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Article

Knut Are Aastveit, James Mitchell, Francesco Ravazzolo, and Herman K. van Dijk

Increasingly, professional forecasters and academic researchers in economics present model-based and subjective or judgment-based forecasts that are accompanied by some measure of uncertainty. In its most complete form this measure is a probability density function for future values of the variable or variables of interest. At the same time, combinations of forecast densities are being used in order to integrate information coming from multiple sources such as experts, models, and large micro-data sets. Given the increased relevance of forecast density combinations, this article explores their genesis and evolution both inside and outside economics. A fundamental density combination equation is specified, which shows that various frequentist as well as Bayesian approaches give different specific contents to this density. In its simplest case, it is a restricted finite mixture, giving fixed equal weights to the various individual densities. The specification of the fundamental density combination equation has been made more flexible in recent literature. It has evolved from using simple average weights to optimized weights to “richer” procedures that allow for time variation, learning features, and model incompleteness. The recent history and evolution of forecast density combination methods, together with their potential and benefits, are illustrated in the policymaking environment of central banks.

Article

Eric Ghysels

The majority of econometric models ignore the fact that many economic time series are sampled at different frequencies. A burgeoning literature pertains to econometric methods explicitly designed to handle data sampled at different frequencies. Broadly speaking these methods fall into two categories: (a) parameter driven, typically involving a state space representation, and (b) data driven, usually based on a mixed-data sampling (MIDAS)-type regression setting or related methods. The realm of applications of the class of mixed frequency models includes nowcasting—which is defined as the prediction of the present—as well as forecasting—typically the very near future—taking advantage of mixed frequency data structures. For multiple horizon forecasting, the topic of MIDAS regressions also relates to research regarding direct versus iterated forecasting.

Article

Silvia Miranda-Agrippino and Giovanni Ricco

Vector autoregressions (VARs) are linear multivariate time-series models able to capture the joint dynamics of multiple time series. Bayesian inference treats the VAR parameters as random variables, and it provides a framework to estimate “posterior” probability distribution of the location of the model parameters by combining information provided by a sample of observed data and prior information derived from a variety of sources, such as other macro or micro datasets, theoretical models, other macroeconomic phenomena, or introspection. In empirical work in economics and finance, informative prior probability distributions are often adopted. These are intended to summarize stylized representations of the data generating process. For example, “Minnesota” priors, one of the most commonly adopted macroeconomic priors for the VAR coefficients, express the belief that an independent random-walk model for each variable in the system is a reasonable “center” for the beliefs about their time-series behavior. Other commonly adopted priors, the “single-unit-root” and the “sum-of-coefficients” priors are used to enforce beliefs about relations among the VAR coefficients, such as for example the existence of co-integrating relationships among variables, or of independent unit-roots. Priors for macroeconomic variables are often adopted as “conjugate prior distributions”—that is, distributions that yields a posterior distribution in the same family as the prior p.d.f.—in the form of Normal-Inverse-Wishart distributions that are conjugate prior for the likelihood of a VAR with normally distributed disturbances. Conjugate priors allow direct sampling from the posterior distribution and fast estimation. When this is not possible, numerical techniques such as Gibbs and Metropolis-Hastings sampling algorithms are adopted. Bayesian techniques allow for the estimation of an ever-expanding class of sophisticated autoregressive models that includes conventional fixed-parameters VAR models; Large VARs incorporating hundreds of variables; Panel VARs, that permit analyzing the joint dynamics of multiple time series of heterogeneous and interacting units. And VAR models that relax the assumption of fixed coefficients, such as time-varying parameters, threshold, and Markov-switching VARs.

Article

Silvia Miranda-Agrippino and Giovanni Ricco

Bayesian vector autoregressions (BVARs) are standard multivariate autoregressive models routinely used in empirical macroeconomics and finance for structural analysis, forecasting, and scenario analysis in an ever-growing number of applications. A preeminent field of application of BVARs is forecasting. BVARs with informative priors have often proved to be superior tools compared to standard frequentist/flat-prior VARs. In fact, VARs are highly parametrized autoregressive models, whose number of parameters grows with the square of the number of variables times the number of lags included. Prior information, in the form of prior distributions on the model parameters, helps in forming sharper posterior distributions of parameters, conditional on an observed sample. Hence, BVARs can be effective in reducing parameters uncertainty and improving forecast accuracy compared to standard frequentist/flat-prior VARs. This feature in particular has favored the use of Bayesian techniques to address “big data” problems, in what is arguably one of the most active frontiers in the BVAR literature. Large-information BVARs have in fact proven to be valuable tools to handle empirical analysis in data-rich environments. BVARs are also routinely employed to produce conditional forecasts and scenario analysis. Of particular interest for policy institutions, these applications permit evaluating “counterfactual” time evolution of the variables of interests conditional on a pre-determined path for some other variables, such as the path of interest rates over a certain horizon. The “structural interpretation” of estimated VARs as the data generating process of the observed data requires the adoption of strict “identifying restrictions.” From a Bayesian perspective, such restrictions can be seen as dogmatic prior beliefs about some regions of the parameter space that determine the contemporaneous interactions among variables and for which the data are uninformative. More generally, Bayesian techniques offer a framework for structural analysis through priors that incorporate uncertainty about the identifying assumptions themselves.

Article

Michael P. Clements and Ana Beatriz Galvão

At a given point in time, a forecaster will have access to data on macroeconomic variables that have been subject to different numbers of rounds of revisions, leading to varying degrees of data maturity. Observations referring to the very recent past will be first-release data, or data which has as yet been revised only a few times. Observations referring to a decade ago will typically have been subject to many rounds of revisions. How should the forecaster use the data to generate forecasts of the future? The conventional approach would be to estimate the forecasting model using the latest vintage of data available at that time, implicitly ignoring the differences in data maturity across observations. The conventional approach for real-time forecasting treats the data as given, that is, it ignores the fact that it will be revised. In some cases, the costs of this approach are point predictions and assessments of forecasting uncertainty that are less accurate than approaches to forecasting that explicitly allow for data revisions. There are several ways to “allow for data revisions,” including modeling the data revisions explicitly, an agnostic or reduced-form approach, and using only largely unrevised data. The choice of method partly depends on whether the aim is to forecast an earlier release or the fully revised values.

Article

Alessandro Casini and Pierre Perron

This article covers methodological issues related to estimation, testing, and computation for models involving structural changes. Our aim is to review developments as they relate to econometric applications based on linear models. Substantial advances have been made to cover models at a level of generality that allow a host of interesting practical applications. These include models with general stationary regressors and errors that can exhibit temporal dependence and heteroskedasticity, models with trending variables and possible unit roots and cointegrated models, among others. Advances have been made pertaining to computational aspects of constructing estimates, their limit distributions, tests for structural changes, and methods to determine the number of changes present. A variety of topics are covered including recent developments: testing for common breaks, models with endogenous regressors (emphasizing that simply using least-squares is preferable over instrumental variables methods), quantile regressions, methods based on Lasso, panel data models, testing for changes in forecast accuracy, factors models, and methods of inference based on a continuous records asymptotic framework. Our focus is on the so-called off-line methods whereby one wants to retrospectively test for breaks in a given sample of data and form confidence intervals about the break dates. The aim is to provide the readers with an overview of methods that are of direct use in practice as opposed to issues mostly of theoretical interest.

Article

During the 18th and 19th centuries, medical spending in the United States rose slowly, on average about .25% faster than gross domestic product (GDP), and varied widely between rural and urban regions. Accumulating scientific advances caused spending to accelerate by 1910. From 1930 to 1955, rapid per-capita income growth accommodated major medical expansion while keeping the health share of GDP almost constant. During the 1950s and 1960s, prosperity and investment in research, the workforce, and hospitals caused a rapid surge in spending and consolidated a truly national health system. Excess growth rates (above GDP growth) were above +5% per year from 1966 to 1970, which would have doubled the health-sector share in fifteen years had it not moderated, falling under +3% in the 1980s, +2% in 1990s, and +1.5% since 2005. The question of when national health expenditure growth can be brought into line with GDP and made sustainable for the long run is still open. A review of historical data over three centuries forces confrontation with issues regarding what to include and how long events continue to effect national health accounting and policy. Empirical analysis at a national scale over multiple decades fails to support a position that many of the commonly discussed variables (obesity, aging, mortality rates, coinsurance) do cause significant shifts in expenditure trends. What does become clear is that there are long and variable lags before macroeconomic and technological events affect spending: three to six years for business cycles and multiple decades for major recessions, scientific discoveries, and organizational change. Health-financing mechanisms, such as employer-based health insurance, Medicare, and the Affordable Care Act (Obamacare) are seen to be both cause and effect, taking years to develop and affecting spending for decades to come.