illustrate the good behavior of the score approach via simulations. The connection with the binary choice and current status linear regression models is also discussed.     

Stable Asymptotics for M‐estimators

We review some first‐order and higher‐order asymptotic techniques for M‐estimators, and we study their stability in the presence of data contaminations. We show that the estimating function (ψ) and its derivative with respect to the parameter play a central role. We discuss in detail the first‐order Gaussian density approximation, saddlepoint density approximation, saddlepoint test, tail area approximation via the Lugannani–Rice formula and empirical saddlepoint density approximation (a technique related to the empirical likelihood method). For all these asymptotics, we show that a bounded ψ (in the Euclidean norm) and a bounded (e.g. in the Frobenius norm) yield stable inference in the presence of data contamination. We motivate and illustrate our findings by theoretical and numerical examples about the benchmark case of one‐dimensional location model.     

THE (NON‐)SIGNIFICANCE OF REPORTING ERRORS IN ECONOMICS: EVIDENCE FROM THREE TOP JOURNALS

We investigate the prevalence and sources of reporting errors in 30,993 hypothesis tests from 370 articles in three top economics journals. We define reporting errors as inconsistencies between reported significance levels by means of eye‐catchers and calculated ‐values based on reported statistical values, such as coefficients and standard errors. While 35.8% of the articles contain at least one reporting error, only 1.3% of the investigated hypothesis tests are afflicted by reporting errors. For strong reporting errors for which either the eye‐catcher or the calculated ‐value signals statistical significance but the respective other one does not, the error rate is 0.5% for the investigated hypothesis tests corresponding to 21.6% of the articles having at least one strong reporting error. Our analysis suggests a bias in favor of errors for which eye‐catchers signal statistical significance but calculated ‐values do not. Survey responses from the respective authors, replications, and exploratory regression analyses indicate some solutions to mitigate the prevalence of reporting errors in future research.     

Bootstrap inference for impulse response functions in factor‐augmented vector autoregressions

In this study, we consider residual‐based bootstrap methods to construct the confidence interval for structural impulse response functions in factor‐augmented vector autoregressions. In particular, we compare the bootstrap with factor estimation (Procedure A) with the bootstrap without factor estimation (Procedure B). Both procedures are asymptotically valid under the condition , where N and T are the cross‐sectional dimension and the time dimension, respectively. However, Procedure A is also valid even when with 0 ≤ c < ∞ because it accounts for the effect of the factor estimation errors on the impulse response function estimator. Our simulation results suggest that Procedure A achieves more accurate coverage rates than those of Procedure B, especially when N is much smaller than T. In the monetary policy analysis of Bernanke et al. (Quarterly Journal of Economics, 2005, 120(1), 387–422), the proposed methods can produce statistically different results.     

On Families of Distributions with Shape Parameters

Univariate continuous distributions are one of the fundamental components on which statistical modelling, ancient and modern, frequentist and Bayesian, multi‐dimensional and complex, is based. In this article, I review and compare some of the main general techniques for providing families of typically unimodal distributions on with one or two, or possibly even three, shape parameters, controlling skewness and/or tailweight, in addition to their all‐important location and scale parameters. One important and useful family is comprised of the ‘skew‐symmetric’ distributions brought to prominence by Azzalini. As these are covered in considerable detail elsewhere in the literature, I focus more on their complements and competitors. Principal among these are distributions formed by transforming random variables, by what I call ‘transformation of scale’—including two‐piece distributions—and by probability integral transformation of non‐uniform random variables. I also treat briefly the issues of multi‐variate extension, of distributions on subsets of and of distributions on the circle. The review and comparison is not comprehensive, necessarily being selective and therefore somewhat personal. © 2014 The Authors. International Statistical Review © 2014 International Statistical Institute     

A central limit theorem for the Hellinger loss of Grenander‐type estimators

We consider Grenander‐type estimators for a monotone function , obtained as the slope of a concave (convex) estimate of the primitive of λ. Our main result is a central limit theorem for the Hellinger loss, which applies to estimation of a probability density, a regression function or a failure rate. In the case of density estimation, the limiting variance of the Hellinger loss turns out to be independent of λ.     

Forecasting stock returns with model uncertainty and parameter instability

We compare several representative sophisticated model averaging and variable selection techniques of forecasting stock returns. When estimated traditionally, our results confirm that the simple combination of individual predictors is superior. However, sophisticated models improve dramatically once we combine them with the historical average and take parameter instability into account. An equal weighted combination of the historical average with the standard multivariate predictive regression estimated using the average windows method, for example, achieves a statistically significant monthly out-of-sample of 1.10% and annual utility gains of 2.34%. We obtain similar gains for predicting future macroeconomic conditions.     

Mixed causal–noncausal autoregressions with exogenous regressors

Mixed causal–noncausal autoregressive (MAR) models have been proposed to model time series exhibiting nonlinear dynamics. Possible exogenous regressors are typically substituted into the error term to maintain the MAR structure of the dependent variable. We introduce a representation including these covariates called MARX to study their direct impact. The asymptotic distribution of the MARX parameters is derived for a class of non-Gaussian densities. For a Student likelihood, closed-form standard errors are provided. By simulations, we evaluate the MARX model selection procedure using information criteria. We examine the influence of the exchange rate and industrial production index on commodity prices.     

Upstream market structure and downstream partial ownership

Existing studies on partial ownership usually overlook the effects of vertically related markets. Our paper highlights the importance of the upstream market on downstream firms' incentives to acquire partial ownership and the consequent welfare implications. In the main model, we assume that there are three firms in the downstream market, two of which may form a partial ownership arrangement. We find several results that are in contrast to those in the literature. First, the two firms will engage in partial ownership if the upstream market is an oligopoly (triopoly or duopoly). Second, partial ownership may raise total production, consumer surplus, and social welfare. This happens when the upstream market consists of a duopoly and the two firms involved in partial ownership are supplied by different suppliers. Third, the outsider, commonly known as a free rider in the literature, may become a victim of partial ownership. Our results are robust to several extensions, including a general $n$-firm framework, product differentiation, and uniform pricing by upstream firms. We also provide the conditions under which the curvature of the demand function and the convexity of the cost function motivate firms to form partial ownership.     

BAYESIAN STATE SPACE MODELS IN MACROECONOMETRICS

State space models play an important role in macroeconometric analysis and the Bayesian approach has been shown to have many advantages. This paper outlines recent developments in state space modelling applied to macroeconomics using Bayesian methods. We outline the directions of recent research, specifically the problems being addressed and the solutions proposed. After presenting a general form for the linear Gaussian model, we discuss the interpretations and virtues of alternative estimation routines and their outputs. This discussion includes the Kalman filter and smoother, and precision-based algorithms. As the advantages of using large models have become better understood, a focus has developed on dimension reduction and computational advances to cope with high-dimensional parameter spaces. We give an overview of a number of recent advances in these directions. Many models suggested by economic theory are either non-linear or non-Gaussian, or both. We discuss work on the particle filtering approach to such models as well as other techniques that use various approximations – to either the time state and measurement equations or to the full posterior for the states – to obtain draws.     

WHAT DETERMINES THE ELASTICITY OF SUBSTITUTION BETWEEN CAPITAL AND LABOR? A LITERATURE REVIEW

This paper provides the first comprehensive review of the empirical and theoretical literature on the determinants of the elasticity of substitution between capital and labor. Our focus is on the two-input constant elasticity of substitution (CES) production function. We start by presenting four concise observations that summarize the empirical literature on the estimation of . Motivated by these observations, the main part of this survey then focuses on potential determinants of capital–labor substitution. We first review several approaches to the microfoundation of production functions where the elasticity of substitution (EOS) is treated as a purely technological parameter. Second, we outline the construction of an aggregate elasticity of substitution (AES) in a multi-sectoral framework and investigate its dependence on underlying intra- and inter-sectoral substitution. Third, we discuss the influence of the institutional framework on the extent of factor substitution. Overall, this survey highlights that the effective elasticity of substitution (EES), which is typically estimated in empirical studies, is generally not an immutable deep parameter but depends on a multitude of technological, non-technological, and institutional determinants. Based on these insights, the final section identifies a number of potential empirical and theoretical avenues for future research.