ABSTRACT
Most econometric models can be viewed only as approximations of the underlying data generating process and are likely to be misspeci…ed. It is therefore desirable to conduct inference for misspeci…ed models which would take into account the additional uncertainty that arises from possible misspeci…- cation. While the analysis of misspeci…ed models in the maximum likelihood framework was originated by White (1982), the methodology for misspeci…ed models de…ned by moment conditions is a relatively recent development and is still an active research area. Moreover, the proposed inference procedures in misspeci…ed moment condition models are not readily adopted by applied researchers. For example, while most asset pricing models are statistically rejected by the data, many empirical studies continue to report standard errors and test hypotheses on the parameters under the assumption of correctly speci…ed models. This creates internal inconsistency in the inference procedure and tends to underestimate the true uncertainty surrounding the parameter estimates. The aim of this chapter is to present some of the existing results in the literature on misspeci…ed models in a unifying and comprehensive framework for statistical inference that is robust to possible model misspeci…cation. Maximum likelihood tends to be a natural estimation framework when the
data generation process is fully parameterized. When the economic theory does not provide enough information about the underlying probabilistic law that governs the data, the econometrician is prone to a risk of misspeci…cation. The Kullback–Leibler information criterion measures the distance between the speci…ed and true densities and is a convenient tool for assessing and properly incorporating this misspeci…cation risk into the estimation and inference procedure. An interesting result that emerges from this analysis is that the misspeci…cation of the true density preserves some desirable properties of the estimator and this quasi-(or pseudo-) maximum likelihood estimator is still consistent and asymptotically normally distributed. This framework also allows us to compare di¤erent misspeci…ed models in order to determine which model is empirically most useful and provides the best approximation to the underlying true data generating mechanism. This parametric framework can be further generalized by adopting a fully
nonparametric view of the true distribution that generates the data, and our knowledge of some characteristics of the model only enters through a set of (conditional or unconditional) moment restrictions. This nonparametric likelihood can handle overidenti…ed moment condition models and is described in greater detail in Chapters 2 and 3. Our use of this framework here is to extend the parametric likelihood comparison of misspeci…ed models to a nonparametric likelihood setup that incorporates explicitly the moment restrictions implied by economic theory. Finally, we point out some problems that arise in the analysis of misspeci…ed moment condition models estimated by GMM and provide a detailed discussion on the estimation, evaluation and model comparison of asset pricing models. This problem serves as a specialized, yet highly relevant practical example of the analysis of inherently misspeci…ed models estimated by GMM with a common weighting matrix.
