Robust Z-Estimators for Semiparametric Moment Condition Models.

In the present paper, we introduce a class of robust Z-estimators for moment condition models. These new estimators can be seen as robust alternatives for the minimum empirical divergence estimators. By using the multidimensional Huber function, we first define robust estimators of the element that realizes the supremum in the dual form of the divergence. A linear relationship between the influence function of a minimum empirical divergence estimator and the influence function of the estimator of the element that realizes the supremum in the dual form of the divergence led to the idea of defining new Z-estimators for the parameter of the model, by using robust estimators in the dual form of the divergence. The asymptotic properties of the proposed estimators were proven, including here the consistency and their asymptotic normality. Then, the influence functions of the estimators were derived, and their robustness is demonstrated.

Robust Model Selection Criteria Based on Pseudodistances.

In this paper, we introduce a new class of robust model selection criteria. These criteria are defined by estimators of the expected overall discrepancy using pseudodistances and the minimum pseudodistance principle. Theoretical properties of these criteria are proved, namely asymptotic unbiasedness, robustness, consistency, as well as the limit laws. The case of the linear regression models is studied and a specific pseudodistance based criterion is proposed. Monte Carlo simulations and applications for real data are presented in order to exemplify the performance of the new methodology. These examples show that the new selection criterion for regression models is a good competitor of some well known criteria and may have superior performance, especially in the case of small and contaminated samples.

Robust Estimation for the Single Index Model Using Pseudodistances.

For portfolios with a large number of assets, the single index model allows for expressing the large number of covariances between individual asset returns through a significantly smaller number of parameters. This avoids the constraint of having very large samples to estimate the mean and the covariance matrix of the asset returns, which practically would be unrealistic given the dynamic of market conditions. The traditional way to estimate the regression parameters in the single index model is the maximum likelihood method. Although the maximum likelihood estimators have desirable theoretical properties when the model is exactly satisfied, they may give completely erroneous results when outliers are present in the data set. In this paper, we define minimum pseudodistance estimators for the parameters of the single index model and using them we construct new robust optimal portfolios. We prove theoretical properties of the estimators, such as consistency, asymptotic normality, equivariance, robustness, and illustrate the benefits of the new portfolio optimization method for real financial data.

Robust Portfolio Optimization Using Pseudodistances.

The presence of outliers in financial asset returns is a frequently occurring phenomenon which may lead to unreliable mean-variance optimized portfolios. This fact is due to the unbounded influence that outliers can have on the mean returns and covariance estimators that are inputs in the optimization procedure. In this paper we present robust estimators of mean and covariance matrix obtained by minimizing an empirical version of a pseudodistance between the assumed model and the true model underlying the data. We prove and discuss theoretical properties of these estimators, such as affine equivariance, B-robustness, asymptotic normality and asymptotic relative efficiency. These estimators can be easily used in place of the classical estimators, thereby providing robust optimized portfolios. A Monte Carlo simulation study and applications to real data show the advantages of the proposed approach. We study both in-sample and out-of-sample performance of the proposed robust portfolios comparing them with some other portfolios known in literature.