目
录
Preface
1Introduction
2Linear
Models
2.1Regression
Models
in
Econometrics
2.2Econometric
Models
2.3The
Reduced
Form
2.4The
Multivariate
Regression
Model
2.5The
Classical
Multivariate
Linear
Regression
Model
2.6The
Generalized
Linear
Regression
Model
3The
Linear
Regression
Model
3.1The
Linear
Model
3.2The
Principle
of
Ordinary
Least
SquaresOLS3.3Geometric
Properties
of
OLS
3.4Best
Linear
Unbiased
Estimation
3.4.1Basic
Theorems
3.4.2Linear
Estimators
3.4.3Mean
Dispersion
Error
3.5EstimationPredictionof
the
Error
Term
e
and
2
3.6Classical
Regression
under
Normal
Errors
3.7Testing
Linear
Hypotheses

......

Chapter 5 is devoted to estimation under exact or stochastic linear re-

strictions. 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 ap-

pear 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 introduc-

ing 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 linear 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 loglinear model are

ofspecial interest. The model choice is demonstrated by means ofexamples.

Categorical regression is integrated into the theory of generalized linear

models.

An independent chapter (Appendix A) on matrix algebra summarizes

standard theorems (including proofs) that are of interest for the book it-

self, but also for linear statistics in general. Of special interest are the

theorems about 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.7l; 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

Linear Models

2.1 Regression Models in Econometrics

The methodology of regression analysis, one of the classical techniques of

mathematical statistics, is an essential part of the 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 the economic reality, have to be

built on appropriate hypotheses about distribution properties of the ran-

dom variables. The specification of such hypotheses is one of the main tasks

of econometric modelling. For the modelling of an economic (or a scientific)

relation, we assume that this relation has a relative constancy over a suffi-

ciently long period of time (that is, over a sufficient length of 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 characteristics 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 probabil-

ity distribution of the random variables, and the parameters of the modei

equations.