Comments on: Model-free model-fitting and predictive distributions
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Comments on: Model-free model-fitting and predictive distributions Ingrid Van Keilegom
Published online: 5 April 2013 © Sociedad de Estadística e Investigación Operativa 2013
I first of all wish to congratulate the author for this inspiring and interesting article. It offers a new, fresh view on how to construct prediction intervals in regression, and it also allows for many extensions beyond the framework of regression. Apart from proposing a novel and promising idea, the article is also a very rich source of references, not only on the topic of prediction intervals, but also on related topics like transformation models, bootstrap, and nonparametric regression. The paper also contains interesting side-comments and links with other areas in the literature, giving hence a very complete picture of the problem considered in the paper. In this comment I would like to discuss briefly an outlook on two possible further topics of research. The first one is on the construction of prediction intervals when endogeneity is present, and the second one is on the theoretical investigation of the proposed intervals.
1 Endogeneity Throughout the paper it is implicitly understood that all independent variables are exogenous. However, in many applications in human or medical sciences, endogeneity is present. It occurs when some of the independent variables in a regression model are correlated with the error term. It can arise when some explanatory variables are subject to measurement error, when relevant explanatory variables are omitted from the model, as a result of sample selection errors or when unobserved subject selection
This comment refers to the invited paper available at doi:10.1007/s11749-013-0317-7. I. Van Keilegom () Université catholique de Louvain, Louvain-la-Neuve, Belgium e-mail: [email protected]
Comments on: Model-free model-fitting and predictive distributions
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occurs in experimental studies. The textbooks by Hayashi (2000) and Wooldridge (2008) are excellent introductions into the problem of endogeneity and how to cope with it in identification, estimation or testing problems. When endogeneity is present, ordinary regression techniques produce biased and inconsistent estimators. Therefore, in the model-based (MB) framework, the classical L2 -based point prediction n−1 ni=1 g(mxf + σxf ei ) of a new response g(Yf ) under model (8) of the paper (see Table 1), is no longer correct under endogeneity. For the same reason, the model-free (MF2 ) optimal point predictor of g(Yf ), namely −1
mean{g(D xf (ui ))}, given in Table 3 in the paper, is not valid when endogeneity is present. In the context of semiparametric transformation models of the form Λθ (Yt ) = μ(Xt ) + εt (t = 1, . . . , n), where Λθ belongs to a parametric family of transformations, Vanhems and Van Keilegom (2013) proposed estimators of θ and of μ(Xt ) when endogeneity is present, and they also studied the asymptotic properties of the proposed estimators. Their results could be used to construct classical model-based predictio
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