The effect of a Durbin–Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin–Watson pretest of the null hypothesis of zero first‐order autocorrelation of the random errors against the alternative hypothesis of positive first‐order autocorrelation. If this null hypothesis is accepted then the confidence interval centered on the ordinary least squares estimator is used; otherwise the confidence interval centered on the feasible generalized least squares estimator is used. For any given design matrix and parameter of interest, we compare the confidence interval resulting from this two‐stage procedure and the confidence interval that is always centered on the feasible generalized least squares estimator, as follows. First, we compare the coverage probability functions of these confidence intervals. Second, we compute the scaled expected length of the confidence interval resulting from the two‐stage procedure, where the scaling is with respect to the expected length of the confidence interval centered on the feasible generalized least squares estimator, with the same minimum coverage probability. These comparisons are used to choose the better confidence interval, prior to any examination of the observed response vector.     

Bayesian Priors from Loss Matching

This paper is concerned with the construction of prior probability measures for parametric families of densities where the framework is such that only beliefs or knowledge about a single observable data point is required. We pay particular attention to the parameter which minimizes a measure of divergence to the distribution providing the data. The prior distribution reflects this attention and we discuss the application of the Bayes rule from this perspective. Our framework is fundamentally non‐parametric and we are able to interpret prior distributions on the parameter space using ideas of matching loss functions, one of which is coming from the data model and the other from the prior.     

Objective Bayes Factors for Inequality Constrained Hypotheses

This paper will present a Bayes factor for the comparison of an inequality constrained hypothesis with its complement or an unconstrained hypothesis. Equivalent sets of hypotheses form the basis for the quantification of the complexity of an inequality constrained hypothesis. It will be shown that the prior distribution can be chosen such that one of the terms in the Bayes factor is the quantification of the complexity of the hypothesis of interest. The other term in the Bayes factor represents a measure of the fit of the hypothesis. Using a vague prior distribution this fit value is essentially determined by the data. The result is an objective Bayes factor.     

A Bayesian Multiple Comparison Procedure for Order-Restricted Mixed Models

A Bayesian hierarchical mixed model is developed for multiple comparisons under a simple order restriction. The model facilitates inferences on the successive differences of the population means, for which we choose independent prior distributions that are mixtures of an exponential distribution and a discrete distribution with its entire mass at zero. We employ Markov Chain Monte Carlo (MCMC) techniques to obtain parameter estimates and estimates of the posterior probabilities that any two of the means are equal. The latter estimates allow one both to determine if any two means are significantly different and to test the homogeneity of all of the means. We investigate the performance of the model-based inferences with simulated data sets, focusing on parameter estimation and successive-mean comparisons using posterior probabilities. We then illustrate the utility of the model in an application based on data from a study designed to reduce lead blood concentrations in children with elevated levels. Our results show that the proposed hierarchical model can effectively unify parameter estimation, tests of hypotheses and multiple comparisons in one setting.     

Critique of p-Values

This paper generalizes the notion of p-value to obtain a system for assessing evidence in favor of an hypothesis. It is not quite a quantification in that evidence is a pair of numbers (the p-value and the p-value with null and alternative interchanged) with evidence for the alternative being claimed when the first number is small and the second is at least moderately large. Traditional significance tests present p-values as a measure of evidence against a theory. This usage is rarely called for since scientists usually wish to accept theories (for the time being) not just not reject them; they are more interested in evidence for a theory. P-values are not just good or bad for this purpose; their efficacy depends on specifics. We find that a single p-value does not measure evidence for a simple hypothesis relative to a simple alternative, but consideration of both p-values leads to a satisfactory theory. This consideration does not, in general, extend to composite hypotheses since there, best evidence calls for optimization of a bivariate objective function. But in some cases, notably one sided tests for the exponential family, the optimization can be solved, and a single p-value does provide an appealing measure of best evidence for a hypothesis. One possible extension of this theory is proposed and illustrated with a practical safety analysis problem involving the difference of two random variables.     

Bayesian model selection using encompassing priors

This paper deals with Bayesian selection of models that can be specified using inequality constraints among the model parameters. The concept of encompassing priors is introduced, that is, a prior distribution for an unconstrained model from which the prior distributions of the constrained models can be derived. It is shown that the Bayes factor for the encompassing and a constrained model has a very nice interpretation: it is the ratio of the proportion of the prior and posterior distribution of the encompassing model in agreement with the constrained model. It is also shown that, for a specific class of models, selection based on encompassing priors will render a virtually objective selection procedure. The paper concludes with three illustrative examples: an analysis of variance with ordered means; a contingency table analysis with ordered odds-ratios; and a multilevel model with ordered slopes.     

Quantifying the uncertainty about the half‐life of deviations from PPP

We propose a Bayesian framework in which the uncertainty about the half‐life of deviations from purchasing power parity can be quantified. Based on the responses to a survey study, we propose a prior probability distribution for the half‐life under the recent float intended to capture widely held views among economists. We derive the posterior probability distribution of the half‐life under this consensus prior and confirm the presence of substantial uncertainty about the half‐life. We provide for the first time a comprehensive formal evaluation of several nonnested hypotheses of economic interest, including Rogoff's ( 1996 ) claim that the half‐life is contained in the range of 3 to 5 years. We find that no hypothesis receives strong support from the data. Copyright © 2002 John Wiley & Sons, Ltd.     

On Bayesian routes to unit roots

This paper is a comment on P. C. B. Phillips, ‘To criticise the critics: an objective Bayesian analysis of stochastic trends’ [Phillips, (1991)]. Departing from the likelihood of an univariate autoregressive model different routes that lead to a posterior odds analysis of the unit root hypothesis are explored, where the differences in routes are due to the different choices of the prior. Improper priors like the uniform and the Jeffreys prior are less suited for Bayesian inference on a sharp null hypothesis as the unit root. A proper normal prior on the mean of the process is analysed and empirical results using extended Nelson-Plosser data are presented.     

Small‐scale Inference: Empirical Bayes and Confidence Methods for as Few as a Single Comparison

Empirical Bayes methods of estimating the local false discovery rate (LFDR) by maximum likelihood estimation (MLE), originally developed for large numbers of comparisons, are applied to a single comparison. Specifically, when assuming a lower bound on the mixing proportion of true null hypotheses, the LFDR MLE can yield reliable hypothesis tests and confidence intervals given as few as one comparison. Simulations indicate that constrained LFDR MLEs perform markedly better than conventional methods, both in testing and in confidence intervals, for high values of the mixing proportion, but not for low values. (A decision‐theoretic interpretation of the confidence distribution made those comparisons possible.) In conclusion, the constrained LFDR estimators and the resulting effect‐size interval estimates are not only effective multiple comparison procedures but also they might replace p‐values and confidence intervals more generally. The new methodology is illustrated with the analysis of proteomics data.     

Estimation strategies for the regression coefficient parameter matrix in multivariate multiple regression

We consider improved estimation strategies for the parameter matrix in multivariate multiple regression under a general and natural linear constraint. In the context of two competing models where one model includes all predictors and the other restricts variable coefficients to a candidate linear subspace based on prior information, there is a need of combining two estimation techniques in an optimal way. In this scenario, we suggest some shrinkage estimators for the targeted parameter matrix. Also, we examine the relative performances of the suggested estimators in the direction of the subspace and candidate subspace restricted type estimators. We develop a large sample theory for the estimators including derivation of asymptotic bias and asymptotic distributional risk of the suggested estimators. Furthermore, we conduct Monte Carlo simulation studies to appraise the relative performance of the suggested estimators with the classical estimators. The methods are also applied on a real data set for illustrative purposes.     

Optimal Adaptive Designs with Inverse Ordinary Differential Equations

Many industrial and engineering applications are built on the basis of differential equations. In some cases, parameters of these equations are not known and are estimated from measurements leading to an inverse problem. Unlike many other papers, we suggest to construct new designs in the adaptive fashion ‘on the go’ using the A‐optimality criterion. This approach is demonstrated on determination of optimal locations of measurements and temperature sensors in several engineering applications: (1) determination of the optimal location to measure the height of a hanging wire in order to estimate the sagging parameter with minimum variance (toy example), (2) adaptive determination of optimal locations of temperature sensors in a one‐dimensional inverse heat transfer problem and (3) adaptive design in the framework of a one‐dimensional diffusion problem when the solution is found numerically using the finite difference approach. In all these problems, statistical criteria for parameter identification and optimal design of experiments are applied. Statistical simulations confirm that estimates derived from the adaptive optimal design converge to the true parameter values with minimum sum of variances when the number of measurements increases. We deliberately chose technically uncomplicated industrial problems to transparently introduce principal ideas of statistical adaptive design.     

Data‐Driven Identification Constraints for DSGE Models

We propose imposing data‐driven identification constraints to alleviate the multimodality problem arising in the estimation of poorly identified dynamic stochastic general equilibrium models under non‐informative prior distributions. We also devise an iterative procedure based on the posterior density of the parameters for finding these constraints. An empirical application to the Smets and Wouters ( 2007 ) model demonstrates the properties of the estimation method, and shows how the problem of multimodal posterior distributions caused by parameter redundancy is eliminated by identification constraints. Out‐of‐sample forecast comparisons as well as Bayes factors lend support to the constrained model.     

Bayes Convolution

A general convolution theorem within a Bayesian framework is presented. Consider estimation of the Euclidean parameter θ by an estimator T within a parametric model. Let W be a prior distribution for θ and define G as the W -average of the distribution of T - θ under θ . In some cases, for any estimator T the distribution G can be written as a convolution G = K * L with K a distribution depending only on the model, i.e. on W and the distributions under θ of the observations. In such a Bayes convolution result optimal estimators exist, satisfying G = K . For location models we show that finite sample Bayes convolution results hold in the normal, loggamma and exponential case. Under regularity conditions we prove that normal and loggamma are the only smooth location cases. We also discuss relations with classical convolution theorems.     

Is infrastructure capital productive? A dynamic heterogeneous approach

This paper offers an evaluation of the output contribution of infrastructure. Using a panel time series approach and a large cross‐country dataset, the paper estimates a long‐run aggregate production function relating gross domestic product to human capital, physical capital, and a synthetic measure of infrastructure comprising transport, power and telecommunications. Tests of the cointegration rank allowing it to vary across countries reveal a common rank with a single cointegrating vector, which we interpret as the long‐run production function. Estimation of its parameters is performed using the pooled mean group (PMG) estimator, which allows for unrestricted short‐run parameter heterogeneity across countries while imposing the (testable) restriction of long‐run parameter homogeneity. The long‐run elasticity of output with respect to the synthetic infrastructure index ranges between 0.07 and 0.10. The estimates are highly significant, both statistically and economically, and robust to alternative dynamic specifications and infrastructure measures. Tests of parameter homogeneity fail to yield evidence that the long‐run parameters differ across countries. Copyright © 2014 John Wiley & Sons, Ltd.     

CHOOSING AMONG ALTERNATIVE LONG‐RUN EVENT‐STUDY TECHNIQUES

This paper reviews the long‐run event‐study debate by outlining the strengths and weakness of the most commonly used alternative techniques. The fist part of the discussion highlights that prior literature has failed to provide a single risk‐adjusted model of long‐run abnormal returns with no biases. Subsequently, the paper provides guidance on how one can choose among pertinent alternative techniques. As a conclusion, researchers ought to choose among alternative techniques after considering issues such as (i) the nature of dataset and market of interest, (ii) the event type (regulatory or corporate), (iii) returns’ time‐interval, (iv) association of the event with accounting data, (v) sample characteristics and prior evidence regarding similar events, as well as (vi) risk changes following the event. Robustness tests are essential, while the road for further research regarding the appropriate technique(s) is open.     

Methods for Default and Robust Bayesian Model Comparison: the Fractional Bayes Factor Approach

In the Bayesian approach to model selection and hypothesis testing, the Bayes factor plays a central role. However, the Bayes factor is very sensitive to prior distributions of parameters. This is a problem especially in the presence of weak prior information on the parameters of the models. The most radical consequence of this fact is that the Bayes factor is undetermined when improper priors are used. Nonetheless, extending the non-informative approach of Bayesian analysis to model selection/testing procedures is important both from a theoretical and an applied viewpoint. The need to develop automatic and robust methods for model comparison has led to the introduction of several alternative Bayes factors. In this paper we review one of these methods: the fractional Bayes factor (O'Hagan, 1995). We discuss general properties of the method, such as consistency and coherence. Furthermore, in addition to the original, essentially asymptotic justifications of the fractional Bayes factor, we provide further finite-sample motivations for its use. Connections and comparisons to other automatic methods are discussed and several issues of robustness with respect to priors and data are considered. Finally, we focus on some open problems in the fractional Bayes factor approach, and outline some possible answers and directions for future research.     

Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

Scattered reports of multiple maxima in posterior distributions or likelihoods for mixed linear models appear throughout the literature. Less scrutinised is the restricted likelihood, which is the posterior distribution for a specific prior distribution. This paper surveys existing literature and proposes a unifying framework for understanding multiple maxima. For those problems with covariance structures that are diagonalisable in a specific sense, the restricted likelihood can be viewed as a generalised linear model with gamma errors, identity link and a prior distribution on the error variance. The generalised linear model portion of the restricted likelihood can be made to conflict with the portion of the restricted likelihood that functions like a prior distribution on the error variance, giving two local maxima in the restricted likelihood. Applying in addition an explicit conjugate prior distribution to variance parameters permits a second local maximum in the marginal posterior distribution even if the likelihood contribution has a single maximum. Moreover, reparameterisation from variance to precision can change the posterior modality; the converse also is true. Modellers should beware of these potential pitfalls when selecting prior distributions or using peak‐finding algorithms to estimate parameters.     

Objective methods for graphical structural learning

Graphical models are used for expressing conditional independence relationships among variables by the means of graphs, whose structure is typically unknown and must be inferred by the data at hand. We propose a theoretically sound Objective Bayes procedure for graphical model selection. Our method is based on the Expected-Posterior Prior and on the Power-Expected-Posterior Prior. We use as input of the proposed methodology a default improper prior and suggest computationally efficient approximations of Bayes factors and posterior odds. In a variety of simulated scenarios with varying number of nodes and sample sizes, we show that our method is highly competitive with, or better than, current benchmarks. We also discuss an application to protein-signaling data, which wieldy confirms existing results in the scientific literature.     

STRATEGIC ASSET ALLOCATION FOR LONG‐TERM INVESTORS: PARAMETER UNCERTAINTY AND PRIOR INFORMATION

We study the effect of parameter uncertainty on the long‐run risk for three asset classes: stocks, bills and bonds. Using a Bayesian vector autoregression with an uninformative prior we find that parameter uncertainty raises the annualized long‐run volatilities of all three asset classes proportionally with the same factor relative to volatilities that are conditional on maximum likelihood parameter estimates. As a result, the horizon effect in optimal asset allocations is much weaker compared to models in which only equity returns are subject to parameter uncertainty. Results are sensitive to alternative informative priors, but generally the term structure of risk for stocks and bonds is relatively flat for investment horizons up to 15 years. Copyright © 2013 John Wiley & Sons, Ltd.     

How to Choose a Working Model for Measuring the Statistical Evidence About a Regression Parameter

Consider using a likelihood ratio to measure the strength of statistical evidence for one hypothesis over another. Recent work has shown that when the model is correctly specified, the likelihood ratio is seldom misleading. But when the model is not, misleading evidence may be observed quite frequently. Here we consider how to choose a working regression model so that the statistical evidence is correctly represented as often as it would be under the true model. We argue that the criteria for choosing a working model should be how often it correctly represents the statistical evidence about the object of interest (regression coefficient in the true model). We see that misleading evidence about the object of interest is more likely to be observed when the working model is chosen according to other criteria (e.g., parsimony or predictive accuracy).