Topics on conditional moment equations models: goodness-of-fit and missing data

Abstract

Models defined by moment conditions are among the most used by statisticians and econometricians. They arise naturally, for example, as regression models when one aims to explain the influence of explanatory (covariate, predictor) variables on output (response, outcome) variables. In this short course we address two aspects related to the models defined by conditional moments: model checks and missingness. 

For model checks, also called goodness-of-fit or specification tests,  the general problem considered is the test of the effect of a Hilbert space valued covariate on a Hilbert space valued response, that is the test of the nullity of the conditional expectation of the response given a general covariate. This general framework includes the model check problem for standard mean and quantile regressions, functional regressions, etc.  against general alternatives. It also includes the problem of testing conditional independence with functional data. The significance test for (functional) regressors in nonparametric regression with general covariates and responses is another example. Several nonparametric tests  based on kernel smoothing are presented. The test statistics are asymptotically standard normal under the null hypothesis provided the smoothing parameter tends to zero at a suitable rate. The one-sided tests are consistent against any fixed alternative and detects local alternatives a la Pitman approaching the null hypothesis. In particular we show that neither the dimension of the outcome nor the dimension of the covariates influences the theoretical power of the tests against such local alternatives. The uniform consistency against special classes of functions of the covariate is also studied. Simulation experiments and a real data application illustrate the performance of the tests with finite samples. 

In the second part of the lectures, we address the problem of missing data. We present an equivalence result for a general class of models defined by moments when some variables are missing at random (MAR). Missingness could affect responses, covariates or any combination of them. We show that the initial model and the missingness mechanism could be equivalently defined under the form of a system of moment equations. The equivalence means the same sets of probability measures for the observed variables. This type of equivalence greatly simplifies the analysis of the initial model under a missingness mechanism satisfying the MAR assumption, for instance for efficiency bounds calculations or efficient estimators construction.