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Model choice is usually an inevitable source of uncertainty in model-based statistical analyses. While the focus of model choice was traditionally on methods for choosing a single model, methods to formally account for multiple models within a single analysis are now accessible to many researchers. The specific technique of model averaging was developed to improve predictive ability by combining predictions from a set of models. However, it is now often used to average regression coefficients across multiple models with the ultimate goal of capturing a variable's overall effect. This use of model averaging implicitly assumes the same parameter exists across models so that averaging is sensible. While this assumption may initially seem tenable, regression coefficients associated with particular explanatory variables may not hold equivalent interpretations across all of the models in which they appear, making explanatory inference about covariates challenging. Accessibility to easily implementable software, concerns about being criticized for ignoring model uncertainty, and the chance to avoid having to justify choice of a final model have all led to the increasing popularity of model averaging in practice. We see a gap between the theoretical development of model averaging and its current use in practice, potentially leaving well-intentioned researchers with unclear inferences or difficulties justifying reasons for using (or not using) model averaging. We attempt to narrow this gap by revisiting some relevant foundations of regression modeling, suggesting more explicit notation and graphical tools, and discussing how individual model results are combined to obtain a model averaged result. Our goal is to help researchers make informed decisions about model averaging and to encourage question-focused modeling over method-focused modeling.

Atmospheric rivers (ARs) are narrow jets of integrated water vapor transport that are important for the global water cycle and also have large impacts on local weather and regional hydrology. Uniformly weighted multi‐model averages have been used to describe how ARs will change in the future, but this type of estimate does not consider skill or independence of the climate models of interest. Here, we utilize information from various model averaging approaches, such as Bayesian model averaging (BMA), to evaluate 21 global climate models from the Coupled Model Intercomparison Project Phase 5. Model ensemble weighting strategies are based on model independence and AR performance skill relative to ERA‐Interim reanalysis data and result in higher accuracy for the historic period, for example, root mean square error for AR frequency (in % of time steps) of 0.69 for BMA versus 0.94 for the multi‐model ensemble mean. Model weighting strategies also result in lower uncertainties in the future estimates, for example, only 20–25% of the total uncertainties seen in the equal weighting strategy. These model averaging methods show, with high certainty, that globally the frequency of ARs is expected to have average relative increases of ~50% (and ~25% in AR intensity) by the end of the century. Plain Language Summary Atmospheric rivers (ARs) are storms of integrated water vapor transport that are important for the global water cycle and also have large impacts on local weather and regional hydrology. An increase in the frequency of ARs is expected to occur by the end of the century throughout most of the globe. Usually, these types of assessments of future climate change rely on simple (i.e., equally weighted) multi‐model averages and do not consider the skill or independence of the climate models of interest. Here, we utilize information from various model averaging approaches to constrain a suite of 21 global climate models from the Coupled Model Intercomparison Project Phase 5. The weighted model combinations are fit to reanalysis data (ERA‐Interim) and are useful because they provide higher skill as well as lower uncertainties compared to equal weighting. This work supports the claim that AR frequency will increase in the future by about ~50% (and intensity will increase by ~25%) globally by the end of the century. Key Points By the end of the century, atmospheric rivers will increase in frequency (~50%) and intensity (~25%) around the globe (CMIP5 RCP8.5 runs) Model ensemble weighting strategies based on atmospheric river performance skill and model independence are considered and compared Averaging methods like Bayesian model averaging are more accurate and have lower uncertainties compared to the multi‐model ensemble mean

A key parameter estimated by lab and field experiments in economics is the individual discount rate—and the results vary widely. We examine the extent to which this variance can be attributed to observable differences in methods, subject pools, and potential publication bias. To address the model uncertainty inherent to such an exercise we employ Bayesian and frequentist model averaging. We obtain evidence consistent with publication bias against unintuitive results. The corrected mean annual discount rate is 0.33. Our findings also suggest that discount rates are independent across domains: people tend to be less patient when health is at stake compared to money. Negative framing is associated with more patience. Finally, the results of lab and field experiments differ systematically, and it also matters whether the experiment relies on students or uses broader samples of the population.

Abiotic conditions, biotic factors, and disturbances can act as filters that control community structure and composition. Understanding the relative importance of these drivers would allow us to understand and predict the causes and consequences of changes in community structure. We used long-term data (1989–2002) from the sagebrush steppe in the state of Washington, USA, to ask three questions: (1) What are the key drivers of community-level metrics of community structure? (2) Do community-level metrics and functional groups differ in magnitude or direction of response to drivers of community structure? (3) What is the relative importance of drivers of community structure? The vegetation in 2002 was expressed as seven response variables: three community-level metrics (species richness, total cover, compositional change from 1989 to 2002) and the relative abundances of four functional groups. We used a multi-model inference framework to identify a set of top models for each response metric beginning from a global model that included two abiotic drivers, six disturbances, a biotic driver (initial plant community), and interactions between the disturbance and biotic drivers. We also used a permutational relative variable importance metric to rank the influence of drivers. Moisture availability was the most important driver of species richness and of native forb cover. Fire was the most important driver of shrub cover and training area usage was important for compositional change, but disturbances, including grazing, were of secondary importance for most other variables. Biotic drivers, as represented by the initial plant communities, were the most important driver for total cover and for the relative covers of exotics and native grasses. Our results indicate that the relative importance of drivers is dependent on the choice of metric, and that drivers such as disturbance and initial plant community can interact.

This paper states that there are exchange rate forecasting gains when combining in-sample data from different models based on economic theory. Data combination is performed using Bayesian model averaging (BMA). Using pooled data by group of countries (developed and emerging economies) generates accuracy gains in an important amount of cases, with respect to forecasts that use country information.Gains are larger for currencies of developed economies, but accuracy decreases as the forecast horizon is extended. BMA models for developed countries tend to be more “sparse” than emerging countries models Este artículo propone la existencia ganancias en la predicción de tipo de cambio cuando se combinan datos in-sample de diferentes modelos basados en la teoría económica. La combinación se realiza mediante Bayesian Model Averaging. Entrenar el modelo con información de otras economías genera ganancias de precisión en una cantidad importante de casos, respecto a pronósticos que utilizan solo información del país. Mayores ganancias de precisión se encuentran para divisas de economías desarrolladas. Los modelos entrenados para países desarrollados tienden a ser más “escasos” que los modelos de países emergentes

Analyses of longitudinal data with non-linear mixed-effects models (NLMEM) are typically associated with high power, but sometimes at the cost of inflated type I error. Approaches to overcome this problem were published recently, such as model-averaging across drug models (MAD), individual model-averaging (IMA), and combined Likelihood Ratio Test (cLRT). This work aimed to assess seven NLMEM approaches in the same framework: treatment effect assessment in balanced two-armed designs using real natural history data with or without the addition of simulated treatment effect. The approaches are MAD, IMA, cLRT, standard model selection (STDs), structural similarity selection (SSs), randomized cLRT (rcLRT), and model-averaging across placebo and drug models (MAPD). The assessment included type I error, using Alzheimer’s Disease Assessment Scale-cognitive (ADAS-cog) scores from 817 untreated patients and power and accuracy in the treatment effect estimates after the addition of simulated treatment effects. The model selection and averaging among a set of pre-selected candidate models were driven by the Akaike information criteria (AIC). The type I error rate was controlled only for IMA and rcLRT; the inflation observed otherwise was explained by the placebo model misspecification and selection bias. Both IMA and rcLRT had reasonable power and accuracy except under a low typical treatment effect.

In high-dimensional data analysis, we propose a sequential model averaging (SMA) method to make accurate and stable predictions. Specifically, we introduce a hybrid approach that combines a sequential screening process with a model averaging algorithm, where the weight of each model is determined by its Bayesian information (BIC) score (Schwarz (1978); Chen and Chen (2008)). The sequential technique makes SMA computationally feasible with high-dimensional data, because the averaging process assures the prediction's accuracy and stability. Results show that SMA not only yields a good model, but also mitigates over-fitting. We demonstrate that SMA provides consistent estimators for the regression coefficients and yields reliable predictions under mild conditions. Simulations and empirical examples are presented to illustrate the usefulness of the proposed method.

The described R package allows to estimate Dynamic Model Averaging (DMA), Dynamic Model Selection (DMS) and Median Probability Model. The original methods, and additionally, some selected modifications of these methods are implemented. For example the user can choose between recursive moment estimation and exponentially moving average for variance updating in the base DMA. Moreover, inclusion probabilities can be computed in a way using “Google Trends” data. The code is written with respect to minimise the computational burden, which is quite an obstacle for DMA algorithm if numerous variables are used. For example, this package allows for parallel computations and implementation of the Occam’s window approach. However, clarity and readability of the code, and possibility for an R-familiar user to make his or her own small modifications in reasonably small time and with low effort are also taken under consideration. Except that, some alternative (benchmark) forecasts can also be quickly performed within this package. Indeed, this package is designed in a way that is hoped to be especially useful for practitioners and researchers in economics and finance.

Multi-model averaging is currently receiving a surge of attention in the atmospheric, hydrologic, and statistical literature to explicitly handle conceptual model uncertainty in the analysis of environmental systems and derive predictive distributions of model output. Such density forecasts are necessary to help analyze which parts of the model are well resolved, and which parts are subject to considerable uncertainty. Yet, accurate point predictors are still desired in many practical applications. In this paper, we compare a suite of different model averaging techniques by their ability to improve forecast accuracy of environmental systems. We compare equal weights averaging (EWA), Bates-Granger model averaging (BGA), averaging using Akaike’s information criterion (AICA), and Bayes’ Information Criterion (BICA), Bayesian model averaging (BMA), Mallows model averaging (MMA), and Granger-Ramanathan averaging (GRA) for two different hydrologic systems involving water flow through a 1950 km 2 watershed and 5 m deep vadose zone. Averaging methods with weights restricted to the multi-dimensional simplex (positive weights summing up to one) are shown to have considerably larger forecast errors than approaches with unconstrained weights. Whereas various sophisticated model averaging approaches have recently emerged in the literature, our results convincingly demonstrate the advantages of GRA for hydrologic applications. This method achieves similar performance as MMA and BMA, but is much simpler to implement and use, and computationally much less demanding.