Jurimetrics: the Methodology of Legal Inquiry
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Instrumental Regression in Partially Linear Models
10-167 Research Group: Econometrics and Statistics September, 2009 Instrumental Regression in Partially Linear Models JEAN-PIERRE FLORENS, JAN JOHANNES AND SEBASTIEN VAN BELLEGEM INSTRUMENTAL REGRESSION IN PARTIALLY LINEAR MODELS∗ Jean-Pierre Florens1 Jan Johannes2 Sebastien´ Van Bellegem3 First version: January 24, 2008. This version: September 10, 2009 Abstract We consider the semiparametric regression Xtβ+φ(Z) where β and φ( ) are unknown · slope coefficient vector and function, and where the variables (X,Z) are endogeneous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of β. An incorrect parameterization of φ may generally lead to an inconsistent estimator of β, whereas even consistent nonparametric estimators for φ imply a slow rate of convergence of the estimator of β. An additional complication is that the solution of the equation necessitates the inversion of a compact operator that has to be estimated nonparametri- cally. In general this inversion is not stable, thus the estimation of β is ill-posed. In this paper, a √n-consistent estimator for β is derived under mild assumptions. One of these assumptions is given by the so-called source condition that is explicitly interprated in the paper. Finally we show that the estimator achieves the semiparametric efficiency bound, even if the model is heteroscedastic. Monte Carlo simulations demonstrate the reasonable performance of the estimation procedure on finite samples. Keywords: Partially linear model, semiparametric regression, instrumental variables, endo- geneity, ill-posed inverse problem, Tikhonov regularization, root-N consistent estimation, semiparametric efficiency bound JEL classifications: Primary C14; secondary C30 ∗We are grateful to R. -
Empire, Trade, and the Use of Agents in the 19Th Century: the “Reception” of the Undisclosed Principal Rule in Louisiana Law and Scots Law
Louisiana Law Review Volume 79 Number 4 Summer 2019 Article 6 6-19-2019 Empire, Trade, and the Use of Agents in the 19th Century: The “Reception” of the Undisclosed Principal Rule in Louisiana Law and Scots Law Laura Macgregor Follow this and additional works at: https://digitalcommons.law.lsu.edu/lalrev Part of the Agency Commons, and the Commercial Law Commons Repository Citation Laura Macgregor, Empire, Trade, and the Use of Agents in the 19th Century: The “Reception” of the Undisclosed Principal Rule in Louisiana Law and Scots Law, 79 La. L. Rev. (2019) Available at: https://digitalcommons.law.lsu.edu/lalrev/vol79/iss4/6 This Article is brought to you for free and open access by the Law Reviews and Journals at LSU Law Digital Commons. It has been accepted for inclusion in Louisiana Law Review by an authorized editor of LSU Law Digital Commons. For more information, please contact [email protected]. Empire, Trade, and the Use of Agents in the 19th Century: The “Reception” of the Undisclosed Principal Rule in Louisiana Law and Scots Law Laura Macgregor* TABLE OF CONTENTS Introduction .................................................................................. 986 I. The Nature and Economic Benefits of Undisclosed Agency ..................................................................... 992 II. The Concept of a “Mixed Legal System” and Agency Law in Mixed Legal System Scholarship ....................... 997 III. Nature and Historical Development of Scots Law ..................... 1002 A. The Reception of Roman Law ............................................. 1002 B. The Institutional Period and Union with England ............... 1004 C. The Development of Scots Commercial Law ...................... 1006 D. When Did Scots Law Become Mixed in Nature? ................ 1008 IV. Undisclosed and Unidentified Agency in English Law ............ -
Arxiv:1502.07813V1 [Cs.LG] 27 Feb 2015 Distributions of the Form: M Pr(X;M ) = ∑ W J F J(X;Θ J) (1) J=1
Minimum message length estimation of mixtures of multivariate Gaussian and von Mises-Fisher distributions Parthan Kasarapu · Lloyd Allison Abstract Mixture modelling involves explaining some observed evidence using a combination of probability distributions. The crux of the problem is the inference of an optimal number of mixture components and their corresponding parameters. This paper discusses unsupervised learning of mixture models using the Bayesian Minimum Message Length (MML) crite- rion. To demonstrate the effectiveness of search and inference of mixture parameters using the proposed approach, we select two key probability distributions, each handling fundamentally different types of data: the multivariate Gaussian distribution to address mixture modelling of data distributed in Euclidean space, and the multivariate von Mises-Fisher (vMF) distribu- tion to address mixture modelling of directional data distributed on a unit hypersphere. The key contributions of this paper, in addition to the general search and inference methodology, include the derivation of MML expressions for encoding the data using multivariate Gaussian and von Mises-Fisher distributions, and the analytical derivation of the MML estimates of the parameters of the two distributions. Our approach is tested on simulated and real world data sets. For instance, we infer vMF mixtures that concisely explain experimentally determined three-dimensional protein conformations, providing an effective null model description of protein structures that is central to many inference problems in structural bioinformatics. The ex- perimental results demonstrate that the performance of our proposed search and inference method along with the encoding schemes improve on the state of the art mixture modelling techniques. Keywords mixture modelling · minimum message length · multivariate Gaussian · von Mises-Fisher · directional data 1 Introduction Mixture models are common tools in statistical pattern recognition (McLachlan and Basford, 1988). -
Nmo Ash University
S ISSN 1440-771X ISBN 0 7326 1048 6 NMO ASH UNIVERSITY, AUSTRALIA t 1 NOV 498 Comparisons of Estimators and Tests Based on Modified Likelihood and Message Length Functions Mizan R. Laskar and Maxwell L. King Working Paper 11/98 August 1998 .DEPARTMENT OF ECONOMETRICS AND BUSINESS STATISTICS COMPARISONS OF ESTIMATORS AND TESTS BASED ON MODIFIED LIKELIHOOD AND MESSAGE LENGTH FUNCTIONS - Mizan R. Laskar and Maxwell L. King Department of Econometrics and Business Statistics Monash University Clayton, Victoria 3168 Australia Summary The presence of nuisance parameters causes unwanted complications in statistical and econometric inference procedures. A number of modified likelihood and message length functions have been developed for better handling of nuisance parameters but they are not equally efficient. In this paper, we empirically compare different modified likelihood and message length functions in the context of estimation and testing of parameters from linear regression disturbances that follow either first-order moving average or first-order autoregressive error processes. The results show that estimators based on the conditional profile likelihood and tests based on the marginal likelihood are best. If there is a minor identification problem, the sizes of the likelihood ratio and Wald tests based on simple message length functions are best. The true sizes of the Lagrange multiplier tests based on message length functions are rather poor because the score functions of message length functions are biased. Key words: linear regression model; marginal likelihood; conditional profile likelihood; first-order moving average errors; first-order autoregressive errors. 1. Introduction Satisfactory statistical analysis of non-experimental data is an important problem in statistics and econometrics. -
A Minimum Message Length Criterion for Robust Linear Regression
A Minimum Message Length Criterion for Robust Linear Regression Chi Kuen Wong, Enes Makalic, Daniel F. Schmidt February 21, 2018 Abstract This paper applies the minimum message length principle to inference of linear regression models with Student-t errors. A new criterion for variable selection and parameter estimation in Student-t regression is proposed. By exploiting properties of the regression model, we de- rive a suitable non-informative proper uniform prior distribution for the regression coefficients that leads to a simple and easy-to-apply criterion. Our proposed criterion does not require specification of hyperparameters and is invariant under both full rank transformations of the design matrix and linear transformations of the outcomes. We compare the proposed criterion with several standard model selection criteria, such as the Akaike information criterion and the Bayesian information criterion, on simulations and real data with promising results. 1 Introduction ′ Consider a vector of n observations y = (y1,...,yn) generated by the linear regression model y = β01n + Xβ + ε , (1) where X = (x ,..., x ) is the design matrix of p explanatory variables with each x Rn, β Rp 1 k i ∈ ∈ is the vector of regression coefficients, β0 R is the intercept parameter, 1n is a vector of 1s of ′ ∈ length n, and ε = (ε1,...,εn) is a vector of random errors. In this paper, the random disturbances εi are assumed to be independently and identically distributed as per a Student-t distribution with location zero, scale τ, and ν degrees of freedom. The Student-t linear regression model finds frequent application in modelling heavy-tailed data [1, 2]. -
Using Epidemiological Evidence in Tort Law: a Practical Guide Aleksandra Kobyasheva
Using epidemiological evidence in tort law: a practical guide Aleksandra Kobyasheva Introduction Epidemiology is the study of disease patterns in populations which seeks to identify and understand causes of disease. By using this data to predict how and when diseases are likely to arise, it aims to prevent the disease and its consequences through public health regulation or the development of medication. But can this data also be used to tell us something about an event that has already occurred? To illustrate, consider a case where a claimant is exposed to a substance due to the defendant’s negligence and subsequently contracts a disease. There is not enough evidence available to satisfy the traditional ‘but-for’ test of causation.1 However, an epidemiological study shows that there is a potential causal link between the substance and the disease. What is, or what should be, the significance of this study? In Sienkiewicz v Greif members of the Supreme Court were sceptical that epidemiological evidence can have a useful role to play in determining questions of causation.2 The issue in that case was a narrow one that did not present the full range of concerns raised by epidemiological evidence, but the court’s general attitude was not unusual. Few English courts have thought seriously about the ways in which epidemiological evidence should be approached or interpreted; in particular, the factors which should be taken into account when applying the ‘balance of probability’ test, or the weight which should be attached to such factors. As a result, this article aims to explore this question from a practical perspective. -
Suboptimal Behavior of Bayes and MDL in Classification Under
Mach Learn (2007) 66:119–149 DOI 10.1007/s10994-007-0716-7 Suboptimal behavior of Bayes and MDL in classification under misspecification Peter Gr¨unwald · John Langford Received: 29 March 2005 / Revised: 15 December 2006 / Accepted: 20 December 2006 / Published online: 30 January 2007 Springer Science + Business Media, LLC 2007 Abstract We show that forms of Bayesian and MDL inference that are often applied to classification problems can be inconsistent. This means that there exists a learning prob- lem such that for all amounts of data the generalization errors of the MDL classifier and the Bayes classifier relative to the Bayesian posterior both remain bounded away from the smallest achievable generalization error. From a Bayesian point of view, the result can be reinterpreted as saying that Bayesian inference can be inconsistent under misspecification, even for countably infinite models. We extensively discuss the result from both a Bayesian and an MDL perspective. Keywords Bayesian statistics . Minimum description length . Classification . Consistency . Inconsistency . Misspecification 1 Introduction Overfitting is a central concern of machine learning and statistics. Two frequently used learning methods that in many cases ‘automatically’ protect against overfitting are Bayesian inference (Bernardo & Smith, 1994) and the Minimum Description Length (MDL) Principle (Rissanen, 1989; Barron, Rissanen, & Yu, 1998; Gr¨unwald, 2005, 2007). We show that, when applied to classification problems, some of the standard variations of these two methods can be inconsistent in the sense that they asymptotically overfit: there exist scenarios where, no matter how much data is available, the generalization error of a classifier based on MDL Editors: Olivier Bousquet and Andre Elisseeff P. -
Efficient Linear Regression by Minimum Message Length
1 Efficient Linear Regression by Minimum Message Length Enes Makalic and Daniel F. Schmidt Abstract This paper presents an efficient and general solution to the linear regression problem using the Minimum Message Length (MML) principle. Inference in an MML framework involves optimising a two-part costing function that describes the trade-off between model complexity and model capability. The MML criterion is integrated into the orthogonal least squares algorithm (MML-OLS) to improve both speed and numerical stability. This allows for the message length to be iteratively updated with the selection of each new regressor, and for potentially problematic regressors to be rejected. The MML- OLS algorithm is subsequently applied to function approximation with univariate polynomials. Empirical results demonstrate superior performance in terms of mean squared prediction error in comparison to several well-known benchmark criteria. I. INTRODUCTION Linear regression is the process of modelling data in terms of linear relationships between a set of independent and dependent variables. An important issue in multi-variate linear modelling is to determine which of the independent variables to include in the model, and which are superfluous and contribute little to the power of the explanation. This paper formulates the regressor selection problem in a Minimum Message Length (MML) [1] framework which offers a sound statistical basis for automatically discriminating between competing models. In particular, we combine the MML criterion with the well- known Orthogonal Least Squares (OLS) [2] algorithm to produce a search (MML-OLS) that is both numerically stable and computationally efficient. To evaluate the performance of the proposed algorithm Enes Makalic and Daniel Schmidt are with Monash University Clayton School of Information Technology Clayton Campus Victoria 3800, Australia. -
Partially Linear Hazard Regression for Multivariate Survival Data
Partially Linear Hazard Regression for Multivariate Survival Data Jianwen CAI, Jianqing FAN, Jiancheng JIANG, and Haibo ZHOU This article studies estimation of partially linear hazard regression models for multivariate survival data. A profile pseudo–partial likeli- hood estimation method is proposed under the marginal hazard model framework. The estimation on the parameters for the√ linear part is accomplished by maximization of a pseudo–partial likelihood profiled over the nonparametric part. This enables us to obtain n-consistent estimators of the parametric component. Asymptotic normality is obtained for the estimates of both the linear and nonlinear parts. The new technical challenge is that the nonparametric component is indirectly estimated through its integrated derivative function from a lo- cal polynomial fit. An algorithm of fast implementation of our proposed method is presented. Consistent standard error estimates using sandwich-type ideas are also developed, which facilitates inferences for the model. It is shown that the nonparametric component can be estimated as well as if the parametric components were known and the failure times within each subject were independent. Simulations are conducted to demonstrate the performance of the proposed method. A real dataset is analyzed to illustrate the proposed methodology. KEY WORDS: Local pseudo–partial likelihood; Marginal hazard model; Multivariate failure time; Partially linear; Profile pseudo–partial likelihood. 1. INTRODUCTION types of dependence that can be modeled, and model -
Analysis of Scientific Research on Eyewitness Identification
ANALYSIS OF SCIENTIFIC RESEARCH ON EYEWITNESS IDENTIFICATION January 2018 Patricia A. Riley U.S. Attorney’s Office, DC Table of Contents RESEARCH DOES NOT PROVIDE A FIRM FOUNDATION FOR JURY INSTRUCTIONS OF THE TYPE ADOPTED BY SOME COURTS OR, IN SOME INSTANCES, FOR EXPERT TESTIMONY ...................................................... 6 RESEARCH DOES NOT PROVIDE A FIRM FOUNDATION FOR JURY INSTRUCTIONS OF THE TYPE ADOPTED BY SOME COURTS AND, IN SOME INSTANCES, FOR EXPERT TESTIMONY .................................................... 8 Introduction ............................................................................................................................................. 8 Courts should not comment on the evidence or take judicial notice of contested facts ................... 11 Jury Instructions based on flawed or outdated research should not be given .................................. 12 AN OVERVIEW OF THE RESEARCH ON EYEWITNESS IDENTIFICATION OF STRANGERS, NEW AND OLD .... 16 Important Points .................................................................................................................................... 16 System Variables .................................................................................................................................... 16 Simultaneous versus sequential presentation ................................................................................... 16 Double blind, blind, blinded administration (unless impracticable ................................................... -
Partially Linear Spatial Probit Models
Partially Linear Spatial Probit Models Mohamed-Salem AHMED University of Lille, LEM-CNRS 9221 Lille, France [email protected] Sophie DABO INRIA-MODAL University of Lille LEM-CNRS 9221 Lille, France [email protected] Abstract A partially linear probit model for spatially dependent data is considered. A triangular array set- ting is used to cover various patterns of spatial data. Conditional spatial heteroscedasticity and non-identically distributed observations and a linear process for disturbances are assumed, allowing various spatial dependencies. The estimation procedure is a combination of a weighted likelihood and a generalized method of moments. The procedure first fixes the parametric components of the model and then estimates the non-parametric part using weighted likelihood; the obtained estimate is then used to construct a GMM parametric component estimate. The consistency and asymptotic distribution of the estimators are established under sufficient conditions. Some simulation experi- ments are provided to investigate the finite sample performance of the estimators. keyword: Binary choice model, GMM, non-parametric statistics, spatial econometrics, spatial statis- tics. Introduction Agriculture, economics, environmental sciences, urban systems, and epidemiology activities often utilize spatially dependent data. Therefore, modelling such activities requires one to find a type of correlation between some random variables in one location with other variables in neighbouring arXiv:1803.04142v1 [stat.ME] 12 Mar 2018 locations; see for instance Pinkse & Slade (1998). This is a significant feature of spatial data anal- ysis. Spatial/Econometrics statistics provides tools to perform such modelling. Many studies on spatial effects in statistics and econometrics using many diverse models have been published; see 1 Cressie (2015), Anselin (2010), Anselin (2013) and Arbia (2006) for a review. -
Probability, Explanation, and Inference: a Reply
Alabama Law Scholarly Commons Working Papers Faculty Scholarship 3-2-2017 Probability, Explanation, and Inference: A Reply Michael S. Pardo University of Alabama - School of Law, [email protected] Ronald J. Allen Northwestern University - Pritzker School of Law, [email protected] Follow this and additional works at: https://scholarship.law.ua.edu/fac_working_papers Recommended Citation Michael S. Pardo & Ronald J. Allen, Probability, Explanation, and Inference: A Reply, (2017). Available at: https://scholarship.law.ua.edu/fac_working_papers/4 This Working Paper is brought to you for free and open access by the Faculty Scholarship at Alabama Law Scholarly Commons. It has been accepted for inclusion in Working Papers by an authorized administrator of Alabama Law Scholarly Commons. Probability, Explanation, and Inference: A Reply Ronald J. Allen Michael S. Pardo 11 INTERNATIONAL JOURNAL OF EVIDENCE AND PROOF 307 (2007) This paper can be downloaded without charge from the Social Science Research Network Electronic Paper Collection: http://ssrn.com/abstract=2925866 Electronic copy available at: https://ssrn.com/abstract=2925866 PROBABILITY, EXPLANATION AND INFERENCE: A REPLY Probability, explanation and inference: a reply By Ronald J. Allen* and Wigmore Professor of Law, Northwestern University; Fellow, Procedural Law Research Center, China Political Science and Law University Michael S. Pardo† Assistant Professor, University of Alabama School of Law he inferences drawn from legal evidence may be understood in both probabilistic and explanatory terms. Consider evidence that a criminal T defendant confessed while in police custody. To evaluate the strength of this evidence in supporting the conclusion that the defendant is guilty, one could try to assess the probability that guilty and innocent persons confess while in police custody.