Model Error (or Ambiguity) and Its Estimation, with Particular Application to Loss Reserving

Abstract

This paper is concerned with the estimation of forecast error, particularly in relation to insurance loss reserving. Forecast error is generally regarded as consisting of three components, namely parameter, process and model errors. The first two of these components, and their estimation, are well understood, but less so model error. Model error itself is considered in two parts: one part that is capable of estimation from past data (internal model error), and another part that is not (external model error). Attention is focused here on internal model error. Estimation of this error component is approached by means of Bayesian model averaging, using the Bayesian interpretation of the LASSO. This is used to generate a set of admissible models, each with its prior probability and likelihood of observed data. A posterior on the model set, conditional on the data, may then be calculated. An estimate of model error (for a loss reserve estimate) is obtained as the variance of the loss reserve according to this posterior. The population of models entering materially into the support of the posterior may turn out to be “thinner” than desired, and bootstrapping of the LASSO is used to increase this population. This also provides the bonus of an estimate of parameter error. It turns out that the estimates of parameter and model errors are entangled, and dissociation of them is at least difficult, and possibly not even meaningful. These matters are discussed. The majority of the discussion applies to forecasting generally, but numerical illustration of the concepts is given in relation to insurance data and the problem of insurance loss reserving.