Discussion
'Model selection for generalized linear models with factor-augmented predictors'
Professors Ando and Tsay should be congratulated for such nice work, which provides an effective statistical method to handle high-dimensional data sets with generalized linear models. In this discussion, we supply and review additional references from the literature of both sufficient dimension reduction and variable selection. Thus, the authors' contribution could be further appreciated under an even larger context, both theoretically and practically.
The fast development of information technology enables us to collect an enormous amount of information at very low cost. Statistically, this means that we might face data sets with ultrahigh-dimensional predictors. As a result, analyzing high dimensional data effectively has become a fundamental and important subject in the last decade; see Fan and Li [1] for excellent discussions. To effectively conduct dimension reduction in regression, one could construct 'factors' so that they attain the optimal predictability. As noted by the authors, the traditional principal component analysis fails to be the mostly effective approach because the role of the response is completely ignored, and the resulting principal components might not be sufficiently predictive to the response. In fact, the same problem was also noted by Li [2], which leads to the seminal theory of effective dimension reduction and an innovative method of sliced inverse regression (SIR). Later, such a theory was further developed by Cook [3], which resulted in the theory of sufficient dimension reduction. Subsequently, various inverse regression methods have been proposed. For a comprehensive review, we refer to Cook [4] and Luo et al. [5]. A close look at the computational details of most inverse regression methods (e.g. SIR) reveals that they are indeed a supervised version of dimension reduction, where the effect of the response has been carefully taken into consideration. Similar philosophy was also shared in the partial least square (PLS) literature [6][7][8][9][10][11].
While inverse regression is a useful method for dimension reduction, Bair et al.
[12] considered the supervised principal component analysis, which can be considered as a two-stage procedure. In the first stage, a set of relevant variables are selected by simple univariate regressions, which leads to a reduced (or supervised) data set. In the second stage, principal components are extracted from the reduced data set and are used for predictions. Recently, such a method was further improved by the method of pre-conditioning [13]. In addition to the inverse regression and supervised principal component approaches, Fan and Lv [14] proposed an innovative and stimulating method to analyze ultra-high-dimensional data. It is namely Sure Independence Screening (SIS), which is based on correlation learning to reduce dimensionality. The theoretical and empirical performances of the above methods have been well documented in the literature.
In this article, Professors Ando and Tsay obtain an information selection criterion for generalized linear models by employing factor analysis to cope with the high-dimensional variable problem. We believe exploring its relationships with aforementioned dimension reduction methods would not only enrich the authors' results but also broaden the applicability of those methods to generalized linear, dynamic, and time series models. Finally, we want to congratulate the authors again for such an interesting work. We look forward to seeing its important applications in the foreseeable future.