Evaluation and Comparison of Econometric Models Using Nonparametric Likelihood and Bootstrap

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This project covers the following three topics in evaluation and comparison of econometric model specifications: use of bootstrap smoothing in predictive inference, use of nonparametric methodology in comparing possibility misspecified dynamic econometric models, and optimal properties of empirical likelihood ratio tests in terms of Hoeffding's measure of asymptotic relative efficiency. These topics are central to major research activity in econometric model evaluation and comparison. This project should make major contributions to econometric theory and applied econometrics.The first project discusses predictive inference, which can be regarded as a validation problem that uses a certain data splitting scheme. To carry out a predictive test of an econo-metric model, a loss function is specified and the data set is divided into the training set and the validation set. Unknown parameters of the model are estimated using the training set, and the estimated model is tested against the validation set, using the expected loss (sometimes called risk) as a criterion. Thus the efficacy of the predictive test depends on the accuracy of the risk estimator. A conventional risk estimator is the validation-sample average of loss function values eval-uated at the estimated parameter value. This procedure can be viewed as a variant of cross validation that is carried out forward, therefore called forward validation (FV). A drawback of FV is that it does not explicitly incorporate parameter estimation uncertainty into risk estimation.This project proposes to use bootstrap smoothing (BS) to remedy this drawback. The BS algorithm repeats the following two steps: (1) a bootstrap draw of an unknown parameter is obtained by resampling the training set (2) as in FV, the validation-sample average of the loss function values is calculated, but this time evaluated at the bootstrap draw from (1). After ap-plying these two steps to (sufficiently many) bootstrap draws, the average of validation-sample averages is taken. This method is similar to Efron's "leave-one-out-bootstrap," or Breiman's "bagging," which are known to be effective for discontinuous loss functions. The first project develops the following theorem: bootstrap smoothing eliminates the effect of parameter uncer-tainty for indicator loss functions in large samples, thereby providing a clear and potentially substantial asymptotic efficiency gain. This is a rather surprising result, but consistent with the good practical performance of "leave-one-out" methods and bagging. Some connections with Bayesian methods are discussed. The BS methodology is applied to Merton's predictability measure of market timing and other empirical examples.The second project is concerned with the comparison of possibly misspecified and possibly non-nested moment condition models. Standard specification tests for moment condition models assume the presence of an exactly correct moment restriction under the null, but sometimes this assumption is not reasonable in practice. Even so, it is still of interest to compare such possibly misspecified models in terms of an overall measure of goodness-of-fit. This project develops a device for this comparison by combining a nonparametric likelihood method and Vuong's (parametric) model comparison test. Furthermore, a local smoothed version of nonparametric likelihood is used to extend the method to comparing possibly misspecified conditional moment restriction models.The third project considers moment condition models. It pursues optimal tests of overidentifying restrictions, which are typically tested by Hansen's "J-test." Recently several alternatives to Hansen's test have been developed using some versions of nonparametric likelihood. While conventional local asymptotic efficiency comparisons cannot distinguish among these competing tests, a global efficiency measure originally proposed by Hoeffding makes it possible to establish an optimality property of empirical likelihood ratio tests for IID samples. Extensions of this analysis to dependent observations are also investigated.