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Abstract:
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Evaluating forecasts in an empirical setting is now expected whenever a new econometric model is proposed. Typically such an evaluation will involve the use of one or more scoring rules, each targeting some aspect of the predictive model. In a time series context, the popular 'log score' rule, used to evaluate predictive densities, corresponds to the log marginal likelihood from an evaluation period, conditional on data from initial estimation period. This equivalence provides an inherently Bayesian justification of the log score approach (Geweke & Amisano, Int J Forecast, 2010). Other desirable forecast quantities, such as quantile or prediction intervals, tend to only be evaluated using purely frequentist methods. In this paper we further consider connections between marginal likelihoods and the log score rule. By utilizing certain transformations of the predictive density we show how to evaluate quantile or interval forecasts in an inherently Bayesian way rather than resort to a fully frequentist approach. We also consider the equivalence between the log score and the log marginal likelihood when the desired forecast quantity is a non-trivial function of the observed variables.
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