Linear model

Author: C.R. Rao et al.
Publisher:
Publish Date: 1998-08-01
Features:
Fragment: Chapter 5 is devoted to estimation under exactor stochastic linear restrictions. The comparison of two biased estimators according to the MDE criterion is based on recent theorems of matrix theory. The results are the outcome of intensive international research over the last ten years and appear here for the first time in a coherent form. This concerns the concept of the weak r-unbiasedness as well. Chapter 6 contains the theory of the optimal linear prediction and gives, in addition to known results, an insight into recent studies about the MDE matrix comparison of optimal and classical predictions according to alternative superiority criteria. Chapter 7 presents ideas and procedures for studying the effect of single data rows on the estimation of . Here, different measures for revealing outliers or influential points, including graphical methods, are incorporated. Some examples illustrate this. Chapter 8 deals with missing data in the design matrix X. After introducing the general problems and defining the various missing data mechanisms according to Rubin, we demonstrate "adjustment by follow-up interviews" for long-term studies with dropout. For the regression model, the method of imputation is described, in addition to the analysis of the loss of efficiency in case of a reduction to the completely observed submodel. The method of weighted mixed estimates is presented for the first time in a textbook on online ARM models. Chapter 9 contains recent contributions to robust statistical inference based on M-estimation. Chapter 10 describes the model extensions for categorical response and explanatory variables. Here, the binary response and the log linear model are of special interest. The model choice is demonstrated by means of examples. Categorical regression is integrated into the theory of generalized linear models. An independent chapter (Appendix A) on matrix algebras summarizes standard theorems (including proofs) that are of interest for the book itself, but also for linear statistics in general. Of special interest are the theorems about the decomposition of matrices (A.30-A.34), definite matrices (A.35-A.59), the generalized inverse, and especially about the definiteness of differences between matrices (Theorem A.71; cf. A.74-A.78). The book offers an up-to-date and comprehensive account of the theory and applications of linear models. Tables for the X- and F-distributions are provided in Appendix B.
2.1 Regression Models in Econometrics
The methodology of regression analysis, one of the classical techniques of mathematical statistics, is an essential part of modern econometric theory. Econometrics combines elements of economics, mathematical economics, and mathematical statistics. The statistical methods used in econometrics are oriented toward specific econometric problems and hence are highly specialized. In economic laws, stochastic variables play a distinctive role. Hence, econometric models adapted to economic reality must be based on appropriate hypotheses about the distribution properties of the random variables. The specification of such hypotheses is one of the main tasks of econometric modeling. For the modeling of an economic (or scientific) relation, we assume that this relation has a relative constancy over a sufficiently long period of time (that is, over a sufficient length of the observation period), since otherwise its general validity would not be ascertainable. We distinguish between two characteristics of a structural relationship: the variables and the parameters. The variables, which we will classify later on, are those characteristics whose values in the observation period can vary. Those that do not vary can be regarded as the structure of the relation. The structure consists of the functional form of the relation, including the relation between the main variables, the type of probability distribution of the random variables, and the parameters of the model equations.