We in this paper employ a penalized moment selection procedure to identify valid and relevant moments for estimating and testing forecast rationality within the flexible loss framework proposed by Elliott et al. (2005). We motivate the selection of moments in a high-dimensional setting, outlining the fundamental mechanism of the penalized moment selection procedure and demonstrating its implementation in the context of forecast rationality, particularly in the presence of potentially invalid moment conditions. The selection consistency and asymptotic normality are established under conditions specifically tailored to economic forecasting. Through a series of Monte Carlo simulations, we evaluate the finite sample performance of penalized moment estimation in utilizing available instrument information effectively within both estimation and testing procedures. Additionally, we present an empirical analysis using data from the Survey of Professional Forecasters issued by the Federal Reserve Bank of Philadelphia to illustrate the practical utility of the suggested methodology. The results indicate that the proposed post-selection estimator for forecaster’s attitude performs comparably to the oracle estimator by efficiently incorporating available information. The power of rationality and symmetry tests leveraging penalized moment estimation is substantially enhanced by minimizing the impact of uninformative instruments. For practitioners assessing the rationality of externally generated forecasts, such as those in the Greenbook, the proposed penalized moment selection procedure could offer a robust approach to achieve more efficient estimation outcomes.

Factor score indeterminacy is a characteristic property of factor analysis (FA) models. This research introduces a novel procedure, regression-based factor score exploration (RFE), which uniquely determines factor scores and simultaneously estimates other parameters of the FA model. RFE uniquely determines factor scores by minimizing a loss function that balances FA and multivariate regression, regulated by a tuning parameter. Theoretical aspects of RFE, including the uniqueness of factor scores, the relationship between observed and latent variables, and rotational indeterminacy, are examined. Additionally, clustering-based factor exploration (CFE) is presented as a variant of RFE, derived by generalizing the penalty term to enable the clustering of factor scores. It is demonstrated that CFE creates cluster structures more accurately than the existing method. A simulation study shows that the proposed procedures accurately recover true parameter matrices even in the presence of error-contaminated data, with lower computational demand compared to existing methods. Real data examples illustrate that the proposed procedures provide interpretable results, demonstrating high relevance to the factor scores obtained by existing methods.

In this article we tackle the problem of inverse non linear ill-posedproblems from a statistical point of view. We discuss the problemof estimating an indirectly observed function, without priorknowledge of its regularity, based on noisy observations. For this we consider two approaches: one based on the Tikhonov regularization procedure, and another one based on model selection methods for both ordered and non ordered subsets. In each case we prove consistency of the estimators and show that their rate of convergence is optimal for the given estimation procedure.