Model selection for nested and overlapping nonlinear dynamic and possibly misspecified models

The literature on model comparison often requires the assumption that the true conditional distribution corresponds to that of one of the competing models. This strong assumption has been extended by the notion of encompassing and in likelihood based model comparisons. This paper takes the latter approach and develops tests for the comparison of competing nonlinear dynamic models, focusing on the nested and overlaping cases. The null hypothesis is that the models are equally close to the data generating process (DGP), according to a certain measure of closeness. The alternative is that one model is closer to the DGP. The models can be correctly specified or not. Their parameters can be estimated by a variety of methods, including (pseudo) maximum likelihood and ordinary least squares. The tests are symmetric and directional. Their asymptotic distribution under the null is either normal or a weighted sum of chi-squared distributions, depending on the nesting characteristics of the competing models. The comparison of nested AR models, and of nested ARMA models with GARCH errors and exogenous forcing variables (ARMAX-GARCH) are discussed as examples.