Introduction to Mixed ModellingExperimental Design and the Analysis of VariancePrimer of Applied Regression & Analysis of VarianceAnalysis of Variance and CovarianceExperiments in EcologyLearning Statistics with RAnalysis of Variance Via Confidence IntervalsMultivariate Analysis of Variance (MANOVA)Statistical Methods for GeographyMultivariate Analysis of Variance and Repeated MeasuresThe Analysis of VarianceApplied Analysis of Variance in Behavioral ScienceLevine's Guide to SPSS for Analysis of VarianceSequential AnalysisAnalysis of Variance, Design, and RegressionStatistics for Health Care ProfessionalsThe Analysis of VarianceThe SAGE Dictionary of Social Research MethodsAnalysis of Variance for Functional DataTwo-Way Analysis of VarianceAnalysis of Variance DesignsAnalysis of Variance for Random ModelsLinear ModelsOnline Statistics EducationAnalysis of VarianceA Student's Guide to Analysis of VarianceApplied Statistics in Agricultural, Biological, and Environmental SciencesData Analysis Using SASEncyclopedia of Survey Research MethodsThe Analysis of VarianceEncyclopedia of Research DesignAnalysis of Variance for Sensory DataStatistics Using Technology, Second EditionAdvanced Analysis of VarianceAnalysis of VarianceA Practical Approach to Using Statistics in Health ResearchThe SAGE Encyclopedia of Communication Research MethodsAnalysis of Variance, Design, and RegressionStatistical Analysis Quick Reference GuidebookIntroduction to Analysis of Variance

Rényi divergence is a natural way to measure the rate of information flow in contexts like Bayesian updating. This chapter shows how Monte Carlo integration can be used to measure Rényi divergence when (as is often the case) only kernels of the relevant probability densities are available. The chapter further demonstrates that Rényi divergence is central to the convergence and efficiency of Monte Carlo integration procedures in which information flow is controlled. It uses this perspective to develop more flexible approaches to the controlled introduction of information; in the limited set of examples considered here, these alternatives enhance efficiency.

In this paper, a nonlinear and/or non-Gaussian smoother utilizing Markov chain Monte Carlo Methods is proposed, where the measurement and transition equations are specified in any general formulation and the error terms in the state-space model are not necessarily normal. The random draws are directly generated from the smoothing densities. For random number generation, the Metropolis-Hastings algorithm and the Gibbs sampling technique are utilized. The proposed procedure is very simple and easy for programming, compared with the existing nonlinear and non-Gaussian smoothing techniques. Moreover, taking several candidates of the proposal density function, we examine precision of the proposed estimator.

Bayesian A/B inference (BABI) is a method that combines subjective prior information with data from A/B experiments to provide inference for lift – the difference in a measure of response in control and treatment, expressed as its ratio to the measure of response in control. The procedure is embedded in stable code that can be executed in a few seconds for an experiment, regardless of sample size, and caters to the objectives and technical background of the owners of experiments. BABI provides more powerful tests of the hypothesis of the impact of treatment on lift, and sharper conclusions about the value of lift, than do legacy conventional methods. In application to 21 large online experiments, the credible interval is 60% to 65% shorter than the conventional confidence interval in the median case, and by close to 100% in a significant proportion of cases; in rare cases, BABI credible intervals are longer than conventional confidence intervals and then by no more than about 10%.

This article develops practical methods for Bayesian inference in the autoregressive fractionally integrated moving average (ARFIMA) model using the exact likelihood function, any proper prior distribution, and time series that may have thousands of observations. These methods utilize sequentially adaptive Bayesian learning, a sequential Monte Carlo algorithm that can exploit massively parallel desktop computing with graphics processing units (GPUs). The article identifies and solves several problems in the computation of the likelihood function that apparently have not been addressed in the literature. Four applications illustrate the utility of the approach. The most ambitious is an ARFIMA(2,d,2) model for the Campito tree ring time series (length 5405), for which the methods developed in the article provide an essentially uncorrelated sample of size 16,384 from the exact posterior distribution in under four hours. Less ambitious applications take as little as 4 minutes without exploiting GPUs.

We establish methods that improve the predictions of macroeconometric modelsdynamic factor models, dynamic stochastic general equilibrium models, and vector autoregressionsusing a quarterly U.S. data set. We measure prediction quality with one-step-ahead probability densities assigned in real time. Two steps lead to substantial improvements: (a) the use of full Bayesian predictive distributions rather than conditioning on the posterior mode for parameters and (b) the use of an equally weighted pool.

There is a one-to-one mapping between the conventional time series parameters of a third-order autoregression and the more interpretable parameters of secular half-life, cyclical half-life and cycle period. The latter parameterization is better suited to interpretation of results using both Bayesian and maximum likelihood methods and to expression of a substantive prior distribution using Bayesian methods. The paper demonstrates how to approach both problems using the sequentially adaptive Bayesian learning algorithm and sequentially adaptive Bayesian learning algorithm (SABL) software, which eliminates virtually of the substantial technical overhead required in conventional approaches and produces results quickly and reliably. The work utilizes methodological innovations in SABL including optimization of irregular and multimodal functions and production of the conventional maximum likelihood asymptotic variance matrix as a by-product.

Efficient investment of personal savings depends on clear risk disclosures. We study the propensity of individuals to violate some implications of expected utility under alternative "mass-market" descriptions of investment risk, using a discrete choice experiment. We found violations in around 25% of choices, and substantial variation in rates of violation, depending on the mode of risk disclosure and participants’ characteristics. When risk is described as the frequency of returns below or above a threshold we observe more violations than for range and probability-based descriptions. Innumerate individuals are more likely to violate expected utility than those with high numeracy. Apart from the very elderly, older individuals are less likely to violate the restrictions. The results highlight the challenges of disclosure regulation.

Christoffersen, Jacobs, and Ornthanalai (2012) (CJO) propose an interesting and useful class of generalized autoregressive conditional heteroskedasticity (GARCH)-like models with dynamic jump intensity, and find evidence that the models not only fit returns data better than some commonly used benchmarks but also provide substantial improvements in option pricing performance. While such models pose difficulties for estimation and analysis, CJO propose an innovative approach to filtering intended to addresses them. However, some statistical issues arise that their approach leaves unresolved, with implications for the option pricing results. This note proposes a solution based on using the filter and estimator proposed by CJO but interpreted in the context of an alternative model. With respect to this model, the estimator is consistent, and likelihood-based model comparisons and hypothesis tests are valid.

There is a one-to-one mapping between the convenetional time series parameters of a third-order autoregression and the more interpretable parameters of secular half-life, cyclical half-life and cycle period. The latter parameterization is better suited to interpretation of results using both Bayesian and maximum likelihood methods and to expression of a substantive prior distribution using Bayesian methods. The paper demonstrates how to approach both problems using the sequentially adaptive Bayesian learnign algorithm and SABL software, which eliminates virtually of the substantial technical overhead required in conventional approaches and produces results quickly and reliably. The work utilizes methodological innovations in SABL incuding optimization of irregular and multimodal functions and production of the coventional maximum likelihood asymptotic variance matrix as a by-product. ∗University of Technology Sydney, John.Geweke@uts.edu.au. The Australian Research Council provided financial support through grant DP130103356 and through the ARC Centre of Excellence for Mathematical and Statistical Frontiers of Big Data, Big Models, New Insights, grant CD140100049.

This paper demonstrates a method for estimating logit choice models for small sample data, including single individuals, that is computationally simpler and relies on weaker prior distributional assumptions compared to hierarchical Bayes estimation. Using Monte Carlo simulations and online discrete choice experiments, we show how this method is particularly well suited to estimating values of choice model parameters from small sample choice data, thus opening this area to the application of choice modeling. For larger sample sizes of approximately 100–200 respondents, preference distribution recovery is similar to hierarchical Bayes estimation of mixed logit models for the examples we demonstrate. We discuss three approaches for specifying the conjugate priors required for the method: specifying priors based on existing or projected market shares of products, specifying a flat prior on the choice alternatives in a discrete choice experiment, or adopting an empirical Bayes approach where the prior choice probabilities are taken to be the average choice probabilities observed in a discrete choice experiment. We show that for small sample data, the relative weighting of the prior during estimation is an important consideration, and we present an automated method for selecting the weight based on a predictive scoring rule.

This paper shows that regular fractional polynomials can approximate regular cost, production and utility functions and their first two derivatives on closed compact subsets of the strictly positive orthant of Euclidean space arbitrarily well. These functions therefore can provide reliable approximations to demand functions and other economically relevant characteristics of tastes and technology. Using canonical cost function data, it shows that full Bayesian inference for these approximations can be implemented using standard Markov chain Monte Carlo methods.

This paper develops a multiway analysis of variance for non-Gaussian multivariate distributions and provides a practical simulation algorithm to estimate the corresponding components of variance. It specifically addresses variance in Bayesian predictive distributions, showing that it may be decomposed into the sum of extrinsic variance, arising from posterior uncertainty about parameters, and intrinsic variance, which would exist even if parameters were known. Depending on the application at hand, further decomposition of extrinsic or intrinsic variance (or both) may be useful. The paper shows how to produce simulation-consistent estimates of all of these components, and the method demands little additional effort or computing time beyond that already invested in the posterior simulator. It illustrates the methods using a dynamic stochastic general equilibrium model of the US economy, both before and during the global financial crisis.

Public policy setting often involves quantitative choices with quantitative outcomes. Yet unqualified statements about the precise consequences of alternative choices characterize much of the policy analysis bearing on these decisions. Public Policy in an Uncertain World: Analysis and Decisions by Charles F. Manski characterizes and richly illustrates the nature of this unwarranted certitude. It details specific constructive alternatives on which the economics profession has achieved varying degrees of consensus. Those in our profession charged with the education of future policy analysts should consider using it and how to round out its presentation of decision making from their own perspective. (JEL D02, D04, D80, E61)

Financial regulators are weighing up the effectiveness of different templates for communicating investment risk to retirement savers since welfare depends on comprehension of risk information. We compare nine standard risk presentations using a discrete choice experiment where subjects choose between three retirement accounts. Switching between graphical or textual presentations, or between formats that emphasize benchmarks rather than return ranges or values at risk, affects predicted choices more than large changes in underlying risk. Innumerate individuals are more susceptible to presentation, and those with weak basic financial literacy are insensitive to increasing risk levels, regardless of presentation. Presentation effects are moderated but not eliminated as financial literacy improves.

Massively parallel desktop computing capabilities now well within the reach of individual academics modify the environment for posterior simulation in fundamental and potentially quite advantageous ways. But to fully exploit these benefits algorithms that conform to parallel computing environments are needed. This paper presents a sequential posterior simulator designed to operate efficiently in this context. The simulator makes fewer analytical and programming demands on investigators, and is faster, more reliable, and more complete than conventional posterior simulators. The paper extends existing sequential Monte Carlo methods and theory to provide a thorough and practical foundation for sequential posterior simulation that is well suited to massively parallel computing environments. It provides detailed recommendations on implementation, yielding an algorithm that requires only code for simulation from the prior and evaluation of prior and data densities and works well in a variety of applications representative of serious empirical work in economics and finance. The algorithm facilitates Bayesian model comparison by producing marginal likelihood approximations of unprecedented accuracy as an incidental by-product, is robust to pathological posterior distributions, and provides estimates of numerical standard error and relative numerical efficiency intrinsically. The paper concludes with an application that illustrates the potential of these simulators for applied Bayesian inference.

Simulated annealing is a well-established approach to optimization that is robust for irregular objective functions. Recently it has been improved using sequential Monte Carlo. This paper presents further improvements that yield the global optimum with accuracy constrained only by the limitations of floating point arithmetic. Performance is illustrated using a standard set of six test problems in which simulated annealing has had mixed success. Our approach reliably finds the exact global optimum in all six cases, and with fewer function evaluations than competing simulated annealing algorithms. This approach is a specific case of the sequentially adaptive Bayesian learning algorithm, which uses feedback from particles to the design of the algorithm. The feature of this algorithm most critical to exact optimization is targeted tempering, a new technique developed in this paper.

This study considers three alternative sources of information about volatility potentially useful in predicting daily asset returns: daily returns, intraday returns, and option prices. For each source of information the study begins with several alternative models, and then works from the premise that all of these models are false to construct a single improved predictive distribution for daily S&P 500 index returns. The prediction probabilities of the optimal pool exceed those of the conventional models by as much as 5.29%. The optimal pools place substantial weight on models using each of the three sources of information about volatility.

This research studies whether individuals make choices consistent with expected utility maximization in allocating wealth between a lifetime annuity and a phased withdrawal account at retirement. The paper describes the construction and administration of a discrete choice experiment to 854 respondents approaching retirement. The experiment fi nds overall rates of inconsistency with the predictions of the standard CRRA utility model of roughly 50%, and variation in consistency rates depending on the characteristics of the respondents. Individuals with poor numeracy and with low engagement with the choice task, as measured by scores on a task-speci fic recall quiz, are more likely to increase allocations to the phased withdrawal as the risk of exhausting it increases. Individuals with higher scores on tests of financial capability and with knowledge of retirement income products are more likely to score high on the engagement measure, but capability and knowledge do not have independent eff ects on consistent choice rates. Results suggest that initiatives to improve speci fic product knowledge and to help individuals engage with decumulation decisions could be a partial solution to the annuity puzzle.