Essays on the optimal selection of series functions - eScholarship
Francisco L. Pascual · 2007
The object of study of the present dissertation is the asymptotic optimality of model selection procedures for choosing an optimal model whose parameters have been estimated using ridge regression. Thus, we have to select simultaneously both the optimal model and the optimal ridge parameter. Our problem focuses on obtaining a forecast of a target variable given a fixed vector of predictors. It is well known that given a quadratic loss function (that we assumed throughout) the optimal forecast is the conditional expectation of the target variable given the predictors, which is an unknown function of the predictors. Therefore our task reduces to finding an appropriate model capable of approximating this unknown function. In particular and following White (2006) we focus on series estimators which are linear in the coefficients. They are as which are linear in the coefficients. They are as flexible as non-linear models and avoid at the same time the computational difficulties that arise in the implementation of the latter. How well series estimators perform (in terms of approximating an unknown function) depends crucially on how optimally the basis or series functions are chosen. In addition, the use of ridge estimation improves the predictive ability of the model (and therefore its approximation capabilities) as long as the shrinkage parameter is optimally chosen. Ideally we would select the series functions and ridge parameter that minimize prediction mean squared error (PMSE) but since this is unknown (depends on the joint distribution of the data) we have to use an estimator. Most model selection procedures can be regarded as either direct or indirect estimates of PMSE and that is why want to study their optimality. The asymptotic optimality property we analyze is known in the literature as asymptotic loss-efficiency. It implies that the accuracy of the estimator based on the selected model is asymptotically the same as that based on the best model in the list (which is unknown). The first chapter studies the conditions delivering the asymptotic loss-efficiency of Mallows CL, leave-1-out cross-validation, generalized cross-validation and GIC[lambda]n, the latter embedding a number of other model selection criteria. We assume stochastic regressors and iid observations. Therefore the chapter constitutes an extension of the work by Li (1986, 1987) and Shao (1997) to cover model selection under ridge estimation, which generalizes OLS. The second chapter studies asymptotic loss-efficiency in an environment with dependent and heterogeneous observations and therefore it allows us to deal with financial and macroeconomic data. We consider a generalized version of Mallows CL that we called GCL (following Andrews (1991)) and which is appropriate under this more general data structure. We also cover leave-1-out cross-validation. As we show there, when the errors of the data generating process (that we define at the outset of chapter one) are correlated, the optimality of these criteria breaks down. The last chapter analyzes a general version of leave-1-out cross-validation which is known as h-block cross-validation and that is robust to error correlation. It concludes with a small simulation showing the relative performance of the different model selection procedures under dependence and heterogeneity