In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, “diffuse” priors on model-specific parameters can lead to quite unexpected consequences. Here we focus on the practically relevant situation where we need to entertain a (large) number of sampling models and we have (or wish to use) little or no subjective prior information. We aim at providing an “automatic” or “benchmark” prior structure that can be used in such cases. We focus on the Normal linear regression model with uncertainty in the choice of regressors. We propose a partly noninformative prior structure related to a Natural Conjugate g-prior specification, where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j. The consequences of different choices for g0j are examined. We investigate theoretical properties, such as consistency of the implied Bayesian procedure. Links with classical information criteria are provided. More importantly, we examine the finite sample implications of several choices of g0j in a simulation study. The use of the MC3 algorithm of Madigan and York (1995), combined with efficient coding in Fortran, makes it feasible to conduct large simulations. In addition to posterior criteria, we shall also compare the predictive performance of different priors. A classic example concerning the economics of crime will also be provided and contrasted with results in the literature. The main findings of the paper will lead us to propose a “benchmark” prior specification in a linear regression context with model uncertainty.

Forecast combinations have frequently been found in empirical studies to produce better forecasts on average than methods based on the ex-ante best individual forecasting model. Moreover, simple combinations that ignore correlations between forecast errors often dominate more refined combination schemes aimed at estimating the theoretically optimal combination weights. In this chapter we analyze theoretically the factors that determine the advantages from combining forecasts (for example, the degree of correlation between forecast errors and the relative size of the individual models’ forecast error variances). Although the reasons for the success of simple combination schemes are poorly understood, we discuss several possibilities related to model misspecification, instability (non-stationarities) and estimation error in situations where thenumbersofmodelsislargerelativetothe available sample size. We discuss the role of combinations under asymmetric loss and consider combinations of point, interval and probability forecasts. Key words: Forecast combinations; pooling and trimming; shrinkage methods; model misspecification, diversification gains

In this paper we outline conditions under which the Diebold and Mariano (DM: 1995) test for predictive ability can be extended to the case of two forecasting models, each of which may include cointegrating relations, when allowing for parameter estimation error. We show that in the cases where either the loss function is quadratic or the length of the prediction period, P, grows at a slower rate than the length of the regression period, R, the standard DM test can be used. On the other hand, in the case of a generic loss function, if P R ! as T ! 1, 0 < < 1, then the asymptotic normality result of West (1996) no longer holds. We also extend the "data snooping" technique of White (2000) for comparing the predictive ability of multiple forecasting models to the case of cointegrated variables. In a series of Monte Carlo experiments, we examine the impact of both short run and cointegrating vector parameter estimation error on DM, data snooping, and related tests. Our results sugge...

In this paper we consider the process of modelling uncertainty. In particular we are concerned with making inferences about some quantity of interest which, at present, has been unobserved. Examples of such a quantity include the probability of ruin of a surplus process, the accumulation of an investment, the level or surplus or deficit in a pension fund and the future volume of new business in an insurance company. Uncertainty in this quantity of interest, y, arises from three sources: . uncertainty due to the stochastic nature of a given model; . uncertainty in the values of the parameters in a given model; . uncertainty in the model underlying what we are able to observe and determining the quantity of interest. It is common in actuarial science to find that the first source of uncertainty is the only one which receives rigorous attention. A limited amount of research in recent years has considered the effect of parameter uncertainty, while there is still considerable scope ...

In 2007 all ECB publications feature a motif taken from the €20 banknote.

In this paper, we make use of state space models to investigate the presence of stochastic trends in economic time series. A model is specified where such a trend can enter either in the autoregressive representation or in a separate state equation. Tests based on the former are analogous to Dickey-Fuller tests of unit roots, while the latter are analogous to KPSS tests of trend-stationarity. We use Bayesian methods to survey the properties of the likelihood function in such models and to calculate posterior odds ratios comparing models with and without stochastic trends. We extend these ideas to the problem of testing for integration at seasonal frequencies and showhow our techniques can be used to carry out Bayesian variants of either the HEGY or Canova-Hansen test. Stochastic integration rules, based on Markov Chain Monte Carlo, as well as deterministic integration rules are used. Strengths and weaknesses of each approach are indicated.

this paper are first converted to real quantities by dividing each variable by a country-specific price index. The variables are then logged, firstdifferenced, and multiplied by 100 to convert to growth rates. Estimation results for the AR(3)LI model in equation (1) are presented in Table 3, and coefficient posterior means and standard deviations for the models in (2) and (3) with coefficients restricted to be equal across countries are presented in Table 4. On computing the roots of the AR(3) process for the countries' output growth rates from equation(2),

This paper proposes a Bayesian procedure for combining forecasts from multivariate forecasting models, e.g. VAR models. Standard applications of Bayesian model averaging suffer from a basic difficulty in this context, when additional variables are included and modelled the connection between the overall measure of fit for the model and the expected forecasting performance for the variables of interest is lost. We circumvent this problem by focusing on the predictive performance for the variables of interest and base the forecast combination on the predictive likelihood. Specifically we consider forecast combination and, indirectly, model selection for VAR models when there is uncertainty about which variables to include in the model in addition to the forecast variables. For this purpose we consider all possible combinations of variables and lag lengths and the models that arise from these. The procedure is evaluated in a small simulation study and found to perform competitively in applications to real world data.

In previous work, we have used Bayesian methods in the analysis of various models to explain past variation and forecast future values of the rates of growth of real GDP for 18 industrialized countries. Using these models, point and turning point forecasts were calculated and found to be reasonably accurate compared to those of benchmark and other models' forecasts. In this paper, Marshallian demand, supply and entry models are employed for major sectors of an economy that can be combined with factor market models for money, labor, capital and bonds to provide a Marshallian macroeconomic model (MMM). Herein, the sectoral models are used to produce sectoral output forecasts which are summed to provide forecasts of annual growth rates of U.S. real gross domestic product (GDP). These disaggregative forecasts are compared to forecasts derived from models implemented with aggregate data. The empirical evidence indicates that it pays to disaggregate, particularly when employing Bayesian shri...

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