Some theoretical aspects of econometric inference with heteroskedastic models

Abstract

This Thesis is concerned with econometric inference in parametric heteroskedastic models. Each moment of the conditional distribution can be seen as a source of information which provides an estimating equation for the parameter vector. Different issues arise in the different moments concerning the identifiability of parameters, the observability of the dependent variable of the estimating equation, and the positivity restrictions implicit in even order moments. Estimators of the identifiable functions of the parameter vector are obtained from orthogonality conditions in each moment. Under symmetry of the distribution, the sources of information corresponding to the first two conditional moments are independent, at least asymptotically, and the information about common parameters is combined in estimation by constructing a matrix weighted average. Estimation procedures under normality are viewed in a maximum likelihood framework, and generalized method of moments estimation provides the setup for the analysis of more general distributions. The separation of the information into its moment source constitutes a basic element for diagnostic testing of the model. The implications of different forms of misspecification are analyzed and robustness properties are established for some leading cases, especially the ARCH class of models. A general framework is presented for diagnostic testing of heteroskedastic models, which includes tests of the coherency of the information contributed by the two moments, a family of 'consistency tests' which concentrates on the assessment of the first two moments, and a family of 'efficiency tests' which concentrates on checking the specification of moments of order three and higher. The consistency and efficiency tests may be constructed without using information external to the model and thus may be reported with standard computer output, but these families also include many LM tests against specific departures by suitable choice of the test parameters. Tests for autocorrelation, dynamics, parameter stability, different types of exogeneity, and normality, are analyzed in particular. The estimation and diagnostic testing framework is extended to the inclusion of la ten t variables in the conditional mean, such as parametric risk measures and varying coefficients, and also to a multivariate setting. Finally, the problem of extracting information from higher order moments is considered by looking a t the information th a t each moment contributes in addition to what has already been contributed by the lower order moments. Information is extracted from orthogonality conditions and a sequential strategy proposed which analyzes the efficiency gains and the coherency of the available information with the new information obtained from incorporating an additional moment into the model.