| Model selection for regression and time series models can be regarded as a special kind of statistical inference. Any inference based on the assumption of a “true” model can be criticized. Model checking can help but often the diagnostics that are used for this purpose also assume a specific parameterization of the model. For this reason, despite their popularity and availability in statistical packages, criteria like the adjusted R-squared, Akaike's AIC, Sawa's BIC or Schwarz's BIC do not always produce the right answer and they should not be expected to. Nevertheless, if we use the definition of regression function as the conditional expectation of the response variable given a set of predictor variables, we can exploit our ability to estimate density functions consistently to derive a test for variable selection which is also consistent. In the dissertation two main approaches are suggested. The first one which uses the original response and predictors and that amounts to a test for conditional independence. The second one, on the contrary, uses the residuals from two nonparametric regression models and compares their estimated kernel density estimates using a distance between functions and is closer in spirit to the approach used in regression graphics. The approach based on the use of residuals, while weaker than the first one, can be easily extended to derive tests to perform model selection for nonnested models or to detect structural breaks in regression models.; None of these techniques can expect to escape the “curse of dimensionality”; nonetheless for the case of relatively simple models, basic simulations suggest that the technique is capable of producing good results. |
