Causal Reasoning from Longitudinal Data*

Abstract. This paper reviews some of the key statistical ideas that are encountered when trying to find empirical support to causal interpretations and conclusions, by applying statistical methods on experimental or observational longitudinal data. In such data, typically a collection of individuals are followed over time, then each one has registered a sequence of covariate measurements along with values of control variables that in the analysis are to be interpreted as causes, and finally the individual outcomes or responses are reported. Particular attention is given to the potentially important problem of confounding. We provide conditions under which, at least in principle, unconfounded estimation of the causal effects can be accomplished. Our approach for dealing with causal problems is entirely probabilistic, and we apply Bayesian ideas and techniques to deal with the corresponding statistical inference. In particular, we use the general framework of marked point processes for setting up the probability models, and consider posterior predictive distributions as providing the natural summary measures for assessing the causal effects. We also draw connections to relevant recent work in this area, notably to Judea Pearl's formulations based on graphical models and his calculus of so‐called do‐probabilities. Two examples illustrating different aspects of causal reasoning are discussed in detail.     

Conditional Likelihood Estimators for Hidden Markov Models and Stochastic Volatility Models

ABSTRACT.  This paper develops a new contrast process for parametric inference of general hidden Markov models, when the hidden chain has a non-compact state space. This contrast is based on the conditional likelihood approach, often used for ARCH-type models. We prove the strong consistency of the conditional likelihood estimators under appropriate conditions. The method is applied to the Kalman filter (for which this contrast and the exact likelihood lead to asymptotically equivalent estimators) and to the discretely observed stochastic volatility models.     

Empirical Bayes estimation in finite population sampling under functional measurement error models

The paper considers simultaneous estimation of finite population means for several strata. A model-based approach is taken, where the covariates in the super-population model are subject to measurement errors. Empirical Bayes (EB) estimators of the strata means are developed and an asymptotic expression for the MSE of the EB estimators is provided. It is shown that the proposed EB estimators are “first order optimal” in the sense of Robbins [1956. An empirical Bayes approach to statistics. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California Press, Berkeley, pp. 157–164], while the regular EB estimators which ignore the measurement error are not.     

A Unified Approach to the Characterization of Equivalence Classes of DAGs, Chain Graphs with no Flags and Chain Graphs

Abstract.  A Markov property associates a set of conditional independencies to a graph. Two alternative Markov properties are available for chain graphs (CGs), the Lauritzen–Wermuth–Frydenberg (LWF) and the Andersson–Madigan– Perlman (AMP) Markov properties, which are different in general but coincide for the subclass of CGs with no flags . Markov equivalence induces a partition of the class of CGs into equivalence classes and every equivalence class contains a, possibly empty, subclass of CGs with no flags itself containing a, possibly empty, subclass of directed acyclic graphs (DAGs). LWF-Markov equivalence classes of CGs can be naturally characterized by means of the so-called largest CGs , whereas a graphical characterization of equivalence classes of DAGs is provided by the essential graphs . In this paper, we show the existence of largest CGs with no flags that provide a natural characterization of equivalence classes of CGs of this kind, with respect to both the LWF- and the AMP-Markov properties. We propose a procedure for the construction of the largest CGs, the largest CGs with no flags and the essential graphs, thereby providing a unified approach to the problem. As by-products we obtain a characterization of graphs that are largest CGs with no flags and an alternative characterization of graphs which are largest CGs. Furthermore, a known characterization of the essential graphs is shown to be a special case of our more general framework. The three graphical characterizations have a common structure: they use two versions of a locally verifiable graphical rule. Moreover, in case of DAGs, an immediate comparison of three characterizing graphs is possible.     

关于外汇期货套期保值比率的实证研究

最佳套期保值比率(OHR)的估计方法一直是金融工程理论研究的核心问题,从最开始的幼稚法到JSE法 以及随后的很多其他改进方法,保值效率都有不同程度的提高。使用包含误差修正结构的GARCH模型估计外汇 (澳大利亚元)期货的套期保值比率。通过效率比较,证实该模型所得到的套期保值比率比起传统方法都具有更好 的降低风险能力。     

A Class of Efficient Non-parametric K-sample Tests for Multivariate Counting Processes with Frailties

A K -sample testing problem is studied for multivariate counting processes with time-dependent frailty. Asymptotic distributions and efficiency of a class of non-parametric test statistics are established for certain local alternatives. The concept of efficiency is to show that for every non-parametric test in this class, there is a parametric submodel for which the optimal test has the same asymptotic power as the non-parametric one. The theory is applied to analyse a diabetic retinopathy study data set. A simulation study is also presented to illustrate the theory     

Comparative probability for conditional events: A new look through coherence

We study, from the standpoint of coherence, comparative probabilities on an arbitrary familyE of conditional events. Given a binary relation ·, coherence conditions on · are related to de Finetti's coherent betting system: we consider their connections to the usual properties of comparative probability and to the possibility of numerical representations of ·. In this context, the numerical reference frame is that of de Finetti's coherent subjective conditional probability, which is not introduced (as in Kolmogoroff's approach) through a ratio between probability measures.Another relevant feature of our approach is that the family & need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence.     

Sensitivity of the Optimum to Perturbations of the Profit or Weight of an Item in the Binary Knapsack Problem

In the binary single constraint Knapsack Problem, denoted KP, we are given a knapsack of fixed capacity c and a set of n items. Each item j, j = 1,...,n, has an associated size or weight w_j and a profit p_j. The goal is to determine whether or not item j, j = 1,...,n, should be included in the knapsack. The objective is to maximize the total profit without exceeding the capacity c of the knapsack. In this paper, we study the sensitivity of the optimum of the KP to perturbations of either the profit or the weight of an item. We give approximate and exact interval limits for both cases (profit and weight) and propose several polynomial time algorithms able to reach these interval limits. The performance of the proposed algorithms are evaluated on a large number of problem instances.     

An algorithm for maximum likelihood estimation in incomplete block variance component models

In the recovery of interblock information to improve the treatment differences estimates in incomplete block designs, the parameter p is usually unknown. Many authors have worked on the problem of estimating it and of studying its properties together with the properties of the treatment differences estimates. In this paper a numerically efficient algorithm is developed which yields the maximum likelihood estimates (MLE) of all the parameters in the mixed incomplete block design model (treatment effects, ρ and variance)     

The performance of alternative estimators for the probit model with first-order serial correlation 1

The purpose of the paper is to evaluate the relative performance of two generalized conditional moment (GCM) estimators in terms of their mean squared errors, for the Probit model with first-order serial correlation. The first estimator is a linearized one-step estimator described by Poirier and Ruud (1988). The second one is defined in the present paper. Monte Car10 experiments suggest that the GCM estimators outperform the ordinary Probit estimator. The two GCM estimators do almost equally well, except that the second one may be easier to calculate, especially in large samples.