Symmetric diagnostics for the analysis of the residuals in regression models

Typical alternative hypotheses in the analysis of residuals of a standard regression model are considered, and for each one a Bayesian diagnostic based on a symmetric form of the Kullback--Leibler divergence is determined. The results include an explicit expression for the diagnostic when the alternative hypothesis is that the errors are generated by an unknown distribution function with a Dirichlet process prior. This expression is immediately interpretable, exactly computable and endowed with important asymptotic connections. A linear approximation of the diagnostic reveals close links with the class of Lagrange multiplier test statistics. When the alternative hypothesis is that the errors are generated by an autoregressive process the linear approximation is proportional to the Box--Pierce statistic or to the Ljung--Box statistic, according to the characteristics of the prior, if the observations have zero mean; it depends on the Durbin--Watson statistic if the errors are first-order autoregressive, and it is related to the Cliff--Ord statistic if they are generated by a first-order spatial autoregression. The sensitivity to the prior of the diagnostic and of its linear approximation is also discussed. Copyright 2005, Oxford University Press.